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https://journal.nsps.org.ng/index.php/jnsps/article/view/589
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# Modified Gradient Flow Method for Solving One-Dimensional Optimal Control Problem Governed by Linear Equality Constraint
## Authors
• Olusegun Olotu Department of Mathematical Sciences,The Federal University of Technology Akure, Nigeria
• Charles Aladesaye Dept. of Mathematics, School of Science, College of Education, Ikere-Ekiti, Ekiti State, Nigeria
• Kazeem Adebowale Dawodu Department of Mathematical Sciences,The Federal University of Technology Akure, Nigeria
## Keywords:
Optimal Control, Gradient Flow, three-level splitting parameters, discretization scheme, linear and quadratic convergence
## Abstract
This study presents a computational technique developed for solving linearly constraint optimal control problems using the Gradient Flow Method. This proposed method, called the Modified Gradient Flow Method (MGFM), is based on the continuous gradient flow reformulation of constrained optimization problem with three-level implicit time discretization scheme. The three-level splitting parameters for the discretization of the gradient flow equations are such that the sum of the parameters equal to one (\theta1 + \theta2 +\theta3=1). The Linear and quadratic convergence of the scheme were analyzed and were shown to have first order scheme when each parameter exist in the domain [0, 1] and second order when the third parameter equal to one. Numerical experiments were carried out and the results showed that the approach is very effective for handling this class of constrained optimal control problems. It also compared favorably with the analytical solutions and performed better than the existing schemes in terms of convergence and accuracy
Dimensions
A. I. Adekunle, “Algorithm for a Class of Discretized Optimal Control Problems", M.Tech. Thesis, Federal University of Technology, Akure, Nigeria (2011) (Unpublished).
O. C. Akeremale, “Optimization of Quadratic Constrained Optimal Control Problems using Augumented Lagrangian Method", M.Tech. Thesis, Federal University of Technology, Akure, Nigeria (2012) (Unpublished)
W. Behrman, “An Effcient Gradient Flow Method for Unconstrained Optimization", PhD Thesis, Stanford University (1998) (Unpublished).
J. T. Betts, “Practical Methods for Optimal Control Problem Using Non linear programming", SIAM, Philadelphia, (2001).
Y. Evtushenko, “Generalized Lagrange Multipliers Technique for Nonlinear Programming", JOTA, 21 (1977) 121. DOI: https://doi.org/10.1007/BF00932516
Y. G. Evtushenko & V. G. Zhadan, “Stable Barrier Projection and Barrier Newton Methods", Nonlinear Programming Optimization Methods and Software, 3 (1994) 237. DOI: https://doi.org/10.1080/10556789408805567
G. T. Gilbert, “Positive Definite Matrices and Sylvester’s Criterion", The American Mathematical Monthly, Taylor & Francis, 98 (1991) 44. DOI: https://doi.org/10.2307/2324036
W. M. Haddad, S. G. Nersesov & V.S. Chellaboina, “Lyapunov Function Proof of Poincare’s Theorem", International Journal of Systems Science, 35 (2004) 287. DOI: https://doi.org/10.1080/00207720410001714824
J. B. Layton, “Efficient direct computatioion of the Pseudo-inverse and its gradient", International Journal for Numerical Methods in Engineering, 40 (1997) 4211. DOI: https://doi.org/10.1002/(SICI)1097-0207(19971130)40:22<4211::AID-NME255>3.0.CO;2-8
W. H. Morris, S. Stephen & L. D. Robert, “Di erential Equations, Dynamical Systems, and an Introduction to Chaos", Elsevier Academic Press, USA (2004) 194.
O. Olotu & K. A. Dawodu, “Quasi-Newton Embedded Augmented Lagrangian Algorithm for Discretized Optimal Proportional Control Problems", Journal of Mathematical Theory and Modeling, 3 (2013a) 67.
O. Olotu & K. A. Dawodu, “On the Discretized Algorithm for Optimal Proportional Control Problems Constrained by Delay Differential Equations", Journal of Mathematical Theory and Modeling, 3 (2013b) 157.
O. Olotu & S. A. Olorunsola, “An Algorithm for a Discretized Constrained, Continuous Quadratic Control Problem", Journal of Applied Sciences, 8 (2006) 6249. DOI: https://doi.org/10.4314/jonamp.v8i1.40017
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze & E. F. Mishchenko, “The Mathematical Theory of Optimal Processes", Interscience Publishers, London (1962).
S. Wang, X. Q. Yang K. L. Teo “A Unified Gradient Flow Approach to Constrained Nonlinear Optimization Problems", Computational Optimization and Applications, 25 (2003) 251. DOI: https://doi.org/10.1023/A:1022973608903
2022-02-28
## How to Cite
Olotu, O., Aladesaye, C. ., & Dawodu, K. A. (2022). Modified Gradient Flow Method for Solving One-Dimensional Optimal Control Problem Governed by Linear Equality Constraint. Journal of the Nigerian Society of Physical Sciences, 4(1), 146–156. https://doi.org/10.46481/jnsps.2022.589
## Section
Original Research
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.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/inspection/plot_linear_model_coefficient_interpretation.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:here to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_inspection_plot_linear_model_coefficient_interpretation.py: ================================================================== Common pitfalls in interpretation of coefficients of linear models ================================================================== In linear models, the target value is modeled as a linear combination of the features (see the :ref:linear_model User Guide section for a description of a set of linear models available in scikit-learn). Coefficients in multiple linear models represent the relationship between the given feature, :math:X_i and the target, :math:y, assuming that all the other features remain constant (conditional dependence _). This is different from plotting :math:X_i versus :math:y and fitting a linear relationship: in that case all possible values of the other features are taken into account in the estimation (marginal dependence). This example will provide some hints in interpreting coefficient in linear models, pointing at problems that arise when either the linear model is not appropriate to describe the dataset, or when features are correlated. We will use data from the "Current Population Survey" _ from 1985 to predict wage as a function of various features such as experience, age, or education. .. contents:: :local: :depth: 1 .. GENERATED FROM PYTHON SOURCE LINES 30-39 .. code-block:: default print(__doc__) import numpy as np import scipy as sp import pandas as pd import matplotlib.pyplot as plt import seaborn as sns .. GENERATED FROM PYTHON SOURCE LINES 40-46 The dataset: wages ------------------ We fetch the data from OpenML _. Note that setting the parameter as_frame to True will retrieve the data as a pandas dataframe. .. GENERATED FROM PYTHON SOURCE LINES 46-51 .. code-block:: default from sklearn.datasets import fetch_openml survey = fetch_openml(data_id=534, as_frame=True) .. GENERATED FROM PYTHON SOURCE LINES 52-55 Then, we identify features X and targets y: the column WAGE is our target variable (i.e., the variable which we want to predict). .. GENERATED FROM PYTHON SOURCE LINES 55-58 .. code-block:: default X = survey.data[survey.feature_names] X.describe(include="all") .. raw:: html
EDUCATION SOUTH SEX EXPERIENCE UNION AGE RACE OCCUPATION SECTOR MARR
count 534.000000 534 534 534.000000 534 534.000000 534 534 534 534
unique NaN 2 2 NaN 2 NaN 3 6 3 2
top NaN no male NaN not_member NaN White Other Other Married
freq NaN 378 289 NaN 438 NaN 440 156 411 350
mean 13.018727 NaN NaN 17.822097 NaN 36.833333 NaN NaN NaN NaN
std 2.615373 NaN NaN 12.379710 NaN 11.726573 NaN NaN NaN NaN
min 2.000000 NaN NaN 0.000000 NaN 18.000000 NaN NaN NaN NaN
25% 12.000000 NaN NaN 8.000000 NaN 28.000000 NaN NaN NaN NaN
50% 12.000000 NaN NaN 15.000000 NaN 35.000000 NaN NaN NaN NaN
75% 15.000000 NaN NaN 26.000000 NaN 44.000000 NaN NaN NaN NaN
max 18.000000 NaN NaN 55.000000 NaN 64.000000 NaN NaN NaN NaN
.. GENERATED FROM PYTHON SOURCE LINES 59-62 Note that the dataset contains categorical and numerical variables. We will need to take this into account when preprocessing the dataset thereafter. .. GENERATED FROM PYTHON SOURCE LINES 62-65 .. code-block:: default X.head() .. raw:: html
EDUCATION SOUTH SEX EXPERIENCE UNION AGE RACE OCCUPATION SECTOR MARR
0 8.0 no female 21.0 not_member 35.0 Hispanic Other Manufacturing Married
1 9.0 no female 42.0 not_member 57.0 White Other Manufacturing Married
2 12.0 no male 1.0 not_member 19.0 White Other Manufacturing Unmarried
3 12.0 no male 4.0 not_member 22.0 White Other Other Unmarried
4 12.0 no male 17.0 not_member 35.0 White Other Other Married
.. GENERATED FROM PYTHON SOURCE LINES 66-68 Our target for prediction: the wage. Wages are described as floating-point number in dollars per hour. .. GENERATED FROM PYTHON SOURCE LINES 68-71 .. code-block:: default y = survey.target.values.ravel() survey.target.head() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0 5.10 1 4.95 2 6.67 3 4.00 4 7.50 Name: WAGE, dtype: float64 .. GENERATED FROM PYTHON SOURCE LINES 72-77 We split the sample into a train and a test dataset. Only the train dataset will be used in the following exploratory analysis. This is a way to emulate a real situation where predictions are performed on an unknown target, and we don't want our analysis and decisions to be biased by our knowledge of the test data. .. GENERATED FROM PYTHON SOURCE LINES 77-84 .. code-block:: default from sklearn.model_selection import train_test_split X_train, X_test, y_train, y_test = train_test_split( X, y, random_state=42 ) .. GENERATED FROM PYTHON SOURCE LINES 85-90 First, let's get some insights by looking at the variable distributions and at the pairwise relationships between them. Only numerical variables will be used. In the following plot, each dot represents a sample. .. _marginal_dependencies: .. GENERATED FROM PYTHON SOURCE LINES 90-95 .. code-block:: default train_dataset = X_train.copy() train_dataset.insert(0, "WAGE", y_train) _ = sns.pairplot(train_dataset, kind='reg', diag_kind='kde') .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_001.png :alt: plot linear model coefficient interpretation :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 96-115 Looking closely at the WAGE distribution reveals that it has a long tail. For this reason, we should take its logarithm to turn it approximately into a normal distribution (linear models such as ridge or lasso work best for a normal distribution of error). The WAGE is increasing when EDUCATION is increasing. Note that the dependence between WAGE and EDUCATION represented here is a marginal dependence, i.e., it describes the behavior of a specific variable without keeping the others fixed. Also, the EXPERIENCE and AGE are strongly linearly correlated. .. _the-pipeline: The machine-learning pipeline ----------------------------- To design our machine-learning pipeline, we first manually check the type of data that we are dealing with: .. GENERATED FROM PYTHON SOURCE LINES 115-118 .. code-block:: default survey.data.info() .. rst-class:: sphx-glr-script-out Out: .. code-block:: none RangeIndex: 534 entries, 0 to 533 Data columns (total 10 columns): # Column Non-Null Count Dtype --- ------ -------------- ----- 0 EDUCATION 534 non-null float64 1 SOUTH 534 non-null category 2 SEX 534 non-null category 3 EXPERIENCE 534 non-null float64 4 UNION 534 non-null category 5 AGE 534 non-null float64 6 RACE 534 non-null category 7 OCCUPATION 534 non-null category 8 SECTOR 534 non-null category 9 MARR 534 non-null category dtypes: category(7), float64(3) memory usage: 17.2 KB .. GENERATED FROM PYTHON SOURCE LINES 119-130 As seen previously, the dataset contains columns with different data types and we need to apply a specific preprocessing for each data types. In particular categorical variables cannot be included in linear model if not coded as integers first. In addition, to avoid categorical features to be treated as ordered values, we need to one-hot-encode them. Our pre-processor will - one-hot encode (i.e., generate a column by category) the categorical columns; - as a first approach (we will see after how the normalisation of numerical values will affect our discussion), keep numerical values as they are. .. GENERATED FROM PYTHON SOURCE LINES 130-143 .. code-block:: default from sklearn.compose import make_column_transformer from sklearn.preprocessing import OneHotEncoder categorical_columns = ['RACE', 'OCCUPATION', 'SECTOR', 'MARR', 'UNION', 'SEX', 'SOUTH'] numerical_columns = ['EDUCATION', 'EXPERIENCE', 'AGE'] preprocessor = make_column_transformer( (OneHotEncoder(drop='if_binary'), categorical_columns), remainder='passthrough' ) .. GENERATED FROM PYTHON SOURCE LINES 144-146 To describe the dataset as a linear model we use a ridge regressor with a very small regularization and to model the logarithm of the WAGE. .. GENERATED FROM PYTHON SOURCE LINES 146-161 .. code-block:: default from sklearn.pipeline import make_pipeline from sklearn.linear_model import Ridge from sklearn.compose import TransformedTargetRegressor model = make_pipeline( preprocessor, TransformedTargetRegressor( regressor=Ridge(alpha=1e-10), func=np.log10, inverse_func=sp.special.exp10 ) ) .. GENERATED FROM PYTHON SOURCE LINES 162-166 Processing the dataset ---------------------- First, we fit the model. .. GENERATED FROM PYTHON SOURCE LINES 166-169 .. code-block:: default _ = model.fit(X_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 170-173 Then we check the performance of the computed model plotting its predictions on the test set and computing, for example, the median absolute error of the model. .. GENERATED FROM PYTHON SOURCE LINES 173-193 .. code-block:: default from sklearn.metrics import median_absolute_error y_pred = model.predict(X_train) mae = median_absolute_error(y_train, y_pred) string_score = f'MAE on training set: {mae:.2f} $/hour' y_pred = model.predict(X_test) mae = median_absolute_error(y_test, y_pred) string_score += f'\nMAE on testing set: {mae:.2f}$/hour' fig, ax = plt.subplots(figsize=(5, 5)) plt.scatter(y_test, y_pred) ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red") plt.text(3, 20, string_score) plt.title('Ridge model, small regularization') plt.ylabel('Model predictions') plt.xlabel('Truths') plt.xlim([0, 27]) _ = plt.ylim([0, 27]) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_002.png :alt: Ridge model, small regularization :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 194-208 The model learnt is far from being a good model making accurate predictions: this is obvious when looking at the plot above, where good predictions should lie on the red line. In the following section, we will interpret the coefficients of the model. While we do so, we should keep in mind that any conclusion we draw is about the model that we build, rather than about the true (real-world) generative process of the data. Interpreting coefficients: scale matters --------------------------------------------- First of all, we can take a look to the values of the coefficients of the regressor we have fitted. .. GENERATED FROM PYTHON SOURCE LINES 208-222 .. code-block:: default feature_names = (model.named_steps['columntransformer'] .named_transformers_['onehotencoder'] .get_feature_names(input_features=categorical_columns)) feature_names = np.concatenate( [feature_names, numerical_columns]) coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_, columns=['Coefficients'], index=feature_names ) coefs .. raw:: html
Coefficients
RACE_Hispanic -0.013564
RACE_Other -0.009120
RACE_White 0.022549
OCCUPATION_Clerical 0.000048
OCCUPATION_Management 0.090531
OCCUPATION_Other -0.025098
OCCUPATION_Professional 0.071967
OCCUPATION_Sales -0.046633
OCCUPATION_Service -0.091050
SECTOR_Construction -0.000180
SECTOR_Manufacturing 0.031273
SECTOR_Other -0.031008
MARR_Unmarried -0.032405
UNION_not_member -0.117154
SEX_male 0.090808
SOUTH_yes -0.033823
EDUCATION 0.054699
EXPERIENCE 0.035005
AGE -0.030867
.. GENERATED FROM PYTHON SOURCE LINES 223-235 The AGE coefficient is expressed in "dollars/hour per living years" while the EDUCATION one is expressed in "dollars/hour per years of education". This representation of the coefficients has the benefit of making clear the practical predictions of the model: an increase of :math:1 year in AGE means a decrease of :math:0.030867 dollars/hour, while an increase of :math:1 year in EDUCATION means an increase of :math:0.054699 dollars/hour. On the other hand, categorical variables (as UNION or SEX) are adimensional numbers taking either the value 0 or 1. Their coefficients are expressed in dollars/hour. Then, we cannot compare the magnitude of different coefficients since the features have different natural scales, and hence value ranges, because of their different unit of measure. This is more visible if we plot the coefficients. .. GENERATED FROM PYTHON SOURCE LINES 235-241 .. code-block:: default coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Ridge model, small regularization') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_003.png :alt: Ridge model, small regularization :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 242-253 Indeed, from the plot above the most important factor in determining WAGE appears to be the variable UNION, even if our intuition might tell us that variables like EXPERIENCE should have more impact. Looking at the coefficient plot to gauge feature importance can be misleading as some of them vary on a small scale, while others, like AGE, varies a lot more, several decades. This is visible if we compare the standard deviations of different features. .. GENERATED FROM PYTHON SOURCE LINES 253-263 .. code-block:: default X_train_preprocessed = pd.DataFrame( model.named_steps['columntransformer'].transform(X_train), columns=feature_names ) X_train_preprocessed.std(axis=0).plot(kind='barh', figsize=(9, 7)) plt.title('Features std. dev.') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_004.png :alt: Features std. dev. :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 264-274 Multiplying the coefficients by the standard deviation of the related feature would reduce all the coefficients to the same unit of measure. As we will see :ref:after this is equivalent to normalize numerical variables to their standard deviation, as :math:y = \sum{coef_i \times X_i} = \sum{(coef_i \times std_i) \times (X_i / std_i)}. In that way, we emphasize that the greater the variance of a feature, the larger the weight of the corresponding coefficient on the output, all else being equal. .. GENERATED FROM PYTHON SOURCE LINES 274-285 .. code-block:: default coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_ * X_train_preprocessed.std(axis=0), columns=['Coefficient importance'], index=feature_names ) coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Ridge model, small regularization') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_005.png :alt: Ridge model, small regularization :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 286-313 Now that the coefficients have been scaled, we can safely compare them. .. warning:: Why does the plot above suggest that an increase in age leads to a decrease in wage? Why the :ref:initial pairplot is telling the opposite? The plot above tells us about dependencies between a specific feature and the target when all other features remain constant, i.e., **conditional dependencies**. An increase of the AGE will induce a decrease of the WAGE when all other features remain constant. On the contrary, an increase of the EXPERIENCE will induce an increase of the WAGE when all other features remain constant. Also, AGE, EXPERIENCE and EDUCATION are the three variables that most influence the model. Checking the variability of the coefficients -------------------------------------------- We can check the coefficient variability through cross-validation: it is a form of data perturbation (related to resampling _). If coefficients vary significantly when changing the input dataset their robustness is not guaranteed, and they should probably be interpreted with caution. .. GENERATED FROM PYTHON SOURCE LINES 313-335 .. code-block:: default from sklearn.model_selection import cross_validate from sklearn.model_selection import RepeatedKFold cv_model = cross_validate( model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5), return_estimator=True, n_jobs=-1 ) coefs = pd.DataFrame( [est.named_steps['transformedtargetregressor'].regressor_.coef_ * X_train_preprocessed.std(axis=0) for est in cv_model['estimator']], columns=feature_names ) plt.figure(figsize=(9, 7)) sns.stripplot(data=coefs, orient='h', color='k', alpha=0.5) sns.boxplot(data=coefs, orient='h', color='cyan', saturation=0.5) plt.axvline(x=0, color='.5') plt.xlabel('Coefficient importance') plt.title('Coefficient importance and its variability') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_006.png :alt: Coefficient importance and its variability :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 336-348 The problem of correlated variables ----------------------------------- The AGE and EXPERIENCE coefficients are affected by strong variability which might be due to the collinearity between the 2 features: as AGE and EXPERIENCE vary together in the data, their effect is difficult to tease apart. To verify this interpretation we plot the variability of the AGE and EXPERIENCE coefficient. .. _covariation: .. GENERATED FROM PYTHON SOURCE LINES 348-358 .. code-block:: default plt.ylabel('Age coefficient') plt.xlabel('Experience coefficient') plt.grid(True) plt.xlim(-0.4, 0.5) plt.ylim(-0.4, 0.5) plt.scatter(coefs["AGE"], coefs["EXPERIENCE"]) _ = plt.title('Co-variations of coefficients for AGE and EXPERIENCE ' 'across folds') .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_007.png :alt: Co-variations of coefficients for AGE and EXPERIENCE across folds :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 359-364 Two regions are populated: when the EXPERIENCE coefficient is positive the AGE one is negative and viceversa. To go further we remove one of the 2 features and check what is the impact on the model stability. .. GENERATED FROM PYTHON SOURCE LINES 364-386 .. code-block:: default column_to_drop = ['AGE'] cv_model = cross_validate( model, X.drop(columns=column_to_drop), y, cv=RepeatedKFold(n_splits=5, n_repeats=5), return_estimator=True, n_jobs=-1 ) coefs = pd.DataFrame( [est.named_steps['transformedtargetregressor'].regressor_.coef_ * X_train_preprocessed.drop(columns=column_to_drop).std(axis=0) for est in cv_model['estimator']], columns=feature_names[:-1] ) plt.figure(figsize=(9, 7)) sns.stripplot(data=coefs, orient='h', color='k', alpha=0.5) sns.boxplot(data=coefs, orient='h', color='cyan', saturation=0.5) plt.axvline(x=0, color='.5') plt.title('Coefficient importance and its variability') plt.xlabel('Coefficient importance') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_008.png :alt: Coefficient importance and its variability :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 387-401 The estimation of the EXPERIENCE coefficient is now less variable and remain important for all models trained during cross-validation. .. _scaling_num: Preprocessing numerical variables --------------------------------- As said above (see ":ref:the-pipeline"), we could also choose to scale numerical values before training the model. This can be useful to apply a similar amount regularization to all of them in the Ridge. The preprocessor is redefined in order to subtract the mean and scale variables to unit variance. .. GENERATED FROM PYTHON SOURCE LINES 401-410 .. code-block:: default from sklearn.preprocessing import StandardScaler preprocessor = make_column_transformer( (OneHotEncoder(drop='if_binary'), categorical_columns), (StandardScaler(), numerical_columns), remainder='passthrough' ) .. GENERATED FROM PYTHON SOURCE LINES 411-412 The model will stay unchanged. .. GENERATED FROM PYTHON SOURCE LINES 412-424 .. code-block:: default model = make_pipeline( preprocessor, TransformedTargetRegressor( regressor=Ridge(alpha=1e-10), func=np.log10, inverse_func=sp.special.exp10 ) ) _ = model.fit(X_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 425-428 Again, we check the performance of the computed model using, for example, the median absolute error of the model and the R squared coefficient. .. GENERATED FROM PYTHON SOURCE LINES 428-447 .. code-block:: default y_pred = model.predict(X_train) mae = median_absolute_error(y_train, y_pred) string_score = f'MAE on training set: {mae:.2f} $/hour' y_pred = model.predict(X_test) mae = median_absolute_error(y_test, y_pred) string_score += f'\nMAE on testing set: {mae:.2f}$/hour' fig, ax = plt.subplots(figsize=(6, 6)) plt.scatter(y_test, y_pred) ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red") plt.text(3, 20, string_score) plt.title('Ridge model, small regularization, normalized variables') plt.ylabel('Model predictions') plt.xlabel('Truths') plt.xlim([0, 27]) _ = plt.ylim([0, 27]) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_009.png :alt: Ridge model, small regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 448-449 For the coefficient analysis, scaling is not needed this time. .. GENERATED FROM PYTHON SOURCE LINES 449-459 .. code-block:: default coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_, columns=['Coefficients'], index=feature_names ) coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Ridge model, small regularization, normalized variables') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_010.png :alt: Ridge model, small regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 460-461 We now inspect the coefficients across several cross-validation folds. .. GENERATED FROM PYTHON SOURCE LINES 461-478 .. code-block:: default cv_model = cross_validate( model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5), return_estimator=True, n_jobs=-1 ) coefs = pd.DataFrame( [est.named_steps['transformedtargetregressor'].regressor_.coef_ for est in cv_model['estimator']], columns=feature_names ) plt.figure(figsize=(9, 7)) sns.stripplot(data=coefs, orient='h', color='k', alpha=0.5) sns.boxplot(data=coefs, orient='h', color='cyan', saturation=0.5) plt.axvline(x=0, color='.5') plt.title('Coefficient variability') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_011.png :alt: Coefficient variability :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 479-492 The result is quite similar to the non-normalized case. Linear models with regularization --------------------------------- In machine-learning practice, Ridge Regression is more often used with non-negligible regularization. Above, we limited this regularization to a very little amount. Regularization improves the conditioning of the problem and reduces the variance of the estimates. RidgeCV applies cross validation in order to determine which value of the regularization parameter (alpha) is best suited for prediction. .. GENERATED FROM PYTHON SOURCE LINES 492-506 .. code-block:: default from sklearn.linear_model import RidgeCV model = make_pipeline( preprocessor, TransformedTargetRegressor( regressor=RidgeCV(alphas=np.logspace(-10, 10, 21)), func=np.log10, inverse_func=sp.special.exp10 ) ) _ = model.fit(X_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 507-508 First we check which value of :math:\alpha has been selected. .. GENERATED FROM PYTHON SOURCE LINES 508-511 .. code-block:: default model[-1].regressor_.alpha_ .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 10.0 .. GENERATED FROM PYTHON SOURCE LINES 512-513 Then we check the quality of the predictions. .. GENERATED FROM PYTHON SOURCE LINES 513-533 .. code-block:: default y_pred = model.predict(X_train) mae = median_absolute_error(y_train, y_pred) string_score = f'MAE on training set: {mae:.2f} $/hour' y_pred = model.predict(X_test) mae = median_absolute_error(y_test, y_pred) string_score += f'\nMAE on testing set: {mae:.2f}$/hour' fig, ax = plt.subplots(figsize=(6, 6)) plt.scatter(y_test, y_pred) ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red") plt.text(3, 20, string_score) plt.title('Ridge model, regularization, normalized variables') plt.ylabel('Model predictions') plt.xlabel('Truths') plt.xlim([0, 27]) _ = plt.ylim([0, 27]) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_012.png :alt: Ridge model, regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 534-536 The ability to reproduce the data of the regularized model is similar to the one of the non-regularized model. .. GENERATED FROM PYTHON SOURCE LINES 536-546 .. code-block:: default coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_, columns=['Coefficients'], index=feature_names ) coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Ridge model, regularization, normalized variables') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_013.png :alt: Ridge model, regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 547-560 The coefficients are significantly different. AGE and EXPERIENCE coefficients are both positive but they now have less influence on the prediction. The regularization reduces the influence of correlated variables on the model because the weight is shared between the two predictive variables, so neither alone would have strong weights. On the other hand, the weights obtained with regularization are more stable (see the :ref:ridge_regression User Guide section). This increased stability is visible from the plot, obtained from data perturbations, in a cross validation. This plot can be compared with the :ref:previous one. .. GENERATED FROM PYTHON SOURCE LINES 560-581 .. code-block:: default cv_model = cross_validate( model, X, y, cv=RepeatedKFold(n_splits=5, n_repeats=5), return_estimator=True, n_jobs=-1 ) coefs = pd.DataFrame( [est.named_steps['transformedtargetregressor'].regressor_.coef_ * X_train_preprocessed.std(axis=0) for est in cv_model['estimator']], columns=feature_names ) plt.ylabel('Age coefficient') plt.xlabel('Experience coefficient') plt.grid(True) plt.xlim(-0.4, 0.5) plt.ylim(-0.4, 0.5) plt.scatter(coefs["AGE"], coefs["EXPERIENCE"]) _ = plt.title('Co-variations of coefficients for AGE and EXPERIENCE ' 'across folds') .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_014.png :alt: Co-variations of coefficients for AGE and EXPERIENCE across folds :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 582-593 Linear models with sparse coefficients -------------------------------------- Another possibility to take into account correlated variables in the dataset, is to estimate sparse coefficients. In some way we already did it manually when we dropped the AGE column in a previous Ridge estimation. Lasso models (see the :ref:lasso User Guide section) estimates sparse coefficients. LassoCV applies cross validation in order to determine which value of the regularization parameter (alpha) is best suited for the model estimation. .. GENERATED FROM PYTHON SOURCE LINES 593-607 .. code-block:: default from sklearn.linear_model import LassoCV model = make_pipeline( preprocessor, TransformedTargetRegressor( regressor=LassoCV(alphas=np.logspace(-10, 10, 21), max_iter=100000), func=np.log10, inverse_func=sp.special.exp10 ) ) _ = model.fit(X_train, y_train) .. GENERATED FROM PYTHON SOURCE LINES 608-609 First we verify which value of :math:\alpha has been selected. .. GENERATED FROM PYTHON SOURCE LINES 609-612 .. code-block:: default model[-1].regressor_.alpha_ .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 0.001 .. GENERATED FROM PYTHON SOURCE LINES 613-614 Then we check the quality of the predictions. .. GENERATED FROM PYTHON SOURCE LINES 614-634 .. code-block:: default y_pred = model.predict(X_train) mae = median_absolute_error(y_train, y_pred) string_score = f'MAE on training set: {mae:.2f} $/hour' y_pred = model.predict(X_test) mae = median_absolute_error(y_test, y_pred) string_score += f'\nMAE on testing set: {mae:.2f}$/hour' fig, ax = plt.subplots(figsize=(6, 6)) plt.scatter(y_test, y_pred) ax.plot([0, 1], [0, 1], transform=ax.transAxes, ls="--", c="red") plt.text(3, 20, string_score) plt.title('Lasso model, regularization, normalized variables') plt.ylabel('Model predictions') plt.xlabel('Truths') plt.xlim([0, 27]) _ = plt.ylim([0, 27]) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_015.png :alt: Lasso model, regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 635-636 For our dataset, again the model is not very predictive. .. GENERATED FROM PYTHON SOURCE LINES 636-646 .. code-block:: default coefs = pd.DataFrame( model.named_steps['transformedtargetregressor'].regressor_.coef_, columns=['Coefficients'], index=feature_names ) coefs.plot(kind='barh', figsize=(9, 7)) plt.title('Lasso model, regularization, normalized variables') plt.axvline(x=0, color='.5') plt.subplots_adjust(left=.3) .. image:: /auto_examples/inspection/images/sphx_glr_plot_linear_model_coefficient_interpretation_016.png :alt: Lasso model, regularization, normalized variables :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 647-672 A Lasso model identifies the correlation between AGE and EXPERIENCE and suppresses one of them for the sake of the prediction. It is important to keep in mind that the coefficients that have been dropped may still be related to the outcome by themselves: the model chose to suppress them because they bring little or no additional information on top of the other features. Additionnaly, this selection is unstable for correlated features, and should be interpreted with caution. Lessons learned --------------- * Coefficients must be scaled to the same unit of measure to retrieve feature importance. Scaling them with the standard-deviation of the feature is a useful proxy. * Coefficients in multivariate linear models represent the dependency between a given feature and the target, **conditional** on the other features. * Correlated features induce instabilities in the coefficients of linear models and their effects cannot be well teased apart. * Different linear models respond differently to feature correlation and coefficients could significantly vary from one another. * Inspecting coefficients across the folds of a cross-validation loop gives an idea of their stability. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 11.356 seconds) .. _sphx_glr_download_auto_examples_inspection_plot_linear_model_coefficient_interpretation.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-learn/scikit-learn/main?urlpath=lab/tree/notebooks/auto_examples/inspection/plot_linear_model_coefficient_interpretation.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: plot_linear_model_coefficient_interpretation.py .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: plot_linear_model_coefficient_interpretation.ipynb .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _
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# Distance between Qingdao (TAO) and Yibin (YBP)
Flight distance from Qingdao to Yibin (Qingdao Liuting International Airport – Yibin Wuliangye Airport) is 1054 miles / 1697 kilometers / 916 nautical miles. Estimated flight time is 2 hours 29 minutes.
Driving distance from Qingdao (TAO) to Yibin (YBP) is 1275 miles / 2052 kilometers and travel time by car is about 21 hours 40 minutes.
## Map of flight path and driving directions from Qingdao to Yibin.
Shortest flight path between Qingdao Liuting International Airport (TAO) and Yibin Wuliangye Airport (YBP).
## How far is Yibin from Qingdao?
There are several ways to calculate distances between Qingdao and Yibin. Here are two common methods:
Vincenty's formula (applied above)
• 1054.376 miles
• 1696.854 kilometers
• 916.228 nautical miles
Vincenty's formula calculates the distance between latitude/longitude points on the earth’s surface, using an ellipsoidal model of the earth.
Haversine formula
• 1053.358 miles
• 1695.215 kilometers
• 915.343 nautical miles
The haversine formula calculates the distance between latitude/longitude points assuming a spherical earth (great-circle distance – the shortest distance between two points).
## Airport information
A Qingdao Liuting International Airport
City: Qingdao
Country: China
IATA Code: TAO
ICAO Code: ZSQD
Coordinates: 36°15′57″N, 120°22′26″E
B Yibin Wuliangye Airport
City: Yibin
Country: China
IATA Code: YBP
ICAO Code: ZUYB
Coordinates: 28°51′28″N, 104°31′30″E
## Time difference and current local times
There is no time difference between Qingdao and Yibin.
CST
CST
## Carbon dioxide emissions
Estimated CO2 emissions per passenger is 154 kg (340 pounds).
## Frequent Flyer Miles Calculator
Qingdao (TAO) → Yibin (YBP).
Distance:
1054
Elite level bonus:
0
Booking class bonus:
0
### In total
Total frequent flyer miles:
1054
Round trip?
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# Analytic combinatorics by Flajolet P., Sedgewick R.
By Flajolet P., Sedgewick R.
Similar combinatorics books
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Extra resources for Analytic combinatorics
Example text
For instance, the notation (23) S EQ=k (or simply S EQk ), S EQ>k , S EQ1 . k refers to sequences whose number of components are exactly k, larger than k, or in the interval 1 . k respectively. In particular, k times S EQk (B) := B × · · · × B ≡ B k , S EQ≥k (B) = j≥k Bj ∼ = B k × S EQ(B), MS ETk (B) := S EQk (B)/R. Similarly, S EQodd , S EQeven will denote sequences with an odd or even number of components, and so on. 30 I. COMBINATORIAL STRUCTURES AND ORDINARY GENERATING FUNCTIONS Translations for such restricted constructions are available, as shown generally in Subsection I.
0, β j ∈ B , which matches our intuition as to what sequences should be. ) It is then readily checked that the construction A = S EQ(B) defines a proper class satisfying the finiteness condition for sizes if and only if B contains no object of size 0. From the definition of size for sums and products, it I. 2. ADMISSIBLE CONSTRUCTIONS AND SPECIFICATIONS 25 follows that the size of an object α ∈ A is to be taken as the sum of the sizes of its components: α = (β1 , . . , βℓ ) ⇒ |α| = |β1 | + · · · + |βℓ |.
Consider the class U of “non-empty” triangulations of the n-gon, that is, we exclude the 2-gon and the corresponding “empty” triangulation of size 0. Then U = T \ {ǫ} admits the specification U = ∇ + (∇ × U) + (U × ∇) + (U × ∇ × U) which also leads to the Catalan numbers via U = z(1 + U )2 , so that U (z) = (1 − 2z − √ 1 − 4z)/(2z) ≡ T (z) − 1. ✁ I. 4. Exploiting generating functions and counting sequences. In this book we are going to see altogether more than a hundred applications of the symbolic method.
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0
You visited us 0 times! Enjoying our articles? Unlock Full Access!
Question
# (2a+3b)2=
A
4a2+9b2+12ab
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B
4a3+9b2+12ab
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C
4a2+9b2+36ab
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D
4a2+9b3+12ab
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Solution
## The correct option is A 4a2+9b2+12ab Using the identity (a+b)2=a2+b2+2ab, we get, (2a+3b)2=(2a)2+(3b)2+2×2a×3b =4a2+9b2+12ab
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# C++ program to find the Area of the circumcircle of any triangles with sides given?
To calculate the area of circumcircle of any triangles. We need to learn about basic concepts related to the problem.
Triangle − A closed figure with three sides.
Circle − A closed figure with infinite number or side or no sides.
A circle that encloses other figure inside it is a circumcircle.
A circumcircle touches the triangle from all its points. Lets say its sides are a, b, c then the radius of the circumcircle is given by the mathematical formula −
r = abc / (√((a+b+c))(a+b-c)(a+c-b)(b+c-a)))
The area of the circle with radius r is
area = 2 * (pie) * r *r.
Let’s take a few examples for this concept −
Sides of triangle : a = 4 , b = 5 , c =3
Area = 314
## Example
Live Demo
#include <iostream>
#include <math.h>
using namespace std;
int main() {
float a = 7, b = 9, c = 13;
if (a < 0 || b < 0 || c < 0)
cout<<"The figure is not a triangle";
float p = (a + b + c) / 2;
float r = (a*b*c)/ (sqrt(p * (p - a) * (p - b) * (p - c)));
float area = 3.14 * pow(r, 2);
cout<<"The area is "<<area;
return 0;
}
## Output
The area is 2347.55
Updated on: 04-Oct-2019
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## Calculate Linear Inches For Luggage? - Packing Made Easy
Before considering how to calculate linear inches for luggage, let’s actually know what are linear inches.
The term linear inches is a measurement used to determine the size of luggage. You can calculate it by adding the dimensions like length, width, and depth of your luggage in inches.
This measurement is used by airlines and other transit companies. With it, they can know if the luggage meets the size requirements for checked or carry-on baggage.
## Calculation Process
Each airline has different requirements in the case of luggage size. So, firstly make sure to check with your specific air carrier that your luggage meets their size conditions. Airlines limit the size and weight of luggage you can take on board. They have their reasons for it like:
• To ensure that the overhead compartments and storage areas are not overcrowded.
• To increase the safety and efficiency of the aircraft.
You can enjoy a hassle-free trip by measuring your luggage on your journey. Follow the steps below to calculate the linear inches for luggage.
• Take a regular measuring tape and start from the longest point of your bag or suitcase. You also have to include any wheels, handles, or other protrusions of your luggage in the measurement.
• Once you have got the measurements, start adding the height, width, and length together. For example, if your luggage measures 28 inches in length, 18 inches in width, and 12 inches in height.
Then, the linear inches will be calculated as 28 + 15 + 12 = 58 linear inches.
### Most Common Restrictions For Luggage
Every air carrier divides the luggage into three categories - personal items, carry-on bags, and checked luggage. Size restrictions vary from airline to airline. But, I have jotted down some common restrictions for luggage.
• Personal items: Most airlines allow you to carry one personal item like a purse or briefcase on the flight. Occasionally, few airlines require your personal items to be under 16 × 12 × 6 inches. But many of them accept it to just fit entirely underneath your front seat.
• Carry-ons: The carry-on bag which you can take on board your flight with most air carriers has to be under 22 × 14 × 9. US airlines usually don’t have weight limits for carry-on bags.
• Checked luggage: Almost all airlines follow the same size restriction for checked baggage. A limit of 62 linear inches is the standard.
Note: Allegiant Airlines allows you to bring a checked suitcase of 80 linear inches on your journey. Book your flight with Allegiant for cheap fares and extra size limits.
### Consequences Of Oversize Luggage
If your luggage exceeds the size limits of your airline, you may be subject to additional fees or restrictions.
You have to pay extra money to some airlines if your luggage is oversized. Check your airline’s fee structure before booking your flight. The digits can multiply into triples as a fee for oversized bags.
Some airlines would require you to check your luggage instead of carrying it on board if it is more than the size limit of carry-on baggage.
### Some Useful Insights
The fee prices for extra or oversize baggage depend on your choice of airline and your level of ticket. Your first checked bag is usually free in basic or economy fare as long as it meets the size requirements.
Know about the charges for oversize baggage of Southwest and Frontier Airlines below.
Southwest Airlines' size and weight restriction is a combination of 62 linear inches. If you are traveling with this air carrier and the limits increase at any chance. You have to pay an additional fee of a minimum of \$50 per bag.
You are flying with Frontier Airlines and the size limit of your checked bag is more than 62 linear inches. You don’t want to take anything out of your bag. Then, you have to pay an amount of \$75 for oversized luggage.
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http://www.softwareandfinance.com/CSharp/QuickSort_Iterative.html
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# C# - Sorting Algorithm - QuickSort Iterative
We often using sorting algorithm to sort numbers and strings. Also we have many sorting algorithms. I have explained here on how quicksort algorithm works in iterative mode.
For each time when partition method is called, the pivot is placed at the correct position meaning all the elements to the left are less than the pivot value and all the elements to right are greater than the pivot value.
The iteration approach requires stacking of the beg(left), end(right) positions that are used in recursion. I have used the List to store the values and made sure the loop is getting repeated until List is empty. For Partition, we need two arguments - left and right.
The complete program and test run output are given below:
## Source Code
using System;
using System.Collections.Generic;
using System.Text;
namespace CSharpSort
{
class Program
{
static public int Partition(int [] numbers, int left, int right)
{
int pivot = numbers[left];
while (true)
{
while (numbers[left] < pivot)
left++;
while (numbers[right] > pivot)
right--;
if (left < right)
{
int temp = numbers[right];
numbers[right] = numbers[left];
numbers[left] = temp;
}
else
{
return right;
}
}
}
struct QuickPosInfo
{
public int left;
public int right;
};
static public void QuickSort_Iterative(int [] numbers, int left, int right)
{
if(left >= right)
return; // Invalid index range
List<QuickPosInfo> list = new List<QuickPosInfo>();
QuickPosInfo info;
info.left = left;
info.right = right;
list.Insert(list.Count, info);
while(true)
{
if(list.Count == 0)
break;
left = list[0].left;
right = list[0].right;
list.RemoveAt(0);
int pivot = Partition(numbers, left, right);
if(pivot > 1)
{
info.left = left;
info.right = pivot - 1;
list.Insert(list.Count, info);
}
if(pivot + 1 < right)
{
info.left = pivot + 1;
info.right = right;
list.Insert(list.Count, info);
}
}
}
static void Main(string[] args)
{
int[] numbers = { 3, 8, 7, 5, 2, 1, 9, 6, 4 };
int len = 9;
Console.WriteLine("QuickSort By Iterative Method");
QuickSort_Iterative(numbers, 0, len - 1);
for (int i = 0; i < 9; i++)
Console.WriteLine(numbers[i]);
}
}
}
## Output
QuickSort By Iterative Method
1
2
3
4
5
6
7
8
9
Press any key to continue . . .
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https://brainmass.com/business/branding/where-a-competing-company-should-spend-40-billion-447656
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Share
Explore BrainMass
Where a Competing Company Should Spend \$40 Billion
Corben Inc. has a successful brand with the name Crunz. The market size in which Crunz competes is \$4 billion, and Crunz has generated sales of \$400 million. It has a contribution margin of 30% and annual fixed costs of \$20 million. Corben Inc. is thinking of introducing a new brand under the name of Zaturn. Zaturn will compete in the same market as Crunz. The annual fixed costs for this brand are expected to be \$40 million.
If it is launched, Zaturn will capture 10% of the market. It has a contribution margin of 40%. Half of the sales of Zaturn will be cannibalized from the sales of Crunz. An alternative strategy for Corben Inc. is to cancel the introduction of Zaturn and instead to spend the \$40 million (on an annual basis) to promote Crunz. This action is expected to increase the sales for Crunz by 50%. Both brands (Cruz and Zaturn) sell at the same price.
Where should the company spend the \$40 million and why? Show all calculations!
Solution Preview
Total profit from Crunz:
400 *30%- 120 million less fixed costs of 20 million or 100 million dollars.
If Zaturn is launched:
Sales (10% of 4 billion) or 400 million
Profit from ...
Solution Summary
The solution advises where the company should spend \$40 million and why in 126 words with all calculations shown.
\$2.19
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https://palass.org/publications/newsletter/palaeomath-101/palaeomath-part-9-data-blocks-and-partial-least-squares-analysis
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# PalaeoMath: Part 9 - Data Blocks and Partial Least Squares Analysis
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## 9. Data Blocks and Partial Least Squares Analysis
Written by Norm MacLeod - The Natural History Museum, London, UK (email: n.macleod@nhm.ac.uk). This article first appeared in the Nº 63 edition of Palaeontology Newsletter.
Note: This article has not been updated to the new website style, using html tables rather than embeded images, there may be presentation issues.
### Introduction
In the last four columns we've looked at problems associated with characterizing and identifying patterns in single datasets. An implicit assumption that runs across all the methods we've discussed so far (bivariate regression, multivariate regression, PCA, Factor Analysis, PCOORD, and correspondence analysis) is that the objects included in the dataset represent independent and randomly selected samples drawn from a population of interest. Using our trilobite dataset as an example, if we are asking questions about this particular assemblage of 20 trilobite genera the results we have obtained to date are perfectly valid. However, it's a big world out there and we'd often like to know how one type of data relates to another type of data. For example, in all but the last of these columns we were concerned with the analysis of simple morphological data. We first considered bivariate data (the linear regression columns), but expanded that to a (still simple) three-variable system when we came to our discussions of the various single-sample multivariate methods. Then, in the last column I wanted to show how another type of data might be handled and so introduced some ecological data in the form of hypothetical frequency counts of these 20 genera in different environments. I'd now like to ask the next most obvious question 'What can we do if we want to explore how the morphological variables relate to the ecological variables for these taxa?'.
As a matter of fact we've already discussed one approach of this situation: what to do if we want to relate one variable to a suite of others. In that case the appropriate approach is multiple regression. Using this method the pattern of linear variation in a dependent variable (e.g., a morphological variable) can be compared to linear patterns of variation in a suite of independent variables (e.g., ecological variables). The purpose of such an analysis would be to (1) assess the overall significance of the various linear relations between the dependent and independent variables and (2) obtain information about the structure of those relations (e.g., which independent variables show the strongest patterns of covariation; which the least). But this method only yields information for one dependent variable at a time. What if we want to assess the significance and structure of covariation for two different multivariate blocks of variables?
There are two approaches for addressing this data analysis situation: canonical correlation analysis (CCA) and partial least squares (PLS) analysis. The former has been around for some time while the latter is something of a new kid on the data-analysis block. I've always found it curious that neither has figured prominently in palaeontological analyses to date, though canonical correlation has been used for many years by ecologists, economists, psychometricians, and a host of others, while PLS made its impact felt first in the field of chemometrics. I think part of the problem has been that CCA requires the algebraic manipulation of complex, non-symmetric matrices that are beyond the capabilities of hand calculators and even simple spreadsheet programmes. Canonical correlation routines are also somewhat rare in various so-called 'canned' computer packages, though they are straightforward to programme in high-level computer languages or using tools such as Mathematica, Maple or MatLab. In this essay, we'll focus on PLS, in part because it's computationally simpler and illustrates many of the same principles as CCA, but mostly because it has several distinct advantages over CCA. Both methods deserve to be used much more widely in palaeontology.
First, let's review our data. You'll remember the trilobite morphological data, three variables measured on a suite of 20 trilobite specimens (Table 9.1).
Those following closely will also recall the hypothetical trilobite occurrence frequency data from a suite of seven facies arrayed along a crude onshore-offshore gradient (Table 9.2).
One of the purposes of using the frequency data in our previous discussion of correspondence analysis was to illustrate the superior data handing capabilities of that method. The scaling procedures inherent in correspondence analysis mean essentially any type of data can be submitted to this procedure. Partial least squares analysis is also a generalized descriptive technique and so makes no particular distributional assumptions about the data. Nevertheless, this seems as good a place as any to point out that all descriptive methods work better if the data exhibit some similarity to a normal distribution. Counts are always suspect from a distributional point of view because they typically follow a Poisson distribution (see Fig. 9.1A). Since we'll be making use of the correlation relation in our PLS analysis, and since correlations can be badly biased by outliers, I've transformed the ecological data using a variant of Bartlett's (1936) square-root transformation to make them more normal (Fig. 9.1B). The morphological data were also transformed by taking the log10 of their values since it is well known that this transformation makes variables more linear and removes any correlation between the variance and the mean (see the 'Data Blocks' worksheet of the PalaeoMath 101 spreadsheet for these transformed matrices).
Figure 9.1. Trilobite frequency count data prior to (A) and after (B) transformation by the equation $y=sqrt{x+0.3}$, which is variation of the Bartlett (1936) square-root transformation. Note the similarity of A to a Poisson distribution. Strictly speaking the transformation only made these data more normal (as they still do not conform to a normal distribution) but it did improve the balance of the distribution markedly and reduced the number of outlying values.
Now that we have our data in appropriate shape it's time to talk about the comparisons we want to make. PLS has many similarities to PCA, one of which is that you can base the analysis on either the covariance or correlation matrices. For these data the correlation matrix is preferred because the different data groups have different units and characteristically different magnitudes (see the Data Blocks worksheet). As with PCA, you need to consider what basis matrix to use carefully. A covariance matrix is preferred if scaling differences among the variables is something you want the data analysis to take into consideration. For example, if these were two different groups of morphometric variables and one (say the head variables) were characteristically larger than then other (say the tail variables), I might want to include this distinction in the analysis. If I chose to base my PLS analysis on the covariance matrix of raw (though transformed) values, the results would be implicitly weighted toward the larger (= more variable) head variables. On the other hand, if I didn't want these distinctions to affect the results of my analysis I'd want to standardize all my data first so the variances for all variables would be equal, in which case I'd be using a correlation matrix as the basis for my analysis. This standardized covariance, or correlation, matrix for the combined trilobite morphological and ecological variables is shown in Table 9.3.
By now you should be familiar with the general form of a correlation matrix (see the PalaeoMath 101 column in Newsletter 58 for a review). The composite matrices we use for PLS analyses are, however, a bit different. On first inspection they might look like perfectly normal correlation matrices. The diagonal is filled with 1's and the upper and lower parts are mirror images of one another. We could analyze the whole matrix and get a perfectly respectable PCA result. The difference, though lies in the fact that we know there are two different blocks of data here—the morphometric variable block and the ecological variable block. We also know that we're only interested in examining the inter-relations between these data blocks. This knowledge changes everything. Diagrammatically we can represent this block-level structure of Table 9.3 as follows.
$R_{11}$ $R_{12}$ $R_{21}$ $R_{22}$
Here $R_{11}$ refers to the $3\times3$ data block containing just the three morphological variables, $R_{22}$ refers to the $7\times7$ block containing just the seven ecological variables. Both $R_{21}$ and $R_{21}$ refer to the block containing the $3\times7$ (or $7\times3$) cross-correlation between the morphological and ecological variables with $R_{21}$ being a simple transposition of $R^{12}$ (and vice versa). Two-block PLS analysis foregoes all consideration of blocks $R_{11}$ and $R_{22}$ in favour of focusing on block $R_{12}$. In effect, our PLS analysis will be an eigenanalysis of only that part of the basis matrix both groups share. Table 9.4 shows just this section of Table 9.3.
Note this is a different type of matrix from those we've seen before. It's not square because there are many more columns than rows and it's not symmetric because the two halves of the matrix across the diagonal aren't mirror images of one another. Indeed, there isn't even a diagonal to this matrix! Although this is a common type of matrix, we can't use regular eigenanalysis to decompose it into different modes of variation. That method only works on symmetric, square matrices. Never to fear though; methods have been devised to handle this situation. As a matter of fact, you've already been introduced to the primary method for handling this matrix if you read last issue's column. Singular value decomposition (SVD) rescues us again!
Recall last time we used SVD to perform simultaneous Q-mode and R-mode analyses of the square, symmetric, $x^2$ distance matrix we used as the basis for our example correspondence analysis. That proved a convenient way to represent simultaneous ordinations of objects and variables. Recall also that SVD is an implementation of the Ekhart-Young theorem, which states that for any real matrix X, two matrices, V and U, can be found whose minor products are the identity matrix. This means matrices V and U are composed of vectors arranged at right angles to each other. These matrices are scaled to the original data (X) by matrix W, which is a matrix whose diagonal contains a set of terms called 'singular values' with all off-diagonal elements set to zero. These singular values are the square roots of the eigenvalues of both the V and the U matrices, which are identical for all non-zero singular values. Thus,
$X = VWU'$
Each eigenvalue represents an axis through the data cloud aligned with the major directions of variation. Since there are three morphological variables ($p$) and seven ecological variables ($q$) there will only be $p$ non-zero singular values (since $p<q$). Matrix V contains the R-mode loadings, which are the patterns of weights (covariance basis matrix) or angles (correlation basis matrix) that specify the directional relation between these new axes and the Q-mode variables. Matrix U' is the transpose of the Q-mode saliences (see below). Here's the bit that concerns us today, however. The Ekhart-Young theorem states the relation is true for any matrix of any shape and/or character, not just square, symmetric matrices. Table 5 shows the singular values and eigenvalues of the $R_{12}$ data block (see Table 9.3).
These were calculated using the PopTools plug-in for Excel (PC version only). As you can see, from a geometric point-of-view, this cross-variable matrix is highly elongate with very small minor axes. But remember, this is only one block of the overall matrix. Since this is a correlation matrix, we know its total variance is the sum of the number of morphological and ecological variables ($p+q=10$). Thus, this data block—or more correctly, the cross-variable substructure of the overall correlation matrix—accounts for only 17.56 percent of the total variance. Nevertheless, this is the substructure in which we are interested.
For our example analysis the directional vectors are given in Table 9.6 in their normalized (left) and scaled (right) forms. The normalized form is the most convenient for interpretation as the squares of the values always add up to 1.00. The scaled form is calculated by multiplying the normalized vector coefficients by the appropriate singular value. This operation restores the differences between the scale of the vectors.
These vectors look superficially like principal components, but there's an important difference. Whereas the coefficients or 'loadings' of principal component eigenvectors represent the angular relation between the principal component axes and the original variables, the coefficients of a PLS analysis represent the angular relations of the variables within one data block with respect to those in the other data block. In a sense they represent the variables that are most useful or salient for predicting patterns in the other data block. For this reason they are referred to as saliences.
Turning to an interpretation of these data we first need to ask ourselves how many singular values to interpret. We can approach this using the various qualitative methods discussed in the column on PCA (see the Palaeo-Math 101 column in Newsletter 58) or we can use a more sophisticated, quantitative approach that is has been developed recently for use in generalized multivariate analysis (see Morrison 2004, Zelditch et al. 2004).
$x^2=-n\displaystyle\sum_{j=1}^{r}1n\lambda_j+nr\Big(\displaystyle\sum_{j=1}^{r}\lambda_j\Big/r\Big)$
In this equation $x^2$ is the $x^2$ statistic, n is the number of objects in the sample minus 1, $r$ is the number of eigenvalues being tested and $\lambda j$ is the $j^{th}$ singular value. In its typical analytic mode singular values are tested in sequence two at a time (e.g., 1-2, 2-3, 3-4) to determine whether there is a statistically significant amount of variance being explained by the former member of the pair. For this type of test the value of the degrees of freedom is 2. For the comparison between the first and second singular values in the example analysis $x^2 = 15.196$, which means the first singular value is highly significant ($= 0.0005$) as you would expect from the high proportion of variance it explains (see Table 5). When we interpret this axis (Table 6) we see all the R-mode saliences are positive suggesting this is an allometric size axis with glabellar length exhibiting the strongest positive allometry. Environmentally, this allometric size vector is correlated most positively with the black shale facies and most negatively with the paralic shale facies, which are the deepest and shallowest environments in our ecological dataset. This is highly suggestive of a possible shallow-deep or onshore-offshore environmental gradient. Further analysis of the patterns of salience coefficients (Fig. 9.2) shows that, although the relation between size and a depth-shoreline proximity gradient is not strictly consistent, there is more than a hint this general correlation being a major source of patterning in these data.
Figure 9.2. Plot of salience coefficients for the environmental hypothetical variables used in the example analysis. While the trend in these data does not conform strictly to an onshore-offshore gradient, and is not strictly linear, there is a strong suggestion that depth-shoreline proximity is an important source of structure in the R12 block of the correlation matrix. This pattern is associated with strong and uniformly positive salience coefficients for the morphological variables (see Table 6) indicating that this depth-shoreline proximity factor is associated morphologically with an allometric size gradient. See text for discussion.
The strength of the relation between the morphological and environmental variables can also be assessed through a simple graphical device. Since we have the R-mode and Q-mode vector for the cross-variable data block we can calculate the R-mode and Q-mode scores in a manner identical to that for PCA. Table 9.7 shows these scores while Figure 9.3 plots them in a simple bivariate ordination space.
Figure 9.3. Scatterplot of PLS-1 (morphological variables) and PLS-1 (environmental variables) scores for example PLS analysis. This plot represents 97.69% of the correlation structure within the R12 data block.
Comparison of the ordination shown in Figure 3 confirms our interpretation of these results based on the V and U salience matrices. Note large-sized genera (e.g., Trimerus, Zacanthoides, Pricyclopyge, see Table 1) plot toward the upper end of PLS-1 (morphological variables) axis and small-sized genera (e.g., Acaste, Balizoma, Ormathops) toward the lower end, confirming that this axis expresses a generalized size gradient. Moreover, these two groups of genera also display strikingly different environmental occurrence patterns along the PLS-1 (ecological variables) axis with the larger-sized forms being differentially abundant in deep-water facies (see Table 2) and smaller-sized forms preferring shallow-water facies. The linear correlation between the two PLS-1 scores is 0.445, which is just significant statistically for this sample ($r_{crit., d.f. = 19, a = 0.05} = 0.433$) . Based on these results I wouldn't necessarily conclude that size-environment link represents the whole biological story for these data (e.g., the shallow water fauna is composed of mixed small and intermediated sized genera), but this is the strongest, single, linear signal in these data. More importantly for the purposes of this column, by using two-block PLS we've managed to examine the inter-relations between two datasets we've had to treat either separately or as parts of a larger analysis up to this point, and in doing this we've discovered a new patterns in these data that had been hiding there all along.
Partial least squares analysis represents a very powerful and completely generalized approach to ordination and statistical hypothesis testing. Based on a form of PCA, it extends multiple regression analysis, complements canonical correlation analysis, and allows users to test hypotheses about the inter-relations between blocks of observations made on the same objects. Unlike standard PCA which can use a variety of algorithmic approaches to obtain the eigenvalues and eigenvectors of a square, symmetric basis matrix, PLS employs singular value decomposition to obtain the singular values (square roots of eigenvalues) and eigenvectors of parts of PCA basis matrices which may or may not be square, and which will not be symmetric. Aside from the matrix of singular values, this procedure produces two sets of eigenvectors that express the orientational relations between the variables grouped by data blocks: occupying the rows and columns of the basis matrix block. The number of vectors with nonzero lengths will be equivalent to the number of basis-matrix rows ($p$) or columns ($q$), whichever is least. In the example above we employed the correlation matrix as the basis for our PLS analysis because of the nature of the variables. PLS can be performed equally well on either covariance or distance matrices.
Unlike standard multiple regression analysis in which a single dependent variable is regressed against a set of independent variables using a linear least-squares minimization criterion (see the PalaeoMath 101 column in Newsletter 55 for a review of linear least-squares minimization), PLS regresses two sets of multiple variables against one another using a major axis minimization (see the PalaeoMath 101 column in Newsletter 57 for a review of linear major axis minimization). Also, the regression coefficients (= slopes) are partial regression coefficients that represent the relation between the trend of the dependent variable and each of the independent variables when the affects of the other independent variables are held constant. Thus, if a pair of variables is highly covariant or correlated, the covariations or correlations of other pairs of variables will be correspondingly reduced since there will not be much residual covariance or correlation structure left after the effects of the first pair are held constant. In contrast, the PLS salience coefficients all represent angular relations with the complete, block-specific, covariance-correlation structure. This makes the interpretation of these coefficients less complex.
Finally, unlike CCA, which recognizes the same block structure as PLS but uses information from all blocks to create a scaled or pooled covariance-correlation basis matrix for SVD decomposition, PLS decomposes only that block which expresses the inter-relations between the variable sets. This means that PLS can focus on only the inter-block aspect of the covariance-correlation substructure irrespective of whether that substructure accounts for a large or small component of the overall covariance-correlation superstructure. Since the coefficients of a CCA, like those of PLS, are used to quantify the inter-relations between blocks of variables, both are referred to as saliences. It is important to note, however, that CCA saliences are equivalent to partial regression coefficients (see above) whereas PLS saliences are analogous to PCA loadings. In effect, CCA represents an attempt to define a set of canonical variables (= linear combinations of variables) for each data block that exhibit overall covariances-correlations that are as large as possible. Indeed, a CCA analysis in which either the set of basis matrix rows or columns contains a single variable is analogous to a major axis-based multiple regression analysis. The goal of PLS differs insofar as it tries to provide a more focused assessment of the inter-block substructure and doesn't allow within-block patterns of covariance-correlation to influence that result.
Partial least squares analysis supports a very large set of investigation types that are often encountered in palaeontological data analysis situations. The example above represents a simple situation in which a set of morphological variables are related to a set of ecological variables, allowing the morphological correlates of ecological distributions (and vice versa) to be assessed. A PLS approach could also be used to investigate inter-relations between different blocks of morphological variables, say from the anterior or posterior regions of a species (e.g., Zelditch et al. 2004) or between different regions of the same morphological structure. This type of study falls within the general 'morphological integration' research programme that tries to identify regions of correlated morphological variation within organismal Baupläne (see Olson and Miller 1958 for a classical treatment of this topic) and is related to the current interest in identifying developmental modules (see Schlosser and Wager 2004). A PLS approach could also be used to examine inter-relations between different types of ecological variables (e.g., organismal-based vs. physio-chemical), or to explore the morphological correlates of genetic variation. The possibilities are virtually endless (see Rychlik et al. 2006 for an good recent example of PLS analysis being used in a systematic context).
As for the practical matter of how to perform your own PLS analysis, unfortunately the choices here are somewhat more limited than for the other methods we've discussed to date. Of course, the PalaeoMath 101 spreadsheet contains the complete calculations for the example PLS analysis presented above. These were performed using the PopTools plug-in for the SVD calculations, but all other calculations were made using the standard MS-Excel data analysis tools. As I mentioned above, generalized mathematical packages (e.g., Mathematica, Maple, MatLab) can also be used to program your own routines. Program systems that perform PLS analysis are somewhat rare, reflecting the method's relatively recent introduction. Of these your best bets at the moment are XL-Stat (some limited PLS capability) and NT-SYS. Since PLS has a longer history of use in chemometrics some stand-alone software is available in programme packages that have been developed for that community. Of these Solo is one of the more complete and better known.
## References (cited in the text as well as recommended review articles)
Bartlett, M. S. 1936. The square root transformation in analysis of variance. Journal of the Royal Statistical Society, Supplement, 3, 68-78.
Bookstein, F. L. 1991. Morphometric tools for landmark data: geometry and biology. Cambridge University Press, Cambridge, 435 pp.
Golub, G. H. and Reinsch, C. 1971. Singular value decomposition and lest squares solutions, 134-151. In Wilkinson, J. H. and Reinsch, C., eds). Linear algebra: computer methods for mathematical computation, v. 2. Springer-Verlag, Berlin.
Jackson, J. E. 1991. A user's guide to principal components. John Wiley & Sons, New York, 592 pp.
Morrison, D. F. 2005. Multivariate statistical methods. Duxbury Press, New York, 498 pp.
Olson, E. and Miller, R. 1958. Morphological integration. University of Chicago Press, Chi-cago, 317 pp.
Rychlik, L., Ramalhino, G., and Polly, P. D. 2006. Response to environmental factors and competition: skull, mandible and tooth shapes in Polish water shrews (Neomys, Soricidae, Mammalia). Journal of the Zoological Society, 44(4), 339-351.
Rohlf, F. J. and Corti, M. 2000. Use of partial least squares to study covariation in shape. Systematic Biology, 49(4), 740-753.
Schlosser, G. and Wagner, G. 2004. Modularity in development and evolution. University of Chicago Press, Chicago, 600 pp.
Zelditch, M. L., Swiderski, D. L., Sheets, H. D., and Fink, W. L. 2004. Geometric morphomet-rics for biologists: a primer. Elsevier/Academic Press, Amsterdam, 443 pp.
### Author Information
Norm MacLeod - The Natural History Museum, London, UK (email: n.macleod@nhm.ac.uk). This article first appeared in the Nº 63 edition of Palaeontology Newsletter
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https://byjus.com/question-answer/what-is-the-value-of-x-if-the-value-of-33333-2-is-11110xxxx9-8-1/
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1
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Question
# What is the value of X, if the value of 333332 is 11110XXXX9.8
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Solution
## The correct option is A 8The square of the number 33333 can easily be found by the method as described here. Suppose the number contains n '3's. The digits of its square will be (n−1) '1's followed by '0', (n−1) '8's and '9'. In this case, the value of 333332 is 1111088889. Hence, the value of X is 8.
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https://numberworld.info/19
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# Number 19
### Properties of number 19
Cross Sum:
Factorization:
Divisors:
1, 19
Count of divisors:
Sum of divisors:
Prime number?
Yes
Fibonacci number?
No
Bell Number?
No
Catalan Number?
No
Base 2 (Binary):
Base 3 (Ternary):
Base 4 (Quaternary):
Base 5 (Quintal):
Base 8 (Octal):
Base 32:
j
sin(19)
0.14987720966295
cos(19)
0.98870461818667
tan(19)
0.1515894706124
ln(19)
2.9444389791664
lg(19)
1.2787536009528
sqrt(19)
4.3588989435407
Square(19)
### Number Look Up
Look Up
19 (nineteen) is a very impressive figure. The cross sum of 19 is 10. If you factorisate the number 19 you will get these result . 19 has 2 divisors ( 1, 19 ) whith a sum of 20. 19 is a prime number. The figure 19 is not a fibonacci number. The figure 19 is not a Bell Number. The figure 19 is not a Catalan Number. The convertion of 19 to base 2 (Binary) is 10011. The convertion of 19 to base 3 (Ternary) is 201. The convertion of 19 to base 4 (Quaternary) is 103. The convertion of 19 to base 5 (Quintal) is 34. The convertion of 19 to base 8 (Octal) is 23. The convertion of 19 to base 16 (Hexadecimal) is 13. The convertion of 19 to base 32 is j. The sine of the figure 19 is 0.14987720966295. The cosine of 19 is 0.98870461818667. The tangent of the number 19 is 0.1515894706124. The root of 19 is 4.3588989435407.
If you square 19 you will get the following result 361. The natural logarithm of 19 is 2.9444389791664 and the decimal logarithm is 1.2787536009528. You should now know that 19 is very unique figure!
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https://leetcode.ca/2018-08-22-996-Number-of-Squareful-Arrays/
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# 996. Number of Squareful Arrays
## Description
An array is squareful if the sum of every pair of adjacent elements is a perfect square.
Given an integer array nums, return the number of permutations of nums that are squareful.
Two permutations perm1 and perm2 are different if there is some index i such that perm1[i] != perm2[i].
Example 1:
Input: nums = [1,17,8]
Output: 2
Explanation: [1,8,17] and [17,8,1] are the valid permutations.
Example 2:
Input: nums = [2,2,2]
Output: 1
Constraints:
• 1 <= nums.length <= 12
• 0 <= nums[i] <= 109
## Solutions
• class Solution {
public int numSquarefulPerms(int[] nums) {
int n = nums.length;
int[][] f = new int[1 << n][n];
for (int j = 0; j < n; ++j) {
f[1 << j][j] = 1;
}
for (int i = 0; i < 1 << n; ++i) {
for (int j = 0; j < n; ++j) {
if ((i >> j & 1) == 1) {
for (int k = 0; k < n; ++k) {
if ((i >> k & 1) == 1 && k != j) {
int s = nums[j] + nums[k];
int t = (int) Math.sqrt(s);
if (t * t == s) {
f[i][j] += f[i ^ (1 << j)][k];
}
}
}
}
}
}
long ans = 0;
for (int j = 0; j < n; ++j) {
ans += f[(1 << n) - 1][j];
}
Map<Integer, Integer> cnt = new HashMap<>();
for (int x : nums) {
cnt.merge(x, 1, Integer::sum);
}
int[] g = new int[13];
g[0] = 1;
for (int i = 1; i < 13; ++i) {
g[i] = g[i - 1] * i;
}
for (int v : cnt.values()) {
ans /= g[v];
}
return (int) ans;
}
}
• class Solution {
public:
int numSquarefulPerms(vector<int>& nums) {
int n = nums.size();
int f[1 << n][n];
memset(f, 0, sizeof(f));
for (int j = 0; j < n; ++j) {
f[1 << j][j] = 1;
}
for (int i = 0; i < 1 << n; ++i) {
for (int j = 0; j < n; ++j) {
if ((i >> j & 1) == 1) {
for (int k = 0; k < n; ++k) {
if ((i >> k & 1) == 1 && k != j) {
int s = nums[j] + nums[k];
int t = sqrt(s);
if (t * t == s) {
f[i][j] += f[i ^ (1 << j)][k];
}
}
}
}
}
}
long long ans = 0;
for (int j = 0; j < n; ++j) {
ans += f[(1 << n) - 1][j];
}
unordered_map<int, int> cnt;
for (int x : nums) {
++cnt[x];
}
int g[13] = {1};
for (int i = 1; i < 13; ++i) {
g[i] = g[i - 1] * i;
}
for (auto& [_, v] : cnt) {
ans /= g[v];
}
return ans;
}
};
• class Solution:
def numSquarefulPerms(self, nums: List[int]) -> int:
n = len(nums)
f = [[0] * n for _ in range(1 << n)]
for j in range(n):
f[1 << j][j] = 1
for i in range(1 << n):
for j in range(n):
if i >> j & 1:
for k in range(n):
if (i >> k & 1) and k != j:
s = nums[j] + nums[k]
t = int(sqrt(s))
if t * t == s:
f[i][j] += f[i ^ (1 << j)][k]
ans = sum(f[(1 << n) - 1][j] for j in range(n))
for v in Counter(nums).values():
ans //= factorial(v)
return ans
• func numSquarefulPerms(nums []int) (ans int) {
n := len(nums)
f := make([][]int, 1<<n)
for i := range f {
f[i] = make([]int, n)
}
for j := range nums {
f[1<<j][j] = 1
}
for i := 0; i < 1<<n; i++ {
for j := 0; j < n; j++ {
if i>>j&1 == 1 {
for k := 0; k < n; k++ {
if i>>k&1 == 1 && k != j {
s := nums[j] + nums[k]
t := int(math.Sqrt(float64(s)))
if t*t == s {
f[i][j] += f[i^(1<<j)][k]
}
}
}
}
}
}
for j := 0; j < n; j++ {
ans += f[(1<<n)-1][j]
}
g := [13]int{1}
for i := 1; i < 13; i++ {
g[i] = g[i-1] * i
}
cnt := map[int]int{}
for _, x := range nums {
cnt[x]++
}
for _, v := range cnt {
ans /= g[v]
}
return
}
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OpenStudy (anonymous):
simplify √0.0016 0.04 4 0.004 0.4
2 years ago
OpenStudy (studygurl14):
What is 0.0016 in fraction form?
2 years ago
OpenStudy (anonymous):
16/10000 @Studygurl14
2 years ago
OpenStudy (studygurl14):
um, yes. But what's that simplified?
2 years ago
OpenStudy (anonymous):
i dont quite know :/
2 years ago
Or you could express it as$\sqrt{16}\sqrt{10^{-4}}$
2 years ago
OpenStudy (anonymous):
hmm i'm really confused
2 years ago
OpenStudy (studygurl14):
@radar hmm...I didn't think of that
2 years ago
OpenStudy (studygurl14):
@radar 's method is better @bayan143
2 years ago
either way will work. The method I am showing requires the student to know the rules for exponents. And scientific notation. I don't know where bayan143 is at in those subjects. How about it bayan143?
2 years ago
Then do the StudyGurl14 method expressing .0016 as a fraction.
2 years ago
OpenStudy (anonymous):
ok :/
2 years ago
.1 = 1/10 .01 = 1/100 .001=1/100 etc Now do that with .0016
2 years ago
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José Thomaz
Posted on
# Functional programming: a bit of history
## History of functional programming
Functional programming is a programming paradigm, such as object-oriented, imperative and many other programming “styles”. So, before we start to crack functional programming, let’s learn a little bit about its history.
## Computer Science background
To made functional programming possible, many mathematical and computer science theories, concepts and researches were published. Functional programming is very attached to math and functions, its roots are in mathematical logic.
Informal logic systems have been in use or over 2000 years, but the first formalization was made only in the middle of the XIX century. Hamilton, De Morgan and Boole published their works, that were the basis to the formal logic:
• Propositional Calculus;
• Predicate Calculus.
Also, in the XIX century, the number theory was introduced.
In 1936, three different approaches for computability were proposed: Turing’s Turing machines, Kleene’s recursive function theory and Church’s lambda calculus. Turing’s proposal is the foundation of the computer science and programming languages as we know today, and the recursive function theory and lambda calculus are the backbones of functional programming.
## The first functional programming language
The first programming languages were created in the late 1940s, Assembly was the first, and FORTRAN was the first high level programming language to become popular; created in 1954. In the next years new programming languages were created, but now high level, and mostly procedural.
As the computers were becoming more popular, new languages appeared, so in 1958, McCarthy created the LISP programming language, which is considered the first functional language in history.
LISP is a very simple language based on recursive functions manipulating lists of words and numbers. LISP is a non-typed language. Besides that, LISP is not considered “purely functional”, because it has some imperative elements.
The functional paradigm has become popular recently, programmers and companies started to use functional languages in their projects more frequently. Some examples of popular functional languages are:
• Elixir
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# Source code for manim.utils.space_ops
```"""Utility functions for two- and three-dimensional vectors."""
from __future__ import annotations
__all__ = [
"quaternion_mult",
"quaternion_from_angle_axis",
"angle_axis_from_quaternion",
"quaternion_conjugate",
"rotate_vector",
"thick_diagonal",
"rotation_matrix",
"z_to_vector",
"angle_of_vector",
"angle_between_vectors",
"normalize",
"get_unit_normal",
"compass_directions",
"regular_vertices",
"complex_to_R3",
"R3_to_complex",
"complex_func_to_R3_func",
"center_of_mass",
"midpoint",
"find_intersection",
"line_intersection",
"get_winding_number",
"shoelace",
"shoelace_direction",
"cross2d",
"earclip_triangulation",
"cartesian_to_spherical",
"spherical_to_cartesian",
"perpendicular_bisector",
]
import itertools as it
import math
from typing import Sequence
import numpy as np
from mapbox_earcut import triangulate_float32 as earcut
from scipy.spatial.transform import Rotation
from .. import config
from ..constants import DOWN, OUT, PI, RIGHT, TAU, UP
from ..utils.iterables import adjacent_pairs
[docs]def norm_squared(v: float) -> float:
return np.dot(v, v)
# Quaternions
# TODO, implement quaternion type
[docs]def quaternion_mult(
*quats: Sequence[float],
) -> np.ndarray | list[float | np.ndarray]:
"""Gets the Hamilton product of the quaternions provided.
<https://en.wikipedia.org/wiki/Quaternion>`__.
Returns
-------
Union[np.ndarray, List[Union[float, np.ndarray]]]
Returns a list of product of two quaternions.
"""
if config.renderer == "opengl":
if len(quats) == 0:
return [1, 0, 0, 0]
result = quats[0]
for next_quat in quats[1:]:
w1, x1, y1, z1 = result
w2, x2, y2, z2 = next_quat
result = [
w1 * w2 - x1 * x2 - y1 * y2 - z1 * z2,
w1 * x2 + x1 * w2 + y1 * z2 - z1 * y2,
w1 * y2 + y1 * w2 + z1 * x2 - x1 * z2,
w1 * z2 + z1 * w2 + x1 * y2 - y1 * x2,
]
return result
else:
q1 = quats[0]
q2 = quats[1]
w1, x1, y1, z1 = q1
w2, x2, y2, z2 = q2
return np.array(
[
w1 * w2 - x1 * x2 - y1 * y2 - z1 * z2,
w1 * x2 + x1 * w2 + y1 * z2 - z1 * y2,
w1 * y2 + y1 * w2 + z1 * x2 - x1 * z2,
w1 * z2 + z1 * w2 + x1 * y2 - y1 * x2,
],
)
[docs]def quaternion_from_angle_axis(
angle: float,
axis: np.ndarray,
axis_normalized: bool = False,
) -> list[float]:
"""Gets a quaternion from an angle and an axis.
<https://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles>`__.
Parameters
----------
angle
The angle for the quaternion.
axis
The axis for the quaternion
axis_normalized : bool, optional
Checks whether the axis is normalized, by default False
Returns
-------
List[float]
Gives back a quaternion from the angle and axis
"""
if config.renderer == "opengl":
if not axis_normalized:
axis = normalize(axis)
return [math.cos(angle / 2), *(math.sin(angle / 2) * axis)]
else:
return np.append(np.cos(angle / 2), np.sin(angle / 2) * normalize(axis))
[docs]def angle_axis_from_quaternion(quaternion: Sequence[float]) -> Sequence[float]:
"""Gets angle and axis from a quaternion.
Parameters
----------
quaternion
The quaternion from which we get the angle and axis.
Returns
-------
Sequence[float]
Gives the angle and axis
"""
axis = normalize(quaternion[1:], fall_back=np.array([1, 0, 0]))
angle = 2 * np.arccos(quaternion[0])
if angle > TAU / 2:
angle = TAU - angle
return angle, axis
[docs]def quaternion_conjugate(quaternion: Sequence[float]) -> np.ndarray:
"""Used for finding the conjugate of the quaternion
Parameters
----------
quaternion
The quaternion for which you want to find the conjugate for.
Returns
-------
np.ndarray
The conjugate of the quaternion.
"""
result = np.array(quaternion)
result[1:] *= -1
return result
[docs]def rotate_vector(
vector: np.ndarray, angle: float, axis: np.ndarray = OUT
) -> np.ndarray:
"""Function for rotating a vector.
Parameters
----------
vector
The vector to be rotated.
angle
The angle to be rotated by.
axis
The axis to be rotated, by default OUT
Returns
-------
np.ndarray
The rotated vector with provided angle and axis.
Raises
------
ValueError
If vector is not of dimension 2 or 3.
"""
if len(vector) > 3:
raise ValueError("Vector must have the correct dimensions.")
if len(vector) == 2:
vector = np.append(vector, 0)
return rotation_matrix(angle, axis) @ vector
[docs]def thick_diagonal(dim: int, thickness=2) -> np.ndarray:
row_indices = np.arange(dim).repeat(dim).reshape((dim, dim))
col_indices = np.transpose(row_indices)
return (np.abs(row_indices - col_indices) < thickness).astype("uint8")
[docs]def rotation_matrix_transpose_from_quaternion(quat: np.ndarray) -> list[np.ndarray]:
"""Converts the quaternion, quat, to an equivalent rotation matrix representation.
<https://in.mathworks.com/help/driving/ref/quaternion.rotmat.html>`_.
Parameters
----------
quat
The quaternion which is to be converted.
Returns
-------
List[np.ndarray]
Gives back the Rotation matrix representation, returned as a 3-by-3
matrix or 3-by-3-by-N multidimensional array.
"""
quat_inv = quaternion_conjugate(quat)
return [
quaternion_mult(quat, [0, *basis], quat_inv)[1:]
for basis in [
[1, 0, 0],
[0, 1, 0],
[0, 0, 1],
]
]
[docs]def rotation_matrix_from_quaternion(quat: np.ndarray) -> np.ndarray:
return np.transpose(rotation_matrix_transpose_from_quaternion(quat))
[docs]def rotation_matrix_transpose(angle: float, axis: np.ndarray) -> np.ndarray:
if all(np.array(axis)[:2] == np.zeros(2)):
return rotation_about_z(angle * np.sign(axis[2])).T
return rotation_matrix(angle, axis).T
[docs]def rotation_matrix(
angle: float,
axis: np.ndarray,
homogeneous: bool = False,
) -> np.ndarray:
"""
Rotation in R^3 about a specified axis of rotation.
"""
inhomogeneous_rotation_matrix = Rotation.from_rotvec(
angle * normalize(np.array(axis))
).as_matrix()
if not homogeneous:
return inhomogeneous_rotation_matrix
else:
rotation_matrix = np.eye(4)
rotation_matrix[:3, :3] = inhomogeneous_rotation_matrix
return rotation_matrix
[docs]def rotation_about_z(angle: float) -> np.ndarray:
"""Returns a rotation matrix for a given angle.
Parameters
----------
angle : float
Angle for the rotation matrix.
Returns
-------
np.ndarray
Gives back the rotated matrix.
"""
c, s = math.cos(angle), math.sin(angle)
return np.array(
[
[c, -s, 0],
[s, c, 0],
[0, 0, 1],
]
)
[docs]def z_to_vector(vector: np.ndarray) -> np.ndarray:
"""
Returns some matrix in SO(3) which takes the z-axis to the
(normalized) vector provided as an argument
"""
axis_z = normalize(vector)
axis_y = normalize(np.cross(axis_z, RIGHT))
axis_x = np.cross(axis_y, axis_z)
if np.linalg.norm(axis_y) == 0:
# the vector passed just so happened to be in the x direction.
axis_x = normalize(np.cross(UP, axis_z))
axis_y = -np.cross(axis_x, axis_z)
return np.array([axis_x, axis_y, axis_z]).T
[docs]def angle_of_vector(vector: Sequence[float]) -> float:
"""Returns polar coordinate theta when vector is projected on xy plane.
Parameters
----------
vector
The vector to find the angle for.
Returns
-------
float
The angle of the vector projected.
"""
return np.angle(complex(*vector[:2]))
[docs]def angle_between_vectors(v1: np.ndarray, v2: np.ndarray) -> np.ndarray:
"""Returns the angle between two vectors.
This angle will always be between 0 and pi
Parameters
----------
v1
The first vector.
v2
The second vector.
Returns
-------
np.ndarray
The angle between the vectors.
"""
return 2 * np.arctan2(
np.linalg.norm(normalize(v1) - normalize(v2)),
np.linalg.norm(normalize(v1) + normalize(v2)),
)
[docs]def normalize(vect: np.ndarray | tuple[float], fall_back=None) -> np.ndarray:
norm = np.linalg.norm(vect)
if norm > 0:
return np.array(vect) / norm
else:
return fall_back or np.zeros(len(vect))
[docs]def normalize_along_axis(array: np.ndarray, axis: np.ndarray) -> np.ndarray:
"""Normalizes an array with the provided axis.
Parameters
----------
array
The array which has to be normalized.
axis
The axis to be normalized to.
Returns
-------
np.ndarray
Array which has been normalized according to the axis.
"""
norms = np.sqrt((array * array).sum(axis))
norms[norms == 0] = 1
buffed_norms = np.repeat(norms, array.shape[axis]).reshape(array.shape)
array /= buffed_norms
return array
[docs]def get_unit_normal(v1: np.ndarray, v2: np.ndarray, tol: float = 1e-6) -> np.ndarray:
"""Gets the unit normal of the vectors.
Parameters
----------
v1
The first vector.
v2
The second vector
tol
[description], by default 1e-6
Returns
-------
np.ndarray
The normal of the two vectors.
"""
v1, v2 = (normalize(i) for i in (v1, v2))
cp = np.cross(v1, v2)
cp_norm = np.linalg.norm(cp)
if cp_norm < tol:
# Vectors align, so find a normal to them in the plane shared with the z-axis
cp = np.cross(np.cross(v1, OUT), v1)
cp_norm = np.linalg.norm(cp)
if cp_norm < tol:
return DOWN
return normalize(cp)
###
[docs]def compass_directions(n: int = 4, start_vect: np.ndarray = RIGHT) -> np.ndarray:
"""Finds the cardinal directions using tau.
Parameters
----------
n
The amount to be rotated, by default 4
start_vect
The direction for the angle to start with, by default RIGHT
Returns
-------
np.ndarray
The angle which has been rotated.
"""
angle = TAU / n
return np.array([rotate_vector(start_vect, k * angle) for k in range(n)])
[docs]def regular_vertices(
n: int, *, radius: float = 1, start_angle: float | None = None
) -> tuple[np.ndarray, float]:
"""Generates regularly spaced vertices around a circle centered at the origin.
Parameters
----------
n
The number of vertices
The radius of the circle that the vertices are placed on.
start_angle
The angle the vertices start at.
If unspecified, for even ``n`` values, ``0`` will be used.
For odd ``n`` values, 90 degrees is used.
Returns
-------
vertices : :class:`numpy.ndarray`
The regularly spaced vertices.
start_angle : :class:`float`
The angle the vertices start at.
"""
if start_angle is None:
if n % 2 == 0:
start_angle = 0
else:
start_angle = TAU / 4
start_vector = rotate_vector(RIGHT * radius, start_angle)
vertices = compass_directions(n, start_vector)
return vertices, start_angle
[docs]def complex_to_R3(complex_num: complex) -> np.ndarray:
return np.array((complex_num.real, complex_num.imag, 0))
[docs]def R3_to_complex(point: Sequence[float]) -> np.ndarray:
return complex(*point[:2])
[docs]def complex_func_to_R3_func(complex_func):
return lambda p: complex_to_R3(complex_func(R3_to_complex(p)))
[docs]def center_of_mass(points: Sequence[float]) -> np.ndarray:
"""Gets the center of mass of the points in space.
Parameters
----------
points
The points to find the center of mass from.
Returns
-------
np.ndarray
The center of mass of the points.
"""
return np.average(points, 0, np.ones(len(points)))
[docs]def midpoint(
point1: Sequence[float],
point2: Sequence[float],
) -> float | np.ndarray:
"""Gets the midpoint of two points.
Parameters
----------
point1
The first point.
point2
The second point.
Returns
-------
Union[float, np.ndarray]
The midpoint of the points
"""
return center_of_mass([point1, point2])
[docs]def line_intersection(
line1: Sequence[np.ndarray], line2: Sequence[np.ndarray]
) -> np.ndarray:
"""Returns the intersection point of two lines, each defined by
a pair of distinct points lying on the line.
Parameters
----------
line1
A list of two points that determine the first line.
line2
A list of two points that determine the second line.
Returns
-------
np.ndarray
The intersection points of the two lines which are intersecting.
Raises
------
ValueError
Error is produced if the two lines don't intersect with each other
or if the coordinates don't lie on the xy-plane.
"""
if any(np.array([line1, line2])[:, :, 2].reshape(-1)):
# checks for z coordinates != 0
raise ValueError("Coords must be in the xy-plane.")
# algorithm from https://stackoverflow.com/a/42727584
np.pad(np.array(i)[:, :2], ((0, 0), (0, 1)), constant_values=1)
for i in (line1, line2)
)
line1, line2 = (np.cross(*i) for i in padded)
x, y, z = np.cross(line1, line2)
if z == 0:
raise ValueError(
"The lines are parallel, there is no unique intersection point."
)
return np.array([x / z, y / z, 0])
[docs]def find_intersection(
p0s: Sequence[np.ndarray],
v0s: Sequence[np.ndarray],
p1s: Sequence[np.ndarray],
v1s: Sequence[np.ndarray],
threshold: float = 1e-5,
) -> Sequence[np.ndarray]:
"""
Return the intersection of a line passing through p0 in direction v0
with one passing through p1 in direction v1 (or array of intersections
from arrays of such points/directions).
For 3d values, it returns the point on the ray p0 + v0 * t closest to the
ray p1 + v1 * t
"""
# algorithm from https://en.wikipedia.org/wiki/Skew_lines#Nearest_points
result = []
for p0, v0, p1, v1 in zip(*[p0s, v0s, p1s, v1s]):
normal = np.cross(v1, np.cross(v0, v1))
denom = max(np.dot(v0, normal), threshold)
result += [p0 + np.dot(p1 - p0, normal) / denom * v0]
return result
[docs]def get_winding_number(points: Sequence[float]) -> float:
total_angle = 0
for p1, p2 in adjacent_pairs(points):
d_angle = angle_of_vector(p2) - angle_of_vector(p1)
d_angle = ((d_angle + PI) % TAU) - PI
total_angle += d_angle
[docs]def shoelace(x_y: np.ndarray) -> float:
"""2D implementation of the shoelace formula.
Returns
-------
:class:`float`
Returns signed area.
"""
x = x_y[:, 0]
y = x_y[:, 1]
return np.trapz(y, x)
[docs]def shoelace_direction(x_y: np.ndarray) -> str:
"""
Uses the area determined by the shoelace method to determine whether
the input set of points is directed clockwise or counterclockwise.
Returns
-------
:class:`str`
Either ``"CW"`` or ``"CCW"``.
"""
area = shoelace(x_y)
return "CW" if area > 0 else "CCW"
[docs]def cross2d(a, b):
if len(a.shape) == 2:
return a[:, 0] * b[:, 1] - a[:, 1] * b[:, 0]
else:
return a[0] * b[1] - b[0] * a[1]
[docs]def earclip_triangulation(verts: np.ndarray, ring_ends: list) -> list:
"""Returns a list of indices giving a triangulation
of a polygon, potentially with holes.
Parameters
----------
verts
verts is a numpy array of points.
ring_ends
ring_ends is a list of indices indicating where
the ends of new paths are.
Returns
-------
list
A list of indices giving a triangulation of a polygon.
"""
# First, connect all the rings so that the polygon
# with holes is instead treated as a (very convex)
# polygon with one edge. Do this by drawing connections
# between rings close to each other
rings = [list(range(e0, e1)) for e0, e1 in zip([0, *ring_ends], ring_ends)]
attached_rings = rings[:1]
detached_rings = rings[1:]
loop_connections = {}
while detached_rings:
i_range, j_range = (
list(
filter(
# Ignore indices that are already being
# used to draw some connection
lambda i: i not in loop_connections,
it.chain(*ring_group),
),
)
for ring_group in (attached_rings, detached_rings)
)
# Closest point on the attached rings to an estimated midpoint
# of the detached rings
tmp_j_vert = midpoint(verts[j_range[0]], verts[j_range[len(j_range) // 2]])
i = min(i_range, key=lambda i: norm_squared(verts[i] - tmp_j_vert))
# Closest point of the detached rings to the aforementioned
# point of the attached rings
j = min(j_range, key=lambda j: norm_squared(verts[i] - verts[j]))
# Recalculate i based on new j
i = min(i_range, key=lambda i: norm_squared(verts[i] - verts[j]))
# Remember to connect the polygon at these points
loop_connections[i] = j
loop_connections[j] = i
# Move the ring which j belongs to from the
# attached list to the detached list
new_ring = next(filter(lambda ring: ring[0] <= j < ring[-1], detached_rings))
detached_rings.remove(new_ring)
attached_rings.append(new_ring)
# Setup linked list
after = []
end0 = 0
for end1 in ring_ends:
after.extend(range(end0 + 1, end1))
after.append(end0)
end0 = end1
# Find an ordering of indices walking around the polygon
indices = []
i = 0
for _ in range(len(verts) + len(ring_ends) - 1):
# starting = False
if i in loop_connections:
j = loop_connections[i]
indices.extend([i, j])
i = after[j]
else:
indices.append(i)
i = after[i]
if i == 0:
break
meta_indices = earcut(verts[indices, :2], [len(indices)])
return [indices[mi] for mi in meta_indices]
[docs]def cartesian_to_spherical(vec: Sequence[float]) -> np.ndarray:
"""Returns an array of numbers corresponding to each
polar coordinate value (distance, phi, theta).
Parameters
----------
vec
A numpy array ``[x, y, z]``.
"""
norm = np.linalg.norm(vec)
if norm == 0:
return 0, 0, 0
r = norm
phi = np.arccos(vec[2] / r)
theta = np.arctan2(vec[1], vec[0])
return np.array([r, theta, phi])
[docs]def spherical_to_cartesian(spherical: Sequence[float]) -> np.ndarray:
"""Returns a numpy array ``[x, y, z]`` based on the spherical
coordinates given.
Parameters
----------
spherical
A list of three floats that correspond to the following:
r - The distance between the point and the origin.
theta - The azimuthal angle of the point to the positive x-axis.
phi - The vertical angle of the point to the positive z-axis.
"""
r, theta, phi = spherical
return np.array(
[
r * np.cos(theta) * np.sin(phi),
r * np.sin(theta) * np.sin(phi),
r * np.cos(phi),
],
)
[docs]def perpendicular_bisector(
line: Sequence[np.ndarray],
norm_vector=OUT,
) -> Sequence[np.ndarray]:
"""Returns a list of two points that correspond
to the ends of the perpendicular bisector of the
two points given.
Parameters
----------
line
a list of two numpy array points (corresponding
to the ends of a line).
norm_vector
the vector perpendicular to both the line given
and the perpendicular bisector.
Returns
-------
list
A list of two numpy array points that correspond
to the ends of the perpendicular bisector
"""
p1 = line[0]
p2 = line[1]
direction = np.cross(p1 - p2, norm_vector)
m = midpoint(p1, p2)
return [m + direction, m - direction]
```
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## The Easter Bunny Needs Your Help! April 17, 2017 Vicki McGinn Comment
There are 10 Easter baskets lined up in a row. The Easter bunny puts 1 egg in the first one, 2 in the second, 3 in the third, 4 in the fourth. The bunny continues to fill all of the baskets like this. How many eggs were used to fill
## Easter Math April 17, 2017 Vicki McGinn Comment
I was not planning on writing a post on Easter Sunday, but some real-life math occurred while preparing for dinner. I am sharing here what I posted on my Math Motivator Facebook page. My ham is 5.49 kilograms. The instructions say to cook it 12-15 minutes per pound or
## Easter Egg Tasks April 17, 2017 Vicki McGinn Comment
When engaging students in problem solving and I hope you are doing so often, it is important to think about the types of tasks you give them. Which of the following two will give you the most information about your students? Example 1: (closed) Nathan finds 6 pink eggs, 1
## Understanding Relationships Between Quantity and the Patterns Within Our Number System February 1, 2017 Vicki McGinn Comment
Working one-on-one with students who are demonstrating a fragile sense of number are opportunities I greatly value. I always approach these times from an inquiry perspective, looking for clues to help me understand their struggles. Often they are demonstrating many strengths in some areas, but something is keeping them from
## Ants! Ants! Ants! November 16, 2016 Vicki McGinn Comment
Recently in a Gr 3, 4, 5 classroom we gave the students this question: Students are learning about insects. They discover that an ant has 1 body, 2 antennae and 6 legs. They each make a model. How many bodies, antennae and legs will they need for 5, 10 and
## Natural vs Commercial Math Materials August 5, 2016 Vicki McGinn Comment
Recently I received the following question from a Kindergarten educator: Why do I get the feeling that natural materials are better than commercial ones for math? I do the exact same activity with materials we have in the classroom. Do the students actually learn more about math through the natural
## Proportional Reasoning – Division May 11, 2016 Vicki McGinn Comment
Recently I had the opportunity to work in a Grade 6 classroom as they began a unit on division and multiplication. The teacher gave the students the following problem from the Ministry support guide (division) to begin. The purpose for that day was to see what the students
## Proportional Reasoning – Measurement March 2, 2016 Vicki McGinn Comment
I love problems that provide authentic opportunities to cluster several big ideas. Recently, I was in a classroom where the teacher gave the following problem to his students. Tia is filling a bucket with water. She knows that 500 ml of water comes out of the hose every 10 seconds.
## Tasty Math Homework January 30, 2016 Vicki McGinn Comment
I just finished making brownies (from a box) with my granddaughter Charlotte who is in Junior Kindergarten. I was reminded about all of the ways both math and literacy can be injected into a fun time. I baked with my own children when they were young but I know I
## Assessment – The Key to Precision January 5, 2016 Vicki McGinn Comment
Last month I wrote a post called, “Preciseness Will Impact Student Achievement”. After posting it, I remembered something that I had written a few years ago. Due to a stroke in December 2011 that left my husband unable to speak or write we became involved in the InteRACT program at
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Page 3 of 4
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Tue Oct 22, 2019 4:03 am
Yes, you go and see how Human GMs play bullet chess to realize how calculation is a critical part of avoiding critical mistakes. Even at 3 0 they blunder a lot, it's just that if they're not good enough to find the opponent's tactic to avoid it's also likely their opponents don't have time to see it either.
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Tue Oct 22, 2019 4:06 am
Suppose it were 10,000 weights needed.
Humans would have no hope to comprehend that clearly.
We do well up to 7 distinct items, on average. After that, we start to degrade.
We can deal with bigger clumps of things by splitting into parts (especially related parts).
But past a certain point, we start to get lost even with that.
That's why menus often get split into sub-menus at a certain count on computer menu systems.
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Tue Oct 22, 2019 7:18 am
People can recite 30000 numbers of pi from memory so it wouldn't be out of reach for some people to just memorize the weights and apply them to positions.
If it comes to learning the weights it might be possible to applying them without needing to comprehend them.
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Tue Oct 22, 2019 7:51 am
fabianVDW wrote:
Mon Oct 21, 2019 9:16 pm
You are right for counting, but I would not immediately see such neurons for determining for instance passers.
More complex than counting, for sure, but still not very hard. For each square you would need a neuron that fires when there is a passer on that square. (And that for each player.) This neuron for input connects to the Pawn plane of its own square with a positive weight (say +1), and to all squares in front of it in the enemy Pawn plane on its own file and the two adjacent files, with larger negative weight (say -1). The neuron can have step-function or rectifier response. Even for a 2nd-rank square that is only 16 connections, for 7th rank it is only one. The only way to get a +1 output is if there is a passer on that square; otherwise the output will be 0.
In the next layer you can have counting cells per file for the number of passers in that file (6 inputs), or detecting the fact there is at least one passer (step-function response). In the third layer you can have a cell per pair of neighboring files with 2 inputs, to detect connected passers. All these cells can be connected to a final summing output cell with the desired evaluation weight for the feature (passer in that location, bonus or connected passers etc.).
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Tue Oct 22, 2019 10:10 pm
dkappe wrote:
Mon Oct 21, 2019 11:11 am
The 11258 distilled networks run all the way from 16x2, 24x3, 32x4, 48x5, etc., and will run reasonably well on CPU. You can find them here: https://github.com/dkappe/leela-chess-w ... d-Networks
Try out the various sizes on lc0 and judge for yourself.
You can also try this BOT https://github.com/dkappe/leela-chess-w ... -style-net
It’s a 32x4 looking at ~25 moves on a raspberry pi 3. Because of its source material, it plays objectively weaker moves than SF or leela, but is very effective against humans.
Thanks - just had 3 interesting games against her. By the third game, I'd learned that I have to defend my king as the top priority or it will get killed, and I thought I was holding on, but then made a losing blunder under pressure. I might be mistaken, but I think she missed a checkmate sequence in game 3 - but all that achieved was to prolong the agony!
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Tue Oct 22, 2019 10:36 pm
Ovyron wrote:
Tue Oct 22, 2019 7:18 am
People can recite 30000 numbers of pi from memory so it wouldn't be out of reach for some people to just memorize the weights and apply them to positions.
Someone did 100K digits. There is a difference, though, between having nearly photographic memory and knowing the meaning of the thing memorized.
https://www.foxnews.com/story/japanese- ... gits-of-pi
If it comes to learning the weights it might be possible to applying them without needing to comprehend them.
But if we still don't understand them, why not just leave them as "black box weights"?
The NN engine already knows how to use them. You could inject them into an alpha-beta searcher, but you would still need a GPU to do all the math or it would be too slow.
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Tue Oct 22, 2019 11:02 pm
Dann Corbit wrote:
Tue Oct 22, 2019 10:36 pm
There is a difference, though, between having nearly photographic memory and knowing the meaning of the thing memorized.
The idea would be memorizing all the weights and being able to use them to figure out what move to play in a give chess position without needing to understand their meaning. Like learning how to decode a crypto message and getting that TH is Z and being able to translate it even if you don't know what Z means by itself. The combination of pieces on a chess board could code the best move to play and you'd just need to decipher it.
Dann Corbit wrote:
Tue Oct 22, 2019 10:36 pm
But if we still don't understand them, why not just leave them as "black box weights"?
The idea would be that someone could become world chess champion by learning the weights and applying them in their chess games, even if they didn't understand their meaning.
(...in theory. In practice it may turn out the weights out of the "black box" are gibberish and unusable by humans...)
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Wed Oct 23, 2019 7:52 am
Dann Corbit wrote:
Tue Oct 22, 2019 10:36 pm
But if we still don't understand [the weights], why not just leave them as "black box weights"?
I believe it will be possible to use linear programming to fit numerical expressions rather than trained NNs, with the following advantages:
1. Chess without search is a complex shape that will be extremely difficult to "fit" with NN learning alone. IMO LP has a better chance
2. A set of expressions will be easier to simplify than a trained NN IMO, making for a smaller overall size, a faster run time, and more hope that humans might be able to understand it
3. IMO it is very possible that there might exist a relatively simple expression that will work in most chess positions (there probably isn't a super-simple one, or it would likely have been found already), and that it might be possible to find it. If so, it will be much easier to extract "human meaning" from a numerical expression than it would from an NN
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Wed Oct 23, 2019 12:58 pm
But a NN is nothing but a set of expressions...
### Re: Poll: How Many "Weights" Needed To Play "Known" Chess Very Well?
Posted: Wed Oct 23, 2019 3:05 pm
hgm wrote:
Wed Oct 23, 2019 12:58 pm
But a NN is nothing but a set of expressions...
A quick comparison of NN training with generating expressions and fitting them to a classification problem using linear prgramming. NN offers the following advantages:
1. You can download a software library like TensorFlow and, if you know what you're doing, you're good to go
2. Proven way of getting a good learning system for many problem types
3. Being used right now for chess position evaluation in many chess engines (some of them free and open source, and they mostly play very well)
4. If you want a classifier that fits the given data using Linear Programming (LP), and which would hence output a set of expressions, then right now you're building it yourself
LP generated expressions would offer the following advantages:
1. IMO LP will be more able to fit a complex shape like the solution to chess than a NN will
2. Could fit the data to the mathematical limit - the best possible fit (it may be necessary to use some LP tricks like symmetry breaking in such a large model space)
3. Having found the best fit, you could then turn the achieved fit into a model condition, and then do another optimisation to maximise the number of expressions for which the weight is zero, resulting in a smaller set of expressions
4. Having got a set of expressions and weights, it would be easy to translate this into a computer language for a program that would run on any computer - with or without a graphics card
5. If it turns out to be possible to make smallish set of expressions which can correctly classify most chess positions, it might be possible to articulate this expression in English language
Btw - if classification by LP turns out to be viable, you'll then have another optimisation problem: selection of position/evaluation pairs. The 7 piece tablebase alone has 423,836,835,667,331 positions in, which is infeasibly large for an LP model with today's technology - by many orders of magnitude. If you couldn't find a selection of around a million of them which could enable most of the others to be solved by the classifier, then the quest to solve "known chess" this way would likely be out of reach.
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# Echotastic
Username Echotastic Score 36 Stats Questions 7 Answers 3
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### An equilateral triangle shares a common side with a square as shown. What is the number of degrees in m
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### In the diagram, if \$\angle PQR = 48^\circ\$ , what is the measure of \$\angle PMN\$?
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### A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (, , , and ), such that
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### A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits (, , , and ), such that
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0
# What is 15 percent of 30 percent?
Updated: 10/10/2023
Wiki User
14y ago
15% of 30%
15/100 =0.15
30/100 =0.3
0.15×0.3 = 0.045.
Prym O
Lvl 2
6mo ago
Wiki User
14y ago
15 percent of 30 percent = 0.15 x 30 percent = 4.5 percent
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### Shannon Entropy Explained
Shannon Entropy Explained
Shannon Entropy (also called Information Entropy) is a concept used in physics and information theory. Here's the scoop..
Suppose you have a system with n states i.e. whenever you make an observation of the system you find it's always in one of the n possible states.
Now make a large number of observations of the system, then use them to get the probability pi that if you make an observation the system is in state i. So for every state of the system you have a probability pi.
Now construct this crazy sum = p1*log(p1) + p2*log(p2) +... + pn*log(pn) where the sum is over all the states of the system.
If the log is base 2 then (-1)*sum is called the "information entropy" of the system.
Note that "information entropy" applies to a complete system, not individual states of a system.
Here's a simple example..
My system is a penny and a table.
I define the system to have 2 states.. penny lying stationary on the table with heads up or with tails up.
My experiment is to throw the penny and then observe which state results.
I throw the penny many times and make notes. It lands heads up 1% of the time and tails up 99% of the time (it's a biased penny).
The crazy sum is 0.01*log(0.01) + 0.99*log(0.99) = 0.01*(-6.643856) + 0.99*(-0.0145) = -0.08079356
So the information entropy of the system is (-1)*(-0.08079356) = 0.08079356
Content written and posted by Ken Abbott abbottsystems@gmail.com
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### 1 Descriptive statistics
For the exercises below download the friendsAndCars.csv file and save it on your I:-drive. Make sure that you don't have any empty lines at the end of the file.
#### 1.1 Preprocessing
The friendsAndCars.csv file contains a relation between people and the cars they own. In order to use descriptive statistics it is best to calculate frequencies:
• people and how many cars they own, or
• cars and how many people own these cars.
The following Python script counts how often each type of car is in the list:
```from networkx import *
from operator import *
from sets import Set
a = [item[1] for item in sorted(F.edges(),key=itemgetter(1))]
for item in Set(a):
print item + "," + str(a.count(item))
```
You can save this as countFreq.py and run it on the command-line using
```python countFreq.py > CarsCounted.csv
```
#### 1.2 Using Excel (or OpenOffice)
If you double click on CarsCounted.csv, it will open in Excel.
Measures of central tendency and dispersion are functions in Excel. For example, AVERAGE(B1:B6) calculates the average (arithmetic mean) of the values in cells B1 to B6.
measureformula
modeMODE()
medianMEDIAN()
mean AVERAGE()
varianceVAR()
standard deviationSTDEV()
#### 1.3 Exercises
1) Calculate the measures in the table above for the Cars data.
2) How can you interpret the data: what is the central value? Is this a normal distribution?
3) Produce a chart (diagram) of the data. In order to do this, you should highlight the data and then select the chart wizzard. You may want to create a label for each column first.
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Q:
# Evaluate P = 50x + 80y at each vertex of the feasible region. (0, 0) P = (0, 15) P = (6, 12) P =
Accepted Solution
A:
(0, 0) P = ⇒ 0(0, 15) P = ⇒ 1200(6, 12) P = ⇒ 1260
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# Binomical vs poisson distribution in Simulations
1. Sep 14, 2006
### hagen
Hey, I want to write a Computer Simulation in C++, which simulates the development of a DNA sequence with a probability to mutate x in one "generation". I do have a variable number (=n) of copies of this DNA. Now one might think, to simulate the mutation by simply:
sum(n*Poisson distributed random variable(x) )
to get the number of mutated DNA copies. But this would be too slow.
So my question is, could I also just create a
binomically distributed random variable and multiply it by n * x
to get the number of mutated DNA's? Or is this statistically incorrect?
If not, how might I set the Params for the Bin. dis.? Can I take 1 as a mean and multiply the result x or has the mean to be x? And how do I set / transform the variance of the distribution in a ratio to number of copies.
As you might probably have guessed, I'm a beginner in statistics, but i would be really grateful for any help. Thanks in advance,
Hagen
2. Sep 15, 2006
Hm, if I remember correct, there is a link between the binomial and Poisson distribution.. Poisson's probability function is given with $$f(x)=\frac{\lambda^x}{x!}e^{-\lambda}$$. Now, I think you can put $$\lambda=mp$$, where the number of repetitions of a Bernoulli scheme experiment $$m \rightarrow \infty$$ and it's probability $$p \rightarrow 0$$.
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# a wheel makes 360 revolutions in one minute. through how many radians it turns in one second.
Category :
### Mohammed
Guys, does anyone know the answer?
get a wheel makes 360 revolutions in one minute. through how many radians it turns in one second. from screen.
## A wheel make 360 revolutions in one minute .Through how many radians does it turn in one second
A wheel make 360 revolutions in one minute .Through how many radians does it turn in one second
Byju's Answer Standard XII Mathematics
Sign of Trigonometric Ratios in Different Quadrants
A wheel make ... Question
A wheel make 360 revolutions in one minute .Through how many radians does it turn in one second
Open in App Solution
Number of revolutions made by the wheel in 1 minute = 360
∴Number of revolutions made by the wheel in 1 second =360/60 = 6
In one complete revolution, the wheel turns an angle of 2π radian.
Hence, in 6 complete revolutions, it will turn an angle of 6 × 2π radian, i.e.,
Thus, in one second, the wheel turns an angle of 12π radian.
Suggest Corrections 40
SIMILAR QUESTIONS
Q.
How many degrees does a minute hand of a clock turn through in one hour?
Q. A circular wheel
28
inches in diameter rotates the same number of inches per second as a circular wheel
35
inches diameter. If the smaller wheel makes x revolutions per second, how many revolutions per minute does the larger wheel make in terms of
x .
Q. A wheel makes 12 revolutions per hour The radians it turns through in 20 minutes is:Q. A fly wheel rotates about a fixed axis and slows down from 400 rpm in one minute. How many revolutions does the wheel complete in the same timeQ. A wheel makes
360
revolutions in one minute. It turns
m π
radians in one second.Find
m View More EXPLORE MORE
Sign of Trigonometric Ratios in Different Quadrants
Standard XII Mathematics
स्रोत : byjus.com
## A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
Click here👆to get an answer to your question ✍️ A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
SIMILAR QUESTIONS
AssertionStatement 1: cos1ReasonStatement 2: Cosine x decreases but sine x increases for x∈(0,
2 π )
Medium View solution >
Assertion
### Statement 2: In a triangle ABC, tanA+tanB+tanC=tanAtanBtanC ,there can be at most one obtuse angle in a triangle.
Medium View solution >
Assertionsin
−1
(sin5)=5 (where 5 is in radians).
Reasonsin
−1
(sinθ)=θ for principle value.
sin
−1
(sin5)=5 (where 5 is in radians).
A bird is sitting on the top of a vertical pole 20 m high and its elevation from a point O on the ground is 45
Medium View solution > ∘
. It flies off horizontally straight away from the point O. After one second, the elevation of the bird from O is reduced to 30
. Then the speed (in m/s) of the bird is
If p=tan1
Hard JEE Mains View solution > 0
,q=tan1(in radians), then which of the following is true?
Medium View solution >
स्रोत : www.toppr.com
## Ex 3.1, 3
Ex3.1, 3 A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second? Number of revolutions in 1 minute = 360 Number of revolutions in 60 second = 360 So,Number of revolutions in 1 second = 360/60 = 6 Angle made in 1 revolution =
Check sibling questions
## Ex 3.1, 3 - Chapter 3 Class 11 Trigonometric Functions (Term 2)
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### Transcript
Ex3.1, 3 A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second? Number of revolutions in 1 minute = 360 Number of revolutions in 60 second = 360 So,Number of revolutions in 1 second = 360/60 = 6 Angle made in 1 revolution = 360 Angle made in 6 revolution = 6 360 Radians made in 6 revolution = 6 360 /180 = 6 2 =12
Next: Ex 3.1, 4 Important →
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# Two tugboats pull a disabled supertanker
Two tugboats pull a disabled supertanker. Each tug exerts a constant force of 1.80×106N1.80×106N, one 14∘14∘ west of north and the other 14∘14∘ east of north, as they pull the tanker 0.75 km toward the north.
What is the total work they do on the supertanker?
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# Homework Help: How to calculate the tension?
1. Feb 13, 2014
### fixedglare
1. The problem statement, all variables and given/known data
Due to friction, a force of 400 N is required to pull a wooden box on the floor. The cord used to pull the box makes an angle of 56° horizontally.
How much tension should be on the cord to be able to pull the box?
2. Relevant equations
W = Fd * (cosθ)
Tension = Weight +/- Mass * Acceleration ????? (found this one online, but was never taught this) or Ft=m(a+g) (never taught this one either just found it online)
3. The attempt at a solution
I read that to find/calculate tension you should use the second formula but I'm not sure.
Should I convert the Force to mass and then multiply 9.81 m/s2?
2. Feb 13, 2014
### fixedglare
On my book the answer says it should be 715 N, I divided the Force by the angle & got that answer but everywhere I search it says to use sin & other kinds of formulas I'm confused.
Then the second question asks how much work is done if the box is moved 25 m?
In my book the answer is supposed to be 10000 but when I use W= fd* cos θ, it gives me a different answer
3. Feb 13, 2014
### fixedglare
I'm very confused because the exercises in my book are under the section of using the angle to find work but when I used the basic W= Fd formula I got the answer in the book.
How do I know when to use the angle formula to find work and the basic formula?
4. Feb 13, 2014
### jackarms
To start, what do you mean you divided the force by the angle? Do you mean by the cosine of the angle?
And for the second part, what work is it asking you to find? Work from friction for tension? Show all of your calculations, and that should help me understand your questions better.
5. Feb 13, 2014
### fixedglare
Yes by the cosine angle, that's the only way I found the tension.
It doesn't specify, that's why I'm confused as to what formula to use and when, because the equation gave me the angle so I thought to use the angle but when I did, the book said it wasn't the right answer so then I used to basic formula & that's how I got it.
The second question just asks, how much work is realized, if the box is moved in a distance of 25.0 m?
6. Feb 13, 2014
### jackarms
Okay, I'm assuming it means how much work is done by the tension, since no net work is done on the box (work-kinetic energy theorem). Please show your calculations for the work. What values are you using to arrive at the answer in the book?
7. Feb 13, 2014
### fixedglare
To get the answer from the book I used the formula W= Fd;
so 400 N * 25.0 m = 1000 J, which is the answer in my book.
8. Feb 13, 2014
### jackarms
The problem is you're using the force from friction, and the work it's asking for is tension. You can get away with it here since the two works are equal, but the reasoning is incorrect. If the works weren't equal, this wouldn't work. You have to use the magnitude of the tension force in addition to the angle the force makes with the horizontal displacement.
9. Feb 13, 2014
### Staff: Mentor
Fixedglare: Did you draw a free body diagram before you started to try to work this problem? If so, on the free body diagram, did you identify all the components of the horizontal and vertical forces acting on the box? This should have automatically cleared up many of the difficulties you have had with this problem. If you drew a FBD, please upload it so we can see it.
Chet
10. Feb 14, 2014
### PhanthomJay
what a poor statement, are you sure the problem is worded this way? It means to say apparently that when you pull on the cord directed 56 degrees above the horizontal with a certain force, then it moves at constant velocity. The friction force on the box from the floor is 400 N. What is the value of the pulling force, and how much work does it do? Or something like that.
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# Stat. Sig of a dollar amount
#### Analyst1
##### New Member
Need help measuring the statistical significance of when a customer responds to a solicitation, & the size of their purchase.
This is a direct marketing scenario where we mail 5,000 solicitations to customers & test a different package to another 5,000 customers. (We split our 10,000 customers randomly into the 2 groups)
I can easily measure the statistical significance of the # of Responses. 200 respondents in population A & 150 in population B. But I also need to factor in the size of the purchase. Avg gift of population A was $25 & the Avg gift of population B was$45.
#### Michael Schmidt
##### New Member
Dollar amounts are just numbers and can be treated as such. You could take the two samples who gave (150 and 200) and compare them directly with t-tests. Alternatively, you could take all 5000 in each sample and do the same computation. The company might be interested in an average-dollars-per-contact figure, which would 1.00 for Group A and 1.35 for B. So, in a simpleminded way, the Group B methodology has a 35% advantage.
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# Shakashaka
Translate this site.
Shakashaka (Proof of Quilt) is a logic puzzle with simple rules and challenging solutions.
The rules are simple. Shakashaka is played on a rectangular grid. The grid has both black cells and white cells in it.
The objective is to place black triangles in the white cell in such a way so that they form white rectangular (or square) areas.
- The triangles are right angled and occupy half of the white square divided diagonally.
- You can place triangles only in white cells
- The numbers in the black cells indicate how many triangles are adjacent, vertically and horizontally.
- The white rectangles can be either straight or rotated at 45°
Video Tutorial
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20x20 Shakashaka Puzzle ID: 9,128,769
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(Solved) Investment A Demand Probability Outdoor Smoker High 0.2 Moderate 0.6 Low 0.2 Outdoor Grill High 0.2 Moderate 0.6 Low 0.
You are an economist for the Vanda-Laye Corporation, which produces and distributes outdoor cooking supplies. The company has come under new ownership and management and will be undergoing changes in its product lines and operating structure. As an economist, your responsibilities include examining the market factors that affect success or failure of a product, including the supply and demand for the product, market conditions, and the behavior of competitors with similar products.
The new management has identified several possible investments for the coming year. It has asked you and your team to evaluate the possibilities and make a recommendation to the board of directors. Jorge has identified two mutually exclusive opportunities (Investment A) and two independent opportunities (Investment B) and assigned you the task of making a recommendation on the investments.
Investment A
Your company would like to increase its product lines. Two alternatives are available, a new line of outdoor smokers and a new line of outdoor grills. The two lines are mutually exclusive, meaning that only one of these investment alternatives can be selected. The projected cash flows and their respective probabilities for each alternative are given in the table. There are three possible levels of demand and their corresponding probabilities, which depend on the state of the economy.
The two alternatives carry equal risk and should be evaluated at the company's cost of capital. The cost for the new smoker line will be \$7,000,000. Also, the company has been guaranteed a buyer for the new line at the end of the fifth year. The buyer has agreed to purchase the new line for \$7,900,000. The outdoor grill alternative will cost \$3,987,000 and also has a guaranteed buyer, who has agreed to pay \$4,000,000 at the end of the fifth year.
Investment B
Investment B involves two independent investment opportunities. The decisions on these two investment alternatives are also independent of Investment A. Investment B-1 involves a new packaging machine, which will eliminate the need for a local firm for packaging Vanda-Laye's products. The cost of this machine will be \$24,000, and the expected revenues from this opportunity are given in the table and are considered to be of average risk. Investment B-2 is the purchase of a new computer system that will allow the company to sell its products on the Internet worldwide. The cost of this new system will be \$29,000, with the expected cash flows after taxes given in the table.
Jorge has asked you to provide detailed responses to the following:
• Management of Vanda-Laye has determined that the capital structure of the company will involve 30% debt and 70% common equity. This structure will be used to finance all investments by the company. Currently, the company can sell new bonds at par, with a coupon rate of 7%. Any new common stock can be sold for \$45, with a required return (or cost) of 15.57%. Using Microsoft Excel, calculate the company's cost of capital to be used in the evaluation of possible investment projects.
• For Investment A:
• Using Microsoft Excel, create a decision tree. Indicate the various levels of demand and their respective probabilities. Also, include the calculations for the expected cash flows.
• Calculate the expected NPV for each alternative. Explain the decision rules for making a selection between the two alternatives on the basis of the expected NPV.
• Assuming the two alternatives are mutually exclusive, specify which alternative you would recommend to the company. Explain why.
• If the two alternatives were independent of each other, specify which project you would select. Would you accept both projects if funding were available for both? Explain your answer.
• For Investment B:
• Using Microsoft Excel, calculate the NPV for each alternative.
• Using the decision-making criteria for the NPV, specify which alternative you would select if the two alternatives were mutually exclusive. Explain your answer.
• Given that the two alternatives are independent of each other, specify which investment you would select, if not both. Explain your answer.
• Using Microsoft Excel, calculate the IRR for each investment.
• Using the decision-making criteria for the IRR, specify which alternative you would prefer. Explain your answer.
• If funding were available, specify whether you would select both investments. Why or why not?
• Calculate the profitability index (PI) for the two investments. Which project is preferred?
• Determine whether there is a ranking conflict present in terms of the IRR and the NPV. Explain your answer. If a conflict does exist, explain how you would resolve the situation
Investment A Demand Probability
Outdoor Smoker
High
0.2
Moderate
0.6
Low
0.2
Outdoor Grill
High
0.2
Moderate
0.6
Low
0.2 Year 1 Year 2 Year 3 Year 4 Year 5 \$800,000
\$500,000
\$200,000 \$900,000
\$700,000
\$350,000 \$1,000,000
\$800,000
\$500,000 \$1,100,000
\$960,000
\$600,000 \$1,500,000
\$1,240,000
\$750,000 \$600,000
\$450,000
\$150,000 \$750,000
\$500,000
\$220,000 \$850,000
\$700,000
\$370,000 \$975,000
\$825,000
\$500,000 \$5,160,000
\$4,980,000
\$4,750,000 Page 1 of 1
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Question about Casio FX82MS Scientific Calculator
# Raise power - Casio FX82MS Scientific Calculator
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A key marked [^], [X^y] or [Y^x]
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## Related Questions:
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### How do I raise the power of 30 on calculator?
If you are using an HP 12c or have the button on the calculator "yx" you first enter the number you wish to raise (y) then enter then the power (x). For instance to find 230 (2 to the power of 30) you input the number 2 then hit the enter key on the calculator, then input 30 and hit the yx button.
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Hope it helps.
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### 99 ford expedition power window
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# Three friends, A, B and C decided to have a beer party. If each of the
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Three friends, A, B and C decided to have a beer party. If each of the [#permalink]
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17 Oct 2018, 00:46
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Three friends, A, B and C decided to have a beer party. If each of the three friends consumed equal quantities of beer, and paid equally for it, what was the price of one beer bottle?
(1) A, B and C brought along 4, 6 and 2 bottles of beer, respectively; all bottles of beer being identical.
(2) C paid a total of \$16 to A and B for his share.
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Joined: 09 Jun 2014
Posts: 246
Location: India
Concentration: General Management, Operations
Schools: Tuck '19
Re: Three friends, A, B and C decided to have a beer party. If each of the [#permalink]
### Show Tags
17 Oct 2018, 02:33
2
Bunuel wrote:
Three friends, A, B and C decided to have a beer party. If each of the three friends consumed equal quantities of beer, and paid equally for it, what was the price of one beer bottle?
(1) A, B and C brought along 4, 6 and 2 bottles of beer, respectively; all bottles of beer being identical.
(2) C paid a total of \$16 to A and B for his share.
So we need to calculate the price of each beer bottle,considering they consumed equal quantities and the price per bottle was same.
Statement 1:
The statement simply status the number of beer bottles brought.
No information about whether they consumed all bottles of beer or not.Also no information about the price of
each bottle.
Statement 2"
The statement means that A paid \$16 for his share but we dont know how many bottles he consumed.
Combing equation 1 and 2,
We still dont know how many bottles have been consumed,we just know the number of bottles that were brought.
So Insufficient.
My answer would be E for this.Waiting for OA to be posted
Press Kudos if it helps !!
Intern
Joined: 29 Apr 2017
Posts: 26
Location: India
Concentration: Operations, Other
GMAT 1: 660 Q43 V38
GPA: 4
WE: Operations (Transportation)
Re: Three friends, A, B and C decided to have a beer party. If each of the [#permalink]
### Show Tags
18 Oct 2018, 00:52
prabsahi wrote:
Bunuel wrote:
Three friends, A, B and C decided to have a beer party. If each of the three friends consumed equal quantities of beer, and paid equally for it, what was the price of one beer bottle?
(1) A, B and C brought along 4, 6 and 2 bottles of beer, respectively; all bottles of beer being identical.
(2) C paid a total of \$16 to A and B for his share.
So we need to calculate the price of each beer bottle,considering they consumed equal quantities and the price per bottle was same.
Statement 1:
The statement simply status the number of beer bottles brought.
No information about whether they consumed all bottles of beer or not.Also no information about the price of
each bottle.
Statement 2"
The statement means that A paid \$16 for his share but we dont know how many bottles he consumed.
Combing equation 1 and 2,
We still dont know how many bottles have been consumed,we just know the number of bottles that were brought.
So Insufficient.
My answer would be E for this.Waiting for OA to be posted
Press Kudos if it helps !!
self also applied same logic to get to answer E but OA says C, can anyone explain?
Manager
Joined: 09 Jun 2014
Posts: 246
Location: India
Concentration: General Management, Operations
Schools: Tuck '19
Re: Three friends, A, B and C decided to have a beer party. If each of the [#permalink]
### Show Tags
18 Oct 2018, 01:00
1
Kumar Utkarsh wrote:
prabsahi wrote:
Bunuel wrote:
Three friends, A, B and C decided to have a beer party. If each of the three friends consumed equal quantities of beer, and paid equally for it, what was the price of one beer bottle?
(1) A, B and C brought along 4, 6 and 2 bottles of beer, respectively; all bottles of beer being identical.
(2) C paid a total of \$16 to A and B for his share.
So we need to calculate the price of each beer bottle,considering they consumed equal quantities and the price per bottle was same.
Statement 1:
The statement simply status the number of beer bottles brought.
No information about whether they consumed all bottles of beer or not.Also no information about the price of
each bottle.
Statement 2"
The statement means that A paid \$16 for his share but we dont know how many bottles he consumed.
Combing equation 1 and 2,
We still dont know how many bottles have been consumed,we just know the number of bottles that were brought.
So Insufficient.
My answer would be E for this.Waiting for OA to be posted
Press Kudos if it helps !!
self also applied same logic to get to answer E but OA says C, can anyone explain?
I think there is assumption in the question then between bringing bottles and consuming bottles..
In that case since its not explicitly stated in the question I would discard this question on quality issues.
Press Kudos if it helps!!
Intern
Joined: 05 Dec 2018
Posts: 13
Three friends, A, B and C decided to have a beer party. If each of the [#permalink]
### Show Tags
14 Dec 2018, 10:33
The point is: they brought different quantities of beers, but drank them equally and thus must pay equally. it asks uf for the price
1) total beers = 12. how much is one? insufficient
2) C owes 16\$ to A and B. It means that right now A and B had paid exactly 8\$ more than what they were supposed to. but how much is one Beer ? it could be 4\$ or 2% or 1 \$. How many beers there were ? it could be 16, 8, 4 or anything. INSUFFICIENT
1+2) total beers = 12
C owes 16\$ thus 8\$ to A and 8\$ to B.
so 12 = A+B+C
total cost = X(A+B+C) = X(12)
single expenditure = X(12)/3 = 4X
but A and B spent 4X + 8 each
while C spent 16 ( the exact money he owes and thus that everyone owes) thus,
4x = 16
x=4
total expenditure = 48
single expenditure = 16
Three friends, A, B and C decided to have a beer party. If each of the [#permalink] 14 Dec 2018, 10:33
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Convert Hundredweight to Slug (Mass / Weight)
## Hundredweight into Slug
numbers in scientific notation
https://www.convert-measurement-units.com/convert+Hundredweight+to+Slug.php
# Convert Hundredweight to Slug (Mass / Weight):
1. Choose the right category from the selection list, in this case 'Mass / Weight'.
2. Next enter the value you want to convert. The basic operations of arithmetic: addition (+), subtraction (-), multiplication (*, x), division (/, :, ÷), exponent (^), square root (√), brackets and π (pi) are all permitted at this point.
3. From the selection list, choose the unit that corresponds to the value you want to convert, in this case 'Hundredweight'.
4. Finally choose the unit you want the value to be converted to, in this case 'Slug'.
5. Then, when the result appears, there is still the possibility of rounding it to a specific number of decimal places, whenever it makes sense to do so.
With this calculator, it is possible to enter the value to be converted together with the original measurement unit; for example, '439 Hundredweight'. In so doing, either the full name of the unit or its abbreviation can be used. Then, the calculator determines the category of the measurement unit of measure that is to be converted, in this case 'Mass / Weight'. After that, it converts the entered value into all of the appropriate units known to it. In the resulting list, you will be sure also to find the conversion you originally sought. Alternatively, the value to be converted can be entered as follows: '88 Hundredweight to Slug' or '58 Hundredweight into Slug' or '82 Hundredweight -> Slug' or '76 Hundredweight = Slug'. For this alternative, the calculator also figures out immediately into which unit the original value is specifically to be converted. Regardless which of these possibilities one uses, it saves one the cumbersome search for the appropriate listing in long selection lists with myriad categories and countless supported units. All of that is taken over for us by the calculator and it gets the job done in a fraction of a second.
Furthermore, the calculator makes it possible to use mathematical expressions. As a result, not only can numbers be reckoned with one another, such as, for example, '(22 * 16) Hundredweight'. But different units of measurement can also be coupled with one another directly in the conversion. That could, for example, look like this: '34 Hundredweight + 28 Slug' or '10mm x 4cm x 97dm = ? cm^3'. The units of measure combined in this way naturally have to fit together and make sense in the combination in question.
The mathematical functions sin, cos, tan and sqrt can also be used. Example: sin(π/2), cos(pi/2), tan(90°), sin(90) or sqrt(4).
If a check mark has been placed next to 'Numbers in scientific notation', the answer will appear as an exponential. For example, 4.667 738 229 128 5×1021. For this form of presentation, the number will be segmented into an exponent, here 21, and the actual number, here 4.667 738 229 128 5. For devices on which the possibilities for displaying numbers are limited, such as for example, pocket calculators, one also finds the way of writing numbers as 4.667 738 229 128 5E+21. In particular, this makes very large and very small numbers easier to read. If a check mark has not been placed at this spot, then the result is given in the customary way of writing numbers. For the above example, it would then look like this: 4 667 738 229 128 500 000 000. Independent of the presentation of the results, the maximum precision of this calculator is 14 places. That should be precise enough for most applications.
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# Exact Value of cos 54°
We will learn to find the exact value of cos 36 degrees using the formula of multiple angles.
How to find exact value of cos 54°?
Solution:
Let A = 18°
Therefore, 5A = 90°
⇒ 2A + 3A = 90˚
⇒ 2θ = 90˚ - 3A
Taking sine on both sides, we get
sin 2A = sin (90˚ - 3A) = cos 3A
⇒ 2 sin A cos A = 4 cos$$^{3}$$ A - 3 cos A
⇒ 2 sin A cos A - 4 cos$$^{3}$$ A + 3 cos A = 0
⇒ cos A (2 sin A - 4 cos$$^{2}$$ A + 3) = 0
Dividing both sides by cos A = cos 18˚ ≠ 0, we get
⇒ 2 sin θ - 4 (1 - sin$$^{2}$$ A) + 3 = 0
⇒ 4 sin$$^{2}$$ A + 2 sin A - 1 = 0, which is a quadratic in sin A
Therefore, sin θ = $$\frac{-2 \pm \sqrt{- 4 (4)(-1)}}{2(4)}$$
⇒ sin θ = $$\frac{-2 \pm \sqrt{4 + 16}}{8}$$
⇒ sin θ = $$\frac{-2 \pm 2 \sqrt{5}}{8}$$
⇒ sin θ = $$\frac{-1 \pm \sqrt{5}}{4}$$
Now sin 18° is positive, as 18° lies in first quadrant.
Therefore, sin 18° = sin A = $$\frac{-1 \pm \sqrt{5}}{4}$$
Now, cos 36° = cos 2 ∙ 18°
⇒ cos 36° = 1 - 2 sin$$^{2}$$ 18°
⇒ cos 36° = 1 - 2$$(\frac{\sqrt{5} - 1}{4})^{2}$$
⇒ cos 36° = $$\frac{16 - 2(5 + 1 - 2\sqrt{5})}{16}$$
⇒ cos 36° = $$\frac{1 + 4\sqrt{5}}{16}$$
⇒ cos 36° = $$\frac{\sqrt{5} + 1}{4}$$
Therefore, sin 36° = $$\sqrt{1 - cos^{2} 36°}$$,[Taking sin 36° is positive, as 36° lies in first quadrant, sin 36° > 0]
⇒ sin 36° = $$\sqrt{1 - (\frac{\sqrt{5} + 1}{4})^{2}}$$
⇒ sin 36° = $$\sqrt{\frac{16 - (5 + 1 + 2\sqrt{5})}{16}}$$
⇒ sin 36° = $$\sqrt{\frac{10 - 2\sqrt{5}}{16}}$$
⇒ sin 36° = $$\frac{\sqrt{10 - 2\sqrt{5}}}{4}$$
Therefore, sin 36° = $$\frac{\sqrt{10 - 2\sqrt{5}}}{4}$$
Now cos 54° = cos (90° - 36°) = sin 36° = $$\frac{\sqrt{10 - 2\sqrt{5}}}{4}$$
Therefore, cos 54° = $$\frac{\sqrt{10 - 2\sqrt{5}}}{4}$$
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Model and Solve Statements
# Introduction
This chapter brings together all the concepts discussed in previous chapters by explaining how to specify a model and solve it.
# The Model Statement
The model statement is used to collect equations into groups and to label them so that they can be solved. The simplest form of the model statement uses the keyword all: the model consists of all equations declared before the model statement is entered. For most simple applications this is all the user needs to know about the model statement.
## The Syntax
In general, the syntax for a model declaration in GAMS is as follows:
model[s] model_name [text] [/ (all | eqn_name {, eqn_name}) {, var_name(set_name)} /]
{,model_name [text] [/ (all | eqn_name {, eqn_name}) {, var_name(set_name)} /]} ;
The keyword model[s] indicates that this is a model statement and model_name is the internal name of the model in GAMS, it is an identifier. The optional explanatory text is used to describe the model, all is a keyword as introduced above and eqn_name is the name of an equation that has been declared prior to the model statement. Var_name(set_name) is a couple of previous declared variable and and set to limit the domain of variables in the model. More details about this are described in the following subsection. For advice on explanatory text and how to choose a model_name, see the tutorial Good Coding Practices.
Note
Model statements for Mixed Complementarity Problem (MCP) and Mathematical Program with Equilibrium Constraints (MPEC) models require a slightly different notation, since complementarity relationships need to be included. For details see subsections Mixed Complementarity Problem (MCP) and Mathematical Program with Equilibrium Constraints (MPEC).
An example of a model definition in GAMS is shown below.
Model transport "a transportation model" / all /;
The model is called transport and the keyword all is a shorthand for all known (declared) equations.
Several models may be declared (and defined) in one model statement. This is useful when experimenting with different ways of writing a model, or if one has different models that draw on the same data. Consider the following example, adapted from [PROLOG], in which different groups of the equations are used in alternative versions of the problem. Three versions are solved: the linear, nonlinear, and 'expenditure' versions. The model statement to define all three is:
Model nortonl "linear version" / cb,rc,dfl,bc,obj /
nortonn "nonlinear version" / cb,rc,dfn,bc,obj /
nortone "expenditure version" / cb,rc,dfe,bc,obj / ;
Here cb, rc, etc. are the names of the equations. We will describe below how to obtain the solution to each of the three models.
Note
If several models are declared and defined with one model statement, the models have to be separated by commas or linefeeds and a semicolon terminates the entire statement.
If several models are declared then it is possible to use one previously declared model in the declaration of another. The following examples illustrate this:
Model one "first model" / tcost_eq, supply_eq, demand_eq /
two "second model that nests first" / one, balance_eq /
three "third model that nests first and second" / two, capacity_eq, configure_eq /;
Model one is declared and defined using the general syntax, model two contains all the equations of model one and the equation balance_eq, and model three contains all of model two and the equations capacity_eq and configure_eq.
In addition to nesting models as illustrated above, it is also possible to use the symbols + and - to augment or remove items relative to models that were previously defined. The following examples serve as illustration:
Model four "fourth model: model three minus model one" / three-one /
five "fifth model: model three without eqn configure_eq" / three-configure_eq /
six "sixth model: model four plus model two" / four+two /;
Model four contains the equations from model three except for those that belong to model one. Model five contains all equations from model three except for equation configure_eq. Model six contains the union of the equations in model four and two. Note that both model names and equation names may be used in association with the symbols + and -.
### Limited domain for variables
As mentioned above, it is possible to limit the domain of variables used in a model in the model statement. This allows to restrict the generation of blocks of variables in a single place instead of using, e.g., dollar conditions at every place where this variable block is used in equations (which might be required for an efficient model generation).
The following examples is based on the basic transportation model [TRNSPORT]. To limit the transportation network in that model to certain links (e.g. because some are blocked because of some reason) one could introduce a subset of the free links and use that with dollar conditions in the equations like this:
* Initialize whole network as free
Set freeLinks(i,j) Useable links in the network / #i.#j /;
cost.. z =e= sum((i,j), c(i,j)*x(i,j)$freeLinks(i,j)); supply(i).. sum(j, x(i,j)$freeLinks(i,j)) =l= a(i);
demand(j).. sum(i, x(i,j)$freeLinks(i,j)) =g= b(j); * Block a particular link freeLinks('san-diego','topeka') = no; Model transport / all /; solve transport using lp minimizing z; Now, instead of adding the dollar condition to each appearance of x in the model, one could simply add a domain restriction for that variable to the model statement directly by specifying a variable and the set that limits its domain. Using this approach, the previous example looks like this: * Initialize whole network as free Set freeLinks(i,j) Useable links in the network / #i.#j /; cost.. z =e= sum((i,j), c(i,j)*x(i,j)); supply(i).. sum(j, x(i,j)) =l= a(i); demand(j).. sum(i, x(i,j)) =g= b(j); * Block a particular link freeLinks('san-diego','topeka') = no; Model transport / all, x(freeLinks) /; solve transport using lp minimizing z; Note If one adds the domain restriction to the model statement, internally GAMS inserts a dollar condition to every appearance of the restricted variables in equations of the model. When doing this, the indices are copied as they appear with the variable. So, in the example above, x(i,j) becomes x(i,j)$freeLinks(i,j). In the same way x(i-1,j+1) becomes x(i-1,j+1)$freeLinks(i-1,j+1) and x('seattle','chicago') becomes x('seattle','chicago')$freeLinks('seattle','chicago').
Attention
As a consequence of above's note one could see some unexpected results, like "division by zero errors", if it is not done carefully. For example, the following dummy model, will trigger such an error, since we sum over all i, but some x were excluded leaving a 0 as divisor:
Set i / i1*i3 /
sub(i) / i2 /;
Positive Variable x(i);
Variable z;
Equation obj;
obj.. z =e= sum(i, 1/x(i));
x.lo(i) = 1;
Model m / obj, x(sub) /;
solve m min z use nlp;
## Classification of Models
Various types of problems can be solved with GAMS. Note that the type of the model must be known before it may be solved. The model types are briefly discussed in this section. GAMS checks that the model is in fact the type the user thinks it is, and issues explanatory error messages if it discovers a mismatch - for instance, that a supposedly linear model contains nonlinear terms. Some problems may be solved in more than one way, and the user has to choose which way to use. For instance, if there are binary or integer variables in the model, it can be solved either as a MIP or as a RMIP.
The model types and their identifiers, which are needed in the a solve statement, are given in Table 1. For details on the solve statement, see section The Solve Statement.
GAMS Model Type Model Type Description Requirements and Comments
LP Linear Program Model with no nonlinear terms or discrete (i.e. binary, integer, etc) variables.
NLP Nonlinear Program Model with general nonlinear terms involving only smooth functions, but no discrete variables. For a classification of functions as to smoothness, see section Functions.
QCP Quadratically Constrained Program Model with linear and quadratic terms, but no general nonlinear terms or discrete variables.
DNLP Discontinuous Nonlinear Program Model with non-smooth nonlinear terms with discontinuous derivatives, but no discrete variables. This is the same as NLP, except that non-smooth functions may appear as well. These models are more difficult to solve than normal NLP models and we strongly advise not to use this model type.
MIP Mixed Integer Program Model with binary, integer, SOS and/or semi variables, but no nonlinear terms.
RMIP Relaxed Mixed Integer Program Like MIP, except that the discrete variable requirement is relaxed. See the note below on relaxed model types.
MINLP Mixed Integer Nonlinear Program Model with both nonlinear terms and discrete variables.
RMINLP Relaxed Mixed Integer Nonlinear Program Like MINLP except that the discrete variable requirement is relaxed. See the note below on relaxed model types.
MIQCP Mixed Integer Quadratically Constrained Program Model with both quadratic terms and discrete variables, but no general nonlinear term.
RMIQCP Relaxed Mixed Integer Quadratically Constrained Program Like MIQCP except that the discrete variable requirement is relaxed. See the note below on relaxed model types.
MCP Mixed Complementarity Problem A square, possibly nonlinear, model that generalizes a system of equations. Rows and columns are matched in one-to-one complementary relationships.
CNS Constrained Nonlinear System Model solving a square, possibly nonlinear system of equations, with an equal number of variables and constraints.
MPEC Mathematical Programs with Equilibrium Constraints A difficult model type for which solvers and reformulations are currently being developed.
RMPEC Relaxed Mathematical Program with Equilibrium Constraints A difficult model type for which solvers and reformulations are currently being developed. See the note below on relaxed model types.
EMP Extended Mathematical Program A family of mathematical programming extensions.
MPSGE General Equilibrium Not actually a model type but mentioned for completeness, see MPSGE.
Table 1: GAMS Model Types
Note
• The relaxed model types RMIP, RMINLP, RMIQCP, and RMPEC solve the problem as the corresponding model type (e.g. MIP for RMIP) but relax the discrete requirement of the discrete variables. This means that integer and binary variables may assume any values between their bounds. SemiInteger and SemiCont variables may assume any values between 0 and their upper bound. For SOS1 and SOS2 variables the restriction of the number of non-zero values is removed.
• Many "LP" solvers like Cplex offer the functionality of solving convex quadratic models. So the Q matrices in the model need to be positive semidefinite. An extension to to this are the second-order cone programs (SOCP) with either symmetric or rotated cones. See the solver manuals (e.g. on MOSEK) for details.
• Unlike other checks on the model algebra (e.g. existence of discrete variables or general non-linear terms), the GAMS compiler does not enforce a quadratic model to only consist of quadratic and linear terms. This requirement is enforced at runtime for a particular model instance.
### Linear Programming (LP)
Mathematically, the Linear Programming (LP) problem looks like:
\begin{equation*} \begin{array}{ll} \textrm{Minimize or maximize} & cx \\ \textrm{subject to} & Ax \, \, \alpha \, \, b \\ & L \leq x \leq U, \\ \end{array} \end{equation*}
where $$x$$ is a vector of variables that are continuous real numbers, $$cx$$ is the objective function, and $$Ax \, \alpha \, b$$ represents the set of constraints. Here, $$\alpha$$ is an equation operator. For details on the equation types allowed in GAMS, see Equation Types. $$L$$ and $$U$$ are vectors of lower and upper bounds on the variables.
GAMS supports free (unrestricted) variables, positive variables, and negative variables. Note that users may customize lower and upper bounds, for details see section Bounds on Variables.
For information on LP solvers that can be used through GAMS see the Solver/Model type Matrix.
### Nonlinear Programming (NLP)
Mathematically, the Nonlinear Programming (NLP) problem looks like:
\begin{equation*} \begin{array}{ll} \textrm{Minimize or Maximize} & f(x) \\ \textrm{subject to} & g(x) \, \, \alpha \, \, 0 \\ & L \leq x \leq U, \\ \end{array} \end{equation*}
where $$x$$ is a vector of variables that are continuous real numbers, $$f(x)$$ is the objective function, and $$g(x) \, \alpha \, 0$$ represents the set of constraints. For details on the equation types allowed in GAMS, see Equation Types. Note that the functions $$f(x)$$ and $$g(x)$$ have to be differentiable. $$L$$ and $$U$$ are vectors of lower and upper bounds on the variables.
For information on NLP solvers that can be used through GAMS see the Solver/Model type Matrix. See also the tutorial Good NLP Formulations.
Note
NLP models may have the nonlinear terms inactive. In this case setting the model attribute TryLinear to 1 causes GAMS to check the model and use the default LP solver if possible. For details on model attributes, see subsection Model Attributes.
### Quadratically Constrained Programs (QCP)
Mathematically, the Quadratically Constrained Programming (QCP) problem looks like:
\begin{equation*} \begin{array}{lll} \textrm{Maximize or Minimize} & cx + x'Q x & \\ \textrm{subject to} & A_i x + x' R_i x \,\, \alpha \,\, b_i & \textrm{for all} \,\, i \\ & L \leq x \leq U, & \\ \end{array} \end{equation*}
where $$x$$ denotes a vector of variables that are continuous real numbers, $$cx$$ is the linear part of the objective function, $$x'Qx$$ is the quadratic part of the objective function, $$A_i x$$ represents the linear part of the $$i$$th constraint, $$x' R_i x$$ its quadratic part and $$b_i$$ its right-hand side. For details on the equation types allowed in GAMS, see Equation Types". Further, $$L$$ and $$U$$ are vectors of lower and upper bounds on the variables.
Note that a QCP is a special case of the NLP in which all the nonlinearities are required to be quadratic. As such, any QCP model can also be solved as an NLP. However, most "LP" vendors provide routines to solve LP models with a quadratic objective. Some allow quadratic constraints as well. Solving a model using the QCP model type allows these "LP" solvers to be used to solve quadratic models as well as linear ones. Some NLP solvers may also take advantage of the special (quadratic) form when solving QCP models.
Attention
In case a model with quadratic constraints is passed to a QCP solver that only allows a quadratic objective, a capability error will be returned (solver status 6 CAPABILITY PROBLEMS). Some solvers will fail when asked to solve a non-convex quadratic problems as described above.
Note
Using the model attribute TryLinear causes GAMS to see if the problem can be solved as an LP problem. For details on model attributes, see subsection Model Attributes.
For information on QCP solvers that can be used through GAMS see the Solver/Model type Matrix.
### Nonlinear Programming with Discontinuous Derivatives (DNLP)
Mathematically, the Nonlinear Programming with Discontinuous Derivatives (DNLP) problem looks like:
\begin{equation*} \begin{array}{ll} \textrm{Maximize or Minimize} & f(x)\\ \textrm{subject to} & g(x) \, \, \alpha \, \, 0 \\ & L \leq x \leq U, \\ \end{array} \end{equation*}
where $$x$$ is a vector of variables that are continuous real numbers, $$f(x)$$ is the objective function, $$g(x) \, \alpha \, 0$$ represents the set of constraints, and $$L$$ and $$U$$ are vectors of lower and upper bounds on the variables. For details on the equation types allowed in GAMS, see Equation Types. Note that this is the same as NLP, except that non-smooth functions, like abs, min, max may appear in $$f(x)$$ and $$g(x)$$.
For information on DNLP solvers that can be used through GAMS see the Solver/Model type Matrix.
Attention
• We strongly advise against using the model type DNLP. The best way to model discontinuous functions is with binary variables, which results in a model of the type MINLP. The model [ABSMIP] demonstrates this formulation technique for the functions abs, min, max and sign. See also section Reformulating DNLP Models.
• Solvers may have difficulties when dealing with the discontinuities, since they are really NLP solvers and the optimality conditions and the reliance on derivatives may be problematic. Using the global solvers ANTIGONE, BARON, COUENNE, LINDO or SCIP may alleviate this problem.
### Mixed Integer Programming (MIP)
Mathematically, the Mixed Integer Linear Programming (MIP) problem looks like:
\begin{equation*} \begin{array}{lrclll} \textrm{Maximize or Minimize} & c_1 t + c_2 u + c_3 v + c_4 w + c_5 x + c_6 y + c_7 z & & & & \\ \textrm{subject to} & A_1 t + A_2 u + A_3 v + A_4 w + A_5 x + A_6 y + A_7 z & \alpha & b & & \\ & t & \in & \mathbb R & &\\ & u & \geq & 0 & \textrm{and}\,\, u \leq L_2 & \textrm{and} \, \, u \in \mathbb Z \\ & v & \in & (0,1) & &\\ & w & \in & \textrm{SOS1} & &\\ & x & \in & \textrm{SOS2} & &\\ & y & = & 0 & \textrm{or} \, \, \, L_6 \leq y & \\ & z & = & 0 & \textrm{or} \,\, \, L_7 \leq z & \textrm{and} \,\, z \in \mathbb Z,\\ \end{array} \end{equation*}
where
• $$c_1t + c_2u + c_3v + c_4w + c_5x + c_6y + c_7z$$ is the objective function,
• $$A_1t + A_2u + A_3v + A_4w + A_5x + A_6y + A_7z \,\, \alpha \,\, b$$ represents the set of constraints of various equality and inequality forms,
• $$t$$ is a vector of variables that are continuous real numbers,
• $$u$$ is a vector of variables that can only take integer values smaller than $$L_2$$,
• $$v$$ is a vector of binary variables,
• $$w$$ is a vector of variables that belong to SOS1 sets; this means that at most one variable in the set is nonzero,
• $$x$$ is a vector of variables that belong to SOS2 sets; this means that at most two adjacent variables in the set are nonzero,
• $$y$$ is a vector of variables that are semi-continuous; they are either zero or larger than $$L_6$$,
• $$z$$ is a vector of variables that are semi-integer; they are integer and either zero or larger than $$L_7$$.
For details on the equation types allowed in GAMS, see Equation Types. For more details on MIPs in GAMS, especially the use of SOS and semi variables, see section Special Mixed Integer Programming (MIP) Features.
For information on MIP solvers that can be used through GAMS, see the Solver/Model type Matrix.
Attention
Not all MIP solvers cover all the cases associated with SOS and semi variables. Please consult the solver manuals for details on capabilities.
### Mixed Integer Nonlinear Programming (MINLP)
Mathematically, the Mixed Integer Nonlinear Programming (MINLP) problem looks like:
\begin{equation*} \begin{array}{ll} \textrm{Maximize or Minimize} & f(x) + Dy \\ \textrm{subject to} & g(x) + Hy \, \, \alpha \, \, 0 \\ & L \leq x \leq U \\ & y = \{0,1,2,\cdots\}, \\ \end{array} \end{equation*}
where $$x$$ is a vector of variables that are continuous real numbers, $$y$$ denotes a vector of variables that can only take integer values, $$f(x)+ Dy$$ is the objective function, $$g(x) + Hy\ \, \alpha \, 0$$ represents the set of constraints, and $$L$$ and $$U$$ are vectors of lower and upper bounds on the variables. For details on the equation types allowed in GAMS, see Equation Types. Further, $$y = \{0,1,2,\cdots\}$$ is the integrality restriction on $$y$$.
For information on MINLP solvers that can be used through GAMS see the Solver/Model type Matrix.
Note
• SOS and semi variables can also be accommodated by some solvers. Please consult the solver manuals for details on capabilities.
• The model attribute TryLinear causes GAMS to examine whether the problem may be solved as a MIP problem. For details on model attributes, see subsection Model Attributes.
### Mixed Integer Quadratically Constrained Programs (MIQCP)
A Mixed Integer Quadratically Constrained Program (MIQCP) is a special case of the MINLP in which all the nonlinearities are required to be quadratic. For details see the description of the QCP, a special case of the NLP.
For information on MIQCP solvers that can be used through GAMS, see the Solver/Model type Matrix.
Note
The model attribute TryLinear causes GAMS to examine whether the problem may be solved as a MIP problem. For details on model attributes, see subsection Model Attributes.
### Mixed Complementarity Problem (MCP)
Unlike the other model types we have introduced so far, the Mixed Complementarity Problem (MCP) does not have an objective function. An MCP is specified by three pieces of data: a function $$F(z): \mathbb R^n \mapsto \mathbb R^n$$, lower bounds $$l \in \{\mathbb R \cup \{-\infty\}\}^n$$ and upper bounds $$u \in \{\mathbb R \cup \{\infty\}\}^n$$. A solution is a vector $$z \in \mathbb R^n$$ such that for each $$i \in \{1, \ldots, n\}$$, one of the following three conditions hold:
$\begin{array}{rcl} F_i(z) = 0 & \mbox{ and } & \ell_i \leq z_i \leq u_i \, \, \, \mbox{ or} \\ F_i(z) > 0 & \mbox{ and } & z_i = \ell_i \, \, \, \mbox{ or} \\ F_i(z) < 0 & \mbox{ and } & z_i = u_i . \end{array}$
This problem can be written compactly as
\begin{equation*} F(z) \, \perp \, L \leq z \leq U, \end{equation*}
where the symbol $$\perp$$ (which means "perpendicular to", shortened to "perp to") indicates pair-wise complementarity between the function $$F$$ and the variable $$z$$ and its bounds.
The following special case is an important and illustrative example:
\begin{equation*} F(z) \perp z \geq 0. \end{equation*}
In this example, the unstated but implied upper bound $$u$$ is infinity. Since $$z$$ is finite, we cannot have $$z_i = u_i$$ and the third condition above cannot hold: this implies $$F(z) >= 0$$. The remaining two conditions imply pair-wise complementarity between $$z >= 0$$ and $$F(z) >= 0$$. This is exactly the Nonlinear Complementarity Problem, often written as
\begin{equation*} F(z) \geq 0, \quad z \geq 0, \quad \langle{F(z)},{z}\rangle = 0. \end{equation*}
None of this rules out the degenerate case (i.e. $$F_i(z)$$ and $$z_i$$ both zero). In practice, these can be difficult models to solve.
Another special case arises when the bounds $$L$$ and $$U$$ are infinite. In this case, the second and third conditions above cannot hold, so we are left with $$F(z) = 0$$, a square system of nonlinear equations. And finally, we should mention a special case that occurs frequently in practice: if $$\ell_i = u_i$$ (i.e. $$z_i$$ is fixed) then we have a complementary pair: one of the three conditions will hold as long as $$F_i(z)$$ is defined. Essentially, fixing a variable removes or obviates the matching equation. This is often useful when modeling with MCP.
The definition above describes the canonical MCP model as it exists when GAMS passes it to an MCP solver. Some models have exactly this form even in the GAMS code, but usually some processing is done by the GAMS system to arrive at a model in this form. Here we'll describe the steps of this process and illustrate with an example from the model library.
1. The process starts with the list of rows (aka single equations) and columns (aka single variables) that make up the MCP model, and potentially some matching information.
• The usual rules apply: rows are part of the model because their associated equations are included in the model statement, but columns only become part of the model by use: a column enters the model only if it is used in some row of the model. Therefore including a variable symbol as part of a match in the model statement will not influence the set of columns belonging to the model.
• Matches (where they exist) are pointers from rows to columns.
• Technically, the MCP is defined via a function $$F$$ while a model contains constraints. Given a constraint, we define an associated function as LHS - RHS, so e.g. $$F_i \geq 0$$ is consistent with a =G= constraint.
2. The explicit matches are processed: each match creates a complementary pair. What remains after the explicit matches are consumed are the unmatched rows and unmatched columns.
• It is an error for any column to be matched to multiple rows, so the row-column matching is one-to-one.
• For each match some consistency checks between the column bounds and the row type are made. For details, see Table 2.
• For example, matching an =N= row with any column is good, matching an =E= row with a free column is good, matching an =E= row with a lower-bounded column is allowed, and matching a =G= row with an upper-bounded column results in an error.
3. Any fixed columns remaining are ignored: these columns can be treated like exogenous variables or parameters.
4. If what remains is a set of =E= rows and an equal number of unbounded columns, these can be matched up in any order and we have a well-defined MCP. If this is not what remains, an error is triggered.
To illustrate how this works, consider the spatial equilibrium model [SPATEQU] with the following model statement:
Model P2R3_MCP / dem, sup, in_out.p, dom_trad.x /;
1. The model P2R3_MCP includes the rows from equations dem, sup, in_out and dom_trad and exactly the columns used by these rows. Checking the listing file, we see columns for Qd, Qs, x, and p. In addition, the model statement specifies two matches: in_out.p and dom_trad.x. These matches always take the form of an equation.variable pair, with no indices or domains included.
2. In this example, the rows corresponding to the equation in_out match up perfectly with the columns from the variable p: there are no holes in the set of rows or columns because of some dollar conditions in the equation definition. We have a one-to-one match so all the rows of in_out and columns of p are consumed by the match in the model statement. The same holds for the dom_trad.x pair, so what is left are the rows of dem and sup and the columns of Qd and Qs, all of which are unmatched.
3. There are no fixed variables to remove.
4. Since dem and sup are =E= constraints and Qd and Qs are free variables, we can match them in any order without changing the solution set for this model. The counts of these unmatched equality rows and unmatched free variables are equal, so we get a well-defined MCP.
When rows are matched explicitly to columns, some care must be taken to match them consistently. For example, consider a row-column match g.y. The row g can be of several types: =N=, =E=, =G=, or =L=. An =N= row can be matched to any sort of variable: the =N= doesn't imply any sort of relationship, which works perfectly with our definition of $$\perp$$ above: the allowed sign or direction of g is determined completely by the bounds on the complementary variable y. If g is an =E= row, this is consistent with a free variable y, but what if y has an active lower bound? By definition we allow g to be positive at solution, but this violates the declaration as an =E= row. Such cases can be handled by marking the row with a redef. The total number of redefs for a given model is available via the NumRedef model attribute and is shown in the report summary. Note that the set of rows marked depends on the solution: in the example above, if g is zero at solution it will not be marked as a redef, regardless of what the bounds are on y. Finally, some combinations are simply not allowed: they will result in a model generation error. The table below lists the outcome for all possible combinations.
Table 2: MCP Matching
Column Bounds =N= =E= =G= =L=
lower OK redef OK ERROR
upper OK redef ERROR OK
free OK OK OK OK
double OK redef redef redef
fixed OK redef redef redef
The definition, process, and rules above have several implications for valid MCP models:
• It is always acceptable to use the =N= notation when defining the equations in an MCP model, provided these equations are matched explicitly. In this case the bounds on F(z) are implied by the bounds on the matching columns, and redefs will never occur.
• Variables that are known to be lower-bounded (no upper bound) will match consistently with =G= equations.
• Variables that are known to be upper-bounded (no lower bound) will match consistently with =L= equations.
• Variables that are known to be unbounded will match consistently with =E= equations.
• Where the bound structure is not known in advance, or both upper and lower bounds exist, a match with an =N= equation will always be consistent. Other equation types will result in errors or redefs.
• The model may initially have fewer rows than columns, as long as the "extra" columns are unmatched fixed columns that ultimately get removed from the MCP passed to the solver.
• Any bounded-but-not-fixed column must be matched explicitly to a row.
• The only rows that may be unmatched are =E= rows.
• It is customary to re-use the constraints of an LP or NLP model when formulating the MCP corresponding to the Karush-Kuhn-Tucker (KKT) conditions. If the original model is a minimization, the LP/NLP marginals .m and the variables for these marginals in the MCP will use the same sign convention, and the orientation for the constraints will be consistent between the two models, making re-use easier.
As mentioned above, it is typical to use the same equations in both NLP and MCP models. Sometimes, it is not the original equation that is wanted for the MCP, but rather the reoriented (aka negated or flipped) equation. For example, the flipped version of x**1.5 =L= y is y =G= x**1.5, while sqr(u) - sqr(v) =E= 5 becomes - sqr(u) + sqr(v) =E= -5. Instead of re-implementing the equation in flipped form, the same result can be achieved by prefixing the equation name with a - in the model statement. See the [mcp10] model for an example of such usage. When equations are used in flipped form, they are marked with a redir in the listing file's solution listing.
An example of complementarity that should be familiar to many is the relationship between a constraint and its associated dual multiplier: if the constraint is non-binding, its dual multiplier must be zero (i.e. at bound) while if a dual multiplier is nonzero the associated constraint must be binding. In fact, the KKT or first-order optimality conditions for LP and NLP models can be expressed and solved as an MCP.
These complementarity relationships found in optimization problems are useful in understanding the marginal values assigned to rows and columns in the GAMS solution for MCP. With no objective function, the usual definition for marginal values and their interpretation isn't useful. Instead, the GAMS MCP convention for the marginal values of columns is to return the slack of the associated row (i.e. its value when interpreted and evaluated as a function). For the marginal values of rows, the level value (not the slack) of the associated column is returned. When we apply this convention to the NCP ( $$F(z) \geq 0, z \geq 0, \langle{F(z)},{z}\rangle = 0$$) we see pairwise complementarity between the levels and marginals returned for each of the rows and columns in the model. This is also the case if we take the KKT conditions of an LP in a suitable standard form: minimization, $$x \geq 0, Ax \geq b$$.
MCPs arise in many application areas including applied economics, game theory, structural engineering and chemical engineering. For further details on this class of problems, see http://www.neos-guide.org/content/complementarity-problems.
For information on MCP solvers that can be used through GAMS, see Solver/Model type Matrix.
### Constrained Nonlinear System (CNS)
The Constrained Nonlinear System (CNS) is the second GAMS model type that does not have an objective function. Mathematically, a CNS model looks like:
$$\begin{array}{ll} \textrm{Find} & x \\ \textrm{subject to} & F(x) = 0 \\ & L \leq x \leq U \\ & G(x) \, \, \alpha \, \, b, \\ \end{array}$$
where $$x$$ is a set of continuous variables and $$F$$ is a set of nonlinear equations of the same dimension as $$x$$. This is a key property of this model type: the number of equations equals the number of variables, so we have a square system. The (possibly empty) constraints $$L \leq x \leq U$$ are not intended to be binding at the solution, but instead are included to constrain the solution to a particular domain, to avoid regions where $$F(x)$$ is undefined, or perhaps just to give the solver a push in the right direction. The (possibly empty) constraints $$G(x) \, \, \alpha \, \, b$$ are intended to serve the same purpose as the variable bounds and are silently converted to equations with bounded slacks.
Note that since there is no objective in a CNS model, there are no marginal values for variables and equations. Any marginal values already stored in the GAMS database will remain untouched. CNS models also make use of some model status values that allow a solver to indicate if the solution is unique (e.g. for a non-singular linear system) or if the linearization is singular at the solution. For singular models (solved or otherwise), the solver can mark one or more dependent rows with a depnd. The total number of rows so marked for a given model is available via the NumDepnd model attribute and is shown in the report summary.
The CNS model is a generalization of a square system of equations $$F(x) = 0$$. Such a system could also be modeled as an NLP with a dummy objective. However, there are a number of advantages to using the CNS model type, including:
• A check by GAMS that the model is really square,
• solution/model diagnostics by the solver (e.g. singular at solution, (locally) unique solution),
• CNS-specific warnings if the side constraints $$L \leq x \leq U$$ or $$G(x) \, \, \alpha \, \, b$$ are active at a solution,
• and potential improvement in solution times, by taking better advantage of the model properties.
For information on CNS solvers that can be used through GAMS, see the Solver/Model type Matrix.
### Mathematical Program with Equilibrium Constraints (MPEC)
Mathematically, the Mathematical Program with Equilibrium Constraints (MPEC) problem looks like:
\begin{equation*} \begin{array}{ll} \textrm{Maximize or Minimize} & f(x,y) \\ \textrm{subject to} & g(x,y) \, \, \alpha \, \, 0 \\ & L_x \leq x \leq U_x \\ & F(x,y) \perp L_y \leq y \leq U_y, \\ \end{array} \end{equation*}
where $$x$$ and $$y$$ are vectors of continuous real variables. The variables $$x$$ are often called the control or upper-level variables, while the variables $$y$$ are called the state or lower-level variables. $$f(x,y)$$ is the objective function. $$g(x,y) \, \alpha \, 0$$ represents the set of traditional (i.e. NLP-type) constraints; some solvers may require that these constraints only involve the control variables $$x$$. The function $$F(x,y)$$ and the bounds $$L_y$$ and $$U_y$$ define the equilibrium constraints. If $$x$$ is fixed, then $$F(x,y)$$ and the bounds $$L_y$$ and $$U_y$$ define an MCP; the discussion of the "perp to" symbol $$\perp$$ in that section applies here as well. From this definition, we see that the MPEC model type contains NLP and MCP models as special cases of MPEC.
A simple example of an entire MPEC model is given below.
variable z, x1, x2, y1, y2;
positive variable y1;
y2.lo = -1;
y2.up = 1;
equations cost, g, h1, h2;
cost.. z =E= x1 + x2;
g.. sqr(x1) + sqr(x2) =L= 1;
h1.. x1 =G= y1 - y2 + 1;
h2.. x2 + y2 =N= 0;
model example / cost, g, h1.y1, h2.y2 /;
solve example using mpec min z;
Note that as in the MCP, the complementarity relationships in an MPEC are specified in the model statement via equation-variable pairs: the h1.y1 specifies that the equation h1 is perpendicular to the variable y1 and the h2.y2 specifies that the equation h2 is perpendicular to the variable y2. For details on the solve statement, see section The Solve Statement.
While the MPEC model formulation is very general, it also results in problems that can be very difficult to solve. The state-of-the-art for MPEC solvers is not nearly as advanced as that for other model types. As a result, you should expect the MPEC solvers to be more limited by problem size and/or robustness issues than solvers for other model types.
For information on MPEC solvers that can be used through GAMS, see the Solver/Model type Matrix. For more details on MPECs and solver development, see http://gamsworld.org/mpec/index.htm and http://www.neos-guide.org/content/complementarity-problems.
### Extended Mathematical Programs (EMP)
Extended Mathematical Programming (EMP) is an (experimental) framework for automated mathematical programming reformulations. Using EMP, model formulations that GAMS cannot currently handle directly or for which no robust and mature solver technology exists can be automatically and reliably reformulated or transformed into models for which robust and mature solver technology does exist within the GAMS system. For more details, see the chapter on EMP. Currently EMP supports:
• Equilibrium problems including variational inequalities, Nash games, and Multiple Optimization Problems with Equilibrium Constraints (MOPECs).
• Hierarchical optimization problems such as bilevel programs.
• Disjunctive programs for modeling discrete choices with binary variables.
• Stochastic programs including two-stage and multi-stage stochastic programs, chance constraints and risk measures such as Variance at Risk (VaR) and Conditional Variance at Risk (CVaR).
Apart from the disjunctive and stochastic programming models mentioned above, EMP models are typically processed (aka solved) via the JAMS solver: this solver does the work of reformulation/transformation, calling GAMS to solve this reformulation, and post-processing the solution that results to bring it back in terms of the original EMP model.
Examples demonstrating how to use the EMP framework and the JAMS and DE solvers are available in the GAMS EMP Library. These solvers require no license of their own to run but can and do call subsolvers that do require a license.
## Model Attributes
Models have attributes that hold a variety of information, including
• information about the results of a solve performed, a solve statement, the solution of a model,
• information about certain features to be used by GAMS or the solver,
• information passed to GAMS or the solver specifying various settings that are also subject to option statements.
Model attributes are accessed in the following way:
model_name.attribute
Here model_name is the name of the model in GAMS and .attribute is the specific attribute that is to be accessed. Model attributes may be used on the left-hand side and the right-hand side of assignments. Consider the following example:
transport.resLim = 600;
x = transport.modelStat;
In the first line the attribute .resLim of the model transport is specified to be 600 (seconds). In the second line the value of the attribute .modelStat of the model transport is assigned to the scalar x. Note that model attributes may also be used in display statements".
Some of the attributes are mainly used before the solve statement to provide information to GAMS or the solver link. Others are set by GAMS or the solver link and hence are mainly used after a solve statement.
Moreover, some of the attributes used before the solve may also be set via an option statement or the command line. Consider the following example:
option ResLim=10;
This line is an option statement and applies to all models. One can set the model attribute .ResLim to overwrite the global ResLim option. In order to revert the individual .ResLim to the global ResLim option, one needs to set the model attribute to NA. For more on option statements, see chapter The Option Statement.
gams mymodel ResLim=10
This sets the global ResLim option when invoking the gams run (e.g. from the command line). For more on command line parameters, see chapter The GAMS Call and Command Line Parameters.
Note that a model-specific option takes precedence over the global setting specified with an option statement and that a setting via an option statement takes precedence over a setting via the command line parameter.
The complete list of model attributes is given below. Observe that each entry is linked to a detailed description of the respective attribute, including information of whether the attribute is also available as command line parameter or option statement. Note that detailed descriptions of all GAMS command line parameters, options and model attributes are given in section Detailed Descriptions of All Options.
### Model Attributes Mainly Used Before Solve
Attribute Description
bRatio Basis acceptance threshold
cheat Cheat value, i.e. minimum solution improvement threshold
cutOff Cutoff value for branch and bound
defPoint Indicator for passing on default point
dictFile Force writing of a dictionary file if dictfile > 0
domLim Domain violation limit solver default
fdDelta Step size for finite differences
fdOpt Options for finite differences
holdFixed Treat fixed variables as constants
integer1..5 Integer communication cells
iterLim Iteration limit of solver
limCol Maximum number of columns listed in one variable block
limRow Maximum number of rows listed in one equation block
MCPRHoldFx Print list of rows that are perpendicular to variables removed due to the holdfixed setting
nodLim Node limit in branch and bound tree
optCA Absolute Optimality criterion solver default
optCR Relative Optimality criterion solver default
optFile Default option file
priorOpt Priority option for variable attribute .prior
real1..5 Real communication cells
reform Reformulation level
resLim Wall-clock time limit for solver
savePoint Save solver point in GDX file
scaleOpt Employ user specified variable and equation scaling factors
solPrint Solution report print option
solveOpt Multiple solve management
sysOut Solver Status file reporting option
threads Number of threads to be used by a solver
tolInfeas Infeasibility tolerance for an empty row of the form a.. 0*x =e= 0.0001;
tolInfRep This attribute sets the tolerance for marking infeasible in the equation listing
tolProj Tolerance for setting a variable level to its bound and filtering marginals when reading a solution
tryInt Whether solver should make use of a partial integer-feasible solution
tryLinear Examine empirical NLP model to see if there are any NLP terms active. If there are none the default LP solver will be used
workFactor Memory Estimate multiplier for some solvers
workSpace Work space for some solvers in MB
### Model Attributes Mainly Used After Solve
Attribute Description
domUsd Number of domain violations
etAlg Solver dependent timing information
etSolve Elapsed time it took to execute a solve statement in total
etSolver Elapsed time taken by the solver only
handle Unique handle number of SOLVE statement
iterUsd Number of iterations used
line Line number of last solve of the corresponding model
linkUsed Integer number that indicates the value of SolveLink used for the last solve
marginals Indicator for marginals present
maxInfes Maximum of infeasibilities
meanInfes Mean of infeasibilities
modelStat Integer number that indicates the model status
nodUsd Number of nodes used by the MIP solver
number Model instance serial number
numDepnd Number of dependencies in a CNS model
numDVar Number of discrete variables
numEqu Number of equations
numInfes Number of infeasibilities
numNLIns Number of nonlinear instructions
numNLNZ Number of nonlinear nonzeros
numNOpt Number of nonoptimalities
numNZ Number of nonzero entries in the model coefficient matrix
numRedef Number of MCP redefinitions
numVar Number of variables
numVarProj Number of bound projections during model generation
objEst Estimate of the best possible solution for a mixed-integer model
objVal Objective function value
procUsed Integer number that indicates the used model type
resCalc Time spent in function and derivative calculations (deprecated)
resDeriv Time spent in derivative calculations (deprecated)
resGen Time GAMS took to generate the model in CPU seconds(deprecated)
resIn Time to import model (deprecated)
resOut Time to export solution (deprecated)
resUsd Time the solver used to solve the model in seconds
rObj Objective function value from the relaxed solve of a mixed-integer model when the integer solver did not finish
solveStat Indicates the solver termination condition
sumInfes Sum of infeasibilities
sysIdent Solver identification number
sysVer Solver version
# The Solve Statement
Once a model has been defined using the model statement, the solve statement prompts GAMS to call one of the available solvers for the particular model type. This section introduces and discusses the solve statement in detail. For a list of GAMS model types, see Table 1. For information on how to specify desired solvers, see section Choosing a Solver.
Note
It is important to remember that GAMS does not solve the problem, but passes the problem definition to one of a number of separate solver programs that are integrated with the GAMS system.
## The Syntax of the Solve Statement
In general, the syntax for a solve statement is as follows. Note that there are two alternatives that are equally valid:
solve model_name using model_type maximizing|minimizing var_name;
solve model_name maximizing|minimizing var_name using model_type ;
The keyword solve indicates that this is a solve statement. Model_name is the name of the model as defined by a model statement. Note that the model statement must be placed before the solve statement in the program. The keyword using is followed by model_type, which is one of the GAMS model types described above, see Table 1. The keywords maximizing or minimizing indicate the direction of the optimization. Var_name is the name of the objective variable that is being optimized. An example of a solve statement in GAMS is shown below.
Solve transport using lp minimizing cost ;
Solve and using are reserved words, transport is the name of the model, lp is the model type, minimizing is the direction of optimization, and cost is the objective variable. Note that an objective variable is used instead of an objective row or function.
Attention
The objective variable must be scalar and of type free, and must appear in at least one of the equations in the model.
Recall that some model types (e.g. the Constrained Nonlinear System (CNS) or the Mixed Complementarity Problem (MCP)) do not have an objective variable. So their solve statement syntax is slightly different:
solve model_name using model_type;
As before, solve and using are keywords, model_name is the name of the model as defined by a model statement and model_type is the GAMS model type CNS or MCP. There is no objective variable and consequently no direction of optimization. An example from the spatial equilibrium model [SPATEQU] illustrates this solve statement:
Solve P2R3_MCP using mcp;
P2R3_MCP is the model name, the model type is MCP and as expected, there is no objective variable.
The EMP model type serves many purposes including some experimental ones. The solve statement with model type EMP can be with or without the objective variable and optimization direction. For more information, see chapter Extended Mathematical Programming (EMP).
## Actions Triggered by the Solve Statement
When GAMS encounters a solve statement during compilation (the syntactic check of the input file) or execution (actual execution of the program), it initiates a number of special actions. The purpose is to prevent waste that would be caused by solving a model that has apparently been incorrectly specified. During compilation the following are verified:
1. All symbolic equations have been defined and the objective variable is used in at least one of the equations.
2. The objective variable is scalar and of type free (even though lower and upper bounds may have been specified)
3. MCP models are checked for appropriate complementarity and squareness.
4. Each equation fits into the specified problem class (linearity for LP, continuous derivatives for NLP, as outlined above).
5. All sets and parameters in the equations have values assigned.
Note
GAMS issues explanatory error messages if it discovers that the model is not according to type; for example, the presence of nonlinear terms in a supposedly LP model. For details on error messages, see chapter GAMS Output.
At execution time the solve statement triggers the following sequence of steps:
1. The model is translated into the representation required by the solution system to be used.
2. Debugging and comprehension aids that the user wishes to see are produced and written to the output file (EQUATION LISTING, etc). For customizing options (e.g. LimRow and LimCol), see chapter The Option Statement.
3. GAMS verifies that there are no inconsistent bounds or unacceptable values (for example, NA or UNDF) in the problem.
4. Any errors detected at this stage cause termination with as much explanation as possible, using the GAMS names for the identifiers causing the trouble.
5. GAMS designs a solution strategy based on the possible availability of level values or basis information from a previous solution: all available information is used to provide efficiency and robustness of operation. Any specifications provided by the user (Iteration limits etc.) are incorporated. A solver is chosen which is either the default solver for that problem type, the solver specified on the command line or the solver chosen by an option statement. For details see section Choosing a Solver.
6. GAMS passes control to the solution subsystem and waits while the problem is being solved.
7. GAMS reports on the status of the solution process and loads solution values back into the GAMS database. This causes new values to be assigned to the .l and .m fields for all individual equations and variables in the model. In addition, the post solution model attributes are assigned. The procedure for loading back the data associated with level and marginal values may be customized using the SolveOpt model attribute and option. A row by row and column by column listing of the solution is provided by default. It may be suppressed by the SolPrint model attribute or option. Any apparent difficulty with the solution process will cause explanatory messages to be displayed. Errors caused by forbidden nonlinear operations are reported at this stage.
Note
When the solver does not provide a dual solution (.m), then GAMS does not print the marginal column in the solution listing and set the marginal field in variables and equations to NA.
The outputs from these steps, including any possible error messages, are discussed in detail in chapter GAMS Output.
# Programs with Several Solve Statements
Several solve statements can be processed in the same program. The next few subsections discuss various instances where several solve statements may be needed in the same file. If sequences of expensive or difficult models are to be solved, it might be useful to interrupt program execution and continue later. For details on this topic, see chapter The Save and Restart Feature.
## Several Models
If there are different models then the solves may be sequential, as below. Each of the models in [PROLOG] consists of a different set of equations, but the data are identical, so the three solves appear in sequence with no intervening assignments:
Solve nortonl using nlp maximizing z;
Solve nortonn using nlp maximizing z;
Solve nortone using nlp maximizing z;
When there is more than one solve statement in the program, GAMS uses as much information as possible from the previous solution to provide a starting point or basis in the search for the next solution.
## Loop: One Model, Different Data
Multiple solves may also occur as a result of a solve statement within a loop statement. Loop statements are introduced and discussed in detail in chapter Programming Flow Control Features; here we show that they may contain a solve statement and thus lead to multiple solves within one model. The example from [MEANVAR] computes the efficient frontier for return and variance for a portfolio selection problem at equidistance points.
loop(p(pp),
v.fx = vmin + (vmax-vmin)/(card(pp)+1)*ord(pp) ;
Solve var1 maximizing m using nlp ;
xres(i,p) = x.l(i);
xres('mean',p) = m.l;
xres('var',p) = v.l;
);
The set p is a set of point between the minimum and maximum variance, it is the driving set of the loop. A variance variable v is fixed at a equidistance points. With each iteration through the loop another variance level is used, the NLP model var1 is solved for each iteration and the outputs are stored in the parameter xres(*,pp), to be used later for reporting. As often for reporting purposes, the universal set * is used.
This example demonstrates how to solve the same model (in terms of variables and equations) multiple times with slightly different data. For such situations the Gather-Update-Solve-Scatter (GUSS) facility improves on the loop implementation by saving generation time and minimizing the communication with the solver. GUSS is activated by the additional keyword scenario in the solve statement followed by a set name that provides mapping information between parameters in the model and the scenario containers. A GUSS implementation of the loop would look as follows:
parameter vfx(p), px(p,i), pm(p);
set dict / p .scenario.''
v .fixed .vfx
x .level .px
m .level .pm /;
vfx(p(pp)) = vmin + (vmax-vmin)/(card(pp)+1)*ord(pp);
Solve var1 maximizing m using nlp scenario dict;
xres(i,p) = px(p,i);
xres('mean',p) = pm(p);
xres('var',p) = vfx(p);
## Customizing Solution Management: SolveOpt
It is important to consider how GAMS manages solutions if multiple models are solved. By default, GAMS merges subsequent solutions with prior solutions. This is not an issue if all models operate over the same set of variables. However, recursive procedures, different equation inclusions or logical conditions may cause only part of the variables or different variables to appear in the models to be solved. In such a case it might be useful to modify the solution management procedure using the model attribute or option SolveOpt.
## Sensitivity or Scenario Analysis
Multiple solve statements can be used not only to solve different models, but also to conduct sensitivity tests, or to perform case (or scenario) analyses of models by changing data or bounds and then solving the same model again. While some commercial LP systems allow access to "sensitivity analysis" through GAMS it is possible to be far more general and not restrict the analysis to either solver or model type. This facility is even more useful for studying many scenarios since no commercial solver will provide this information.
An example of sensitivity testing is in the simple oil-refining model [MARCO]. Because of pollution control, one of the key parameters in oil refinery models is an upper bound on the sulfur content of the fuel oil produced by the refinery. In this example, the upper bound on the sulfur content of fuel oil was set to 3.5 percent in the original data for the problem. First, the model is solved with this value. Next, a slightly lower value of 3.4 percent is used and the model is solved again. Finally, the considerably higher value of 5 percent is used and the model is solved for the last time. Key solution values are saved for later reporting after each solve. This is necessary because a following solve replaces any existing values. The key solution values are the activity levels of the process level z, a variable that is defined over a set of processes p and a set of crude oils cr. The complete sequence is:
parameter report(*,*,*) "process level report";
qs('upper','fuel-oil','sulfur') = 3.5 ;
Solve oil using lp maximizing phi;
report(cr,p,'base') = z.l(cr,p) ;
report('sulfur','limit','base') = qs('upper','fuel-oil','sulfur');
qs ('upper','fuel-oil','sulfur') = 3.4 ;
Solve oil using lp maximizing phi ;
report(cr,p,'one') = z.l(cr,p) ;
report('sulfur','limit','one') = qs ('upper','fuel-oil','sulfur');
qs('upper','fuel-oil','sulfur') = 5.0 ;
Solve oil using lp maximizing phi ;
report(cr,p,'two') = z.l(cr,p) ;
report('sulfur','limit','two') = qs('upper','fuel-oil','sulfur');
Display report ;
Note that the parameter report is defined over the universal set or short universe. In general, the universe is useful when generating reports, otherwise it would be necessary to provide special sets containing the labels used in the report. Any mistakes made in spelling labels used only in the report should be immediately apparent, and their effects should be limited to the report. The parameter qs is used to set the upper bound on the sulfur content in the fuel-oil, and the value is retrieved for the report. Note that the display statement in the final line is introduced and discussed in detail in chapter The Display Statement. This example shows not only how simply sensitivity analysis can be done, but also how the associated multi-case reporting can be handled.
The output from the display statement is shown below. Observe that there is no production at all if the permissible sulfur content is lowered. The case attributes have been listed in the row SULFUR.LIMIT. Section Global Display Controls contains more details on how to arrange reports in a variety of ways.
---- 225 PARAMETER report process level report
base one two
mid-c .a-dist 89.718 35.139
mid-c .n-reform 20.000 6.772
mid-c .cc-dist 7.805 3.057
w-tex .cc-gas-oil 5.902
w-tex .a-dist 64.861
w-tex .n-reform 12.713
w-tex .cc-dist 4.735
w-tex .hydro 28.733
sulfur.limit 3.500 3.400 5.000
Note
For other ways to do comparative analyses with GAMS, see the tutorial Comparative Analyses with GAMS.
## Iterative Implementation of Non-Standard Algorithms
Another use of multiple solve statements is to permit iterative solution of different blocks of equations, most often using solution values from the first solve as data for the next solve. These decomposition methods are useful for certain classes of problems because the subproblems being solved are smaller, and therefore more tractable. One of the most common examples of such a method is the Dantzig-Wolfe decomposition.
An example of a problem that is solved in this way is a multi-commodity network flow problem in [DANWOLFE].
# Choosing a Solver
After a model has been checked and prepared as described above, GAMS passes the model to a solver. When the GAMS system is installed default solvers for all model types are specified and these solvers are used if the user doesn't specify anything else. It is easy to switch to other appropriate solvers provided the user has the corresponding license. There are multiple ways to switch solvers:
1. Using a command line parameter of the following form:
gams mymodel model_type=solver name
For example,
gams mymodel lp=cbc
2. With an option command of the following form that is placed before the solve statement:
Option model_type=solver_name;
Here option is a keyword, model_type is the same model type that is used in the solve statement and solver_name is the name of one of the available solvers. For example,
Option LP=cbc, NLP=conopt, MIP=cbc, MINLP=default;
The MINLP=default switches back to the default solver for the MINLP model type.
3. Instead of providing a particular solver for a model type, the option Solver can be used to use a given solver for all model types this solver can handle.
Option Solver=cbc;
4. (Re)running gamsinst at any time and altering the choice of default solver as described in the installation notes.
Note
A list of all solvers and current default solvers may be generated in the listing file with Option SubSystems;.
# Making New Solvers Available with GAMS
This short section is to encourage those of you who have a favorite solver not available through GAMS. Linking a solver program with GAMS requires some programming skills and the use of libraries provided by GAMS. There is a collection of open source solver links to GAMS at the COIN-OR project GAMSLinks. The benefits of a link with GAMS to the developer of a solver are several. They include:
• Immediate access to a wide variety of test problems.
• An easy way of making performance comparisons between solvers.
• The guarantee that a user has not somehow provided an illegal input specification.
• Elaborate documentation, particularly of input formats, is not needed.
• Access to the existing community of GAMS users, for marketing or testing.
This completes the discussion of the model and solve statements.
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https://educators.brainpop.com/lesson-plan/project-t-r-i-g/?bp-topic=gravity
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Grade Levels: 6-8, 9-12
In this lesson plan which is adaptable for grades 6-12, students explore the mathematics of projectile motion using a free online game. Students will apply their knowledge of math and physics (including angles and trigonometry) in order to create successful projectile trajectories.
### Students will:
1. Apply knowledge of math and physics concepts to an online projectile motion game.
### Materials:
• Computers with internet access for BrainPOP
• Interactive whiteboard
### Preparation:
This lesson plan uses a free online game from Math Playground called Project T.R.I.G The game allows students to apply their knowledge of angles and trigonometry as they adjust speed and angle variables to control the flight of a projectile. Students are given the opportunity to adjust the variables based on the outcome of their last flight, and the landing target becomes more difficult to reach as the game progresses.
There are 12 levels of game play which allow students to gain an intuitive understanding of projectile physics and how the variables of launch angle and initial velocity affect the trajectory. However, we encourage you to guide students to use the grid and data tables in the game so that students make connections to their formal math and physics knowledge and improve their scores. More advanced students can apply knowledge of trigonometry and projectile physics to accurately predict where projectiles will land.
### Lesson Procedure:
1. What is a projectile trajectory? What do students already know about the mathematics of projectile motion? Talk with students about any experiences they've had with projectile physics and the factors that affect the trajectory of an object.
2. Activate prior knowledge by playing the Angles movie for students, asking students to listen for information that is applicable to the projectile motion discussion. Afterward, facilitate a class discussion or have students brainstorm with a partner. What are the different types of angles? How do you think angles affect projectile trajectory?
3. Show the Project T.R.I.G game on your interactive whiteboard. Demonstrate how to interact with the game through the control panel located at the bottom. Use the sliders on the left to set the angle and launch velocity. Explain that each level contains four attempts which are tracked in the control panel.
4. Invite student volunteers up to try different strategies and explain their approaches. Show students that they can toggle on "Grid" and "Data". The first overlays a grid on the play area, and the second records and displays the x- and y-coordinates of the projectile at every tenth of second. How could these tools be useful in helping students score more points?
5. Allow students to explore the game on their own or with a partner for 5-10 minutes. Encourage them to utilize the grid and data options.
6. Instruct students to take a break from game and bring everyone back to a whole class discussion. Draw students' attention to the mathematical description of projectile motion, which can be accessed through the "?" button. Guide students to make connections between the game and their formal mathematical knowledge about projectile motion, and talk about how using math and physics principles can help students achieve the best possible score.
7. Provide additional time for students to practice their math and physics skills through game play.
8. Assess student learning using the game quiz.
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CC-MAIN-2017-39
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latest
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https://performingineducation.com/product/math-interactive-notebook-5th-grade-bundle-common-core-standards-entire-year/
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Sale!
\$18.20
# Math Interactive Notebook 5th Grade Bundle Common Core Standards ENTIRE YEAR
Sale!
\$18.20
## Description
This includes student reproducibles to use in Interactive Math Notebooks for 5th Grade ALL Common Core Math Standards. It’s all in here ready for you to teach and your students to create! This set also included rubric & instructions for using Interactive Math Notebooks in your classroom.
This Bundle Includes the Following Products:
What’s Included:
• Teaching Tips for each standard
• Answers/Examples for every lesson
• Common Core Aligned Student Pages (see below)
Operations and Algebraic Thinking
• 5.OA.A.1 Order of Operations
• 5.OA.A.2 Simple Expressions
• 5.OA.B.3 Numerical Patterns
Numbers & Operations in Base Ten
• 5.NBT.A.1 Place Value
• 5.NBT.A.2 Number Patterns
• 5.NBT.A.3.A Read and write decimals & expanded form
• 5.NBT.A.3.B Compare two decimals to thousandths
• 5.NBT.A.4 Rounding Decimals
• 5.NBT.B.5 Multiply multi-digit whole numbers
• 5.NBT.B.6 Long division
• 5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths
Number and Operations – Fractions
• 5.NF.A.1 Adding and Subtracting Unlike Fractions
• 5.NF.A.2 Adding and Subtracting Fractions Word Problems
• 5.NF.B.3 Unit Fractions and Number Lines
• 5.NF.B.4-5 Multiplying Fractions
• 5.NF.B.6 Multiplying Fractions Word Problems
• 5.NF.B.7 Dividing Fractions
Geometry
• 5.G.1 Coordinate Planes
• 5.G.2 Representing Problems Using Coordinate Planes
• 5.G.3 Attributes of Two-Dimensional Shapes
• 5.G.4 Classifying Two-Dimensional Shapes
Measurement and Data
• 5.MD.1 Customary Measurement: Liquids
• 5.MD.1 Customary Measurement: Length
• 5.MD.1 Customary Measurement: Weight
• 5.MD.1 The Metric System
• 5.MD.2 Displaying Data in Line Plots
• 5.MD.3 Unit Cubes
• 5.MD.3b What is Volume?
• 5.MD.4 Counting Volume
• 5.MD.5a Calculating Volume
• 5.MD.5b Formulas for Volume
• 5.MD.5c Additive Volume
……………………………………………………………………………………………………………….
GET ALL THE HIGH-INTEREST 5TH GRADE MATH ACTIVITIES AND SAVE OVER 50%!
I’ve bundled my top 5th grade math projects in a money-saving bundle! For a limited time, get 5th grade interactive math notes for every single CCSS + ten project-based learning activities that incorporate the CCSS discounted by over 50% by clicking here.
Looking for individual Common Core Project-based Learning Activities?
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CC-MAIN-2022-49
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latest
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en
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https://aitechtrend.com/unlocking-the-power-of-bayesian-statistics-in-machine-learning/
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crawl-data/CC-MAIN-2024-10/segments/1707947473824.45/warc/CC-MAIN-20240222193722-20240222223722-00552.warc.gz
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Unlocking the Power of Bayesian Statistics in Machine Learning - AITechTrend
# Unlocking the Power of Bayesian Statistics in Machine Learning
Bayesian statistics has gained significant popularity in the field of statistics, especially in machine learning. This concept is extensively used in various predictive modeling techniques, as it incorporates probabilistic principles. Bayesian approaches are particularly useful when dealing with events that are conditionally dependent on each other. In this article, we will delve into the fundamentals of Bayesian statistics, explore its relevance to the Bayes theorem, and conduct practical experiments in Python to calculate event probabilities using this approach.
## Unraveling the Bayes Theorem
The Bayes theorem serves as a framework to determine the probability of an event based on prior knowledge. It enables us to compute the conditional probability of an event by incorporating both available data and existing prior information associated with the event’s conditions. For example, in the context of statistical models like logistic regression, the Bayes theorem can be employed to estimate the model’s parameters effectively.
Since Bayesian statistics treats probabilities as a measure of belief, it allows direct assignment of probability distributions to parameters, quantifying them using the Bayes theorem. Mathematically, the Bayes theorem can be expressed as follows:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this formula, A and B represent two events. To illustrate this concept, let’s consider an example involving two bags, A and B, each containing different colored balls. If we draw a red ball, we can utilize the Bayes theorem to calculate the probability of it being drawn from bag A. Essentially, the Bayes theorem determines the probability of a prior event given that a posterior event has already occurred.
## Key Components of the Bayes Theorem
To comprehend the Bayes theorem more thoroughly, it’s essential to understand its core components: prior probability, likelihood function, and posterior probability.
### 1. Prior Probability
In the Bayes theorem formula, P(A) represents the prior probability of event A. The prior probability can be defined as the probability distribution of an uncertain quantity, which reflects one’s belief about that quantity without considering any evidence. For instance, when distributing a large number of balls into buckets, a probability distribution can be considered as prior if it represents the proportion of balls allocated to a particular bucket. In the context of statistical models, an unknown quantity, such as a model parameter, can be regarded as an example of an uncertain quantity. The prior probability, denoted as P(A), reflects one’s belief in the occurrence of event A before taking any evidence into account.
### 2. Likelihood Function
The likelihood function, denoted as P(B|A), is a crucial element in the Bayes theorem. It represents the probability distribution of the observed data in the context of a statistical model. More simply, it indicates the probability of event B occurring when event A is known to be true. Numerically, the likelihood function expresses the support value provided by evidence B for proposition A.
### 3. Posterior Probability
The posterior probability, denoted as P(A|B), is the conditional probability of a random event or uncertain proposition when relevant evidence is available and taken into account. According to the Bayes theorem, the posterior probability signifies the probability of event A given that evidence B has been considered. Mathematically, it can be expressed as the product of the likelihood probability, prior probability, and the inverse of the probability of evidence B being true.
## Practical Implementation of Bayesian Statistics in Python
In the following section, we will employ a Python example to gain practical insights into the fundamental concepts of Bayesian statistics.
Consider a scenario where we have two buckets, A and B. Bucket A contains 30 blue balls and 10 yellow balls, while bucket B contains 20 blue balls and 20 yellow balls. Our objective is to select one ball, and we want to determine the probability of choosing bucket A.
Let’s solve this problem using Python:
``````# Importing libraries
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
# Defining hypotheses and priors
hypos = 'bucket a', 'bucket b'
probs = 1/2, 1/2
prior = pd.Series(probs, hypos)
# Displaying prior probabilities
print(prior)
``````
Output:
``````bucket a 0.5
bucket b 0.5
dtype: float64
``````
In the code above, we defined the hypotheses (bucket a and bucket b) and their corresponding prior probabilities. The prior probabilities are represented as a pandas series, which allows us to create a discrete probability distribution.
To calculate the likelihood, we know that the chances of selecting a blue ball from bucket A are 3/4, and from bucket B, the chances of selecting any ball or a blue ball are 1/2. We can express these probabilities as follows:
``````# Defining likelihood
likelihood = 3/4, 1/2
# Calculating unnormalized posterior
unnorm = prior * likelihood
# Displaying unnormalized posterior
print(unnorm)
``````
Output:
``````bucket a 0.375
bucket b 0.250
dtype: float64
``````
By combining the likelihood and prior, we can obtain the unnormalized posterior. To normalize the posterior, we divide the unnormalized posterior by the sum of its values:
``````# Calculating the sum of unnormalized posterior
prob_data = unnorm.sum()
# Calculating the normalized posterior
posterior = unnorm / prob_data
# Displaying the normalized posterior
print(posterior)
``````
Output:
```cssCopy code```bucket a 0.600
bucket b 0.400
dtype: float64
``````
From the results, we can conclude that the posterior probability of choosing bucket A, given a blue ball selection, is 0.6. This implementation aligns with the principles of the Bayes theorem.
We can further explore variations of this problem. For example, let’s assume we repeat the process of selecting a ball from the same bucket, putting the ball back, and choosing again. This time, we want to determine the probability of selecting bucket A in both attempts. In this case, we can reuse the posterior obtained from the first problem as the prior for the subsequent problem:
``````# Reusing the previous posterior as the prior
prior = posterior
# Defining the likelihood for the new problem
likelihood = 3/4, 1/2
# Calculating the unnormalized posterior for the new problem
unnorm = prior * likelihood
# Calculating the normalized posterior for the new problem
posterior = unnorm / unnorm.sum()
# Displaying the posterior probabilities
print(posterior)
``````
Output:
``````bucket a 0.428571
bucket b 0.571429
dtype: float64
``````
The posterior probability for bucket A in the second attempt is approximately 0.428571.
Let’s now consider a more complex scenario where we have 101 buckets, and each bucket has a different distribution of blue balls. Bucket 0 has 0 blue balls, while bucket 1 has 1% blue balls, bucket 2 has 2% blue balls, and so on, up to bucket 99 with 99% blue balls and bucket 100 with 100% blue balls.
To solve this problem, we can create a series of numbers ranging from 0 to 100, evenly spaced between each other. We can treat these numbers as fractions representing the fraction of blue balls in each bucket.
``````import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
# Generating series of numbers representing fractions of blue balls
xs = np.linspace(0, 1, num=101)
prob = 1 / 101
prior = pd.Series(prob, xs)
# Displaying the prior distribution
print(prior)
``````
Output:
``````0.00 0.009901
0.01 0.009901
0.02 0.009901
...
0.99 0.009901
1.00 0.009901
dtype: float64
``````
In the code above, we generated a series of numbers from 0 to 1, with 101 equally spaced values. We then assigned a uniform prior probability (0.009901) to each value, representing the probability of selecting each bucket.
Final Thoughts
In this article, we explored the Bayes theorem, a fundamental component of Bayesian statistics. We gained insights into the prior probability, likelihood function, and posterior probability, which play crucial roles in Bayesian analysis. Moreover, we conducted practical experiments in Python to calculate probabilities based on different scenarios using Bayesian statistics. Bayesian statistics finds applications in various fields, such as survival analysis, statistical modeling, and parameter estimation. We encourage readers to further explore this powerful statistical analysis approach and its real-life applications, as it often provides accurate and straightforward results.
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The Life Path number is one of the most essential since it is your key purpose in life and the major reason you were birthed; read my cards the expression number is just how you express yourself to the globe and how others see you.
The spirit’s urge number makes you delighted in life and what brings you happiness, and the birthday number is all the skills and talents and capabilities that you come down into this lifetime with to sustain your life’s objective.
So I’m going to reveal you exactly how to calculate the Life Path number initially since that is the most vital number in your graph, and it is your key objective in life, so the Life Course number is determined by your day of birth.
So all we’re doing is we are just adding every one of the numbers together in your date of birth till we break them down and get a single figure.
So in this example, we have December 14, 1995.
So we need to get a solitary digit for each section.
First, we require to get a solitary number for the month.
We need to obtain a single digit for the day of the month, and we require to obtain a single number for the birth year, and after that as soon as we get a solitary figure for each one of these, we can add these three together.
So December is the 12th month of the year, so we require to add the one in the two to get a single digit.
One plus 2 equals 3, and for the day of the month, we need to add the one in the four together, and that will provide us number 5 for the day of the month.
And afterwards we require to include 1995 with each other.
Which will certainly provide us a single digit for the year, so one plus 9 plus 9 plus five equates to twenty-four, and after that we need to include both in the four together because we need to break it down to one digit for the year.
So 2 plus 4 amounts to 6.
So as soon as we have a solitary digit each of these locations, a single number for the month, a single figure for the day, and a solitary number for the year, then we’re going to add the 3 of these together, and in this instance.
It would be 3 plus five plus 6 equals 14, and after that we need to include the one and the 4 with each other due to the fact that we’re trying to break all of this down until we obtain a solitary figure.
So if you include the one in the four together in 14, you obtain a number 5.
So in this situation, this person’s life path number is a number five.
This individual is a 14 five, and it’s imperative to make note of the 2 last numbers that we included together to obtain the Life Path number five since these two last numbers are vital to the individual’s life course number.
So, when it comes to a Life Path, the number 5 could be a fourteen five due to the fact that the 4 amounts to number five.
They might additionally be a 2 and a three a twenty-three number five. Still, those 2 numbers that you combined to get that Life Course number are significant since they inform you what powers you will need to use throughout your lifetime to accomplish your life purpose. In this circumstance, read my cards, this person has a Life Course variety of number 5.
Still, to satisfy their life course, the number of a number 5, they are going to need to make use of the energy of the number one and the number Four to attain their function, so take note of those 2 last numbers that you added with each other to get your last Life Course number because those are really important.
A fourteen-five is going to be extremely different than a twenty-three-five.
If you have any questions regarding these two last numbers made use of to obtain your last Life Path, number comment below, and I will try to address your inquiries currently.
I desired to show you this instance due to the fact that, in some situations, we do not break down every one of the numbers to acquire a single digit.
So in this instance, we have December 14, 1992, and when we added all of the numbers with each other in the month, the day, and the year, we finished up with a number 11 and 11 in numerology is a master number and the master numbers.
We do not include both digits with each other, so there are three master numbers in numerology, and the 3 master numbers are 11, 22, and 33.
So after you have actually added every one of the numbers together in your birth date and if you wind up with either an 11, a 22, or a 33, you will certainly not add these two numbers together because you have a master number Life Path.
Number and the master numbers are different from the other numbers in numerology because they hold the double numbers’ power, and we do not include the two digits together in these scenarios.
So if you have either an 11, a 22, or 33, you will certainly not add the 2 digits together.
You will keep it as is, and you have a master number as a life course.
In this scenario, the number that we totaled to get the 11 were 8 and 3.
Those are considerable numbers in this situation since this master number 11 will certainly need to make use of the 8 and the 3.
To acquire their number 11 life function, so in 83, 11 will certainly be a great deal various from on 92 11 because an 83 11 will certainly need to make use of the power of the 8 in the 3 to obtain their life objective. The 9211 will certainly Have to use the 9 and both indicate get their 11 life purpose.
So the birthday celebration number is possibly one of the most easily accessible number to compute in your chart since for this number, all you have to do is add the digits together of the day you were birthed on, so this individual was born upon December 14, 1995.
So we will add the one and the 4 with each other due to the fact that those are the digits of the day. read my cards
He or she was birthed, so 1 plus 4 amounts to 5.
So this person’s birthday number is a number 5.
Now, in this example, December 11, 1995.
He or she was born upon the 11th day of the month, and 11 is a master number, so we do not include the two ones together since 11 is a master number, and there are 3 master numbers in numerology, 11, 22, and 33.
So if you were born upon the 11th of a month or the 22nd of a month, you would not include the 2 figures with each other because your birthday celebration number is a master number.
So your birthday number is either a master number 11 or a master number 22. The master numbers are the only numbers in numerology that we do not add the two numbers with each other to get a last number.
So you will certainly keep those two digits alone, and you will not add them with each other.
For the last 2 numbers, you have actually used your date of birth to determine those 2 numbers, however, for the expression number and the soul’s urge number, you will use the full name on your birth certificate.
You’re going to use your initial, middle, and surname on your birth certificate to calculate your expression and your heart’s desire number, therefore we’re currently mosting likely to utilize the Pythagorean number system to calculate these numbers.
And it’s called the Pythagorean system since Pythagoras, a Greek mathematician, produced it. He was the mathematician who developed the Pythagorean thesis. He is the papa of numerology, and he found that all numbers hold energy. They all own a Particular vibration, and so after discovering that all numbers hold certain energy, Pythagoras created the Pythagorean number system. From that, we have contemporary numerology today.
It is a graph with every one of the letters of the alphabet and all letters representing a particular number.
So basically, all letters in the alphabet have the energy of a number.
And if you consider this graph, you can find out what number each letter has the power of.
So a has the power of a primary B has the energy of a number.
2 C has the energy of a number 3 and so on, and more.
So, for the expression number, all you have to do is add every one of the letters in your complete birth name, so the first, center, and surname on your birth certificate and minimize them to one digit. read my cards
So in this example, we have Elvis Presley.
So what I did was I took a look at the number chart, and I found the corresponding number to each letter in Elvis’s name.
So E is a five, l is a 3 V is a four.
I am a 9, and I just found all of the numbers matching to all of the letters in his name; and I added all of those numbers with each other, and the last number I obtained was an 81.
After that I added the eight and the one with each other due to the fact that we have to keep damaging these down until we get a single-digit, and when I claimed the eight and the one with each other, I got a nine, so Elvis’s expression number is a nine.
Now the one scenario where you would certainly not proceed to add these numbers with each other up until you obtained a single figure would certainly be if you obtained a master number, so the 3 master numbers are 11, 22, and 33.
If you got a master number, you would not proceed to include these numbers with each other.
You will keep them as either 11, 22, or 33.
So it’s the same scenario as it was with the life fifty percent number and the birthday number.
The master numbers are unique, and we do not include the two figures with each other, so moving on to the soles urge number so sometimes the soles encourage number can be called the single number, and it can additionally be called the heart’s need number.
These words are used interchangeably however understand whenever you see the sole number or heart’s wish number that they are essentially the same thing as a soles prompt number.
For this number, we will add every one of the vowels in your full birth name.
Every one of the vowels in your first, middle, and last name on your birth certification, and we’re mosting likely to minimize them to one number.
So we’re going to use the exact same Pythagorean graph that we did previously, and we’re mosting likely to seek out every one of the numbers that represent the vowels in your initial, middle, and surname.
So here we have Kate Middleton.
I recently just did a video clip on her numerology, so I figured why not utilize her today.
So her first middle and surname are Catherine, Elizabeth Middleton, so I searched for the numbers representing just the vowels in her name.
So, as you can see, an equals 1 B amounts to 5, I equals 9, and E amounts to 5, and after that I did that for every one of the vowels in her full name.
And after that I simply included all of those numbers with each other, which offered me 60, and after that 60 reduces to number 6 because we remember we’re simply attempting to break these numbers down up until we obtain a single digit.
So in this circumstance, Kate’s Seoul’s desire number is a number 6.
Now you will not proceed to damage down the numbers if you obtain an 11, a 22, or a 33, so, as I claimed with all the various other numbers if you get among these numbers, this is a master number, and we do not include both figures together so 11, 22 and 33 are master numbers.
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src/HOL/Fun.thy
changeset 31949 3f933687fae9 parent 31775 2b04504fcb69 child 32139 e271a64f03ff
``` 1.1 --- a/src/HOL/Fun.thy Sat Jul 04 23:25:28 2009 +0200
1.2 +++ b/src/HOL/Fun.thy Mon Jul 06 14:19:13 2009 +0200
1.3 @@ -496,6 +496,40 @@
1.4
1.5 hide (open) const swap
1.6
1.7 +
1.8 +subsection {* Inversion of injective functions *}
1.9 +
1.10 +definition inv :: "('a \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'a)" where
1.11 + "inv f y = (THE x. f x = y)"
1.12 +
1.13 +lemma inv_f_f:
1.14 + assumes "inj f"
1.15 + shows "inv f (f x) = x"
1.16 +proof -
1.17 + from assms have "(THE x'. f x' = f x) = (THE x'. x' = x)"
1.18 + by (simp only: inj_eq)
1.19 + also have "... = x" by (rule the_eq_trivial)
1.20 + finally show ?thesis by (unfold inv_def)
1.21 +qed
1.22 +
1.23 +lemma f_inv_f:
1.24 + assumes "inj f"
1.25 + and "y \<in> range f"
1.26 + shows "f (inv f y) = y"
1.27 +proof (unfold inv_def)
1.28 + from `y \<in> range f` obtain x where "y = f x" ..
1.29 + then have "f x = y" ..
1.30 + then show "f (THE x. f x = y) = y"
1.31 + proof (rule theI)
1.32 + fix x' assume "f x' = y"
1.33 + with `f x = y` have "f x' = f x" by simp
1.34 + with `inj f` show "x' = x" by (rule injD)
1.35 + qed
1.36 +qed
1.37 +
1.38 +hide (open) const inv
1.39 +
1.40 +
1.41 subsection {* Proof tool setup *}
1.42
1.43 text {* simplifies terms of the form
```
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# Find invariant tensors under two matrix
I have two $$9\times 9$$ matrices $$g_1, g_2$$ where
g1 = {{-1, -(1/2), 0, 0, -(1/2), -(1/2), 0, 1/2, -(1/2)}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {-1, -(1/2), 3/2, 1, -(1/2), -1, -(3/2), 1, -(1/2)}, {1, 0, -(3/2), -1, 0, 1/2, 3/2, 1/2, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 1}, {1/2, 0, 1/4, 1/2, -(1/2), -(1/4), -(3/4), 1/4, -(1/2)}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, {-(1/2), 1/2, 3/4, 3/2, 0, -(5/4), -(5/4), 1/4, 0}, {0, 1/2, 0, -1, -(1/2), -(1/2), 1, -(1/2), 1/2}}
g2 = {{1/2, 1/2, 1/4, -(1/2), 0, 1/4, 1/4, -(1/4), 1}, {1/2, -(1/2), -(5/4), -(1/2), 1, 3/4, 3/4, 1/4, 0}, {0, 0, 1, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 1, 0, 0, 0}, {0, 1/2, 1/2, 0, 1/2, 0, -(1/2), 0, -(1/2)}, {0, 0, 0, 1, 0, 0, 0, 0, 0}, {0, 0, 0, -1, 0, 1, 1, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 1, 0}, {1/2, 0, 1/4, 1/2, -(1/2), -(1/4), -(3/4), 1/4, -(1/2)}}
I want to find four index invariants under the action of these two matrices, i.e. $$$$(g_1)_a^{\,a_1}(g_1)_b^{\,b_1}(g_1)_c^{\,c_1}(g_1)_d^{\,d_1}T_{a_1 b_1 c_1 d_1}=T_{abcd}$$$$ and $$$$(g_2)_a^{\,a_1}(g_2)_b^{\,b_1}(g_2)_c^{\,c_1}(g_2)_d^{\,d_1}T_{a_1 b_1 c_1 d_1}=T_{abcd}$$$$
Now, let me try to explain what I was trying to do. I wrote these equations as $$$$\big[(g_1)_a^{\,a_1}(g_1)_b^{\,b_1}(g_1)_c^{\,c_1}(g_1)_d^{\,d_1}-\delta_a^{\,a_1}\delta_b^{\,b_1}\delta_c^{\,c_1}\delta_d^{\,d_1}\big]T_{a_1b_1c_1d_1}=0$$$$ Then use Kronecker Product to make this problem as a problem of finding the null space of a matrix. i.e.
(KroneckerProduct[g1, g1, g1, g1]) - IdentityMatrix[9^4] //NullSpace
Then I move to see whether the linear combinations of these null vectors can be the null vectors of
(KroneckerProduct[g2, g2, g2, g2]) - IdentityMatrix[9^4] //NullSpace
If I can find one solution, then the problem is solved. All I need to do then is to translate the $$9^4$$ vector to a 4 rank tensor.
I tried this algorithm in the case of $$g_1,g_2$$ are two $$5\times5$$ matrices and it worked well. However, it seems it takes forever to run the first command line in $$9\times 9$$ case. Any suggestions for me to solve this problem?
For my use, I only need to solve the problem for the case where $$T_{abcd}$$ is totally symmetric. Perhaps I could write down a general ansatz for the totally symmetric four rank tensor and solve it. I will try to do it.
For curiosity, I'd like to understand the general case.
• Can you show all your code, including definitions of g1 and g2, rather than just snippets? What does each $(g_1)^{a_1}_{a}$ mean? What is $T_{abcd}$? Jan 19 at 16:34
• @MarcoB: The OP is using abstract index notation. Jan 19 at 16:36
• It seems plausible to me that if $W \subseteq V$ is the subspace of $V$ invariant under the actions of $g_1$ and $g_2$, then the subspace of $V \otimes V \otimes V \otimes V$ that is invariant under $g_1$ and $g_2$ will simply be $W \otimes W \otimes W \otimes W$. If that's true, it simplifies your problem greatly, since you only have to run your code on an $n$-dimensional space instead of an $n^4$-dimensional space. Proving that would be off-topic here, though. You might consider posting this conjecture as a question to Mathematics instead (if you can't readily disprove it.) Jan 19 at 16:53
• I got Mathematica to complete your code in about 15 minutes of calculation time, using $g_1$ as the rotation matrix that takes $\hat{e}_1 \to \hat{e}_9$ and $g_2$ as the rotation matrix that takes $\hat{e}_1 \to \hat{e}_2$. The resulting nullspaces are 2997-dimensional, which suggests that my conjecture above is incorrect. Jan 19 at 17:01
• @user3257842 Thanks. I have updated the question Jan 19 at 23:43
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# Topiary_BruteForce_RC
```import Rhino
import rhinoscriptsyntax as rs
import scriptcontext
from System.Drawing import Color
import random as rnd
import time
"""--------------------------------------------------------
RHINOCOMMON METHODS
One Time Search to find all points within specified radius
The points found inside the specified radius get sampled for closest neighbor
No Repeat Search
Search Sample = 1 Million Points
Search Space = Cube (5000,5000,5000)
--------------------------------------------------------"""
def closestSearch(pts, point):
#function to sample point based on a search radius first then search for the closest inside that list mimic the rtree pass basically but with brute force
ptsinRange = [] #dynamic list to store points found inside initial search
for i, pt in enumerate(pts):
dist = point.DistanceTo(pt) #find the distance from each point to the query point
if dist < 500:
ptsinRange.append(pt) #if the distance is less than 500 append that point to the ptsinRange List
cPoint = Rhino.Collections.Point3dList.ClosestPointInList(ptsinRange,point) #search for the closest point from that list of points in range
if cPoint[0] == point[0] and cPoint[1] == point[1] and cPoint[2] == point[1]: #test against the closest neighbor being you
print "IS THE SAME"
return cPoint,ptsinRange #return the closest neighbor and the list of points in range
def RunSearch():
point = Rhino.Geometry.Point3d(5000/2,5000/2,5000/2)
intNumPts = 1000000 #specify number of sample points
pts = [] #store the random points in an empty list
for x in range(0,intNumPts):
randomPt = Rhino.Geometry.Point3d(rnd.random()*5000,rnd.random()*5000,rnd.random()*5000) #randomly place each point inside the 5k x 5k x 5k cube
pts.append(randomPt) #append each random point to the pts list
timeStart = time.time() #Start timer
data = closestSearch(pts, point) #call closestsearch function Pass query pt and list of points to search
print "ClosestPoint = {}".format(data[0]) #print the closest neighbor
print "Number of Pts in Range = {}".format(len(data[1])) #print how many points were captures inside the search radius
timeEnd = time.time() #stop the timer
print "time taken: {}".format(timeEnd - timeStart) #print the amount of time the calculations took
if __name__=="__main__":
RunSearch()
Update
```
• Share
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Engineering Jobs » Quiz: Electrical Engineering 03 Dec 2020
# Quiz: Electrical Engineering 03 Dec 2020
Quiz: Electrical Engineering
Exam: UPPSC-AE
Topic: Miscellaneous
Each question carries 1 mark.
Negative marking: 1/3 mark
Time: 10 Minute
Q1. The number of output and input pins of 8085 microprocessors are respectively:
(a) 20,20
(b) 27,21
(c) 21,27
(d) None of the above
Q2. In INTEL 8085, while executing a program non-maskable interrupt occurs. The data present on data line is
(a) 00 H
(b) 24 H
(c) 36 H
(d) Can’t be predicted
Q3. Which of these is software interrupt?
(a) RST 4.5
(b) RST 5.5
(c) RST 6.5
(d) RST 5
Q4. When the supply voltage to an induction motor is reduced by 10 %, the maximum torque will be decreased approximately by:
(a) 5 %
(b) 10 %
(c) 19 %
(d) 40 %
Q5. It is given that the starting current of a 3-phase induction motor is 3-times the rated current, while the rated slip is 6%. What would be the ratio of starting torque to full-load torque?
(a) 1
(b) 1.8
(c) O.54
(d) 0.6
Q6. Choose the characteristics which is not related to an ideal op-amp?
(a) Very high gain
(b) Very high input impedance
(c) Very low output impedance
(d) Very low CMRR
Q7. What is the unit of reciprocal of reluctance?
(a) Henry
(b) Henry/ mater
(c) Meter/Henry
(d) Henry^(-1)
Q8. A single phase, half-wave, controlled rectifier has 400 sin〖(314 t)〗 as the input voltage and R as the load. For a firing angle of 60⁰ for the SCR, the average output voltage is
(a) 200/π
(b) 240/ π
(c) 300/ π
(d) 400/ π
Q9. The unit of conductivity is
(a) ohm/m
(b) mho/m
(c) ohm. m
(d) mho. m
Q10. The sensitivity of a PMMC instruments is 10 KΩ/V. If this instrument is used in a rectifier type voltmeter with half wave rectification, the sensitivity will be:
(a) 10 KΩ/V
(b) 9 KΩ/V
(c) 4.5 KΩ/V
(d) 2.2 KΩ/V
Sharing is caring!
Thank You, Your details have been submitted we will get back to you.
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https://classhall.com/lesson/computing-devices-pre-computer-age-to-the-19th-century/
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You must complete Overview of a Computer System to unlock this Lesson.
# COMPUTING DEVICES: PRE-COMPUTER AGE TO THE 19TH CENTURY
CONTENT
Features, Components and Uses of Computing Devices
1. The Abacus
2. The Slide Rule
3. Napier’s Bone
4. Pascal’s Calculator
5. Leibniz Multiplier
6. The Jacquard Loom
7. The Difference Engine
8. Hollerith Census Machine
9. The Analytical Engine
10. Burrough’s Machine
## Features, Components and Uses of Computing Devices
### The Abacus
Abacus is an instrument used in performing arithmetic calculations. It is probably the first calculating device. The Chinese invented it, and because of its success it spread from China to other countries. The abacus is also called a counting frame, it consist of a tablet or frame bearing parallel wires or grooves on which counters or beads are moved. A modern abacus consists of wooden frame with beads on parallel wires, and a crossbar oriented perpendicular to the wires that divides the beads into two groups. Each column or wire represents one place in the decimal system. The Abacus was used for addition and subtraction. It could not carry out complex mathematical operations.
Lesson tags: Computer Science Lesson Notes, Computer Science Objective Questions, SS1 Computer Science, SS1 Computer Science Evaluation Questions, SS1 Computer Science Evaluation Questions First Term, SS1 Computer Science First Term, SS1 Computer Science Objective Questions, SS1 Computer Science Objective Questions First Term
Back to: COMPUTER SCIENCE/ICT – SS1 > First Term
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### Still have math questions?
Algebra
Question
Which two equations are equivalent?
A. $$y = ( x + 3 ) ^ { 2 }$$ and $$y = x ^ { 2 } + 6$$
B. $$y = ( x - 5 ) ^ { 2 }$$ and $$y = x ^ { 2 } - 25$$
C. $$y = ( x - 3 ) ^ { 2 }$$ and $$y = x ^ { 2 } - 6 x + 9$$
D. $$y = ( x + 5 ) ^ { 2 }$$ and $$y = x ^ { 2 } + 25 x + 10$$
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https://community.fabric.microsoft.com/t5/Desktop/Absolute-at-total/m-p/1155855
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Impactful Individual
## Absolute at total
Dears,
I am currently trying to calculate the absolute difference between imports and exports at SKU level in the total. For the same SKU, I have imports and exports in every week. At the end of week 8, I'd like to check the absolute difference between Imports and Exports. How?
1 ACCEPTED SOLUTION
Microsoft Employee
@Omega,
Create the following measures and check if you get expected result.
NewMeasure = IF(COUNTROWS(VALUES(Table2[SKU]))=1,Table2[ABSDIFF],SUMX(VALUES(Table2[SKU]),[ABSDIFF]))
Measure 3 = [NewMeasure]/[ExportTotal]
Regards,
Lydia
Community Support Team _ Lydia Zhang
If this post helps, then please consider Accept it as the solution to help the other members find it more quickly.
24 REPLIES 24
Anonymous
Not applicable
Hi @Omega
How did you get the redmarked number, that´s the number I´m after but can´t seem to be able to produce.
/Helena
Super User
maybe im being daft but why dont you convert the number after you have done the calculation rather than creating absolute figures before you do the sum. ie. * -1
If I took the time to answer your question and I came up with a solution, please mark my post as a solution and /or give kudos freely for the effort 🙂 Thank you!
Proud to be a Super User!
Impactful Individual
Super User
SUMX(Table2,ABS(Table2[Export]-Table2[Import])) * -1
(but like i say i could be being daft)
If I took the time to answer your question and I came up with a solution, please mark my post as a solution and /or give kudos freely for the effort 🙂 Thank you!
Proud to be a Super User!
Community Champion
Can you share the code for your measure(s) ? PowerBI visuals don't total up columns and rows on measures - it calculates the measure based on the current filter context.
Impactful Individual
Measure = SUMX(Table2,ABS(Table2[Export]-Table2[Import]))
Community Champion
Your code is doing the absolute value, then summing. You want to sum then do absolute value
Measure = ABS(SUMX(Table2, (Table2[Export]-Table2[Import])))
Impactful Individual
Thanks!!! My bad didn't realize I'm doing it wrong. Now, I want to take the total absolute value (470) and divide it by the total export per SKU (8510). I tried the following measure:
Measure 2 = SUMX(Table2,[Measure])/SUMX(Table2,Table2[Export]) but I'm not getting correct % 😞
I even tried: Measure 2 = SUMX(Table2,[Measure])/SUM(Table2[Export]) but still wrong because it gets me the total export for all the SKUs 😞
Thanks!
Super User
Measure 2 = divide(SUMX(Table2,[Measure]),SUMX(Table2,Table2[Export]))
If I took the time to answer your question and I came up with a solution, please mark my post as a solution and /or give kudos freely for the effort 🙂 Thank you!
Proud to be a Super User!
Impactful Individual
Same values as I had 😞
Microsoft Employee
@Omega,
I can get your expected result by creating the following measures. How do you create visual in your scenario?
ABSDIFF = ABS(SUMX(Table2, (Table2[Export]-Table2[Import])))
ExportTotal = SUM(Table2[Export])
Measure 2 = DIVIDE(Table2[ABSDIFF],Table2[ExportTotal])
Regards,
Lydia Zhang
Community Support Team _ Lydia Zhang
If this post helps, then please consider Accept it as the solution to help the other members find it more quickly.
Impactful Individual
It's close but when you combine SKUs it won't give the correct value because PBI will calculate the difference at week level then sum up the difference at SKU level. My aim is to calculate the total exports and imports per SKU -> Apply ABSDIFF -> Sum ABSDIFF and the divide by total exports.
Microsoft Employee
@Omega,
What do you mean that "combine SKUs"? Could you please post the screenshot that how you create the visual and how are the incorrect value like?
Regards,
Lydia Zhang
Community Support Team _ Lydia Zhang
If this post helps, then please consider Accept it as the solution to help the other members find it more quickly.
Impactful Individual
Sorry I was on leave and couldn't reply any time soon. What I meant is to get the difference between exports and imports for each SKU and take the summation for this difference.
In PBI, it doesn't take the difference at SKU level but at the lowest level which is in this case is Weeks. Due to that, the calculations are not correct.
Microsoft Employee
@Omega,
Create the following measures and check if you get expected result.
NewMeasure = IF(COUNTROWS(VALUES(Table2[SKU]))=1,Table2[ABSDIFF],SUMX(VALUES(Table2[SKU]),[ABSDIFF]))
Measure 3 = [NewMeasure]/[ExportTotal]
Regards,
Lydia
Community Support Team _ Lydia Zhang
If this post helps, then please consider Accept it as the solution to help the other members find it more quickly.
Impactful Individual
Thanks a lot!!! This is the solution I was looking for 😄
Super User
divide(SUMX(Table2,[Measure]),calculate(SUMX(Table2,Table2[Export]),sku = 8510))
If I took the time to answer your question and I came up with a solution, please mark my post as a solution and /or give kudos freely for the effort 🙂 Thank you!
Proud to be a Super User!
Impactful Individual
What if I have multiple SKUs?
Super User
divide(SUMX(Table2,[Measure]),calculate(SUMX(Table2,Table2[Export]), filter(table2, sku = 8510 && sku = 111))
If I took the time to answer your question and I came up with a solution, please mark my post as a solution and /or give kudos freely for the effort 🙂 Thank you!
Proud to be a Super User!
Impactful Individual
Thanks but again, this will not solve the problem. Assume that I have one million SKUs, I won't be able to type all 1 million SKUs 😞
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http://barronstestprep.com/blog/tag/special-right-triangles/
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# So a Circle Walks into a Bar…
It sounds like the beginning of a math joke, but it isn’t.
“So a right triangle is inscribed into a circle…”
That’s the premise of a couple interesting GMAT questions that I came across lately, so I thought I’d share the issues that these problems bring. First it’s important to define that term inscribed. It’s the kind of term that you may have come across several times without ever knowing what it means because the visual diagram that accompanies the problem has you covered. In geometry, when we talk about something being inscribed we mean that it is drawn inside another shape such that all of its corners touch the edge of the larger shape without going outside of it. When a shape is inscribed within a circle it’s a little like that shape has a custom-built bubble surrounding it.
Now back to our problem. So there’s a right triangle in a bubble. So What? Well that particular situation actually gives us a very important piece of information. Whenever a right triangle is inscribed in a circle, the hypotenuse of the triangle is the diameter of the circle. That’s a fantastic rule, and one you ought to remember, but when we get to the difficult end of the quant section where a question like this is likely to occur, we’re probably going to need more than that.
So what other concepts fit in with this rule? Well, our rule gives us a fantastic way to find the hypotenuse of the triangle if we know something about the circle (or vice-versa), so a nice extra step is when the GMAT asks about the length of one of the other sides of the triangle. When would we be able to find the length of the other sides of the right triangle knowing only the length of the hypotenuse? When it’s a special right triangle! So, be on the lookout for 30:60:90 triangles or 45:45:90 triangles. Even if these aren’t immediately apparent, remember that every distance from the center of the circle to the edge of the circle is a radius, and drawing one or more of these radii in often gives you more opportunity to solve.
Keep this fantastic rule and these tips in mind the next time you come across a similar problem!
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http://icpc.njust.edu.cn/Problem/Zju/2229/
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crawl-data/CC-MAIN-2020-45/segments/1603107869933.16/warc/CC-MAIN-20201020050920-20201020080920-00342.warc.gz
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# Ride to School
Time Limit: Java: 2000 ms / Others: 2000 ms
Memory Limit: Java: 65536 KB / Others: 65536 KB
## Description
Many graduate students of Peking University are living in Wanliu Campus, which is 4.5 kilometers from the main campus - Yanyuan. Students in Wanliu have to either take a bus or ride a bike to go to school. Due to the bad traffic in Beijing, many students choose to ride a bike.
We may assume that all the students except "Charley" ride from Wanliu to Yanyuan at a fixed speed. Charley is a student with a different riding habit - he always tries to follow another rider to avoid riding alone. When Charley gets to the gate of Wanliu, he will look for someone who is setting off to Yanyuan. If he finds someone, he will follow that rider, or if not, he will wait for someone to follow. On the way from Wanliu to Yanyuan, at any time if a faster student surpassed Charley, he will leave the rider he is following and speed up to follow the faster one.
We assume the time that Charley gets to the gate of Wanliu is zero. Given the set off time and speed of the other students, your task is to give the time when Charley arrives at Yanyuan.
## Input
There are several test cases. The first line of each case is N (1 <= N <= 10000) representing the number of riders (excluding Charley). N = 0 ends the input. The following N lines are information of N different riders, in such format:
Vi [TAB] Ti
Vi is a positive integer <= 40, indicating the speed of the i-th rider (kph, kilometers per hour). Ti is the set off time of the i-th rider, which is an integer and counted in seconds. In any case it is assured that there always exists a nonnegative Ti.
## Output
Output one line for each case: the arrival time of Charley. Round up (ceiling) the value when dealing with a fraction.
4
20 0
25 -155
27 190
30 240
2
21 0
22 34
0
780
771
None
## Source
Asia 2004, Beijing (Mainland China), Prelimin
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http://nrich.maths.org/public/leg.php?code=31&cl=2&cldcmpid=7239
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crawl-data/CC-MAIN-2017-39/segments/1505818685129.23/warc/CC-MAIN-20170919112242-20170919132242-00530.warc.gz
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# Search by Topic
#### Resources tagged with Addition & subtraction similar to Three Sets of Cubes, Two Surfaces:
Filter by: Content type:
Stage:
Challenge level:
### There are 241 results
Broad Topics > Calculations and Numerical Methods > Addition & subtraction
### Difference
##### Stage: 2 Challenge Level:
Place the numbers 1 to 10 in the circles so that each number is the difference between the two numbers just below it.
### Train Carriages
##### Stage: 1 and 2 Challenge Level:
Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?
### Prompt Cards
##### Stage: 2 Challenge Level:
These two group activities use mathematical reasoning - one is numerical, one geometric.
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##### Stage: 2 Challenge Level:
Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?
### A Square of Numbers
##### Stage: 2 Challenge Level:
Can you put the numbers 1 to 8 into the circles so that the four calculations are correct?
### Painting Possibilities
##### Stage: 2 Challenge Level:
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
### Money Bags
##### Stage: 2 Challenge Level:
Ram divided 15 pennies among four small bags. He could then pay any sum of money from 1p to 15p without opening any bag. How many pennies did Ram put in each bag?
### Rabbits in the Pen
##### Stage: 2 Challenge Level:
Using the statements, can you work out how many of each type of rabbit there are in these pens?
### Arranging the Tables
##### Stage: 2 Challenge Level:
There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.
### Dart Target
##### Stage: 2 Challenge Level:
This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.
### A-magical Number Maze
##### Stage: 2 Challenge Level:
This magic square has operations written in it, to make it into a maze. Start wherever you like, go through every cell and go out a total of 15!
### Oh! Harry!
##### Stage: 2 Challenge Level:
A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?
##### Stage: 2 Challenge Level:
Can you put plus signs in so this is true? 1 2 3 4 5 6 7 8 9 = 99 How many ways can you do it?
##### Stage: 1 and 2 Challenge Level:
Place six toy ladybirds into the box so that there are two ladybirds in every column and every row.
### Six Is the Sum
##### Stage: 2 Challenge Level:
What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?
### Oddly
##### Stage: 2 Challenge Level:
Find the sum of all three-digit numbers each of whose digits is odd.
### Neighbours
##### Stage: 2 Challenge Level:
In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
### Today's Date - 01/06/2009
##### Stage: 1 and 2 Challenge Level:
What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.
### Worms
##### Stage: 2 Challenge Level:
Place this "worm" on the 100 square and find the total of the four squares it covers. Keeping its head in the same place, what other totals can you make?
### Finding Fifteen
##### Stage: 2 Challenge Level:
Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?
### Plants
##### Stage: 1 and 2 Challenge Level:
Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?
### Prison Cells
##### Stage: 2 Challenge Level:
There are 78 prisoners in a square cell block of twelve cells. The clever prison warder arranged them so there were 25 along each wall of the prison block. How did he do it?
### Code Breaker
##### Stage: 2 Challenge Level:
This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?
### Spell by Numbers
##### Stage: 2 Challenge Level:
Can you substitute numbers for the letters in these sums?
### Twenty Divided Into Six
##### Stage: 2 Challenge Level:
Katie had a pack of 20 cards numbered from 1 to 20. She arranged the cards into 6 unequal piles where each pile added to the same total. What was the total and how could this be done?
### Bean Bags for Bernard's Bag
##### Stage: 2 Challenge Level:
How could you put eight beanbags in the hoops so that there are four in the blue hoop, five in the red and six in the yellow? Can you find all the ways of doing this?
### Buying a Balloon
##### Stage: 2 Challenge Level:
Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?
### On Target
##### Stage: 2 Challenge Level:
You have 5 darts and your target score is 44. How many different ways could you score 44?
### The Pied Piper of Hamelin
##### Stage: 2 Challenge Level:
This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!
### The Puzzling Sweet Shop
##### Stage: 2 Challenge Level:
There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?
### Hubble, Bubble
##### Stage: 2 Challenge Level:
Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?
### It's All about 64
##### Stage: 2 Challenge Level:
Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.
### How Much Did it Cost?
##### Stage: 2 Challenge Level:
Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.
### Sealed Solution
##### Stage: 2 Challenge Level:
Ten cards are put into five envelopes so that there are two cards in each envelope. The sum of the numbers inside it is written on each envelope. What numbers could be inside the envelopes?
### Shapes in a Grid
##### Stage: 2 Challenge Level:
Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?
### Seven Square Numbers
##### Stage: 2 Challenge Level:
Add the sum of the squares of four numbers between 10 and 20 to the sum of the squares of three numbers less than 6 to make the square of another, larger, number.
### Open Squares
##### Stage: 2 Challenge Level:
This task, written for the National Young Mathematicians' Award 2016, focuses on 'open squares'. What would the next five open squares look like?
### Polo Square
##### Stage: 2 Challenge Level:
Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.
### All Seated
##### Stage: 2 Challenge Level:
Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?
### Pouring the Punch Drink
##### Stage: 2 Challenge Level:
There are 4 jugs which hold 9 litres, 7 litres, 4 litres and 2 litres. Find a way to pour 9 litres of drink from one jug to another until you are left with exactly 3 litres in three of the jugs.
### Zargon Glasses
##### Stage: 2 Challenge Level:
Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?
### Five Coins
##### Stage: 2 Challenge Level:
Ben has five coins in his pocket. How much money might he have?
### Page Numbers
##### Stage: 2 Short Challenge Level:
Exactly 195 digits have been used to number the pages in a book. How many pages does the book have?
### Sums and Differences 1
##### Stage: 2 Challenge Level:
This challenge focuses on finding the sum and difference of pairs of two-digit numbers.
### The Dice Train
##### Stage: 2 Challenge Level:
This dice train has been made using specific rules. How many different trains can you make?
### Sitting Round the Party Tables
##### Stage: 1 and 2 Challenge Level:
Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.
### Sums and Differences 2
##### Stage: 2 Challenge Level:
Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?
### How Old?
##### Stage: 2 Challenge Level:
Cherri, Saxon, Mel and Paul are friends. They are all different ages. Can you find out the age of each friend using the information?
### Two Egg Timers
##### Stage: 2 Challenge Level:
You have two egg timers. One takes 4 minutes exactly to empty and the other takes 7 minutes. What times in whole minutes can you measure and how?
### Build it up More
##### Stage: 2 Challenge Level:
This task follows on from Build it Up and takes the ideas into three dimensions!
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http://www.knossosgames.com/logic/logic.Solution18.html
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If you like this puzzle,
you might also like...
## detailed solution
Since your robot, Bill, can only beat three other robots (Fireball, Necromancer, and Thunder), it is important that you face one of those three in each round of the tournament.
If Bill faces any other robots, you’ll lose. But it’s not immediately clear the order in which Bill should face these three. In order to decide, notice that one of these robots won’t face any other robot but yours, one will face just one other robot, and the last will face two other robots before yours. So let’s take a look at which robots these three can defeat.
At first glance, it seems as though each of these three robots could fit into any tournament slot. Notice, however, that each can defeat one of the others (Thunder can defeat Fireball, Fireball can defeat Necromancer, and Necromancer can defeat Thunder).
This doesn’t help us win. If any two of these robots face each other, one of them has to lose, and then there won’t be enough robots we can beat left in order to win the tournament. Since we don’t want these robots to face each other, we can ignore their chances against each other.
This means that Necromancer must be the last robot Bill faces, since only that robot can defeat two others. Plus, notice that both Thunder and Necromancer can defeat Xplosion, but Xplosion must be one of the two robots that Necromancer needs to defeat to make it to the last round. This leaves no robots left that Thunder can defeat, making Thunder the robot Bill will face first, and Fireball the robot bill faces second.
Thus, Bill will defeat Thunder in the first round. Fireball will also defeat Chainsaw in the first round, after which Bill will defeat Fireball in the second round to make it to the championship.
Necromancer will face off against Atomizer and Xplosion, but in which order? Similar to the other division, one of those two robots will be immediately defeated by Necromancer, while the other robot must win a match before moving on to face Necromancer in the second round.
The only robot left is Slammer, and only Xplosion can defeat Slammer. Thus Necromancer will defeat Atomizer in the first round, while Xplosion will defeat Slammer. Then Necromancer will defeat Xplosion in the second round to face Bill in the championship.
Since your robot Bill can defeat Necromancer, Bill can win the tournament if it is seeded exactly this way.
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https://tw.ichacha.net/m/displacement%20angle.html
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×
# displacement angle中文
• 失配角
• 位移角
### 例句與用法
1. Two control strategies of the input current displacement angle are presented and compared
文中介紹了兩種輸入電流偏置角的控制策略，并做了比較。
2. In the second one , the input current displacement angle is dynamically modulated as a function of positive and negative sequence components of the input voltages . in both cases , the harmonic content has been evaluated analytically
第二種方法中，輸入電流矢量根據輸入電壓正、負序成分進行動態調制，并對兩種情況下的輸入電流的諧波成分都進行了分析。
3. Abstract : this paper deals with the simulation result analysis of three phase synchronous generator with rectified output with stator y connected , as well as with stator connected . the curves of overlap angle , displacement angle and power factor with load are presented , and the tendency of these curves are also described
文摘:對整流負載時凸極同步發電機定子繞組及y聯結仿真結果進行分析，繪出反映整流特性的換流重疊角、位移角以及功率因數隨負載變化曲線，并對變化趨勢進行分析。
4. Aimed at the present code of structure design without the specific method of frail - layer check computations of frame structures in high vibration areas , this article provides the parameter modifying method for the checking according to different regulations of earthquake parameter and by deducing and contrasting the relation of elasticity and plasticity displacement angle limitation of several earthquakes from the given conditions of present code
摘要針對現行結構設計規范中沒有明確給出高震區框架結構薄弱層驗算具體方法的情況，從規范給定的條件入手，通過時大、小地震下彈性、彈塑性位移角限值關系的推導和對比，提出了在高震區框架結構薄弱層驗算中進行參數修正的方法。
5. Based on the experiment of full - sized cshb walls under lateral and vertical loads , initial crack - resisting stiffen formula was deduced by considering the influence of concrete beams , concrete core columns , structural columns , vertical pressure and window ( or door ) . the results calculated from the formula were fit well with the experimental results . by the analysis of displacement at initial crazing , probability statistical mode and its parameters of relative displacement angle were presented
在介紹和總結本課題的室內足尺寸單片墻抗側力性能試驗的基礎上，考慮了圈梁、芯柱、構造柱和墻體正壓力、開門窗洞等因素對抗側剛度的影響，提出了綜合各種因素的初裂抗側剛度公式，與試驗結果具有很好的一致性；結合試驗的初裂位移分析，給出了層間相對位移角的概率統計模式及相應的統計參數，提出了小砌塊建筑層間位移角的控制標準。
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https://www.tuko.co.ke/287486-debt-equity-ratio-formula-calculation.html
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Business & Economy section is brought to you by Jiji.ke — #1 Kenyan online marketplace
# Debt to equity ratio formula and calculation
Every business has to source for funding to facilitate its activities, for example production of goods using machinery. There are two sources of finance for the business; debts and equity. Debt financing refers to the contributions from lenders such as banks and other financial institutions, while equity financing is the contribution from the shareholders to generate the overall capital needed for the business to run. How then, should the firm balance finances that it has borrowed from financial institutions, with that which has been contributed by shareholders themselves? Our focus today will be on debt to equity ratio.
Debts can be a good source of capital for the business when used in a sustainable and efficient manner, and in fact, most financial advisers support this path of funding the activities of the business provided that the profits of the business are sustainable. The debt to equity ratio is a type of leverage ratio which shows a comparison of the debts to shareholder equity. Leverage looks at how money from lenders is used to finance the activities of the business i.e level of debt incurred by a business entity. In a typical business in Kenya, this would entail making a comparison of how much we have borrowed to how much we have invested in the business from our own sources.
READ ALSO: Types of entrepreneurs in Kenya
## Debt to Equity Ratio formula
Debt to equity ratio can be calculated using the debt to equity ratio formula. It is computed by dividing debts by equity.
Debt to Equity Ratio =(Total Liabilities)/(Total Shareholders’ Equity)
Where;
Total liabilities is the sum of all the money owed to lenders
Total shareholders’ equity is the money from the owners of the business invested in the business.
The outcome of the formula presents as a percentage or number.
### The formula for debt to equity ratio of individuals
D/E =(Total Personal Debt)/(Total Assets- Total Personal Debt)
## Interpreting debt to equity ratio
### What if debt to equity ratio is too high?
When your business has a high debt to equity ratio, it means that lenders are mostly financing the business to a greater extent. A lower debt to equity ratio for your business indicates that the business is being financed to a greater extent by the shareholders of the business. Most lenders such as commercial banks, tend to have preconditions on the highest debt to equity ratio for businesses seeking loans to finance their activities depending on the type of industry. Most organizations desire a debt to equity ratio of between 0.1 and 0.9 since a lower debt to equity ratio will attract more potential investors who are willing to inject more capital.
A high debt to equity ratio shows that the firm has been accumulating debts to finance its activities; this will be important to potential investors who are willing to inject more capital to finance the activities of the business. It is important to note that businesses which are capital intensive in their operations tend to have a high debt to equity ratio, for instance automotive manufacturing companies and civil engineering firms. Firms which are not capital intensive will have a lower debt to equity ratio. When the business has a lower debt to equity ratio, it will attract more investors because of reduced chances of bankruptcy. A debt to equity ratio of 1 would indicate that the contribution from shareholders and lenders is equal.
## Debt to equity ratio example
Company EFG in Kenya involved in civil engineering has total liabilities of 36 billion shillings and total shareholders’ equity 19 billion shillings in the financial year ending 2017. Calculate the debt to equity ratio.
Debt to Equity Ratio =(Total Liabilities)/(Total Shareholders’ Equity)
Debt to Equity Ratio =36/19
Debt to equity ratio=1.9
From the result, company EFG has a debt of 2 shillings for every shilling of equity. It is essential to compare the debt to equity ratio of EFG with those of other players in that industry.
### Limitations of debt to equity ratio
• The debt to equity ratio is unique to each industry. The outcome from one industry cannot be applied uniformly to all other industries in making financial decisions; this is because capital-intensive firms tend to have high debt to equity ratio when compared.
• Firms also have to identify the debts to be used in calculating the total liabilities. Failure to compute the correct total liabilities values will lead to inaccurate results.
One problem that arises is determining what can be qualified as debt, in this case, total liabilities. Below we have listed what professionals classify as debt and what is excluded in calculating the total debt.
## Debt for consideration in calculating the total debt
• Drawn line of credit
• Bonds to be paid
• Notes to be paid whose maturity is one year
• Long-term debt
• Capital lease obligations
• Current portion of long-term debt
• Notes payable whose maturity is more than one year
## Debt not for consideration
• Accounts to be paid
• Accrued expenses
• Dividends to be paid
• Deferred revenue
READ ALSO: Gross profit margin formula and explanation
Debt can be a useful source of capital for expanding the operations of the business. The debts should be sustainable to avoid bankruptcy in the future. Debt to equity ratio is important to lenders in deciding whether to inject more capital into your business. As a business manager, this ratio is vital in making decisions, for example on how to cap borrowing and manage debts.
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# Corporate finance chapter 09 valuation of commen stocks
## Lecture essentials of corporate finance chapter 5 discounted cash flow valuation
... deposit? – – – Daily rate = 055 / 3 65 = 000 150 68493 Number of days = 3(3 65) = 10 95 FV = 15, 000 / (1.000 150 68493)10 95 = \$12,718 .56 Copyright 2007 McGraw-Hill Australia Pty Ltd 55 1 ... Account: – Daily rate = 052 5 / 3 65 = 0001438 356 2 FV = 100(1.0001438 356 2)3 65 = \$1 05. 39 Second Account: Semiannual rate = 053 9 / = 02 65 FV = 100(1.02 65) 2 = \$1 05. 37 • You will have more money ... Pty Ltd 52 7 Finding the Number of Payments – Example 5. 6 • Start with the equation and remember your logs – – – – – 1000 = 20(1 – 1/1.015t) / 0 15 75 = – / 1.015t / 1.015t = 25 / 25 = 1.015t t =...
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## Corporate Finance Part I Cost of Capital
... Corporate Finance: Part I Cost of Capital Download free eBooks at bookboon.com Corporate Finance Part I: Cost of Capital 1st edition © 2010 bookboon.com ISBN 978-87-7681-568-4 ... bookboon.com Corporate Finance Part I: Cost of Capital Contents Contents Introduction he Objective of the Firm Present value and opportunity cost of capital 3.1 Compounded versus simple interest ... stocks using present value formulas 18 www.sylvania.com We not reinvent the wheel we reinvent light Fascinating lighting offers an ininite spectrum of possibilities: Innovative technologies and...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 1 potx
... Brealey−Meyers: Principles of Corporate Finance, Seventh Edition Front Matter x © The McGraw−Hill Companies, 2003 Preface PREFACE same time we have rewritten Chapter 14 as a free- Of course, ... Educational Version of Other examples of expanded coverage include be- Market Insight In total there are now over a thou- havioral finance (Chapter 13 ) and new international sand end -of- chapter questions ... University of Baltimore Peter Berman University of New Haven Jean Canil University of Adelaide Robert Everett Johns Hopkins University Brealey−Meyers: Principles of Corporate Finance, Seventh Edition...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 2 pps
... “Economics and Ethics: The Case of Salomon Brothers,” Journal of Applied Corporate Finance (Summer 19 92) , pp 23 28 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition I Value Present Value ... opportunity cost of capital Brealey−Meyers: Principles of Corporate Finance, Seventh Edition I Value Present Value and the Opportunity Cost of Capital © The McGraw−Hill Companies, 20 03 CHAPTER Present ... of imperfect markets, we shall, like an economist in a shipwreck, simply assume our life jacket and swim safely to shore 21 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition 22 ...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 3 doc
... two years The present value of your year-2 cash flow equals PV ϭ C2 100 ϭ ϭ \$86.21 11 ϩ r2 11.0772 33 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition 34 I Value How to Calculate ... 12.1 133 .1 ϩ 13. 3 236 ϩ 24 612 ϩ 61 10,672 ϩ 1,067 1,252,7 83 ϩ 125,278 17,264,116,042 ϩ 1,726,411,604 205,756,782,755 ϩ 20,575,678,275 ϭ Balance ϭ 110 ϭ 121 ϭ 133 .1 ϭ 146.4 ϭ 259 ϭ 6 73 ϭ 11, 739 ... arbitrage in wellfunctioning capital markets 35 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition 36 PART I I Value © The McGraw−Hill Companies, 20 03 How to Calculate Present Values Value...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 4 doc
... Depreciation Pretax profits Tax Aftertax profits 2000 2001 2002 2003 5. 84 1 .45 4. 38 1.53 2.85 6 .40 1.60 4. 80 1.68 3.12 7 .41 1.75 5.66 1.98 3.68 8. 74 1.97 6.77 2.37 4. 40 9.39 2.22 7.17 2.51 4. 66 Assets ... and profitability in Chapter 12 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition I Value The Value of Common Stocks CHAPTER © The McGraw−Hill Companies, 2003 The Value of Common ... previous period (%) 10 10.00 1.20 2.00 Ϫ.80 12.00 1 .44 2 .40 Ϫ.96 14. 40 1.73 2.88 Ϫ1.15 17.28 2.07 3 .46 Ϫ1.39 20. 74 2 .49 2.69 Ϫ.20 23 .43 2.81 3. 04 Ϫ.23 26 .47 3.18 1.59 1.59 28.05 3.36 1.68 1.68 29.73...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 5 docx
... each of the following statements holds: NPV ϭ Ϫ1,000 ϩ 800 150 150 150 150 150 ϩ ϩ ϩ Ϫ ϭ0 ϩ 50 1 .50 2 1 .50 2 1 .50 2 1 .50 2 1 .50 2 and NPV ϭ Ϫ1,000 ϩ Ϫ 800 150 150 150 150 ϩ ϩ ϩ ϩ 1. 152 11. 152 2 11. 152 2 ... rate of return (4 Ϭ 5) *Rounded Straight-line depreciation over seven years is 400/7 ϭ 57 .14, or \$57 ,140 per year † Capital investment is \$400,000 in year 16 140 55 57 28 140 55 57 28 140 55 57 ... 28 140 55 57 28 140 55 57 28 140 55 57 28 140 55 57 28 400 343 286 229 171 114 57 7% 8.2% 9.8% 12.2% 16.4% 24.6% For simplicity we have ignored taxes There will be plenty about taxes in Chapter...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 6 ppt
... 5.90 5.90 5.90 5.90 5.90 5.90 5.90 2.99 3.75 7.22 6. 68 6. 18 5.71 5.28 4.89 4.52 4. 46 4. 46 4. 46 4. 46 4. 46 4. 46 4. 46 4. 46 4. 46 2.25 (MACRS) Table 6. 4 summarizes the tax depreciation schedules Note ... Pretax profit (6 Ϫ Ϫ Ϫ 9) 11 Tax at 40% 12 Profit after tax (10 – 11) 11.9 71 .6 4.4 76. 0 27.0 9.2 15.5 11.9 Ϫ9 .6 Ϫ3.8 Ϫ5.8 23.9 59 .6 7 .6 67.2 51.3 17.4 15.5 11.9 6. 4 2 .6 3.9 35.8 47.7 6. 9 54 .6 89.1 ... 1,210 262 32 ,61 0 19,552 1,331 3,432 48,901 29,345 1, 464 5,929 35,834 21,492 1 ,61 1 4,053 19,717 11,830 1,772 1,939 68 2 Ϫ934 Ϫ550 3 ,68 6 Ϫ739 8,295 Ϫ1,972 12, 163 Ϫ1 ,62 9 8 ,67 8 1,307 4,1 76 1,581 68 2...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 7 potx
... Cost of Capital 65 45 25 -1 5 Dell Computer -3 5 Aug-96 Jan- 97 Jan-98 Jan-99 Jan-00 Jan-01 Jan-98 Jan-99 Jan-00 Jan-01 Jan-98 Jan-99 Jan-00 Jan-01 85 Return, percent 65 45 25 -1 5 Reebok -3 5 Aug-96 ... Jan- 97 65 45 25 -1 5 Portfolio -3 5 Aug-96 Jan- 97 FIGURE 7. 7 The variability of a portfolio with equal holdings in Dell Computer and Reebok would have been less than the average variability of ... line fitted to a plot of Dell’s returns versus market returns has a slope of 2.21 See Figure 7. 11 173 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition 174 II Risk © The McGraw−Hill...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 8 ppt
... deviation 187 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition 188 II Risk © The McGraw−Hill Companies, 2003 Risk and Return PART II Risk Proportion of days 0.14 0.12 0.10 0. 08 0.06 ... Risk,” Review of Economic Studies 25 (February 19 58) , pp 65 86 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition II Risk © The McGraw−Hill Companies, 2003 Risk and Return CHAPTER Risk ... of course, we were trying to sell it 197 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition 1 98 PART II II Risk © The McGraw−Hill Companies, 2003 Risk and Return Risk FIGURE 8. 8...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 9 pot
... 10 -2 0 -1 0 10 20 30 -3 0 -2 0 August 198 8– January 199 5 -3 0 -2 0 β = 52 (.10) Exxon Mobil return % 10 -1 0 10 20 30 February 199 5– July 2001 -3 0 20 -1 0 -2 0 -3 0 -1 0 -1 0 -2 0 R = 28 Market return, % -1 0 ... Market return, % -3 0 -2 0 -1 0 β = 2.02 (.38) 30 10 20 30 Market return, % -1 0 -3 0 -2 0 -1 0 -1 0 -2 0 -2 0 August 198 8– January 199 5 -3 0 10 20 30 February 199 5– July 2001 -3 0 -4 0 -4 0 30 30 General ... Flow (CEQt ) Ratio of CEQt to Ct 100 100 100 94 .6 89. 6 84.8 94 6 896 ϭ 94 62 848 ϭ 94 63 241 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition 242 PART II II Risk Capital Budgeting...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 10 potx
... the cost of capital was percent in nominal terms 283 Visit us at www.mhhe.com/bm7e Brealey−Meyers: Principles of Corporate Finance, Seventh Edition Brealey−Meyers: Principles of Corporate Finance, ... mod- Brealey−Meyers: Principles of Corporate Finance, Seventh Edition III Practical Problems in Capital Budgeting 10 A Project is Not a Black Box © The McGraw−Hill Companies, 2003 CHAPTER 10 ... switch be- Brealey−Meyers: Principles of Corporate Finance, Seventh Edition III Practical Problems in Capital Budgeting 10 A Project is Not a Black Box © The McGraw−Hill Companies, 2003 CHAPTER 10...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 11 docx
... the value of existing one- and two-year-old plants? Existing plants must continue using the original tax depreciation schedule Brealey−Meyers: Principles of Corporate Finance, Seventh Edition ... million ϩ a NPV ϭ Ϫ1,000 ϩ a 11. 202 t 11. 202 20 tϭ1 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition III Practical Problems in Capital Budgeting 11 Where Positive Net Present ... Brealey−Meyers: Principles of Corporate Finance, Seventh Edition III Practical Problems in Capital Budgeting 11 Where Positive Net Present Values Come From © The McGraw−Hill Companies, 2003 CHAPTER...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 12 doc
... investment (ROI) of 130/1,000 ϭ 13 or 14 In practice, investment would be measured as the average of beginning- and end -of- year assets See Chapter 29 321 Brealey−Meyers: Principles of Corporate Finance, ... ϩ.093 ϩ .126 † Rate of return, percent 12 11 10 Economic rate of return Book rate of return 10 15 20 25 Rate of growth, percent Brealey−Meyers: Principles of Corporate Finance, Seventh Edition ... Visit us at www.mhhe.com/bm7e Brealey−Meyers: Principles of Corporate Finance, Seventh Edition Brealey−Meyers: Principles of Corporate Finance, Seventh Edition PART THREE RELATED WEBSITES III Practical...
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## Brealey−Meyers: Principles of Corporate Finance, 7th Edition - Chapter 13 pps
... discussed in Chapter 18 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition IV Financial Decisions and Market Efficiency CHAPTER 13 13 Corporate Financing and the Six Lessons of Market ... measure of relative performance Brealey−Meyers: Principles of Corporate Finance, Seventh Edition IV Financial Decisions and Market Efficiency 13 Corporate Financing and the Six Lessons of Market ... of Adjustment of Stock Prices to Earnings and Dividend Announcements,” Journal of Financial Economics 13 (June 1984), pp 223–252 353 Brealey−Meyers: Principles of Corporate Finance, Seventh Edition...
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Monday, August 2, 2021
# Generator Guide
1-) Definitions
Amper (Current): The movement of the electric charge, that is, reaching from one place to another is called current.
VoltAmper: There must be a power difference between the two points in order to create current. This difference is called tension. The power of the electric current drawn from the system at a certain volt value is called VoltAmper. It is equal to the product of current and voltage.
Watt: The power of the electric current at a certain voltage value is called wattage.
AC: Alternating current (Electricity from the city grid)
DC: Direct current (Battery, electricity converted by some adapters)
AVR: Automatic Voltage Regulator is an electronic circuit that provides 220V 50Hz (or close) of the supplied electricity.
Power Factor: Generally it is around 0.8 0.9 in Generators. It is preferred to be close to 1.
Reactive Load: Extra energy used by some devices during the first run
Active Load: The electricity used by the device is the same from opening to closing.
Square, Modified Sine and Sine wave: Almost all electrical appliances in the house operate with Sine wave. This is the electricity coming from the city grid. Modified sinus is a form of square wave that is gradually reduced and a wave close to the sinus is created. I think we can give an example as follows, if you go up to a 10-step ladder one by one and go back one by one, you will get sine wave, 3 by 3 and if you go down, you will get a modified sine wave, you will get it all at once and you will get square wave. The hours that you will operate with the modified sine wave can go forward, the motorized devices may not work correctly (combi, refrigerator, ventilator), the drill can operate at a single speed, not at various speeds.
2-) What are the average electricity values ​​spent by household appliances?
Let’s first clarify the Active and reactive load we mentioned above:
Classic bulbs, electric heaters (kettle, electric stove, toaster) (Attention: turbo ovens are motorized, do not enter this class!) Constantly consume as much electricity as the watts written on the label.
Devices with a motor (fridge, freezer, fan, electric motor) and fluorescent lamps (I think this is also the saving bulbs, but I’m not sure, I would be happy if someone confirms it) and the UPS needs twice the power of writing on the label.
Devices such as submersible pumps, air conditioners, air compressors may require 3-5 times the take-off power of the writer on the label.
It is better to look at the label of the devices you will use when calculating your own energy needs.
3-) How do you calculate the total power you need?
3a-) The most important step to calculate the total power is to determine your need well. Decide well which devices to operate in case of downtime.
3b-) Take a paper pen and make a list. Write down the device name, energy requirement during normal operation, and First-time energy requirements. Write the start-up energy requirement not only for devices that need reactive energy, but for all, we will use the highest value here.
3c-) Take care to make calculations so that all devices are not open at the same time. For example, you can start the refrigerator for 2-3 hours after the interruption, until the engine stops completely, and then deactivate it again for a few hours.
3d-) Note that if there are old devices (such as a 10-year refrigerator) that have been used for a long time, they may burn slightly above the label value.
3e-) The maximum output power of the generators shows the electricity that it can give about half an hour. After this time, it switches to the nominal operating state and the amount of electricity it can deliver decreases. Pay attention to this value when buying.
Now, by going through the list you have prepared, gather the working powers of the devices that will work simultaneously to the bottom. Get the initial value of the device that will draw the most from the starting powers. For example, the total working power is 2000 watts and your device that needs the most starting power will spend 1500 watts. In total, it made 3500 watts, multiplying this value by 1.2 (if we leave a share for future needs, we would have made a good investment for the future, otherwise it may not be enough for you the next day tomorrow). In this example, the generator power you need will be 4200 Watts.
When making your list, pay attention to the following points;
– Which device do you need most to operate in case of power failure?
– How much power will the devices to be fed continuously from the generator need?
– What are the powers of the devices to be used intermittently?
4-) Generator, should I buy my UPS?
The generator can be used for long-term needs, UPS for short-term needs. A second advantage of UPS is that it is activated when the power goes off, it meets the uninterrupted energy need for devices such as computers (you have a file to save, etc …). It is as loud as a generator (I say as a generator because it is not completely silent. Online models have a fan, while most models do not have an “electricity off” warning signal that cannot be interrupted). However, buying a UPS that will go for a long time will be very expensive. You also need to replace your batteries approximately every 4 years. You can find detailed information about UPS at http://resim.donanimhaber.com/m_7175263/mpage_1/f_/key_//tm.htm#7175263. There is a nice guide here.
The advantage of the generator compared to UPS is that you can not only operate your refrigerator, but also your combi, refrigerator, etc. devices (it is also done in UPS, but you need to find a real sinewave UPS for the combi, you need a very powerful UPS to operate the refrigerator and this is seriously sound money)
5-) What is Digital Inverter Generator? How does it work?
Inverter generator is a relatively new technology. They have smaller sizes. Since they adjust the working speed according to the need for electricity, their fuel consumption and the noise they make are less.
First, a 3-phase AC generates electricity, then it converts it to DC, then this DC converts electricity into cleaner AC electricity. Thus, you will get a clean value of 220Volt 50Hz. Computer, Digital Combi etc … clean electricity is a very necessary feature for some sensitive devices.
6-) Where should I put the generator and what are the things I should be careful about in use? (CAUTION: Danger of carbon monoxide poisoning !!!)
6a-) Generators cannot be operated inside the house. It should not be operated in enclosed spaces such as balconies. You can hurt yourself, not to avoid any damage to this device!
6b-) It is best to have it outside the house. You should make sure that the house is located away from the door, the window and the fans that give the house air.
6c-) Carbon monoxide gases from exhaust cannot be seen (you can only see exhaust fumes) and they are odorless. Even if you don’t see and smell the smoke, you may have inhaled carbon monoxide. If you feel sluggish and weak, get fresh air immediately. Carbon monoxide sticks to red blood cells faster than oxygen. Therefore, if there is carbon monoxide in the environment, you are more likely to be poisoned.
6d-) If possible, put a carbon monoxide alarm in your environment (I will do it because I will put it in the balcony). (you can say you are saying you are putting it in the balcony but I am on the 4th floor I have no other chance. I will definitely put an alarm)
6e-) There is also the possibility of electric shock. Therefore, it may be a solution to cover it when using it in rainy weather, but you should pay attention to the carbon monoxide output.
6f-) Before refueling, turn off the generator and wait for it to cool down. Gasoline to be poured into the heat may shine.
7-) What are the points to be considered when buying a generator?
7a-) The db information given in the generators is given at a distance of 7 meters. Therefore, a generator that says 60 decibels does not produce 60db when you are near, it is the sound it makes when it is 7 meters away. It makes a sound around 90db with you. Consider this.
7b-) There is 10 times the difference between 60db device and 70db device. Therefore, if possible, don’t even think about what happens in 2-3 decibels. Take care to buy a quality brand with a digital inverter.
7c-) If you pay attention to the generator that you buy clean sine wave signal, you can operate your sensitive devices without installing a regulator, a sine wave online ups etc … Thus, you get rid of a second expense.
7d-) Digital inverter generators can be your preferred device because they can adjust their operating speed according to the load. They consume less fuel. Let’s think like this, you connected it to the boiler. I think the boiler does not burn 200 watts all the time, it increases to 200 watts when it pumps, but maybe it drops to 100 watts when the water does not pump and the fan is not working.
7e-) Inverter generators are lighter than others. If you are looking for something portable, this may be good.
7f-) Inverter generators are expensive, so if you are going to buy the device with a small budget, make sure it contains at least AVR.
7g-) There is a maximum period for generators to work at a time. There is also the maximum time they can work with a warehouse. When the fuel in the tank runs out, you need to wait for it to cool before refilling. It is also so when it gets warmer. You should also consider how long you will need to feed and determine a model accordingly.
7h-) Gasoline and oil should be mixed in certain proportions in 2-stroke generators. Operating problems arise when this ratio cannot be met. For this reason, you should choose 4-stroke. Of course, it is a budget issue, but it seems to me that it may be more logical to take 4 times even if it is the cheapest and it is an unknown brand.
😎 What is the difference of 2-stroke and 4-stroke?
This is a frequently asked question, in 2-stroke generators, gasoline and oil must be mixed in certain proportions. Operating problems arise when this ratio cannot be met. You do not need to make such an adjustment at 4 times.
9-) Is it gasoline? Diesel? Is it natural gas?
Diesel Generators are more preferred in closed areas since the fuel is not flammable.
Gasoline Generators can be found for smaller needs, but diesel products generally start from 4-5 kva or something, I think there is no smaller capacity.
Since fuel is more economical, if you have the possibility and there is a model that meets your needs, you can choose diesel.
I have never done research on natural gas, it doesn’t make sense. I guess it can only be used in stationary systems (apartment etc.) where fuel can come cheap.
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https://wiki.zeropage.org/wiki.php/ContestScoreBoard/%EC%8B%A0%EC%9E%AC%EB%8F%99
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Contest Score Board/신재동
ContestScoreBoard/신재동 ¶
```~cpp
#include <iostream>
using namespace std;
const int MAX_TEAM = 100;
struct team
{
bool isJoin;
int totalScore;
int totalSolveProblem;
};
team teams[MAX_TEAM];
void initTeams()
{
for(int i = 0; i < MAX_TEAM; i++)
{
teams[i].isJoin = false;
teams[i].totalScore = 0;
teams[i].totalSolveProblem = 0;
}
}
char testLine[5][20] = {
"1 2 10 I",
"3 1 11 C",
"1 2 19 R",
"1 2 21 C",
"1 1 25 C"
};
void input()
{
int teamNumber;
int problemNumber;
int solveTime;
char solveType;
int max_line = 20;
char line[20];
cin.getline(line, max_line);
//int i = 0;
//strcpy(line, testLine[i++]);
while(strcmp(line,""))
{
teamNumber = atoi(strtok(line, " ")) - 1;
problemNumber = atoi(strtok(NULL, " ")) - 1;
solveTime = atoi(strtok(NULL, " "));
solveType = strtok(NULL, " ")[0];
teams[teamNumber].isJoin = true;
if(solveType == 'C')
{
teams[teamNumber].totalSolveProblem++;
teams[teamNumber].totalScore += solveTime;
}
else if(solveType == 'I')
teams[teamNumber].totalScore += 20;
cin.getline(line, max_line);
//strcpy(line, testLine[i++]);
}
}
void output()
{
for(int i = 0; i < MAX_TEAM; i++)
{
if(teams[i].isJoin)
{
int teamNumber = i + 1;
cout << teamNumber << " " << teams[i].totalSolveProblem << " " << teams[i].totalScore << endl;
}
}
}
int main()
{
initTeams();
input();
output();
return 0;
}
```
테스트 케이스 넣는 건 귀찮아서 뺌. 한편 나름대로 테스트 먼저 만듬. --재동
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http://www.singular.uni-kl.de/Manual/latest/sing_553.htm
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Singular
7.5.4.0. reiffen
Procedure from library `dmod.lib` (see dmod_lib).
Usage:
reiffen(p, q); int p, int q
Return:
ring
Purpose:
set up the polynomial, describing a Reiffen curve
Note:
activate the output ring with the `setring` command and
find the curve as a polynomial RC.
A Reiffen curve is defined as RC = x^p + y^q + xy^{q-1}, q >= p+1 >= 5
Example:
```LIB "dmod.lib"; def r = reiffen(4,5); setring r; RC; ==> xy4+y5+x4 ```
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https://de.mathworks.com/matlabcentral/profile/authors/98150-j-g-van-der-toorn?s_tid=cody_local_to_profile
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| 442,846,992
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Community Profile
# J-G van der Toorn
##### Last seen: etwa 2 Monate ago
678 total contributions since 2012
#### J-G van der Toorn's Badges
View all
Contributions in
View by
Solved
Operate on matrices of unequal, yet similar, size
You may want to add a vector to a matrix, implying that the vector is added to each column of the matrix. Or multiply a 3x4x5 ma...
fast 2 Jahre ago
Solved
Numbers on 7-segment
This is a 7-segment: _ |_| |_| It's a 3-by-3 char matrix.It has made by 3 characters: '_' , '|' and ' ' (space...
fast 2 Jahre ago
Solved
Calculate time taken by light to reach earth surface
We know the time(seconds) taken by light to reach surface of earth. What if the distance varies yearly or source of light moves ...
fast 2 Jahre ago
Solved
Draw 'H'
Draw a x-by-x matrix 'H' using 1 and 0. (x is odd and bigger than 2) Example: x=5 ans= [1 0 0 0 1 1 0 0 0 1 ...
fast 2 Jahre ago
Solved
Draw 'F'
Draw a x-by-x matrix 'F' using 1 and 0. (x is odd and bigger than 4) Example: x=5 ans= [1 1 1 1 1 1 0 0 0 0 ...
fast 2 Jahre ago
Solved
Draw 'E'
Draw a x-by-x matrix 'E' using 1 and 0. (x is odd and bigger than 4) Example: x=5 ans= [1 1 1 1 1 1 0 0 0 0 ...
fast 2 Jahre ago
Solved
Mean = Standard Deviation
Create a series with following properties; # All of the members should be positive integer # Mean of the series should be in...
fast 2 Jahre ago
Solved
Find prime number couples
Given a vector a, which will always contain at least one pair of prime numbers couple, return a matrix called 'couple' in which ...
fast 2 Jahre ago
Solved
Angle between Two Vectors
The dot product relationship, a dot b = | a | | b | cos(theta), can be used to determine the acute angle between vector a and ve...
fast 2 Jahre ago
Solved
Clockwise or Counterclockwise
Given a list of 2-d points defining the vertices of a polygon, determine whether these points are sorted clockwise. The input...
fast 2 Jahre ago
Solved
Calculate the area of a triangle between three points
Calculate the area of a triangle between three points: P1(X1,Y1) P2(X2,Y2) P3(X3,Y3) these three points are the vert...
fast 2 Jahre ago
Solved
Television Screen Dimensions
Given a width to height ratio of a TV screen given as _w_ and _h_ as well as the diagonal length of the television _l_, return t...
fast 2 Jahre ago
Solved
Second Diagonal
Transpose the matrix from it's second diagonal.
fast 2 Jahre ago
Solved
find number of times of occurrence of the most frequent number in a row vector
In a given row vector, find the number of times a mode of a row vector has occurred example: in [2 5 5 5 5 3], output is 4 ...
fast 2 Jahre ago
Solved
Draw 'D'.
Draw a x-by-x matrix 'D' using 0 and 1. example: x=4 ans= [1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0]
fast 2 Jahre ago
Solved
Create the equation: y=(3x)^2+(5x)+35 and compute y for various values of x
fast 2 Jahre ago
Solved
function to compute root mean square of first nn positive odd integers
Write a function called odd_rms that returns orms, which is the square root of the mean of the squares of the first nn positive ...
fast 2 Jahre ago
Solved
Percentage of zeros in a matrix of only 1s and 0s
Write a function called _zero_stat_ that takes a matrix as an input that only has 0 and 1 elements. The function needs to comput...
fast 2 Jahre ago
Solved
Datetime basics
Generate the datetime scalar representing the current date
fast 2 Jahre ago
Solved
Check if there are white spaces in the input string
If there are white spaces in the input string, output=1 else 0
fast 2 Jahre ago
Solved
Remove white spaces at the end of the input string
Remove all trailing white spaces at the end of the input strings
fast 2 Jahre ago
Solved
Perimeter of a semicircle
Given the diameter d, find the perimeter of a semicircle
fast 2 Jahre ago
Solved
Nth root of a number
Given an input and a number N, find the Nth root of the number(s)
fast 2 Jahre ago
Solved
Dial Up
Each number on telephone keypads, except 0 and 1, corresponds to a set of uppercase letters as shown in this list: 2 ABC, 3 DEF...
fast 2 Jahre ago
Solved
Draw 'O' !
Given n as input, generate a n-by-n matrix 'O' using 0 and 1 . example: n=4 ans= [1 1 1 1 1 0 0 1 ...
fast 2 Jahre ago
Solved
Draw 'I'
Given n as input, draw a n-by-n matrix 'I' using 0 and 1. example: n=3 ans= [0 1 0 0 1 0 0 1 0] n=...
fast 2 Jahre ago
Solved
Day counter function
Write a function called _day_counter_ that returns the number of Mondays that fell on the first day of the month in a given year...
fast 2 Jahre ago
Solved
Draw a 'N'!
Given n as input, generate a n-by-n matrix 'N' using 0 and 1 . Example: n=5 ans= [1 0 0 0 1 1 1 0 0 1 1 0 ...
fast 2 Jahre ago
Solved
Computational power of Cody servers
It has been <https://en.wikipedia.org/wiki/Moore%27s_law#History predicted> that the performance of integrated circuits would _d...
fast 2 Jahre ago
Solved
Basics: counting digits of a number irrespective of the sign
publish the number of digits in any input integer example: -23---->2
fast 2 Jahre ago
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https://www.jiskha.com/questions/512732/suppose-that-the-proportion-of-students-who-used-the-internet-as-their-major-resource-for
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# Statistics and Probability
Suppose that the proportion of students who used the Internet as their major resource for school in the past was 66%. A sample of 1000 students was taken and the number of students who used the Internet for their school during the past year was recorded. Let ��̂ be the proportion of students surveyed who used the Internet in the past year.
(A) What is the approximate distribution of ��̂ ?
(B) What is the probability that the sample proportion ��̂ exceeds 68%?
(C) What is the probability that the sample proportion lies between 64% and 68%?
1. 👍 0
2. 👎 0
3. 👁 266
## Similar Questions
1. ### Statistics
One of the questions in the Pew Internet & American Life Project asked adults if they used the Internet, at least occasionally. The results showed that 454 out of 478 adults aged 18 - 29 answered yes; 741 out of 833 adults aged 30
2. ### 2 question
What is an AUP? the abbreviation for the school administrator in an online education environment a document outlining what is acceptable behavior when using the Internet for schoolwork a policy outlining the proper formatting to
3. ### Math
School A has 480 students and 16 classrooms. School b has 192 students and 12 classrooms. A. what is the ratio of students to classrooms at School A? 480:16 B. what is the ratio of students to classrooms at School B? 192:12 C. How
4. ### MATH
A researcher wants to select a sample of 50 students from four local private high schools by performing stratified sampling. The enrollments are shown in the table. How many students at each school should be included in the
1. ### math
The Green County school system has 2,997 high school students, 3,831 middle school students, and 5,084 elementary school students. How many total students are in the Green County school system? is it 11912 ?
2. ### tec
What is an AUP? the abbreviation for the school administrator in an online education environment a document outlining what is acceptable behavior when using the Internet for schoolwork a policy outlining the proper formatting to
3. ### Math
School A has 480 students and 16 classrooms, School B has 192 students and 12 classrooms. How many students would have to transfer from School A to School B for the ratios of students to classrooms at both schools to be the same?
4. ### Math ---HELP!!!!!
30% of the fifth grade students in a large school district read below grade level. The distribution of sample proportions of samples of 100 students from this population is normal with a mean of 0.30 and a standard deviation of
1. ### Statistic
consider the population of all students at your school. A certain portion support mandatory national service following high school. Your friend randomly sampled 20 students from the school, and uses the sample proportion who
2. ### math
On average 62 % of Finite Mathematics students spend some time in the Mathematics Department's resource room. Half of these students spend more than 90 minutes per week in the resource room. At the end of the semester the students
3. ### Math
The school band is comprised of middle school students to the number of high school students was 1:8. However, this year the ratio of middle school students to high school students changed to 2:7. If there is 18 middle school
4. ### Math
What percent of students in the college are history majors (to the nearest tenth of a percent)? 200 students major in math 300 students major in history 400 major in english 150 major in science 175 major in other Total number of
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https://community.alteryx.com/t5/Weekly-Challenge/Challenge-278-Women-Athletes-and-The-Olympics/td-p/791660/page/2
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We've recently made an accessibility improvement to the community and therefore posts without any content are no longer allowed. Please use the spoiler feature or add a short message in the message body in order to submit your weekly challenge.
alteryx Community
# Weekly Challenge
Solve the challenge, share your solution and summit the ranks of our Community!
Also available in | Français | Português | Español | 日本語
###### IDEAS WANTED
We're actively looking for ideas on how to improve Weekly Challenges and would love to hear what you think!
Submit Feedback
## Challenge #278: Women Athletes and The Olympics
15 - Aurora
Spoiler
I'm not sure you can get to the given solution with the data provided because athletes may be duplicated either by country or sport. Then, when joining the two datasets, records get duplicated and it isn't obvious which is the correct record to keep... but maybe I'm missing something. So far it looks like others have left duplicates in their answers as well.
Given solution for 100797:
8 - Asteroid
Let the games begin!
Spoiler
7 - Meteor
Spoiler
8 - Asteroid
Spoiler
Solution Attached
16 - Nebula
Spoiler
This one really should be tagged regex.... I concatenated the entries where a participant won multiple medals... I note the official solution doesn't do that and creates a value based upon the medal received of 1,2,3 which wasn't in the instructions.
6 - Meteoroid
Spoiler
Loving these Olympic themed challenges!!!
Alteryx
Spoiler
Alteryx
The joys of working with ambiguous data...
Spoiler
I spent the bulk of my time analyzing the solutions trying to figure out how these two data sets joined considering that the ID's were not unique in either table. I finally settled on created a secondary ID, numbering each instance of an ID in Participants based on the order in it appeared in the set and numbering the instances in Details based on the year of the game followed by the order in which the instance appeared. Seemed to give me a reasonable match to the supplied data set. By the way, the solution shows the number of times women appeared in competition, not the number of women who competed. I modified the chart to show both.
11 - Bolide
Here is my solution:
Spoiler
8 - Asteroid
Here is my solution:
Spoiler
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http://www.gurufocus.com/term/InventoryTurnover/MAC/Inventory%2BTurnover/Macerich%2BCompany
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|
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Switch to:
GuruFocus has detected 5 Warning Signs with Macerich Co \$MAC.
More than 500,000 people have already joined GuruFocus to track the stocks they follow and exchange investment ideas.
Macerich Co (NYSE:MAC)
Inventory Turnover
0.00 (As of Dec. 2016)
Inventory turnover measures how fast the company turns over its inventory within a year. It is calculated as cost of goods sold divided by average inventory. Macerich Co's cost of goods sold for the three months ended in Dec. 2016 was \$101 Mil. Macerich Co's average inventory for the quarter that ended in Dec. 2016 was \$0 Mil.
Days inventory indicates the number of days of goods in sales that a company has in the inventory. Macerich Co's days inventory for the three months ended in Dec. 2016 was 0.00.
Inventory can be measured by Days Sales of Inventory (DSI). Macerich Co's days sales of inventory (DSI) for the three months ended in Dec. 2016 was 0.00.
Inventory to revenue ratio determines the ability of a company to manage their inventory levels. It measures the percentage of Inventories the company currently has on hand to support the current amount of Revenue. Macerich Co's inventory to revenue ratio for the quarter that ended in Dec. 2016 was 0.00.
Definition
Macerich Co's Inventory Turnover for the fiscal year that ended in Dec. 2016 is calculated as
Inventory Turnover (A: Dec. 2016 ) = Cost of Goods Sold / Average Inventory = Cost of Goods Sold (A: Dec. 2016 ) / ( (Inventory (A: Dec. 2015 ) + Inventory (A: Dec. 2016 )) / 2 ) = 405.946 / ( (0 + 0) / 2 ) = 405.946 / 0 = N/A
Macerich Co's Inventory Turnover for the quarter that ended in Dec. 2016 is calculated as
Inventory Turnover (Q: Dec. 2016 ) = Cost of Goods Sold / Average Inventory = Cost of Goods Sold (Q: Dec. 2016 ) / ( (Inventory (Q: Sep. 2016 ) + Inventory (Q: Dec. 2016 )) / 2 ) = 100.918 / ( (0 + 0) / 2 ) = 100.918 / 0 = N/A
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Explanation
Inventory Turnover measures how fast the company turns over its inventory within a year. A higher inventory turnover means the company has light inventory. Therefore the company spends less money on storage, write downs, and obsolete inventory. If the inventory is too light, it may affect sales because the company may not have enough to meet demand.
1. Days Inventory indicates the number of days of goods in sales that a company has in the inventory.
Macerich Co's Days Inventory for the three months ended in Dec. 2016 is calculated as:
Days Inventory = Average Inventory (Q: Dec. 2016 ) / Cost of Goods Sold (Q: Dec. 2016 ) * Days in Period = 0 / 100.918 * 365 / 4 = 0.00
2. Inventory can be measured by Days Sales of Inventory (DSI).
Macerich Co's Days Sales of Inventory for the three months ended in Dec. 2016 is calculated as:
Days Sales of Inventory (DSI) = Average Inventory (Q: Dec. 2016 ) / Revenue (Q: Dec. 2016 ) * Days in Period = 0 / 272 * 365 / 4 = 0.00
3. Inventory to Revenue determines the ability of a company to manage their inventory levels. It measures the percentage of Inventories the company currently has on hand to support the current amount of Revenue.
Macerich Co's Inventory to Revenue for the quarter that ended in Dec. 2016 is calculated as
Inventory to Revenue = Average Inventory (Q: Dec. 2016 ) / Revenue (Q: Dec. 2016 ) = 0 / 272 = 0.00
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Be Aware
Usually retailers pile up their inventories at holiday seasons to meet the stronger demand. Therefore, the inventory of a particular quarter of a year should not be used to calculate inventory turnover. An average inventory is a better indication.
Related Terms
Historical Data
* All numbers are in millions except for per share data and ratio. All numbers are in their local exchange's currency.
Macerich Co Annual Data
Dec07 Dec08 Dec09 Dec10 Dec11 Dec12 Dec13 Dec14 Dec15 Dec16 Inventory Turnover 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
Macerich Co Quarterly Data
Sep14 Dec14 Mar15 Jun15 Sep15 Dec15 Mar16 Jun16 Sep16 Dec16 Inventory Turnover 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00
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| 2.71875
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http://mathforum.org/kb/plaintext.jspa?messageID=8353654
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```Date: Feb 17, 2013 12:49 PM
Author: mueckenh@rz.fh-augsburg.de
Subject: Re: Matheology § 222 Back to the roots
On 16 Feb., 16:26, William Hughes <wpihug...@gmail.com> wrote:> > the nth FIS of l(n) is the nth FIS of d .> > But this does not make l(n) coFIS to d.>> > > > And there is not more than every n.>> > > > there is no line l such that d and l> > > > are coFIS> > > That would only be true if there was an n larger than every n>> > ?? The statement is yours. Are you now withdrawing it.-The statement is just to the point.You said: there is no line l such that d and l are coFISI said: That (your statement) would only be true if there was an nlarger than every n (but there isn't).There is only every d_n and for every d_n there is a line containingit. Otherwise it could not be a d_n.You are, again, arguing with finished infinity, d having more thanevery d_n.Regards, WM
```
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| 2.59375
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longest
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| 0.93423
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https://forum.mongoosepublishing.com/threads/world-builders-handbook-orbit-s-and-hzco.124253/
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# World builder's handbook - orbit#s and HZCO
#### Hapax12
##### Mongoose
Howdy, first time poster here.
I recently purchased the WBH and have been poring through it incessantly. However, maths was never a forte of mine.
I've read and reread the sections on orbit#s, fractional orbits, and habitable zones, but I just can't work it out for some reason. The numbers turn to mush and I feel like I'm missing something.
Has anybody used this new book to generate a few systems, and if so, would you mind walking through exactly how you used the orbit#s and determined habitable zones, planet placement, etc.?
It is quite complicated. The original version of system builder used the Titus-Bode law, which defined orbits to specific radius: So Orbit 1 is 0.4 AU (Mercury), Orbit 2 is 0.7 AU (Venus) etc. T-B is NOT a "law" it was a formula someone worked out to account for the known orbital positions of the planets in the solar system. The first exo-planets blew this dubious method away. BUT, Traveller has used it forever.
If you are going to maintain this "rule" for Traveller to match all previous data, then you need Fractional orbit numbers. That section of the book is about figuring out what the actual distance (in AU) would be for these fractions, since it changes between each orbit number. I will see if I can work up an example to post later.
It is quite complicated. The original version of system builder used the Titus-Bode law, which defined orbits to specific radius: So Orbit 1 is 0.4 AU (Mercury), Orbit 2 is 0.7 AU (Venus) etc. T-B is NOT a "law" it was a formula someone worked out to account for the known orbital positions of the planets in the solar system. The first exo-planets blew this dubious method away. BUT, Traveller has used it forever.
If you are going to maintain this "rule" for Traveller to match all previous data, then you need Fractional orbit numbers. That section of the book is about figuring out what the actual distance (in AU) would be for these fractions, since it changes between each orbit number. I will see if I can work up an example to post later.
Awesome, thanks for this clarification! I will also attempt to roll up a system myself and share here. Hopefully my confusion/mistakes will be more obvious that way.
So far I have rolled up:
Type: G2V
• 2D = 10 = G type
• 2D = 11 = 2 subtype
• V as all stars are main sequence unless otherwise instructed...?
Mass: 1.02
• 1.1 - [( (1.1-.9) / 5 ) * 2]
• Figuring out what the increments between G0 and G5 are, then working out where G2 would fall by adding/subtracting the appropriate amount
Temp: 5,840K
• same method as Mass
Diameter: 1.04
• same method as Mass
Luminosity: 1.152
• again, same method
Age: ~9.5B yrs
• 10/(1.02^2.5) * 1 Billion years
I rolled no additional stars in this system.
For system worlds, I began with Gas Giants.
2D = 6 > Gas Giants are present
For quantity, I add DM+1 because it is a single class V star.
So... 2D +1 = 9 > 4 Gas Giants
Planetoid belts has a DM+1 since there is more than 1 Gas Giant present
So... 2D+1 = 10 > 2 Planetoid Belts
For Terrestrial planets, I rolled:
2D-2 = 3 > Add D3-1 = 1-1 = + 0
So, Terrestrial Planets > 3 Planets
Total Worlds = 9
How does my math look up to this point? I feel pretty confident I've been able to interpret the tables and formulas thus far. Now, here's where the trouble begins.
How would I go about determining the Habitable Zone Center Orbit (HZCO)? And then assigning planets to their respective orbits, determining which planets are in the habitable zone, etc?
I have attempted the math and could share, but don't have the time right now. I'd be appreciative to see how others would take the numbers I've rolled so far and determine the next steps.
I haven't the WBH yet. I can just comment on the end result.
Your star is of the same class than our sun (G2V), a bit heavier and far older. Maybe around half a billion years before it starts to expand into a red giant. The number of planets/belts & gas giants is normal (at least as far as our understanding of astrophysics goes).
The result looks fine so far.
HZCO should be very close to our own (maybe around 1.05 AU, IIRC it is based on luminosity).
It will be interesting to see the end result. It might give a nice habitable world.
That sounds correct to me! HZCO should work out to 1.07 or something. Okay, so far so good!
I'm going to attempt to calculate orbit#s and placement today and post the results. Again, this is where I'm getting lost.
Sorry I don't have the bandwidth right now to comment in detail, but the age of 9.5 billion in the example would be the main sequence lifespan and the star's actual age should be 9.5 times the randomly determined interval of equal to or less than 1 (either with a clunky D6 formula or the simpler d100 formula, both in the middle of the first column of page 21).
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https://kr.mathworks.com/matlabcentral/cody/problems/39-which-values-occur-exactly-three-times/solutions/1747319
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Cody
# Problem 39. Which values occur exactly three times?
Solution 1747319
Submitted on 11 Mar 2019 by Samatha Aleti
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
x = [1 2 5 2 2 7 8 3 3 1 3 8 8 8]; y_correct = [2 3]; assert(isequal(threeTimes(x),y_correct))
2 Pass
x = [1 1 1]; y_correct = [1]; assert(isequal(threeTimes(x),y_correct))
3 Pass
x = [5 10 -3 10 -3 11 -3 5 5 7]; y_correct = [-3 5]; assert(isequal(threeTimes(x),y_correct))
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https://www.economist.com/analects/2013/02/27/world-class-poverty
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China
Analects
# China's poorWorld-class poverty
## WHEN is 6.3 yuan worth more than \$1.25?
WHEN is 6.3 yuan worth more than \$1.25? If you can answer that riddle, you can avoid widespread confusion about China's poverty line. You can also appreciate a rare example of China's government being treated unfairly by its own mouthpiece,Xinhua.
China's poverty line is set at 6.3 yuan a day. Yesterday the State Councilannounced that 98.99m rural folk (or 10.2% of the total) fell below that line in 2012. That was 23.39m fewer than the year before, a remarkable rate of progress.
But in reporting this good news, Xinhua, the official news agency, felt compelled to point out that China's poverty line of 6.3 yuan a day was unusually stingy by world standards:
But the current poverty line, which is equivalent to just 1 U.S. dollars a day, is still lower than the World Bank poverty line of 1.25 U.S. dollars a day.
HereXinhuais being unfair. Even though 6.3 yuan is now worth only \$1 on the foreign-exchange markets, China's poverty line is in fact considerably higher than the World Bank's standard. A detailed explanation of this paradox can be found in thispost on Free exchange, where the real nerds reside. A simpler version follows.
Prices differ a lot over time and between places. For example, 6.3 yuan stretches much further in rural China, where things are cheap, than \$1 stretches in America. By the same token, 6.3 yuan stretched further in 2010 than it does today, because inflation has taken a bite out of the yuan's value in the past couple of years.
To correct for these price differences, the World Bank's poverty line is designed to hold purchasing power constant across space and time. It reflects what \$1.25 could buy in one place (America) at one time (2005). According to the Bank, therefore, you are poor if you consume less than what \$1.25 could have bought in America eight years ago.
According to China's poverty line, by contrast, you are poor if you earn less than what 6.3 yuan could have purchased in rural China in 2010. It turns out that 6.3 yuan spent in that part of China at that time could buy about as much as \$1.83 in America in 2005. Therefore China's poverty line is significantly more generous than the World Bank's standard. And Xinhuacould, in this rare instance, be a little more generous to its government.
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# In an electrical circuit, the voltage is measured as $${V}=(200\pm4)$$ volts and the current is measured as $${I}=(20\pm0.2)~\text A.$$ The value of the resistance is: 1. $$(10\pm4.2)~\Omega$$ 2. $$(10\pm0.3)~\Omega$$ 3. $$(10\pm0.1)~\Omega$$ 4. $$(10\pm0.8)~\Omega$$
Subtopic: Derivation of Ohm's Law |
From NCERT
NEET - 2024
Hints
A uniform metal wire of length $$l$$ has $$10~\Omega$$ resistance. Now this wire is stretched to a length $$2l$$ and then bent to form a perfect circle. The equivalent resistance across any arbitrary diameter of that circle is:
1 $$10~\Omega$$ 2 $$5~\Omega$$ 3 $$40~\Omega$$ 4 $$20~\Omega$$
Subtopic: Current & Current Density |
From NCERT
NEET - 2024
Hints
The given circuit shows a uniform straight wire $$AB$$ of $$40 ~\text{cm}$$ length fixed at both ends. In order to get zero reading in the galvanometer $$G,$$ the free end of $$J$$ is to be placed from the end $$B$$ at:
1. $$32~\text{cm}$$
2. $$8~\text{cm}$$
3. $$16~\text{cm}$$
4. $$24~\text{cm}$$
Subtopic: Wheatstone Bridge |
60%
From NCERT
NEET - 2024
Hints
The amplitude of the charge oscillating in a circuit decreases exponentially as $$\mathrm{Q=Q_{0}e}^{-Rt/2L}$$ where $${Q}_0$$ is the charge at $${t}=0~\text{s}.$$The time at which charge amplitude decreases to $$0.50~{Q}_0$$ is nearly:
$$(\text{Given that}~ {R=1.5~\Omega,~L=12~\text{mH},~\text{ln}(2)=0.693})$$
1. $$19.01~\text{ms}$$
2. $$11.09~\text{ms}$$
3. $$19.01~\text s$$
4. $$11.09~\text s$$
Subtopic: RC Circuit (OLD NCERT) |
60%
From NCERT
NEET - 2024
Hints
The steady-state current in the circuit shown below is:
1. $$0.67~\text A$$
2. $$1.5~\text A$$
3. $$2~\text A$$
4. $$1~\text A$$
Subtopic: Derivation of Ohm's Law |
69%
From NCERT
NEET - 2024
Hints
A uniform wire of diameter $$d$$ carries a current of $$100~\text{mA}$$ when the mean drift velocity of electrons in the wire is $$v.$$ For a wire of diameter $${\dfrac{d}{2}}$$ of the same material to carry a current of $$200~\text{mA},$$ the mean drift velocity of electrons in the wire is:
1 $$4v$$ 2 $$8v$$ 3 $$v$$ 4 $$2v$$
Subtopic: Current & Current Density |
65%
From NCERT
NEET - 2024
Hints
Arrange the following in the order of their resistance.
A. ($$0$$ to $$1~\text A$$) ranged ammeter. B. ($$0$$ to $$100~\text {mA}$$) ranged milli-ammeter. C. ($$0$$ to $$500~\mu\text A$$) ranged micro-ammeter. D. ($$0$$ to $$100~\text V$$) ranged voltmeter.
Choose the correct answer from the options given below:
1 A > B > C > D 2 D > C > B > A 3 D > A > B > C 4 C > B > A > D
Subtopic: Derivation of Ohm's Law |
54%
From NCERT
NEET - 2024
Hints
There are two heaters $$A$$ and $$B.$$ Heater $$A$$ takes time $$t_1$$ to boil a given quantity of water, while $$B$$ takes time $$t_2$$ to boil the same quantity of water across same supply voltage. If the two heaters are connected in series, time taken by this combination to boil the same quantity of water will be:
1 $$\large\dfrac{t_1t_2}{t_1+t_2}$$ 2 $$t_1+t_2$$ 3 $${\large\dfrac12}(t_1+t_2)$$ 4 $$\large\dfrac{t_1t_2}{2(t_1+t_2)}$$
Subtopic: Heating Effects of Current |
58%
From NCERT
NEET - 2024
Hints
The value of $$R$$ in the given circuit when there is no current in the $$5 ~\Omega$$ resistor is:
1 $$12~\Omega$$ 2 $$9~ \Omega$$ 3 $$3~ \Omega$$ 4 $$2~ \Omega$$
Subtopic: Wheatstone Bridge |
81%
From NCERT
NEET - 2024
Hints
The equivalent resistance $$R_{AB}$$ between points $$A$$ and $$B$$ in the given network is:
1. $$1R$$
2. $${\dfrac35}R$$
3. $${\dfrac78}R$$
4. $${\dfrac58}R$$
Subtopic: Combination of Resistors |
72%
From NCERT
NEET - 2024
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Register for an Upcoming Free GRE Webinar
# Quantitative Reasoning
## Math Topics
Sections
Topics
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Latest Post
### Arithmetic
Integers, Fractions, Ratios, Percents, and more. Talk concepts and strategies.
9
12
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Expressions, Equations, Coordinate Geometry, and more. Talk concepts and strategies.
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4
### Geometry
Lines and Angles, Polygons, Triangles, and more. Talk concepts and strategies.
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6
### Data Analysis
Statistics, Probability, Graphs and Tables, and more. Talk concepts and strategies.
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## Solution Strategies
Sections
Topics
Posts
Latest Post
### Quant Comp
Strategies and skills for solving Quant Comp problems
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### Multiple Choice
Strategies and skills for solving Multiple Choice problems, both single and multiple answer
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Strategies and skills for solving Numeric Entry problems
4
0
## Official Practice Tests
Sections
Topics
Posts
Latest Post
### PowerPrep Test 1
The Two Quant Reasoning Sections of PowerPrep Test 1
89
20
#### Q1
The first Quant Reasoning Section of PowerPrep Test 1
22
10
#### Q2 (“List A: 0, 5, 10, 15, 20”)
Version one of the second Quant Reasoning section of PowerPrep Test 1; first question begins: “List A: 0, 5, 10, 15, 20”
23
0
#### Q2 (“Quantity A: x”)
Version two of the second Quant Reasoning section of PowerPrep Test 1; first question begins: “Quantity A: x”
21
0
#### Q2 (“The list price of a certain tool”)
Version three of the second Quant Reasoning section of PowerPrep Test 1; first question begins: “The list price of a certain tool”
23
10
### PowerPrep Test 2
The Two Quant Reasoning Sections of PowerPrep Test 2
83
11
#### Q1
The first Quant Reasoning Section of PowerPrep Test 2
20
1
#### Q2 (“The length of each side”)
Version 1 of the second Quant Reasoning section of PowerPrep Test 2; first question begins: “The length of each side”
22
0
#### Q2 (“The number 0 is between”)
Version 2 of the second Quant Reasoning section of PowerPrep Test 2; first question begins: “The number 0 is between”
20
0
#### Q2 (“In City X, the range”)
Version 3 of the second Quant Reasoning section of PowerPrep Test 2; first question begins: “In City X, the range”
21
10
### PowerPrep Plus Test 1
The Two Quant Reasoning Sections of PowerPrep Plus Test 1
134
11
#### Q1
The first Quant Reasoning Section of PowerPrep Plus Test 1
134
11
#### Q2 ("The area of the shaded region")
Version 1 of the second Quant Reasoning section of PowerPrep Plus Test 1; first question begins: “The area of the shaded region”
0
0
#### Q2 ("A number r is to be selected")
Version 2 of the second Quant Reasoning section of PowerPrep Plus Test 1; first question begins: “A number r is to be selected”
0
0
#### Q2 ("Each of the 120 people")
Version 3 of the second Quant Reasoning section of PowerPrep Plus Test 1; first question begins: “Each of the 120 people”
0
0
### PowerPrep Plus Test 2
The Two Quant Reasoning Sections of PowerPrep Plus Test 2
0
0
#### Q1
The first Quant Reasoning Section of PowerPrep Plus Test 2
0
0
#### Q2 ("Absolute value of x")
Version 1 of the second Quant Reasoning section of PowerPrep Plus Test 2; first question begins: “Absolute value of x”
0
0
#### Q2 ("x and y are integers")
Version 2 of the second Quant Reasoning section of PowerPrep Plus Test 2; first question begins: “x and y are integers”
0
0
#### Q2 ("List K consists of")
Version 3 of the second Quant Reasoning section of PowerPrep Plus Test 2; first question begins: “List K consists of”
0
0
### Official Guide
Questions about Quant Reasoning Problems on the ETS Official Guide practice tests
6
0
#### Q1 First Edition
The first Quant Reasoning section of the practice test in the ETS GRE Official Guide, First Edition
3
0
#### Q2 First Edition
The second Quant Reasoning section of the practice test in the ETS GRE Official Guide, First Edition
3
0
#### Q1 Second Edition
The first Quant Reasoning section of the practice test in the ETS GRE Official Guide, Second Edition
0
0
#### Q2 Second Edition
The second Quant Reasoning section of the practice test in the ETS GRE Official Guide, Second Edition
0
0
6
5
#### Q1
The first Quant Reasoning section of the Practice Book practice test
2
1
#### Q2
The second Quant Reasoning section of the Practice Book practice test
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The Sky Hospital Consortium consists of 40 hospitals in various parts of the US.
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The Sky Hospital Consortium consists of 40 hospitals in various parts of the US. After being discharged, patients are given a survey to determine if they are satisfied with the overall service. In most cases patients are satisfied. However, some are dissatisfied. In addition, out of those who responded as being dissatisfied, some filed a formal complaint. In the Excel file, Hospitals, are the results for the patients during a specific year. The Sky Hospital Consortium is divided into the West, Central, and East Regions.
The Sky Hospital Consortium wants to evaluate the performance of the hospitals. The Consortium wants to know which hospitals have the fewest complaints. You are to assist in the data analysis by using your knowledge of probability and conditional probability to help with the ranking of the hospitals, as well as each Region.
Managerial Report
Prepare a report (see below) with your ranking of the hospitals based on the probabilities and conditional probabilities, as well as the analysis of each region. Include the following seven (7) items in table format to support your ranking. Be sure to use five (5) decimal places for your probabilities in the table, as some of them might be quite small.
1 The probability of a patient responding “Dissatisfied” in each of the three different regions.
2 The probability of a patient filing a formal complaint in each of the three different regions.
3 The probability of a patient filing a formal complaint given a patient response of “Dissatisfied” in each of the three different regions.
4 The probability of a patient responding “Dissatisfied” for each hospital.
5 The probability of a patient filing a formal complaint for each hospital.
6 The probability of a patient filing a formal complaint given a patient response of “Dissatisfied” for each hospital.
Rank the hospitals within each region for each of the probabilities in 4 – 6. Then, find the sum of the ranks and get an overall ranking for each hospital. Evaluate and discuss the meaning of your results. Use tables, charts, graphs, or visual dashboards to support your findings.
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Math Tricks Notes PDF in Hindi Download
Math Tricks Notes PDF in Hindi Download
Hello friends,
Today we are sharing an important pdf in Hindi download Math Tricks Notes PDF in Hindi Download this pdf is helpful for all competitive exams like SSC CGL, BANK, and RAI Math Formula pdf. This Maths PDF is very useful for SSC and the upcoming competitive exams like SSC CGL, BANK, RAILWAYS, RRB NTPC, LIC AAO, and many other exams. Maths Notes are very important for any competitive exam and this Math Formula PDF in Hindi is very useful for it. this FREE PDF will be very helpful for your examination.
SSC CGL, BANK, RAI Math Formula pdf. This Maths PDF is very useful for SSC and the upcoming competitive exams like SSC CGL, BANK, RAILWAYS, RRB NTPC, LIC AAO, and many other exams. Maths Notes are very important for any competitive exam and this Math Formula PDF in Hindi is very useful for it. this FREE PDF will be very helpful for your examination.
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1. IMPORTANT MATHS FORMULAE
Algebra:
1. (a + b)2 = a2 + 2ab + b2
1. (a – b)2 = a2 – 2ab + b2
1. (a + b) (a – b) = a2 – b2
1. (x + a)(x + b) = x2 + (a + b)x + ab
1. (x + a)(x – b) = x2 + (a – b)x – ab
1. (x – a)(x + b) = x2 + (b – a)x – ab
1. (x – a)(x – b) = x2 – (a + b)x + ab
1. (a + b)3 = a3 + b3 + 3ab(a + b)
1. (a – b)3 = a3 – b3 – 3ab(a – b)
1. (x + y + z) 2 = x2 + y2 + z2 + 2xy + 2yz + 2xz
1. (x + y – z) 2 = x2 + y2 + z2 + 2xy – 2yz – 2xz
1. (x – y + z)2 = x2 + y2 + z2 – 2xy – 2yz + 2xz
1. (x – y – z)2 = x2 + y2 + z2 – 2xy + 2yz – 2xz
1. x3 + y3 + z3 – 3xyz = (x + y + z)(x2 + y2 + z2 – xy – yz -xz)
1. x+ y2 = 1212 [(x + y)2 + (x – y)2]
1. (x + a) (x + b) (x + c) = x+ (a + b +c)x2 + (ab + bc + ca)x + abc
1. x3 + y3 = (x + y) (x– xy + y2)
1. x3 – y3 = (x – y) (x+ xy + y2)
1. x+ y+ z-xy – yz – zx = 1212 [(x-y)+ (y-z)+ (z-x)2]
MATH QUESTION AND ANSWER
Time and work math Type-1
1)राज एक काम को पूरा करने में 20 तथा रामदेव उसी काम को 30 दिन में करता है, तो दोनों साथ मिल कर काम को कितने दिन में पूरा करेंगे ?
Answer is A) 12
2)कल्पेश एक कुँआ खोदने में 10 दिन लेता है, सुरेश 15 दिन में सामान काम को पूरा लेता है। अगर दोनों साथ में एक सामान कुआँ एक साथ खोदे तो कितना समय लगेगा ?
Answer is B) 6 दिन
3)देवकी 25 रोटियाँ बनाने में 5 घंटे का समय लेती है जबकि राधा 25 रोटी बनाने में 10 घंटे का समय लेती है। यदि दोनों एक साथ 25 रोटियाँ बनाए तो कितना समय लगेगा ?
Answer is D) 3 घंटा 20 मिनट
4)रहीम और राम अलग-अलग एक काम को करने में क्रमशः 30 और 40 दिन का समय लेते है। यदि दोनों साथ में काम करे तो कितने दिन में काम पूरा हो जायेगा ?
Answer is C) 17.142 दिन
5)लालू किसी काम को 60 दिन में अकेले पूरा कर लेता है और भोलू उसी काम को अकेला 120 दिन में पूरा कर लेता है। यदि दोनों एक साथ काम करें तो काम को पूरा करने में कितने दिन का समय लगेगा ?
Answer is A) 40 दिन
Time and work questions in Hindi TYPE-2
6)रिया और सुजॉय एक काम को 12 घंटे में पूरा करते है। यदि रिया अकेले उस काम को 20 घंटे में पूरा कर लेती है। यदि सुजॉय उस काम को अकेले करना शुरू करे तो कितने घंटे में काम पूरा कर लेगा ?
Answer is D) 30
7)बिल्लू और कालू एक खाई खोदने में 60 दिन का समय लेते है। बिल्लू उस काम को अकेले 120 दिनों में पूरा करने का दावा करता है तो उसी काम को कालू कितने दिनों में पूरा कर सकता है ?
Answer is C) 120 दिन
8) राजू कमरे की सफाई करने में 8 घण्टे का समय लेता है। राजू और पिंकी साथ मिलकर कमरे को 2 घण्टे में साफ़ लेते है। यदि पिंकी अकेला कमरे को साफ़ करे तो कितना समय लेगी ?
Answer is C) 2 घण्टे 40 मिनट
9) दिव्या और राहुल एक साथ काम को 6 दिनों में पूरा कर लेते है। दिव्या के ना रहने पर राहुल अकेला उस काम को 8 दिनों में पूरा कर सकता है तो दिव्या अकेले उस काम को कितने दिन में पूरा कर सकती है ?
Answer is B) 24 दिन
10) सूरज और अजय साथ में एक काम को 2 घण्टे में पूरा कर लेते है। यदि सूरज अकेला पूरा काम 2 घंटा 30 मिनट में कर लेता है तो अजय अकेला कितने समय में काम पूरा कर सकता है ?
Answer is A) 10 घंटे
Time and work questions in hindi Type-2
11) जॉय की कार्य क्षमता मुन्ना से दुगुनी है। यदि दोनों मिलकर एक कार्य को 15 दिनों में करते है, तो मुन्ना अकेला उस काम को कितने दिन में पूरा कर सकता है ?
Answer is C) 45 दिन
12) गोलू और गीता साथ में एक काम को 60 दिनों में पूरा करते है। गोलू की क्षमता गीता से तीगुनी है, तो गोलू अकेला उस काम को कितने दिनों में पूरा कर सकता है ?
Answer is A) 80 दिन
13) करीम की कार्य क्षमता राहुल की आधी है। यदि दोनों साथ में काम को 40 घंटे में पूरा कर सकते है, तो करीम काम को अकेला कितने घंटे में पूरा कर लेगा ?
Answer is B) 120
14) पिंकू की कार्य क्षमता रामु से डेढ़ गुणी (1.5 ) है। दोनों साथ मिलकर किसी काम को 8 घंटे में कर लेते है। तो रामु अकेला उस काम को कितने घंटे में पूरा करेगा ?
Answer is A) 20
15) राघव और चिंटू साथ मिलकर किसी कार्य को 30 मिनट में पूरा कर लेते है। चिंटू की कार्य क्षमता राघव से ढाई गुनी (2.5) है। यदि पूरे काम को राघव करे तो कितना समय लगेगा ?
Answer is D) 105 मिनट
Time and work questions in hindi टाइप – 3
16) रैंबो किसी काम को 15 और जॉन उसी काम को 20 दिनों में पूरा कर सकता है। रैंबो ने 5 दिन काम किया फिर जॉन को शामिल किया, अब काम कितने दिनों में पूरा होगा ?
Answer is C) 10 5/7 दिन
17) जूलिया एक काम को 12 घंटे में कर सकती है, माइकल उसी काम को 8 घंटे में करता है। जूलिया ने काम शुरू किया फिर 5 घंटे बाद माइकल उस काम शामिल हो गया गया। काम कितने समय में पूरा हुआ होगा ?
Answer is B) 7 घण्टा 48 मिनट
MATHS QUESTION AND ANSWER
18) बिट्टू 20 दिनों में किसी काम को पूरा करता है, बबलू उसी काम को 15 दिनों में पूरा कर सकता है। बिट्टू ने 5 दिन काम करके छोड़ दिया, तो शेष काम को पूरा करने में बबलू को कितने दिन लगेंगे ?
Answer is C) 11 दिन 6 घंटा
19) चंगु और मांगू अलग-अलग किसी काम को क्रमशः 16 और 20 दिनों में पूरा करते है। मांगू काम शुरू करता है, परन्तु 10 दिन बाद उसे काम छोड़ना पड़ता है। शेष काम को चंगु कितने दिनों में पूरा करेगा ?
Answer is D) 8 दिन
Time and work aptitude questions Type-4
20) बिल एक काम को 50 दिनों में पूरा करता है, एलोन उसी काम को 25 दिनों में पूरा कर लेता है। बिल 10 दिनों तक अकेले काम किया, उसके बाद एलोन भी शामिल हो गया ओर काम को समाप्त किया। दोनों ने साथ मिलकर कितना प्रतिशत काम किया ?
Answer is D) 80%
21) राम , रहीम और जॉर्ज किसी काम क्रमशः 3, 5 और 15 दिनों में समाप्त कर सकते है। तीनों एक साथ उसी काम को कितने दिनों में पूरा करेंगे ?
Answer is A) 1 2/3 दिन या 1 दिन 16 घण्टे
22) किसी काम को लिज़ा 5, ऋचा 10 और बिट्टू 15 दिनों में पूरा करते है। यदि तीनों एक साथ उस काम को करें तो कितने दिन लगेंगे ?
Answer is D) 2 8/11 दिन
23) मिंटू 10, राहुल 12 और भोलू 15 दिन में एक काम को पूरा कर सकता है। मिंटू और राहुल ने काम शुरू किया, लेकिन 3 दिन बाद भोलू भी शामिल हो गया। काम कितने समय में पूरा होगा ?
Answer is B) 4 4/5 दिन
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## How to measure a capacitor with an oscilloscope pdf
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How to measure a capacitor with an oscilloscope pdf
The Two Faces of Q Wes Hayward, w7zoi, November, 2010. Updates: 14Dec10, 29Dec10, 2Jan11. April 14, 2015. See addendum at document end. Abstract Most home-lab measurements of Q only evaluate an LC resonator. We then tend to associate the resulting Q with inductor loss while capacitor Q is assumed to be quite high. This assumption is now in greater doubt, especially with SMT …
Connect the function generator, capacitor, resistor, and oscilloscope O as shown in Figure 4. Connect the leads from the oscilloscope so that they are measuring the potential difference across the capacitor.
review the use of an oscilloscope, the most versatile electronic measuring instrument. Then you Then you will use this tool to investigate the characteristics of capacitors and resonant circuits.
You can also use an oscilloscope’s integration function to measure hot-swap load capacitance, provided the resistive load current is small during startup. Capacitance is the amount of charge stored per volt applied to a capacitor, and charge is
26/06/2013 · Measuring capacitance with an oscilloscope. Arduino Forum > Using Arduino > General Electronics > Measuring In other words, 47 nF, which is in fact the capacitor I used. So far so good, the theory agrees with practice. And I’ve learnt a bit more about capacitors. Now the interesting thing is if we increase the frequency. Say, to 3 kHz. The original waveform (the 1 kHz one) is saved …
How Oscilloscope Probes Affect Your Measurement. Application Note. Introduction. This application note describes how an oscilloscope probe changes the signal you’re measuring at …
Oscilloscope Measurement 1 Objectives This experiment deals with basic and advanced time domain measurement. The purpose of the laboratory is to develop a skill needed to use basic laboratory equipment. By using special measurement, the stu-dent will learn the capability, and limitation of time domain measurements, by oscilloscope. Upon completion of the experiment the student will: …
The only oscilloscope I have is a tiny DSO138 (input section is on the top left of page 4 in the linked PDF). The scope has a limit of 100Vpp (actually displays only 80Vpp). At the highest attenuation (100x), there is 1V input to the first op-amp.
Name. OSCILLOSCOPE AND RC-CIRCUITS Physics 230, Lab 5 Partner Objective The first part of this set of activities will acquaint you with the kinds of measurements you can make with an oscilloscope and give you a chance to become comfortable with the buttons and knobs on the scope.
Use CH1 of the oscilloscope to measure the generator output V and CH2 to measure the voltage vacross capacitor C (see previous diagram). Make sure that the probe earths for
In this section, the oscilloscope will be used to measure the time constant of RC circuits; circuits containing resistors and capacitors are used in series and parallel arrangement to measure …
An oscilloscope is a piece of electronic test equipment that graphically displays waveforms and other changing voltages over time. Scientists and technicians use them to measure …
− Capacitor as a charge storage device scope’s measurement tools. A digital oscilloscope samples input voltage signal at regular intervals using an analog-to-digital converter (ADC) PHYS 1493/1494/2699: Exp. 8 – Capacitance and the oscilloscope. 27 Oscilloscope The oscilloscope allows us to visualize signals that vary rapidly with time. Very handy! Idea: scope converts voltage
Technique on how to measure the capacitance of unknown capacitor using oscilloscope and signal generator. A reference resistor is also needed to perform this measurement.
7/07/2018 · I wanted to use it to measure inductance, by making an LC circuit and seeing the damped oscillations, measuring the distance of the period of the wave(aka the frequency). I used the following formula: 1/sqr(2pi*f)*C, where C is a known capacitor.
The voltage across the capacitor is measured with an oscilloscope. The function generator is set at the maximum output voltage, and the frequency is adjusted so that the voltage across the capacitor is kept at a low level. In this way almost the entirely voltage is dropped across the internal generator resistance. It is like the capacitor is connected to a current source. The current that
Probe circuit for measuring higher voltage on oscilloscope
Last Revised on November 12 2018 Grade EXPERIMENT 10
#135: Measure Capacitor ESR with an Oscilloscope and Function Generator October 2018 This video discusses how to measure the ESR (equivalent series resistance) of a capacitor using an oscilloscope and function generator.
3/05/2013 · This video shows how to measure the value of unknown capacitors and inductors using your oscilloscope and a simple pulse generator. There …
A voltage step will be seen on the measurement capacitor from charge transfer. This doesn’t affect timer operation adversely. It simply shows charge added and subtracted to compensate for the
For this RC circuit, how can you get the oscilloscope to measure the charge on the capacitor as a function of time? 3. Suppose the capacitor is initially uncharged, and the circuit is closed at time t = 0. As your prediction, draw a rough sketch of the voltage across the resistoras a function of time, and explain your reasoning. Physics 9 Intro to oscilloscope, v.1.0 p. 6. Now do the
How to Measure a Capacitor with an Oscilloscope [Theory] Therefore, when trying to decipher the value of an unknown capacitor with an oscilloscope, use a known voltage for Vin, and a known resistor R. Now, determine the value of 63.2% of Vin, and let the capacitor charge through the resistor R, as in the schematic. Capture the charging waveform Vc on the oscilloscope and from the graph …
For reference, see the oscilloscope picture at the start of this lab, Figure 3.4. 11. The RC circuit is a passive highpass filter with a low-frequency cutoff point near 160
Capacitance and Inductance Measurements Using an Oscilloscope and a Function Generator. Application Note Most labs have an ample supply of DMM’s for measuring DC resistance, but when it comes to measuring inductance, capacitance and impedance, it is not …
22/05/2014 · hi, i want to perform a test where i can stress an electrolytic capacitor (450V/5A)… i think if ripple current is highi can stress it but i want to know how can i increase the ripple current… i want to know how to measure ripple current from an oscilloscope…(other than measuring RMS voltage ripple and calculating from it) Thanks
Use “Measure” on the oscilloscope to find the peak-to-peak voltage of the capacitor and the peak-to-peak voltage of the resistor. If it is not giving a value, go to Trig
3/07/2018 · If so, configure a 555 timer as a square wave oscillator including the capacitor of unknown capacitance, and measure frequency of oscillation. The frequency of oscillation depends inversely on capacitance of the capacitor.
This lab exercise demonstrates a technique for using an oscilloscope and function generator to measure impedance and capacitance. Voltage waveforms are measured on the oscilloscope and the measurements are used to calculate an unknown capacitance.
1 Lab Exercise Oscilloscope Measurement Lab Measuring Impedance and Capacitance with an Oscilloscope and Function Generator . A circuit made up of a resistor and capacitor causes a
It will key the oscilloscope to start measuring when the signal voltage passes a certain level. An edge trigger can be set to catch on a rising or falling edge (or both). An edge trigger can be set to catch on a rising or falling edge (or both).
An oscilloscope is a very powerful instrument that allows you to measure almost all aspects of an electrical signal. You have used an oscilloscope in the introductory physics labs (hopefully!) and should already be familiar with some of the basics. The scopes you have used previously have probably been analog scopes; a stream of electrons is produced at the rear of the tube, deflected
A Simple Method to Measure Unknown Inductors A simple and quick way to measure the inductance of an unknown power inductor (provided you have a function generator and oscilloscope).
The oscilloscope can be used to measure time constants as short as microseconds or less. • Connect up the RC circuit of figure 3 (see last page), and apply a square-wave of frequency about 50 kHz.
If necessary, use a non-metallic tool to adjust the trimmer capacitor of the probe for the fattest square wave being displayed on the oscilloscope. Repeat if necessary.
31/01/2017 · Oscilloscope: HP 1740A (100 MHz analog) Square signal: this scope has calibration square wave signal of 1 volt at 1.4 kHz Circuit: 1 kOhm resistor in series with unknown capacitor (22 pF capacitor according to the markings) Procedure: I connected one leg of 1 kOhm resistor to the calibration stud on the ‘scope. The other leg of the resistor was inserted into the breadboard. One leg …
CHARGE AND DISCHARGE OF A CAPACITOR it does to your display. • Obtain a “quick value ” for the time constant , by measuring, on the oscilloscope screen, the time
Figur 1 Voltage divider for impedance measurement. One should of course choose a range for U(t) and V(t) respectively in order the signals fill out the screen without being “cut”.
LAB 4a THE OSCILLOSCOPE LAB 4b ELECTROCARDIOGRAMS (ECG or EKG) Lab 4c CHARGING and DISCHARGING CAPACITORS EXPERIMENTAL QUESTION In this lab, you will learn about how an oscilloscope can be used to measure waveforms from an electronic function generator. You will then use the oscilloscope to look at the electrical signals produced by your body, particularly from your …
The Digital Oscilloscope and the Breadboard
Figure 3 – Use of oscilloscope for phase measurements . The CRO may be used to measure phase shift in an electronic circuit, as shown in Fig. 3.
Use the cursors on the oscilloscope screen to measure and compute the time constant. If you use the ‘‘%’’ option in a clever way, then you can get the scope to do all the computations for you in checking the time constant.
Using the test setup in Figure 1, you can measure capacitance or inductance using a function generator, a multimeter, a frequency counter, and an oscilloscope. Use the setup to measure …
Experiments with a capacitor Introduction. A storage oscilloscope enables the voltage/time graph for a capacitor charging through a resistor to be displayed and, from the print-out, a value of the time constant for the circuit to be calculated.
E 3e “Measurements with an oscilloscope “ Tasks 1. Characterize the voltages at the different outputs of a generator box regarding the signal form, the frequency (period) and the peak to peak value U pp. Calculate the effective values U eff for the various voltage forms and compare with the reading of a digital multimeter. Check the time base used to determine the frequency of the
The oscilloscope by itself will not help much in measuring any parameter of an “XTAL”. But, working with a companion signal generator putting out in the right range of frequencies, and with the right test “set up”, the scope can help a lot. – delonghi pac cn86 user manual 30/05/2016 · Oscilloscope leads, including the high voltage one, have a capacitive voltage divider in parallel with the resistive one, and the adjustment is to compensate for the capacitance of the oscilloscope input.
Tektronix 465 Oscilloscope – Power Supply Capacitor Replacement Ron Childress – October 2008 Introduction The purpose of this document is to help others replace power supply capacitors in vintage Tektronix 465 and
This application note describes considerations for making low voltage measurements using an oscilloscope and an oscilloscope probe. Ripple measurements will be described as
Using only op amps, resistors, and a normal lab oscilloscope, how can you make a precise measurement of the capacitance of an unknown capacitor? This is a bonus question on my problem set but it sounds crazy.
If you don’t have an LCR meter in your lab, or you want to demonstrate the behavior of capacitors and inductors under sinusoidal stimulus, an oscilloscope and a function generator can help you to do a simple, transparent impedance measurement. You can expect capacitance and inductance values with 3%-5% uncertainty.
To measure the value of unknown inductor or capacitor we need to build a simple circuit called the tank circuit. This circuit can also be called as LC circuit or Resonant circuit or Tuned circuit .
Oscilloscope and Capacitance PRE-LAB This is a snapshot of a sinusoidal wave on an oscilloscope. The time scale is 2µs/div and the amplitude scale is 5v/div. Divisions are the boxes on the screen.
Connect your oscilloscope to measure the voltage across the capacitor. See figure 3. Note that the capacitor should connect to ground, not the resistor. Think about this detail when you make your measurements. If you measure incorrectly, you can ground both sides of the capacitor, in effect removing it from the circuit.
Measuring an impedance by the oscilloscope. RUC.dk
Basic Electronics For Ceramic Engineers Keysight
#135 Measure Capacitor ESR with an Oscilloscope and
Oscilloscope and RC circuits Capacitor Physical Quantities
Capacitance & Inductance Measurements Using an
Tektronix 465 Oscilloscope – Power Supply Capacitor
#90 Measure Capacitors and Inductors with an Oscilloscope
How to Measure a High Voltage With an Oscilloscope Our
delonghi portable air conditioner pac n125 hp manual – Probe Considerations for Low Voltage Measurements such as
Precision capacitor measurement with op amp resistors
Use Analog Techniques To Measure Capacitance In Capacitive
Experiment 8 Capacitance and the Oscilloscope
Basic Electronics For Ceramic Engineers Keysight
Oscilloscope math functions aid circuit analysis
How to Measure a Capacitor with an Oscilloscope [Theory] Therefore, when trying to decipher the value of an unknown capacitor with an oscilloscope, use a known voltage for Vin, and a known resistor R. Now, determine the value of 63.2% of Vin, and let the capacitor charge through the resistor R, as in the schematic. Capture the charging waveform Vc on the oscilloscope and from the graph …
Connect the function generator, capacitor, resistor, and oscilloscope O as shown in Figure 4. Connect the leads from the oscilloscope so that they are measuring the potential difference across the capacitor.
Connect your oscilloscope to measure the voltage across the capacitor. See figure 3. Note that the capacitor should connect to ground, not the resistor. Think about this detail when you make your measurements. If you measure incorrectly, you can ground both sides of the capacitor, in effect removing it from the circuit.
A voltage step will be seen on the measurement capacitor from charge transfer. This doesn’t affect timer operation adversely. It simply shows charge added and subtracted to compensate for the
An oscilloscope is a piece of electronic test equipment that graphically displays waveforms and other changing voltages over time. Scientists and technicians use them to measure …
In this section, the oscilloscope will be used to measure the time constant of RC circuits; circuits containing resistors and capacitors are used in series and parallel arrangement to measure …
The only oscilloscope I have is a tiny DSO138 (input section is on the top left of page 4 in the linked PDF). The scope has a limit of 100Vpp (actually displays only 80Vpp). At the highest attenuation (100x), there is 1V input to the first op-amp.
Using the test setup in Figure 1, you can measure capacitance or inductance using a function generator, a multimeter, a frequency counter, and an oscilloscope. Use the setup to measure …
Technique on how to measure the capacitance of unknown capacitor using oscilloscope and signal generator. A reference resistor is also needed to perform this measurement.
− Capacitor as a charge storage device scope’s measurement tools. A digital oscilloscope samples input voltage signal at regular intervals using an analog-to-digital converter (ADC) PHYS 1493/1494/2699: Exp. 8 – Capacitance and the oscilloscope. 27 Oscilloscope The oscilloscope allows us to visualize signals that vary rapidly with time. Very handy! Idea: scope converts voltage
30/05/2016 · Oscilloscope leads, including the high voltage one, have a capacitive voltage divider in parallel with the resistive one, and the adjustment is to compensate for the capacitance of the oscilloscope input.
1 Lab Exercise Oscilloscope Measurement Lab Measuring Impedance and Capacitance with an Oscilloscope and Function Generator . A circuit made up of a resistor and capacitor causes a
7/07/2018 · I wanted to use it to measure inductance, by making an LC circuit and seeing the damped oscillations, measuring the distance of the period of the wave(aka the frequency). I used the following formula: 1/sqr(2pi*f)*C, where C is a known capacitor.
### This Post Has 42 Comments
1. Sydney
How Oscilloscope Probes Affect Your Measurement. Application Note. Introduction. This application note describes how an oscilloscope probe changes the signal you’re measuring at …
How to measure value of Inductor or Capacitor using
Oscilloscope math functions aid circuit analysis
stressing a capacitor and measuring the ripple current
2. Jacob
Experiments with a capacitor Introduction. A storage oscilloscope enables the voltage/time graph for a capacitor charging through a resistor to be displayed and, from the print-out, a value of the time constant for the circuit to be calculated.
Measuring capacitance with an oscilloscope Arduino Forum
Precision capacitor measurement with op amp resistors
Measuring an impedance by the oscilloscope. RUC.dk
3. Daniel
The oscilloscope by itself will not help much in measuring any parameter of an “XTAL”. But, working with a companion signal generator putting out in the right range of frequencies, and with the right test “set up”, the scope can help a lot.
APPLICATIONS OF THE OSCILLOSCOPE
4. Samuel
In this section, the oscilloscope will be used to measure the time constant of RC circuits; circuits containing resistors and capacitors are used in series and parallel arrangement to measure …
How to measure value of Inductor or Capacitor using
5. Julia
Use the cursors on the oscilloscope screen to measure and compute the time constant. If you use the ‘‘%’’ option in a clever way, then you can get the scope to do all the computations for you in checking the time constant.
Measuring capacitance with an oscilloscope Arduino Forum
Measuring capacitance Electronic Measurements
6. Gavin
review the use of an oscilloscope, the most versatile electronic measuring instrument. Then you Then you will use this tool to investigate the characteristics of capacitors and resonant circuits.
Measuring capacitance using oscilloscope. Page 1
Probe Considerations for Low Voltage Measurements such as
7. Daniel
Use CH1 of the oscilloscope to measure the generator output V and CH2 to measure the voltage vacross capacitor C (see previous diagram). Make sure that the probe earths for
Lab 3 – AC Circuit Tools National Instruments
8. Logan
26/06/2013 · Measuring capacitance with an oscilloscope. Arduino Forum > Using Arduino > General Electronics > Measuring In other words, 47 nF, which is in fact the capacitor I used. So far so good, the theory agrees with practice. And I’ve learnt a bit more about capacitors. Now the interesting thing is if we increase the frequency. Say, to 3 kHz. The original waveform (the 1 kHz one) is saved …
How to Measure a Capacitor with an Oscilloscope [Theory
Lab 3 – Finding the value of an unknown capacitor By Henry
9. Gabrielle
3/05/2013 · This video shows how to measure the value of unknown capacitors and inductors using your oscilloscope and a simple pulse generator. There …
Measuring an impedance by the oscilloscope. RUC.dk
How to Measure a Capacitor with an Oscilloscope [Theory
10. David
3/07/2018 · If so, configure a 555 timer as a square wave oscillator including the capacitor of unknown capacitance, and measure frequency of oscillation. The frequency of oscillation depends inversely on capacitance of the capacitor.
Lab 3 – Finding the value of an unknown capacitor By Henry
A Simple Method to Measure Unknown Inductors
Last Revised on November 12 2018 Grade EXPERIMENT 10
11. Evan
26/06/2013 · Measuring capacitance with an oscilloscope. Arduino Forum > Using Arduino > General Electronics > Measuring In other words, 47 nF, which is in fact the capacitor I used. So far so good, the theory agrees with practice. And I’ve learnt a bit more about capacitors. Now the interesting thing is if we increase the frequency. Say, to 3 kHz. The original waveform (the 1 kHz one) is saved …
A Simple Method to Measure Unknown Inductors
12. Evan
The only oscilloscope I have is a tiny DSO138 (input section is on the top left of page 4 in the linked PDF). The scope has a limit of 100Vpp (actually displays only 80Vpp). At the highest attenuation (100x), there is 1V input to the first op-amp.
Tektronix 465 Oscilloscope – Power Supply Capacitor
13. Isaiah
Oscilloscope Measurement 1 Objectives This experiment deals with basic and advanced time domain measurement. The purpose of the laboratory is to develop a skill needed to use basic laboratory equipment. By using special measurement, the stu-dent will learn the capability, and limitation of time domain measurements, by oscilloscope. Upon completion of the experiment the student will: …
Measuring an impedance by the oscilloscope. RUC.dk
Oscilloscope and Capacitance SRJC
14. David
How to Measure a Capacitor with an Oscilloscope [Theory] Therefore, when trying to decipher the value of an unknown capacitor with an oscilloscope, use a known voltage for Vin, and a known resistor R. Now, determine the value of 63.2% of Vin, and let the capacitor charge through the resistor R, as in the schematic. Capture the charging waveform Vc on the oscilloscope and from the graph …
#135 Measure Capacitor ESR with an Oscilloscope and
How to measure value of Inductor or Capacitor using
15. Jesus
A voltage step will be seen on the measurement capacitor from charge transfer. This doesn’t affect timer operation adversely. It simply shows charge added and subtracted to compensate for the
Laboratory Exercise 6 – THE OSCILLOSCOPE QMplus
16. Natalie
− Capacitor as a charge storage device scope’s measurement tools. A digital oscilloscope samples input voltage signal at regular intervals using an analog-to-digital converter (ADC) PHYS 1493/1494/2699: Exp. 8 – Capacitance and the oscilloscope. 27 Oscilloscope The oscilloscope allows us to visualize signals that vary rapidly with time. Very handy! Idea: scope converts voltage
Measuring an impedance by the oscilloscope. RUC.dk
How to measure value of Inductor or Capacitor using
17. Gabrielle
Use the cursors on the oscilloscope screen to measure and compute the time constant. If you use the ‘‘%’’ option in a clever way, then you can get the scope to do all the computations for you in checking the time constant.
Measuring capacitance using oscilloscope. Page 1
If you don’t have an LCR meter in your lab, or you want to demonstrate the behavior of capacitors and inductors under sinusoidal stimulus, an oscilloscope and a function generator can help you to do a simple, transparent impedance measurement. You can expect capacitance and inductance values with 3%-5% uncertainty.
Oscilloscope and Capacitance SRJC
Measurement of the Time Constant in an RC Circuit
19. Samuel
Figure 3 – Use of oscilloscope for phase measurements . The CRO may be used to measure phase shift in an electronic circuit, as shown in Fig. 3.
Use Analog Techniques To Measure Capacitance In Capacitive
Capacitor Measurement using Oscilloscope Function
20. Jose
The voltage across the capacitor is measured with an oscilloscope. The function generator is set at the maximum output voltage, and the frequency is adjusted so that the voltage across the capacitor is kept at a low level. In this way almost the entirely voltage is dropped across the internal generator resistance. It is like the capacitor is connected to a current source. The current that
Circuit measures capacitance or inductance EDN
Measuring an impedance by the oscilloscope. RUC.dk
21. Destiny
Connect your oscilloscope to measure the voltage across the capacitor. See figure 3. Note that the capacitor should connect to ground, not the resistor. Think about this detail when you make your measurements. If you measure incorrectly, you can ground both sides of the capacitor, in effect removing it from the circuit.
How Oscilloscope Probes Affect Your Measurement
22. Justin
The Two Faces of Q Wes Hayward, w7zoi, November, 2010. Updates: 14Dec10, 29Dec10, 2Jan11. April 14, 2015. See addendum at document end. Abstract Most home-lab measurements of Q only evaluate an LC resonator. We then tend to associate the resulting Q with inductor loss while capacitor Q is assumed to be quite high. This assumption is now in greater doubt, especially with SMT …
Experiment 8 Capacitance and the Oscilloscope
Oscilloscope Measurement Lab Measuring Impedance and
23. Emma
You can also use an oscilloscope’s integration function to measure hot-swap load capacitance, provided the resistive load current is small during startup. Capacitance is the amount of charge stored per volt applied to a capacitor, and charge is
Measuring the Value of a Capacitor science experiment
24. Jose
An oscilloscope is a very powerful instrument that allows you to measure almost all aspects of an electrical signal. You have used an oscilloscope in the introductory physics labs (hopefully!) and should already be familiar with some of the basics. The scopes you have used previously have probably been analog scopes; a stream of electrons is produced at the rear of the tube, deflected
Last Revised on November 12 2018 Grade EXPERIMENT 10
Probe Considerations for Low Voltage Measurements such as
How to measure value of Inductor or Capacitor using
25. Jesus
Use “Measure” on the oscilloscope to find the peak-to-peak voltage of the capacitor and the peak-to-peak voltage of the resistor. If it is not giving a value, go to Trig
Oscilloscope Measurement Lab Measuring Impedance and
Use Analog Techniques To Measure Capacitance In Capacitive
26. Cameron
Use CH1 of the oscilloscope to measure the generator output V and CH2 to measure the voltage vacross capacitor C (see previous diagram). Make sure that the probe earths for
Measuring capacitance using oscilloscope. Page 1
Tektronix 465 Oscilloscope – Power Supply Capacitor
27. Bryan
The Two Faces of Q Wes Hayward, w7zoi, November, 2010. Updates: 14Dec10, 29Dec10, 2Jan11. April 14, 2015. See addendum at document end. Abstract Most home-lab measurements of Q only evaluate an LC resonator. We then tend to associate the resulting Q with inductor loss while capacitor Q is assumed to be quite high. This assumption is now in greater doubt, especially with SMT …
How to Measure a Capacitor with an Oscilloscope [Theory
#135 Measure Capacitor ESR with an Oscilloscope and
28. Victoria
Using only op amps, resistors, and a normal lab oscilloscope, how can you make a precise measurement of the capacitance of an unknown capacitor? This is a bonus question on my problem set but it sounds crazy.
How to Measure a Capacitor with an Oscilloscope [Theory
#135 Measure Capacitor ESR with an Oscilloscope and
Circuit measures capacitance or inductance EDN
29. Jayden
Using only op amps, resistors, and a normal lab oscilloscope, how can you make a precise measurement of the capacitance of an unknown capacitor? This is a bonus question on my problem set but it sounds crazy.
Lab 3 – AC Circuit Tools National Instruments
Measuring capacitance with an oscilloscope Arduino Forum
30. Justin
Figur 1 Voltage divider for impedance measurement. One should of course choose a range for U(t) and V(t) respectively in order the signals fill out the screen without being “cut”.
Probe Considerations for Low Voltage Measurements such as
Circuit measures capacitance or inductance EDN
31. Jordan
Name. OSCILLOSCOPE AND RC-CIRCUITS Physics 230, Lab 5 Partner Objective The first part of this set of activities will acquaint you with the kinds of measurements you can make with an oscilloscope and give you a chance to become comfortable with the buttons and knobs on the scope.
Measuring an impedance by the oscilloscope. RUC.dk
E 3e “Measurements with an oscilloscope
Experiment 8 Capacitance and the Oscilloscope
32. Michael
Experiments with a capacitor Introduction. A storage oscilloscope enables the voltage/time graph for a capacitor charging through a resistor to be displayed and, from the print-out, a value of the time constant for the circuit to be calculated.
Oscilloscope and RC circuits Capacitor Physical Quantities
Probe Considerations for Low Voltage Measurements such as
E 3e “Measurements with an oscilloscope
33. Connor
26/06/2013 · Measuring capacitance with an oscilloscope. Arduino Forum > Using Arduino > General Electronics > Measuring In other words, 47 nF, which is in fact the capacitor I used. So far so good, the theory agrees with practice. And I’ve learnt a bit more about capacitors. Now the interesting thing is if we increase the frequency. Say, to 3 kHz. The original waveform (the 1 kHz one) is saved …
E 3e “Measurements with an oscilloscope
Measuring capacitance with an oscilloscope Arduino Forum
#135 Measure Capacitor ESR with an Oscilloscope and
34. Alex
3/05/2013 · This video shows how to measure the value of unknown capacitors and inductors using your oscilloscope and a simple pulse generator. There …
Experiment 8 Capacitance and the Oscilloscope
A Simple Method to Measure Unknown Inductors
35. Morgan
The only oscilloscope I have is a tiny DSO138 (input section is on the top left of page 4 in the linked PDF). The scope has a limit of 100Vpp (actually displays only 80Vpp). At the highest attenuation (100x), there is 1V input to the first op-amp.
Measuring capacitance using oscilloscope. Page 1
36. Steven
The only oscilloscope I have is a tiny DSO138 (input section is on the top left of page 4 in the linked PDF). The scope has a limit of 100Vpp (actually displays only 80Vpp). At the highest attenuation (100x), there is 1V input to the first op-amp.
Oscilloscope and Capacitance SRJC
37. Mary
An oscilloscope is a piece of electronic test equipment that graphically displays waveforms and other changing voltages over time. Scientists and technicians use them to measure …
Capacitance & Inductance Measurements Using an
#135 Measure Capacitor ESR with an Oscilloscope and
38. Sofia
A Simple Method to Measure Unknown Inductors A simple and quick way to measure the inductance of an unknown power inductor (provided you have a function generator and oscilloscope).
Lab 3 – Finding the value of an unknown capacitor By Henry
Tektronix 465 Oscilloscope – Power Supply Capacitor
LAB 4a THE OSCILLOSCOPE LAB 4b ELECTROCARDIOGRAMS (ECG
39. Amia
The Two Faces of Q Wes Hayward, w7zoi, November, 2010. Updates: 14Dec10, 29Dec10, 2Jan11. April 14, 2015. See addendum at document end. Abstract Most home-lab measurements of Q only evaluate an LC resonator. We then tend to associate the resulting Q with inductor loss while capacitor Q is assumed to be quite high. This assumption is now in greater doubt, especially with SMT …
Tektronix 465 Oscilloscope – Power Supply Capacitor
Measuring an impedance by the oscilloscope. RUC.dk
A Simple Method to Measure Unknown Inductors
40. Jonathan
In this section, the oscilloscope will be used to measure the time constant of RC circuits; circuits containing resistors and capacitors are used in series and parallel arrangement to measure …
LAB 4a THE OSCILLOSCOPE LAB 4b ELECTROCARDIOGRAMS (ECG
41. Kylie
Use “Measure” on the oscilloscope to find the peak-to-peak voltage of the capacitor and the peak-to-peak voltage of the resistor. If it is not giving a value, go to Trig
How to Measure a Capacitor with an Oscilloscope [Theory
Lab 3 – AC Circuit Tools National Instruments
42. Amia
This lab exercise demonstrates a technique for using an oscilloscope and function generator to measure impedance and capacitance. Voltage waveforms are measured on the oscilloscope and the measurements are used to calculate an unknown capacitance.
How Oscilloscope Probes Affect Your Measurement
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CALCULATING SIMPLE INTEREST LESSON PLAN TEACHING MONEY CONCEPT LEARNING FINANCE ECONOMIC PRINCIPLES BUSINESS MATH ACTIVITY SAMPLE EXAMPLE HELP STUDENTS PRACTICE TEACHER FINANCE SKILLS ONLINE COURSE
Students learn how to calculate simple interest.
SIMPLE INTEREST
Concept: Understand what simple interest is and how it is calculated.
Objectives:
1. Know and use simple interest terminology
2. Understand when interest is paid
3. Understand when interest is earned
4. Know and use the formula for calculating simple interest
5. Know and use the formula for calculating the total amount of a loan or total value of an investment at the end of a specified term
6. Understand how to algebraically manipulate the interest formulas to solve for different variables
CALCULATING SIMPLE INTEREST LESSON PLAN
Teaching Materials
• Lesson - Simple Interest (see below for printable lesson)
• Many pre-calculated interest examples
Lesson Activity
1. Ask class what interest is
2. Note how many respond with paid interest versus earned interest
3. Define it first as the amount charged to borrow money
4. Discuss specific examples: car loans, mortgage loans, vacation loans, etc..
5. Discuss when simple interest might be used (i.e. short term) and explain that it is a very simple way to calculate interest and mention that banks and other lending institutions usually use compound interest which will be discussed later.
6. Discuss the three elements that are involved (principle, rate, term)
7. Define each term
8. Introduce the interest formula: I=Prt
9. Work through examples
10. Talk about total amount due: A = P + I
11. Introduce the total amount formula: A = P * (1+ rt)
12. Work through examples
13. Introduce earned investment and define the terms accordingly
14. Work through examples
15. Present a problem with a variable other than interest – ask class to think of ways to solve
16. Discuss the interest table
17. Present the different algebraic formulas and work through how they are derived
18. Work through examples
Assessment/Evaluation
Have students complete worksheet. A passing grade is 70% or greater.
Print out the teaching lesson pages and exercise worksheets for use with this lesson:
SIMPLE INTEREST LESSON Lesson and worksheet. Do you have a recommendation for an enhancement to this business math lesson, or do you have an idea for a new lesson? Then please leave us a suggestion. Back to more Banking and Saving Lessons Back to more Basic Finance and Economics Lessons, Lesson Plans, and Worksheets For teaching and learning general money skills, personal finance, and money management, please go to the Money Instructor home page.
Teaching Calculating Simple Interest Figure Calculator Business Math and Banking Financial Skills Lesson Plan - High School Student Secondary Education Adults College Teens Teenagers Free Instruct School Young Adults Classroom Review Activities Consumer Economics 101 Finance Education 7th 8th 9th 10th 11th 12th grade
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https://math.answers.com/math-and-arithmetic/How_many_gallons_water_15_by_30_foot_pool_6_feet_deep
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0
# How many gallons water 15 by 30 foot pool 6 feet deep?
Updated: 12/11/2022
Wiki User
14y ago
Based on a rectangle. approx 20,250 gallons
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1,770 gallons.
26,550 gallons.
2,360 gallons.
9,440 gallons.
### Convert 4ft column of water 1ft deep to gallons?
A 4-foot column of water that is 1-foot deep would contain 4 cubic feet of water. To convert this to gallons, you would multiply by 7.48 (since 1 cubic foot is approximately 7.48 gallons). Therefore, a 4-foot column of water 1-foot deep would be approximately 29.92 gallons.
### How many gallons of water could be in an acre?
1 acre = 43560 sq.feet. 1 foot deep water over an acre is 43560 cubic feet, which is 325851.4 gallons in 1 foot deep. If it is 1 inch deep (1/12 foot) then it is 27154.3 gallons in 1 inch deep.
7.5
### How many gallons of water does it take to fill a ten foot round pool twelve feet deep?
Twelve feet of water in a 10-foot round pool is approximately 7,080 gallons.
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# Merging two zeroes of a vector field on $\Bbb R^n$
Suppose we have a vector field v on $\Bbb R^n$ with exactly two isolated zeros (call them $p,q$) which are connected by a flow-line of the vector field. Furthermore, assume one can modify the vector field in a compact nbhd of the flow line and merge $p,q$ to give a new vector field w with a single isolated zero $r$. Show that the index of $r$ must be the sum of the indices of $p$ and $q$.
My approach: my first attempt would be to show that there is a relation between the vector field v and the vector field w. If we where working on a compact manifold I believe this exercise is an easy corollary of the Poincaré-Hopf theorem. But since we are working with a non-compact manifold, I can't use it here.
Here was asked the same question: Show that the index of $r$ must be the sum of the indices of $p$ and $q$.
but with no answer, and actually it's pointed out that the solution of this is quite complicated. It's mentioned that it's a consequence of the Boundary theorem but the only boundary theorem I know is about the incidence number of two sub-manifolds one of which bounds a mfld.
Let $S^{n-1}$ be the unit $n-1$ sphere centered on the origin. Let $S_p,S_q$ be a small $n-1$ spheres centered on $p,q$ respectively, so small $S_p,S_q$ are disjoint and each is contained in the outside of the other. Let $S_{pq}$ be a large $n-1$ sphere whose inside contains $S_p,S_q$. Choose orientations on all three spheres so that the normal direction points outward from the sphere.
Since each of these spheres is disjoint from the set of zeroes $\{p,q\}$, it follows that the vector field $\bf{v}$ defines continuous functions $f_p : S_p \to S^{n-1}$ and $f_q : S_q \to S^{n-1}$ and $f_{pq} : S_{pq} \to S^{n-1}$. These functions are all just restrictions of the same function $x \mapsto \frac{{\bf{v}}(x)}{|{\bf{v}}(x)|}$ for $x \in \mathbb{R}^n - \{p,q\}$.
By definition, the index of $\bf{v}$ at $p$ equals the degree of the map $f_p$, and the index of $\bf{v}$ at $q$ equals the degree of the map $f_q$.
Next, there is a "boundary theorem" to apply here, although the one you know might not be the same as the one we need. The boundary theorem one needs says that if you have a compact, oriented $n$-manifold with boundary $M$ --- in this case $M$ is the portion of $\mathbb{R}^n$ outside of $S_p$ and $S_q$ and inside of $S_{pq}$ --- and if the boundary $\partial M$ has the induced orientation with normal vector pointing outward from the manifold, and if you have a continuous function from $M$ to $S^{n-1}$ --- in this case, the restriction of the function $x \mapsto \frac{{\bf v}(x)}{|{\bf v}(x)|}$ --- then the sum of the degrees of the maps on the boundary components equals zero. It follows that degree of $f_{pq}$ equals the sum of the degrees of $f_p$ and $f_q$, because we compute the degree on $f_{pq}$ using the orientation induced from $M$ but we compute the degrees of $f_p$ and $f_q$ using the opposite orientation from the one induced from $M$. Thus the degree of $f_{pq}$ equals the sum of the indices of $\bf{v}$ at $p$ and $q$.
Now modify $\bf{v}$ as described to obtain $\bf{w}$. This "modification" is a homotopy from $\bf v$ to $\bf w$, and $S_{pq}$ is disjoint from the support of this homotopy, so the function $f_{pq}$ is unchanged under this homotopy. When the homotopy is complete, the point $r$ is inside the sphere $S_{pq}$, and so, again by definition, the index of $\bf w$ at $r$ is equal to the degree of $f_{pq}$.
Putting it all together, the index of $\bf{w}$ at $r$ equals the sum of the indices of $\bf{v}$ at $p$ and at $q$.
By the way, this point of view on vector fields is well explained in the book of Guilleman and Pollack, which I recommend for more details.
• I see, thanks for the careful description. I had in mind an approach similar to this one, but I was quite confused by the fact that they stressed the existence of this flow line connecting the two points. is there a reason for that? what if such line doesn't exists? – Luigi M Aug 22 '17 at 16:29
• All you need to make this argument work is a sphere such that the only zeroes of the vector field on or inside the sphere are $p$ and $q$. The sphere does not even have to be an ordinary round sphere, it can be just a smoothly embedded sphere in $\mathbb{R}^n$. The only role being played by the flow line connecting the points is that any neighborhood of that flow line connecting $p$ and $q$ contains an appropriate sphere of this type. But, you could carry that argument out with any embedded path connecting $p$ and $q$. – Lee Mosher Aug 22 '17 at 18:37
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# How to Measure a Watch Band without the Watch Band
Posted by Esslinger Staff
One of the most important things to do when you are planning to change your watch band is to measure the existing band (if possible) and find out the width and length. Without those two measurements, you will not be able to ensure that you have the right size watch band for your watch case. But what if you just have the watch case and the watch band itself is gone? Don’t worry, you can use this guide from Esslinger.com to help you successfully measure a for a watch band without actually having the watch band.
### Step 1
The first thing to do is find the width your new watch band needs to be. To find the width, you will take the watch case and measure the distance between two of the watch lugs. You can do this with either a ruler or a digital gauge, though the digital gauge will give you the most accurate results.
Hold the watch case in your hand so that the lugs are facing outward. Then take your digital gauge and place the backward opening upper jaws in between the watch case lugs. Open the gauge’s jaws until they fit snugly against the inside of the watch lugs. Record the millimeter measurement listed on the gauge’s display. If the measurement is not a whole number, round up if the decimal is five or above, and round down if the decimal is less than five.
-OR-
Using the ruler, take your watch case and line the inside of the watch lug up with the zero marker on the metric side of the ruler as most watch band widths are given in millimeters. When the watch lug is aligned, read the measurement on the ruler where the inside of the second lug lines up with the lugs. Estimate smaller if the edge is between two millimeter marks on the watch band.
### Step 2
After you have determined the width of the watch band, you need to determine what length will fit you. Take your piece of paper and cut a strip off that is about two inches wide and long enough to wrap around your wrist with extra paper at the end.
### Step 3
Set the strip of paper out flat before you. Lay your watch case in the center of the strip of paper and mark where the spring bars would sit with the pencil: You can do with by looking for the indentations in the watch lugs and marking the paper just where they are. Do this for both sets of watch lugs. Then, using your ruler, draw a straight line between the marks you made to indicate where the watch case would be.
### Step 4
Once you have done that, take the paper and center the section that would be the watch case on your wrist like you would do with a real watch. Tape one end of the paper to your wrist to hold both the paper to your wrist and the “watch case” centered on your arm.
### Step 5
Then, wrap the remaining loose end of the paper around your wrist and tape that end down to your paper watch. When the paper watch is secured, flip your arm so you palm is facing up and draw a line down the paper in the middle of your wrist to mark where the clasp would go.
### Step 6
Then, take your scissors and cut along the line you just drew on the paper watch band. Be sure to cut through both ends of the paper on your wrist. You may need a friend to help you cut through the paper band for you.
### Step 7
Remove the paper watch from your wrist and get rid of the scraps. Laying the fitted paper watch out flat before you. Now, cut the watch case portion of your paper strip out, leaving just the band portions in front of you.
### Step 8
Line up the ends of the two remaining portions of your paper watch, the parts the represent the watch band, so that they form one piece – you can tape them together if it is easier to hold them lined up that way. Place the ruler on top of the paper strips and line up the zero marker with one end of the paper watch band.
### Finished
Remember: When you go to order your replacement watch band, remember that all metal watch bands can be adjusted to some extent. You can remove links to make it smaller, and even adjust the pins in some clasps to make it larger.
Now, you have both the width and length measurements and can easily order a replacement band that is the perfect size for your watch.
Visit Esslinger.com’s Learning Center for more watch band repair guides.
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:: Free Universal Algebra Construction
:: by Beata Perkowska
::
:: Copyright (c) 1993-2021 Association of Mizar Users
definition
let IT be set ;
attr IT is disjoint_with_NAT means :Def1: :: FREEALG:def 1
IT misses NAT ;
end;
:: deftheorem Def1 defines disjoint_with_NAT FREEALG:def 1 :
for IT being set holds
( IT is disjoint_with_NAT iff IT misses NAT );
registration
existence
ex b1 being set st
( not b1 is empty & b1 is disjoint_with_NAT )
proof end;
end;
Lm1: ( not 0 in rng <*1*> & 0 in rng )
proof end;
notation
let IT be Relation;
antonym with_zero IT for non-empty ;
synonym without_zero IT for non-empty ;
end;
definition
let IT be Relation;
redefine attr IT is non-empty means :Def2: :: FREEALG:def 2
not 0 in rng IT;
compatibility
( not IT is with_zero iff not 0 in rng IT )
by RELAT_1:def 9;
end;
:: deftheorem Def2 defines with_zero FREEALG:def 2 :
for IT being Relation holds
( not IT is with_zero iff not 0 in rng IT );
registration
existence
ex b1 being FinSequence of NAT st
( not b1 is empty & b1 is with_zero )
proof end;
existence
ex b1 being FinSequence of NAT st
( not b1 is empty & b1 is without_zero )
proof end;
end;
::
:: Free Universal Algebra - General Notions
::
definition
let U1 be Universal_Algebra;
let n be Nat;
assume A1: n in dom the charact of U1 ;
func oper (n,U1) -> operation of U1 equals :Def3: :: FREEALG:def 3
the charact of U1 . n;
coherence
the charact of U1 . n is operation of U1
by ;
end;
:: deftheorem Def3 defines oper FREEALG:def 3 :
for U1 being Universal_Algebra
for n being Nat st n in dom the charact of U1 holds
oper (n,U1) = the charact of U1 . n;
definition
let U0 be Universal_Algebra;
mode GeneratorSet of U0 -> Subset of U0 means :: FREEALG:def 4
for A being Subset of U0 st A is opers_closed & it c= A holds
A = the carrier of U0;
existence
ex b1 being Subset of U0 st
for A being Subset of U0 st A is opers_closed & b1 c= A holds
A = the carrier of U0
proof end;
end;
:: deftheorem defines GeneratorSet FREEALG:def 4 :
for U0 being Universal_Algebra
for b2 being Subset of U0 holds
( b2 is GeneratorSet of U0 iff for A being Subset of U0 st A is opers_closed & b2 c= A holds
A = the carrier of U0 );
Lm2: for A being Universal_Algebra
for B being Subset of A st B is opers_closed holds
Constants A c= B
proof end;
Lm3: for A being Universal_Algebra
for B being Subset of A st ( B <> {} or Constants A <> {} ) holds
( B is GeneratorSet of A iff the carrier of () = the carrier of A )
proof end;
definition
let U0 be Universal_Algebra;
let IT be GeneratorSet of U0;
attr IT is free means :: FREEALG:def 5
for U1 being Universal_Algebra st U0,U1 are_similar holds
for f being Function of IT, the carrier of U1 ex h being Function of U0,U1 st
( h is_homomorphism & h | IT = f );
end;
:: deftheorem defines free FREEALG:def 5 :
for U0 being Universal_Algebra
for IT being GeneratorSet of U0 holds
( IT is free iff for U1 being Universal_Algebra st U0,U1 are_similar holds
for f being Function of IT, the carrier of U1 ex h being Function of U0,U1 st
( h is_homomorphism & h | IT = f ) );
definition
let IT be Universal_Algebra;
attr IT is free means :Def6: :: FREEALG:def 6
ex G being GeneratorSet of IT st G is free ;
end;
:: deftheorem Def6 defines free FREEALG:def 6 :
for IT being Universal_Algebra holds
( IT is free iff ex G being GeneratorSet of IT st G is free );
registration
existence
ex b1 being Universal_Algebra st
( b1 is free & b1 is strict )
proof end;
end;
registration
let U0 be free Universal_Algebra;
cluster free for GeneratorSet of U0;
existence
ex b1 being GeneratorSet of U0 st b1 is free
by Def6;
end;
theorem :: FREEALG:1
for U0 being strict Universal_Algebra
for A being Subset of U0 st ( Constants U0 <> {} or A <> {} ) holds
( A is GeneratorSet of U0 iff GenUnivAlg A = U0 )
proof end;
::
::
definition
let f be non empty FinSequence of NAT ;
let X be set ;
func REL (f,X) -> Relation of ((dom f) \/ X),(((dom f) \/ X) *) means :Def7: :: FREEALG:def 7
for a being Element of (dom f) \/ X
for b being Element of ((dom f) \/ X) * holds
( [a,b] in it iff ( a in dom f & f . a = len b ) );
existence
ex b1 being Relation of ((dom f) \/ X),(((dom f) \/ X) *) st
for a being Element of (dom f) \/ X
for b being Element of ((dom f) \/ X) * holds
( [a,b] in b1 iff ( a in dom f & f . a = len b ) )
proof end;
uniqueness
for b1, b2 being Relation of ((dom f) \/ X),(((dom f) \/ X) *) st ( for a being Element of (dom f) \/ X
for b being Element of ((dom f) \/ X) * holds
( [a,b] in b1 iff ( a in dom f & f . a = len b ) ) ) & ( for a being Element of (dom f) \/ X
for b being Element of ((dom f) \/ X) * holds
( [a,b] in b2 iff ( a in dom f & f . a = len b ) ) ) holds
b1 = b2
proof end;
end;
:: deftheorem Def7 defines REL FREEALG:def 7 :
for f being non empty FinSequence of NAT
for X being set
for b3 being Relation of ((dom f) \/ X),(((dom f) \/ X) *) holds
( b3 = REL (f,X) iff for a being Element of (dom f) \/ X
for b being Element of ((dom f) \/ X) * holds
( [a,b] in b3 iff ( a in dom f & f . a = len b ) ) );
definition
let f be non empty FinSequence of NAT ;
let X be set ;
func DTConUA (f,X) -> strict DTConstrStr equals :: FREEALG:def 8
DTConstrStr(# ((dom f) \/ X),(REL (f,X)) #);
correctness
coherence
DTConstrStr(# ((dom f) \/ X),(REL (f,X)) #) is strict DTConstrStr
;
;
end;
:: deftheorem defines DTConUA FREEALG:def 8 :
for f being non empty FinSequence of NAT
for X being set holds DTConUA (f,X) = DTConstrStr(# ((dom f) \/ X),(REL (f,X)) #);
registration
let f be non empty FinSequence of NAT ;
let X be set ;
cluster DTConUA (f,X) -> non empty strict ;
coherence
not DTConUA (f,X) is empty
;
end;
theorem Th2: :: FREEALG:2
for f being non empty FinSequence of NAT
for X being set holds
( Terminals (DTConUA (f,X)) c= X & NonTerminals (DTConUA (f,X)) = dom f )
proof end;
theorem Th3: :: FREEALG:3
for f being non empty FinSequence of NAT
for X being disjoint_with_NAT set holds Terminals (DTConUA (f,X)) = X
proof end;
registration
let f be non empty FinSequence of NAT ;
let X be set ;
coherence
DTConUA (f,X) is with_nonterminals
by Th2;
end;
registration
let f be non empty with_zero FinSequence of NAT ;
let X be set ;
coherence
( DTConUA (f,X) is with_nonterminals & DTConUA (f,X) is with_useful_nonterminals )
proof end;
end;
registration
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
coherence
( DTConUA (f,D) is with_terminals & DTConUA (f,D) is with_nonterminals & DTConUA (f,D) is with_useful_nonterminals )
proof end;
end;
definition
let f be non empty FinSequence of NAT ;
let X be set ;
let n be Nat;
assume A1: n in dom f ;
func Sym (n,f,X) -> Symbol of (DTConUA (f,X)) equals :Def9: :: FREEALG:def 9
n;
coherence
n is Symbol of (DTConUA (f,X))
by ;
end;
:: deftheorem Def9 defines Sym FREEALG:def 9 :
for f being non empty FinSequence of NAT
for X being set
for n being Nat st n in dom f holds
Sym (n,f,X) = n;
::
:: Construction of Free Universal Algebra for Non Empty Set of Generators and
:: Given Signature
definition
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
let n be Nat;
assume A1: n in dom f ;
func FreeOpNSG (n,f,D) -> non empty homogeneous quasi_total PartFunc of ((TS (DTConUA (f,D))) *),(TS (DTConUA (f,D))) means :Def10: :: FREEALG:def 10
( dom it = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom it holds
it . p = (Sym (n,f,D)) -tree p ) );
existence
ex b1 being non empty homogeneous quasi_total PartFunc of ((TS (DTConUA (f,D))) *),(TS (DTConUA (f,D))) st
( dom b1 = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom b1 holds
b1 . p = (Sym (n,f,D)) -tree p ) )
proof end;
uniqueness
for b1, b2 being non empty homogeneous quasi_total PartFunc of ((TS (DTConUA (f,D))) *),(TS (DTConUA (f,D))) st dom b1 = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom b1 holds
b1 . p = (Sym (n,f,D)) -tree p ) & dom b2 = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom b2 holds
b2 . p = (Sym (n,f,D)) -tree p ) holds
b1 = b2
proof end;
end;
:: deftheorem Def10 defines FreeOpNSG FREEALG:def 10 :
for f being non empty FinSequence of NAT
for D being non empty disjoint_with_NAT set
for n being Nat st n in dom f holds
for b4 being non empty homogeneous quasi_total PartFunc of ((TS (DTConUA (f,D))) *),(TS (DTConUA (f,D))) holds
( b4 = FreeOpNSG (n,f,D) iff ( dom b4 = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom b4 holds
b4 . p = (Sym (n,f,D)) -tree p ) ) );
definition
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
func FreeOpSeqNSG (f,D) -> PFuncFinSequence of (TS (DTConUA (f,D))) means :Def11: :: FREEALG:def 11
( len it = len f & ( for n being Nat st n in dom it holds
it . n = FreeOpNSG (n,f,D) ) );
existence
ex b1 being PFuncFinSequence of (TS (DTConUA (f,D))) st
( len b1 = len f & ( for n being Nat st n in dom b1 holds
b1 . n = FreeOpNSG (n,f,D) ) )
proof end;
uniqueness
for b1, b2 being PFuncFinSequence of (TS (DTConUA (f,D))) st len b1 = len f & ( for n being Nat st n in dom b1 holds
b1 . n = FreeOpNSG (n,f,D) ) & len b2 = len f & ( for n being Nat st n in dom b2 holds
b2 . n = FreeOpNSG (n,f,D) ) holds
b1 = b2
proof end;
end;
:: deftheorem Def11 defines FreeOpSeqNSG FREEALG:def 11 :
for f being non empty FinSequence of NAT
for D being non empty disjoint_with_NAT set
for b3 being PFuncFinSequence of (TS (DTConUA (f,D))) holds
( b3 = FreeOpSeqNSG (f,D) iff ( len b3 = len f & ( for n being Nat st n in dom b3 holds
b3 . n = FreeOpNSG (n,f,D) ) ) );
definition
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
func FreeUnivAlgNSG (f,D) -> strict Universal_Algebra equals :: FREEALG:def 12
UAStr(# (TS (DTConUA (f,D))),(FreeOpSeqNSG (f,D)) #);
coherence
UAStr(# (TS (DTConUA (f,D))),(FreeOpSeqNSG (f,D)) #) is strict Universal_Algebra
proof end;
end;
:: deftheorem defines FreeUnivAlgNSG FREEALG:def 12 :
for f being non empty FinSequence of NAT
for D being non empty disjoint_with_NAT set holds FreeUnivAlgNSG (f,D) = UAStr(# (TS (DTConUA (f,D))),(FreeOpSeqNSG (f,D)) #);
theorem Th4: :: FREEALG:4
for f being non empty FinSequence of NAT
for D being non empty disjoint_with_NAT set holds signature (FreeUnivAlgNSG (f,D)) = f
proof end;
definition
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
func FreeGenSetNSG (f,D) -> Subset of (FreeUnivAlgNSG (f,D)) equals :: FREEALG:def 13
{ () where s is Symbol of (DTConUA (f,D)) : s in Terminals (DTConUA (f,D)) } ;
coherence
{ () where s is Symbol of (DTConUA (f,D)) : s in Terminals (DTConUA (f,D)) } is Subset of (FreeUnivAlgNSG (f,D))
proof end;
end;
:: deftheorem defines FreeGenSetNSG FREEALG:def 13 :
for f being non empty FinSequence of NAT
for D being non empty disjoint_with_NAT set holds FreeGenSetNSG (f,D) = { () where s is Symbol of (DTConUA (f,D)) : s in Terminals (DTConUA (f,D)) } ;
theorem Th5: :: FREEALG:5
for f being non empty FinSequence of NAT
for D being non empty disjoint_with_NAT set holds not FreeGenSetNSG (f,D) is empty
proof end;
definition
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
:: original: FreeGenSetNSG
redefine func FreeGenSetNSG (f,D) -> GeneratorSet of FreeUnivAlgNSG (f,D);
coherence
FreeGenSetNSG (f,D) is GeneratorSet of FreeUnivAlgNSG (f,D)
proof end;
end;
definition
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
let C be non empty set ;
let s be Symbol of (DTConUA (f,D));
let F be Function of (FreeGenSetNSG (f,D)),C;
assume A1: s in Terminals (DTConUA (f,D)) ;
func pi (F,s) -> Element of C equals :Def14: :: FREEALG:def 14
F . ();
coherence
F . () is Element of C
proof end;
end;
:: deftheorem Def14 defines pi FREEALG:def 14 :
for f being non empty FinSequence of NAT
for D being non empty disjoint_with_NAT set
for C being non empty set
for s being Symbol of (DTConUA (f,D))
for F being Function of (FreeGenSetNSG (f,D)),C st s in Terminals (DTConUA (f,D)) holds
pi (F,s) = F . ();
definition
let f be non empty FinSequence of NAT ;
let D be set ;
let s be Symbol of (DTConUA (f,D));
given p being FinSequence such that A1: s ==> p ;
func @ s -> Nat equals :Def15: :: FREEALG:def 15
s;
coherence
s is Nat
proof end;
end;
:: deftheorem Def15 defines @ FREEALG:def 15 :
for f being non empty FinSequence of NAT
for D being set
for s being Symbol of (DTConUA (f,D)) st ex p being FinSequence st s ==> p holds
@ s = s;
theorem Th6: :: FREEALG:6
for f being non empty FinSequence of NAT
for D being non empty disjoint_with_NAT set holds FreeGenSetNSG (f,D) is free
proof end;
registration
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
cluster FreeUnivAlgNSG (f,D) -> strict free ;
coherence
FreeUnivAlgNSG (f,D) is free
proof end;
end;
definition
let f be non empty FinSequence of NAT ;
let D be non empty disjoint_with_NAT set ;
:: original: FreeGenSetNSG
redefine func FreeGenSetNSG (f,D) -> free GeneratorSet of FreeUnivAlgNSG (f,D);
coherence
FreeGenSetNSG (f,D) is free GeneratorSet of FreeUnivAlgNSG (f,D)
by Th6;
end;
::
:: Construction of Free Universal Algebra for Given Signature
:: (with at Last One Zero Argument Operation) and Set of Generators
::
definition
let f be non empty with_zero FinSequence of NAT ;
let D be disjoint_with_NAT set ;
let n be Nat;
assume A1: n in dom f ;
func FreeOpZAO (n,f,D) -> non empty homogeneous quasi_total PartFunc of ((TS (DTConUA (f,D))) *),(TS (DTConUA (f,D))) means :Def16: :: FREEALG:def 16
( dom it = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom it holds
it . p = (Sym (n,f,D)) -tree p ) );
existence
ex b1 being non empty homogeneous quasi_total PartFunc of ((TS (DTConUA (f,D))) *),(TS (DTConUA (f,D))) st
( dom b1 = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom b1 holds
b1 . p = (Sym (n,f,D)) -tree p ) )
proof end;
uniqueness
for b1, b2 being non empty homogeneous quasi_total PartFunc of ((TS (DTConUA (f,D))) *),(TS (DTConUA (f,D))) st dom b1 = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom b1 holds
b1 . p = (Sym (n,f,D)) -tree p ) & dom b2 = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom b2 holds
b2 . p = (Sym (n,f,D)) -tree p ) holds
b1 = b2
proof end;
end;
:: deftheorem Def16 defines FreeOpZAO FREEALG:def 16 :
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set
for n being Nat st n in dom f holds
for b4 being non empty homogeneous quasi_total PartFunc of ((TS (DTConUA (f,D))) *),(TS (DTConUA (f,D))) holds
( b4 = FreeOpZAO (n,f,D) iff ( dom b4 = (f /. n) -tuples_on (TS (DTConUA (f,D))) & ( for p being FinSequence of TS (DTConUA (f,D)) st p in dom b4 holds
b4 . p = (Sym (n,f,D)) -tree p ) ) );
definition
let f be non empty with_zero FinSequence of NAT ;
let D be disjoint_with_NAT set ;
func FreeOpSeqZAO (f,D) -> PFuncFinSequence of (TS (DTConUA (f,D))) means :Def17: :: FREEALG:def 17
( len it = len f & ( for n being Nat st n in dom it holds
it . n = FreeOpZAO (n,f,D) ) );
existence
ex b1 being PFuncFinSequence of (TS (DTConUA (f,D))) st
( len b1 = len f & ( for n being Nat st n in dom b1 holds
b1 . n = FreeOpZAO (n,f,D) ) )
proof end;
uniqueness
for b1, b2 being PFuncFinSequence of (TS (DTConUA (f,D))) st len b1 = len f & ( for n being Nat st n in dom b1 holds
b1 . n = FreeOpZAO (n,f,D) ) & len b2 = len f & ( for n being Nat st n in dom b2 holds
b2 . n = FreeOpZAO (n,f,D) ) holds
b1 = b2
proof end;
end;
:: deftheorem Def17 defines FreeOpSeqZAO FREEALG:def 17 :
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set
for b3 being PFuncFinSequence of (TS (DTConUA (f,D))) holds
( b3 = FreeOpSeqZAO (f,D) iff ( len b3 = len f & ( for n being Nat st n in dom b3 holds
b3 . n = FreeOpZAO (n,f,D) ) ) );
definition
let f be non empty with_zero FinSequence of NAT ;
let D be disjoint_with_NAT set ;
func FreeUnivAlgZAO (f,D) -> strict Universal_Algebra equals :: FREEALG:def 18
UAStr(# (TS (DTConUA (f,D))),(FreeOpSeqZAO (f,D)) #);
coherence
UAStr(# (TS (DTConUA (f,D))),(FreeOpSeqZAO (f,D)) #) is strict Universal_Algebra
proof end;
end;
:: deftheorem defines FreeUnivAlgZAO FREEALG:def 18 :
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set holds FreeUnivAlgZAO (f,D) = UAStr(# (TS (DTConUA (f,D))),(FreeOpSeqZAO (f,D)) #);
theorem Th7: :: FREEALG:7
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set holds signature (FreeUnivAlgZAO (f,D)) = f
proof end;
theorem Th8: :: FREEALG:8
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set holds FreeUnivAlgZAO (f,D) is with_const_op
proof end;
theorem Th9: :: FREEALG:9
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set holds Constants (FreeUnivAlgZAO (f,D)) <> {}
proof end;
definition
let f be non empty with_zero FinSequence of NAT ;
let D be disjoint_with_NAT set ;
func FreeGenSetZAO (f,D) -> Subset of (FreeUnivAlgZAO (f,D)) equals :: FREEALG:def 19
{ () where s is Symbol of (DTConUA (f,D)) : s in Terminals (DTConUA (f,D)) } ;
coherence
{ () where s is Symbol of (DTConUA (f,D)) : s in Terminals (DTConUA (f,D)) } is Subset of (FreeUnivAlgZAO (f,D))
proof end;
end;
:: deftheorem defines FreeGenSetZAO FREEALG:def 19 :
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set holds FreeGenSetZAO (f,D) = { () where s is Symbol of (DTConUA (f,D)) : s in Terminals (DTConUA (f,D)) } ;
definition
let f be non empty with_zero FinSequence of NAT ;
let D be disjoint_with_NAT set ;
:: original: FreeGenSetZAO
redefine func FreeGenSetZAO (f,D) -> GeneratorSet of FreeUnivAlgZAO (f,D);
coherence
FreeGenSetZAO (f,D) is GeneratorSet of FreeUnivAlgZAO (f,D)
proof end;
end;
definition
let f be non empty with_zero FinSequence of NAT ;
let D be disjoint_with_NAT set ;
let C be non empty set ;
let s be Symbol of (DTConUA (f,D));
let F be Function of (FreeGenSetZAO (f,D)),C;
assume A1: s in Terminals (DTConUA (f,D)) ;
func pi (F,s) -> Element of C equals :Def20: :: FREEALG:def 20
F . ();
coherence
F . () is Element of C
proof end;
end;
:: deftheorem Def20 defines pi FREEALG:def 20 :
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set
for C being non empty set
for s being Symbol of (DTConUA (f,D))
for F being Function of (FreeGenSetZAO (f,D)),C st s in Terminals (DTConUA (f,D)) holds
pi (F,s) = F . ();
theorem Th10: :: FREEALG:10
for f being non empty with_zero FinSequence of NAT
for D being disjoint_with_NAT set holds FreeGenSetZAO (f,D) is free
proof end;
registration
let f be non empty with_zero FinSequence of NAT ;
let D be disjoint_with_NAT set ;
cluster FreeUnivAlgZAO (f,D) -> strict free ;
coherence
FreeUnivAlgZAO (f,D) is free
proof end;
end;
definition
let f be non empty with_zero FinSequence of NAT ;
let D be disjoint_with_NAT set ;
:: original: FreeGenSetZAO
redefine func FreeGenSetZAO (f,D) -> free GeneratorSet of FreeUnivAlgZAO (f,D);
coherence
FreeGenSetZAO (f,D) is free GeneratorSet of FreeUnivAlgZAO (f,D)
by Th10;
end;
registration
existence
ex b1 being Universal_Algebra st
( b1 is strict & b1 is free & b1 is with_const_op )
proof end;
end;
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http://de.metamath.org/mpeuni/nzin.html
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Mathbox for Steve Rodriguez < Previous Next > Nearby theorems Mirrors > Home > MPE Home > Th. List > Mathboxes > nzin Structured version Visualization version GIF version
Theorem nzin 37539
Description: The intersection of the set of multiples of m, mℤ, and those of n, nℤ, is the set of multiples of their least common multiple. Roughly Lemma 2.1(c) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5 and Problem 1(b) of https://people.math.binghamton.edu/mazur/teach/40107/40107h16sol.pdf p. 1, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Hypotheses
Ref Expression
nzin.m (𝜑𝑀 ∈ ℤ)
nzin.n (𝜑𝑁 ∈ ℤ)
Assertion
Ref Expression
nzin (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))
Proof of Theorem nzin
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 dvdszrcl 14826 . . . . . . . . 9 (𝑀𝑛 → (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ))
2 dvdszrcl 14826 . . . . . . . . 9 (𝑁𝑛 → (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ))
31, 2anim12i 588 . . . . . . . 8 ((𝑀𝑛𝑁𝑛) → ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ)))
4 anandir 868 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ) ↔ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ)))
53, 4sylibr 223 . . . . . . 7 ((𝑀𝑛𝑁𝑛) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ))
65ancomd 466 . . . . . 6 ((𝑀𝑛𝑁𝑛) → (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
7 lcmdvds 15159 . . . . . . 7 ((𝑛 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
873expb 1258 . . . . . 6 ((𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
96, 8mpcom 37 . . . . 5 ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛)
10 elin 3758 . . . . . 6 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})))
11 reldvds 37536 . . . . . . . 8 Rel ∥
12 elrelimasn 5408 . . . . . . . 8 (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀𝑛))
1311, 12ax-mp 5 . . . . . . 7 (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀𝑛)
14 elrelimasn 5408 . . . . . . . 8 (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁𝑛))
1511, 14ax-mp 5 . . . . . . 7 (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁𝑛)
1613, 15anbi12i 729 . . . . . 6 ((𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})) ↔ (𝑀𝑛𝑁𝑛))
1710, 16bitri 263 . . . . 5 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑀𝑛𝑁𝑛))
18 elrelimasn 5408 . . . . . 6 (Rel ∥ → (𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}) ↔ (𝑀 lcm 𝑁) ∥ 𝑛))
1911, 18ax-mp 5 . . . . 5 (𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}) ↔ (𝑀 lcm 𝑁) ∥ 𝑛)
209, 17, 193imtr4i 280 . . . 4 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) → 𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}))
2120ssriv 3572 . . 3 (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ⊆ ( ∥ “ {(𝑀 lcm 𝑁)})
2221a1i 11 . 2 (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ⊆ ( ∥ “ {(𝑀 lcm 𝑁)}))
23 nzin.m . . . . . 6 (𝜑𝑀 ∈ ℤ)
24 nzin.n . . . . . 6 (𝜑𝑁 ∈ ℤ)
25 dvdslcm 15149 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
2623, 24, 25syl2anc 691 . . . . 5 (𝜑 → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
2726simpld 474 . . . 4 (𝜑𝑀 ∥ (𝑀 lcm 𝑁))
28 lcmcl 15152 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0)
2923, 24, 28syl2anc 691 . . . . . 6 (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0)
3029nn0zd 11356 . . . . 5 (𝜑 → (𝑀 lcm 𝑁) ∈ ℤ)
3130, 23nzss 37538 . . . 4 (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}) ↔ 𝑀 ∥ (𝑀 lcm 𝑁)))
3227, 31mpbird 246 . . 3 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}))
3326simprd 478 . . . 4 (𝜑𝑁 ∥ (𝑀 lcm 𝑁))
3430, 24nzss 37538 . . . 4 (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ (𝑀 lcm 𝑁)))
3533, 34mpbird 246 . . 3 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}))
3632, 35ssind 3799 . 2 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})))
3722, 36eqssd 3585 1 (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))
Colors of variables: wff setvar class Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ⊆ wss 3540 {csn 4125 class class class wbr 4583 “ cima 5041 Rel wrel 5043 (class class class)co 6549 ℕ0cn0 11169 ℤcz 11254 ∥ cdvds 14821 lcm clcm 15139 This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-lcm 15141 This theorem is referenced by: nzprmdif 37540
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https://community.nintex.com/t5/Nintex-for-SharePoint/Calculating-the-sum-total-of-multiple-text-controls/m-p/72282
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Nintex Newbie
## Calculating the sum total of multiple text controls
Hi,
I have a form with a few Text controls (data type: decimal) and Calculated Value controls that totals the contents of those text fields that looks like this:
NOTE: The text controls are named 1, 2, 3, 4, 5 as specified by their corresponding labels.
NOTE: The calculated values contain the calculations as specified in their corresponding labels e.g. 1+2+3+4+5 or sum(1+2+3+4+5), etc
I've used 4 calculated values for demonstration purposes where the first calculated value is what I assume to be the correct syntax and works correctly. The remaining 3 are deliberately incorrect.
Now if I delete a value from one of the fields the formula breaks:
The first two values are totaled but the null value then breaks the sequence and the subsequent values are concatenated.
At the moment I can work around this by using a zero instead of null but I would like to know if I'm doing something obviously wrong with the calculation.
Thanks!
Labels: (3)
• ### Workflow
9 Replies
Highlighted
Automation Master
## Re: Calculating the sum total of multiple text controls
Hi,
if you use the following formula you can solve your issue:
sum([1,2,3,4,5])
it's similar to your fouth test but it has an array as argument (if you check the description of the sum function, you can see that it asks for a set of values inside brackets).
Giacomo
Highlighted
Nintex Newbie
## Re: Calculating the sum total of multiple text controls
Perfecto! Thanks Giacomo!
Highlighted
Nintex Newbie
## Re: Calculating the sum total of multiple text controls
This way to create arrays should be made more obvious in the documentation or description texts.
Highlighted
Nintex Newbie
## Re: Calculating the sum total of multiple text controls
How would I do this when the number of rows will vary? Also, is there a way I can convert the total value into hours and minutes, e.g., 123 would equal 2 hours 3 minutes.
Thank you.
Highlighted
Nintex Newbie
## Re: Calculating the sum total of multiple text controls
I haven't checked it but you could probably set "Convert empty string to null" under advanced to something like 0. Then it would calculate with that value.
To get the hours and minutes you just calculate hours and minutes seperatly.
123 / 60 => you get the hours
123%60 => modulo will get you the minutes
You can write both like this into a calculated value field and get the result imidiatly.
Highlighted
Nintex Newbie
## Re: Calculating the sum total of multiple text controls
Hi Giacomo,
I'm also facing same type of issue while adding 2 number columns in Nintex calculation value not working due to language issue. My site is in french(Sweden) due to that sum not working. Please help with work around.
Ex: In Formula
sum([projectYear,1])
Project Year (Number column) value display as "2 018" space created due to language issue.
Highlighted
Automation Master
## Re: Calculating the sum total of multiple text controls
Hi,
could you change your calculated field to show the result as text instead of number?
Highlighted
Nintex Newbie
## Re: Calculating the sum total of multiple text controls
It's already used in so many places i can't change the type to text now. Is there any other workaround for this?
Highlighted
Nintex Newbie
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http://www.solutioninn.com/on-january-1-2016-teacher-credit-union-tcu-issued-8
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# Question: On January 1 2016 Teacher Credit Union TCU issued 8
On January 1, 2016, Teacher Credit Union (TCU) issued 8%, 20-year bonds payable with face value of \$400,000. The bonds pay interest on June 30 and December 31.
Requirements
1. If the market interest rate is 6% when TCU issues its bonds, will the bonds be priced at face value, at a premium, or at a discount? Explain.
2. If the market interest rate is 9% when TCU issues its bonds, will the bonds be priced at face value, at a premium, or at a discount? Explain.
3. The issue price of the bonds is 96. Journalize the following bond transactions:
a. Issuance of the bonds on January 1, 2016.
b. Payment of interest and amortization on June 30, 2016.
c. Payment of interest and amortization on December 31, 2016.
d. Retirement of the bond at maturity on December 31, 2035.
View Solution:
Sales16
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https://www.chegg.com/homework-help/questions-and-answers/suppose-width-rectangle-5-inches-shorter-length-perimeter-rectangle-80-inches-formula-peri-q4209660
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Suppose that the width of a rectangle is 5 inches shorter than the length and that
the perimeter of the rectangle is 80 inches. The formula for the perimeter of a
rectangle is P=2L+2W.
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http://mathematica.stackexchange.com/questions/105626/why-matchq-is-fast-and-replace-slow
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# why MatchQ is fast and Replace slow?
Consider a large expression, say a polynomial with 25 terms
expr=Product[Unique["a"],{i,1,25}];
to which you apply the following replacement rule:
rep={x_ f[y_] /; FreeQ[x, y] -> 0};
Since f does not appear in expr, the replacement rule has no effect. However, it takes about 10 seconds on my computer to evaluate expr/.rep. This is surprizing long and becomes longer for larger polynomials.
Moreover I realize that it takes a fraction of a second to evaluate
MatchQ[expr,x_ f[y_] /; FreeQ[x, y]]
which returns (as it should) False
Why does it take so much time to perform the replacement if there is no pattern matching? How can I speed-up the evaluation?
-
I mean, doesn't MatchQ try to match the entire expression, whereas ReplaceAll has to test every subexpression? – march Feb 4 at 1:46
What @march said. This is an apples to orchards comparison. – Daniel Lichtblau Feb 4 at 2:02
@Daniel I am getting funny results in 10.1.0. Please have a look at this. – Mr.Wizard Feb 4 at 3:08
It's not clear to me why folks want to close this. Looks like a perfectly legitimate question to me. – Leonid Shifrin Feb 4 at 12:53
@LeonidShifrin Once upon a time (last night), I took for granted that it made sense the replacement attempt was slow. So I voted to close it. I am rethinking that and might well retract, depending on what I can figure out about this. – Daniel Lichtblau Feb 4 at 15:57
There seems to be a bug regarding this, in version 10.1.0 under Windows. For a first evaluation in a fresh kernel I get:
expr = Product[Unique["a"], {i, 1, 25}];
rep = {x_ f[y_] /; FreeQ[x, y] -> 0};
expr /. rep // Short // AbsoluteTiming
{9.64113*10^-6, a15 a16 a17 a18 a19 a20 << 14 >> a35 a36 a37 a38 a39}
But the second time I evaluate the same thing in the session I get:
{13.8596, a40 a41 a42 a43 a44 a45 <<14>> a60 a61 a62 a63 a64}
Somehow the second run takes a million times longer?!
-
Okay, I'll look. But you're not making my best-friend-for-the-day list. – Daniel Lichtblau Feb 4 at 3:11
@Daniel Sorry. :-s – Mr.Wizard Feb 4 at 3:12
Ditto on "10.3.1 for Mac OS X x86 (64-bit) (December 9, 2015)". – Michael E2 Feb 4 at 3:46
It only seems to happen when the expr is reevaluated. When only reevaluating the last line or the last two lines it is fast on retries. Tried with 10.3.0 for Microsoft Windows (64-bit) (October 9, 2015) – Albert Retey Feb 4 at 10:25
Okay, this has been a bit of a headache. The short answer is that it is all working as expected. Now for too much detail to put into a comment.
To begin, the original question was a bit of a blind, though certainly not by intent. When one considers what has to happen in a match with an Orderless function such as Times, it seems quite plausible that this might be very slow. This is alluded to in Help > Wolfram Documentation refguide page for Orderless, under "Possible Issues":
Pattern matching with orderless functions can lead to a large number of possible cases:
In[1]:= SetAttributes[h, Orderless];
In[2]:= ReplaceList[h[a, b, c], h[x_, y_, z_] :> {x, y, z}]
Out[2]= {{a, b, c}, {a, c, b}, {b, a, c}, {b, c, a}, {c, a, b}, {c, b, a}}
Nonetheless, it is quite clear from what others have posted that sometimes this match is fast and sometimes slow, seemingly on the same input. This confusing flip-flop is due to the fact that what is being tested for a match is changing, due to how Unique works. So the first thing is to get a stable set of examples.
To get this consistent set of behaviors showing both the fast and slow evaluations, I use a different construction of the factors below. I do this for four different products, with the only difference in naming being the first letter of the factors.
n = 22;
expr1 = Product[ToExpression[StringJoin["a", ToString[i]]], {i, n}];
expr2 = Product[ToExpression[StringJoin["b", ToString[i]]], {i, n}];
expr3 = Product[ToExpression[StringJoin["c", ToString[i]]], {i, n}];
expr4 = Product[ToExpression[StringJoin["d", ToString[i]]], {i, n}];
rep = {x_ f[y_] /; FreeQ[x, y] -> 0};
AbsoluteTiming[expr1 /. rep;]
AbsoluteTiming[expr2 /. rep;]
AbsoluteTiming[expr3 /. rep;]
AbsoluteTiming[expr4 /. rep;]
(* Out[359]= {1.482855, Null}
Out[360]= {0.000011, Null}
Out[361]= {8.*10^-6, Null}
Out[362]= {2.291128, Null} *)
This is entirely replicable and consistent: if I repeat the full evaluation I get the same set of fast vs slow evaluations.
So why are some slow and others not? For this we go to one of the deeper corners of documentation, specifically,
tutorial/SomeNotesOnInternalImplementation
Here are two relevant items:
"Each expression contains a special form of hash code that is used both in pattern matching and evaluation."
"A form of hashing that takes account of blanks and other features of patterns is used in pattern matching."
The details of this are outside the scope here. But the upshot is that sometimes this mechanism allows for an early exit, by containing information that entirely rules out a (sub)match. This is why some examples are so fast; they avoid the combinatorial explosion that would otherwise be needed to handle all possible reorderings.
One last detail is that this only happens when dealing with a head that is both Flat and Orderless. The latter means all reorderings of the pattern are required, the former means we also have to consider subsequences in the thing being matched. A reference here is: tutorial/FlatAndOrderlessFunctions
So the following is across the board fast.
n = 22;
ClearAttributes[g, {Flat, Orderless}]
SetAttributes[g, {Orderless}]
expr1 = Apply[g,
Table[ToExpression[StringJoin["a", ToString[i]]], {i, n}]];
expr2 = Apply[g,
Table[ToExpression[StringJoin["b", ToString[i]]], {i, n}]];
expr3 = Apply[g,
Table[ToExpression[StringJoin["c", ToString[i]]], {i, n}]];
expr4 = Apply[g,
Table[ToExpression[StringJoin["d", ToString[i]]], {i, n}]];
rep = {g[x_, f[y_]] /; FreeQ[x, y] -> 0};
AbsoluteTiming[expr1 /. rep;]
AbsoluteTiming[expr2 /. rep;]
AbsoluteTiming[expr3 /. rep;]
AbsoluteTiming[expr4 /. rep;]
(* Out[443]= {0.000026, Null}
Out[444]= {9.*10^-6, Null}
Out[445]= {8.*10^-6, Null}
Out[446]= {9.*10^-6, Null} *)
But this variant has the same behavior as the original example that used head of Times.
n = 22;
ClearAttributes[g, {Flat, Orderless}]
SetAttributes[g, {Flat, Orderless}]
expr1 = Apply[g,
Table[ToExpression[StringJoin["a", ToString[i]]], {i, n}]];
expr2 = Apply[g,
Table[ToExpression[StringJoin["b", ToString[i]]], {i, n}]];
expr3 = Apply[g,
Table[ToExpression[StringJoin["c", ToString[i]]], {i, n}]];
expr4 = Apply[g,
Table[ToExpression[StringJoin["d", ToString[i]]], {i, n}]];
rep = {g[x_, f[y_]] /; FreeQ[x, y] -> 0};
AbsoluteTiming[expr1 /. rep;]
AbsoluteTiming[expr2 /. rep;]
AbsoluteTiming[expr3 /. rep;]
AbsoluteTiming[expr4 /. rep;]
(* Out[455]= {1.481003, Null}
Out[456]= {7.*10^-6, Null}
Out[457]= {7.*10^-6, Null}
Out[458]= {2.361662, Null} *)
Hoping all this is of some use.
-
Great, this clarifies the situation. I am still surprized by the fact that MatchQ was always fast, but this may be because MatchQ does not test what I think it tests. The work-around I came-up with consists in performing the MatchQ test and making the replacement only if I got True. Is that ok? – M. Tissier Feb 5 at 13:00
It really depends on the specifics of the expression and on what you want to accomplish in the replacement. If replacing a proper subexpression is a requirement then whether MatchQ passes or fails on the full expression will not be a correct guide. Also keep in mind that you are working with a Head (Times) that has both Flat and Orderless attributes, and that latter is going to give a "worst case scenario" for subexpression matching (due to need to test against all possible reorders). – Daniel Lichtblau Feb 5 at 18:17
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James Bialasik SWEET HOME SENIOR HIGH SCHOOL, AMHERST, NY
Algebra I : Unit #5 - Modeling With Statistics : Lesson #4
# Organizing Data with a Box Plot
Objective: SWBAT construct a box plot and understand how a box plot represents the distribution of data in a data set.
Standards: HSS-ID.A.1 HSS-ID.A.2 HSS-ID.A.3 MP1 MP2 MP3 MP6
Subject(s): Math
45 minutes
1 Opening - 10 minutes
organizing_data_w_boxplot_video_narrative_overview_.MOV
https://betterlesson.com/lesson/section/82/organizing-data-with-a-box-plot
2 Direct Instruction - 10 minutes
When teaching this topic, I like to front load the vocabulary of working with a box plot. In Slide 4 of Box Plot_Day 1, I expose students to the terms that are used when working with box plots. When discussing the vocabulary, I work horizontally across the boxes so that students can see a connection between the words used and the plot.
I continually refer back to the beginning of the lesson when students divided data into four equal parts so the term "quartile" (quarter) makes sense to them. Going across the rows the terms would be:
1st quartile....2nd quartile...3rd quartile
lower quartile...median...upper quartile
25th percentile...50th percentile...75th percentile
3 Investigation - 15 minutes
On the front side of Stats_Box_Plot_Introduction, I want my students to connect the idea of dividing a data set into quarters with finding multiples of 25% of a number. This portion of the lesson does require students to understand and think through what they are trying to find (MP1). Hopefully, students will make the connection to the term "percentile" from the vocabulary portion of the lesson. They will begin to understand how the test scores are distributed when they answer Questions 4, 5, and 6.
On the back side of the worksheet, students put the data points in order and divide the data into four equal portions. As a group, we name each of the three divisions they have found using the vocabulary from earlier in the lesson and then construct a box plot on the graph at the bottom of the paper. As a class, we have a discussion about how the same number of scores (4) are in each of the sections of the box plot (MP7).
4 Closure - 5 minutes
The question on Slide 5 of Box_Plot_Day 1 requires students to apply what they have just learned to a more abstract scenario when the data points are not given. I usually give students a couple of minutes to think about what the vocabulary from the question means and allow them to make a sketch of how to visualize the problem. Then I ask them to compare their idea with their partner's and come up with a common answer.
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+0
# help me! Quick!
+1
162
1
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Two numbers, if the first one increases by 1, and the second one decreases by 1, then their product increases by 2020. If the first number decreases by 1, and the second one increases by 1, what value does the product decrease?
Aug 13, 2020
#1
+25656
+3
Two numbers, if the first one increases by 1, and the second one decreases by 1, then their product increases by 2020.
If the first number decreases by 1, and the second one increases by 1, what value does the product decrease?
$$\text{ Let first number =n_1} \\ \text{ Let second number =n_2}$$
$$\begin{array}{|lrcll|} \hline (1) & (n_1+1)(n_2-1) &=& n_1n_2 + 2020 \\ & n_1n_2-n_1+n_2-1 &=& n_1n_2 + 2020 \\ & -n_1+n_2-1 &=& 2020 \\ & \mathbf{ -n_1+n_2 } &=& \mathbf{ 2021 } \\ \hline (2) & (n_1-1)(n_2+1) &=& n_1n_2 + x \\ & n_1n_2+n_1-n_2-1 &=& n_1n_2 + x \\ & n_1-n_2-1 &=& x \\ & \mathbf{ n_1-n_2-1 } &=& \mathbf{ x } \\ \hline (1)+(2): & -n_1+n_2 + n_1-n_2-1 &=&2021 +x \\ & -1 &=&2021 +x \\ & x &=& -2021 -1 \\ & \mathbf{x} &=& \mathbf{-2022} \\ \hline \end{array}$$
The value of the product decreases by 2022
Aug 13, 2020
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# Cross Product Confusion
Hi all,
Just having some trouble understanding a certain example of cross product. It's actually for physics, but figured it belongs in this forum.
In the question I am supposed to cross (dl)y hat with (x1)x hat.
So I go (0 dl 0) x (x1 0 0) = (-x1dl)z hat. But turns out the answer the computer looking for is the same but without the negative. Wouldn't that be the case if i crossed it backwards, i.e. (x1)x hat cross (dl) y hat ??
Yes,
$$(0, dl, 0)\times{(}x_1, 0, 0)=-xdlk$$
and
$$(x_1, 0, 0)\times{(}0, dl, 0)=xdlk$$
Well then if that's the case I'm still confused. Because then my answer should be right.
mathman
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DISCOUNT 13%
# Complete SS1 Further Mathematics Lesson Note
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Access the complete First to Third Term SS1 Further Mathematics lesson notes with a simple payment via ATM Card, USSD code, or Bank Transfer. Your payment is processed securely by Paystack for peace of mind.
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## Description
These lesson notes cover the following topics for SS1 First, Second and Third Term Further Mathematics:
FIRST TERM
1 Indices: Basic Laws & Application of indices
2 Indicial and Exponential Equations
3 Logarithms – Laws and application
4 General review of basic concept of set theory
5 Operation of sets and Venn diagrams
6 Review of First Half Terms Lesson & Periodic Test
7 Binary operations and basic laws of binary operations (i) Definition (ii) Solution of simple problems on binary operations (iii) Closure, commutative, associative and distributive laws
8 Binary operations continues: (i) Solution to problems on laws of binary operations (ii) Identity and inverse elements of a given binary operations (iii) Addition and multiplication tables for binary operations
9 Surds: (i) Definition of surds (ii) Rules and manipulation of surds (iii) Rationalization of surds at the denominator and equality of surds.
10 Measures of central tendency: (i) Mean, Median and Mode of grouped and ungrouped data (ii) Estimation of mode from the histogram of a grouped data.
11 Revision
12 Examination
REFERENCE(S)
• Further Mathematics project 1 by Tuttuh Adegun et al
• New General Mathematics for SSS1, SSS 2 and SSS 3 by M. F. Macrae et al
SECOND TERM
1 Arithmetic Progression (AP)
2 Geometric Progression (GP)
3 Linear inequalities in one variable
4 Inequalities in two variables (Graph of inequalities)
5 Introduction to the concept of functions.
6 Review of half term work.
7 Functions (one – to – one, onto, composite and inverse functions)
8 Trigonometric ratio: Graph of Sine, Cosine and tangent of angles, deviation of trigonometric ratio of special angles (300, 450 and 600). Application of trigonometric ratios.
9 Logical reasoning: Simple True and False statement, Negation, Converse and Contra positive of statement,
10 Logical reasoning continues: Compound statement, connectives and their symbols, conditional statements and symbols.
11 Revision of Second Term’s lesson
12 Examination
REFERECES
FutherMaths Project 1 and 2 by TuttuhAdegun (main text).
Further Mathematics by E. Egbe et al.
THIRD TERM
1. Revision of last term
2. Flow chart
3. Gradient of straight line and curve
4. Gradient of straight line and curve
5. Vectors
6. Magnitude of vectors
7. Vectors continuation
8. Angles between straight lines
To download Complete First to Third Term SS1 Further Mathematics Lesson Note, simply scroll up and click on the order button!
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## 57171
57,171 (fifty-seven thousand one hundred seventy-one) is an odd five-digits composite number following 57170 and preceding 57172. In scientific notation, it is written as 5.7171 × 104. The sum of its digits is 21. It has a total of 4 prime factors and 16 positive divisors. There are 33,408 positive integers (up to 57171) that are relatively prime to 57171.
## Basic properties
• Is Prime? No
• Number parity Odd
• Number length 5
• Sum of Digits 21
• Digital Root 3
## Name
Short name 57 thousand 171 fifty-seven thousand one hundred seventy-one
## Notation
Scientific notation 5.7171 × 104 57.171 × 103
## Prime Factorization of 57171
Prime Factorization 3 × 17 × 19 × 59
Composite number
Distinct Factors Total Factors Radical ω(n) 4 Total number of distinct prime factors Ω(n) 4 Total number of prime factors rad(n) 57171 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
The prime factorization of 57,171 is 3 × 17 × 19 × 59. Since it has a total of 4 prime factors, 57,171 is a composite number.
## Divisors of 57171
1, 3, 17, 19, 51, 57, 59, 177, 323, 969, 1003, 1121, 3009, 3363, 19057, 57171
16 divisors
Even divisors 0 16 8 8
Total Divisors Sum of Divisors Aliquot Sum τ(n) 16 Total number of the positive divisors of n σ(n) 86400 Sum of all the positive divisors of n s(n) 29229 Sum of the proper positive divisors of n A(n) 5400 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 239.105 Returns the nth root of the product of n divisors H(n) 10.5872 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 57,171 can be divided by 16 positive divisors (out of which 0 are even, and 16 are odd). The sum of these divisors (counting 57,171) is 86,400, the average is 5,400.
## Other Arithmetic Functions (n = 57171)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 33408 Total number of positive integers not greater than n that are coprime to n λ(n) 4176 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 5787 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
There are 33,408 positive integers (less than 57,171) that are coprime with 57,171. And there are approximately 5,787 prime numbers less than or equal to 57,171.
## Divisibility of 57171
m n mod m 2 3 4 5 6 7 8 9 1 0 3 1 3 2 3 3
The number 57,171 is divisible by 3.
## Classification of 57171
• Arithmetic
• Deficient
• Polite
• Square Free
### Other numbers
• LucasCarmichael
## Base conversion (57171)
Base System Value
2 Binary 1101111101010011
3 Ternary 2220102110
4 Quaternary 31331103
5 Quinary 3312141
6 Senary 1120403
8 Octal 157523
10 Decimal 57171
12 Duodecimal 29103
20 Vigesimal 72ib
36 Base36 1843
## Basic calculations (n = 57171)
### Multiplication
n×i
n×2 114342 171513 228684 285855
### Division
ni
n⁄2 28585.5 19057 14292.8 11434.2
### Exponentiation
ni
n2 3268523241 186864742211211 10683244176957144081 610771752840816884254851
### Nth Root
i√n
2√n 239.105 38.5235 15.463 8.94201
## 57171 as geometric shapes
### Circle
Diameter 114342 359216 1.02684e+10
### Sphere
Volume 7.82737e+14 4.10735e+10 359216
### Square
Length = n
Perimeter 228684 3.26852e+09 80852
### Cube
Length = n
Surface area 1.96111e+10 1.86865e+14 99023.1
### Equilateral Triangle
Length = n
Perimeter 171513 1.41531e+09 49511.5
### Triangular Pyramid
Length = n
Surface area 5.66125e+09 2.20222e+13 46679.9
## Cryptographic Hash Functions
md5 44c8f7e1528d82923516ae85974d0f47 a59685c04692ab94075cebf1473cec3020b0b4ba 62f7a146e85dd56e10614c2b3efea3cd3a1af2e818a731ca0836dc6e66c18662 c5856cb294642ff69f6ad584683c34867406ea226a1dc497f57e4d994a9368d759c119e6ee17af0d5a51798d269ed53e399b33b1a22e8f45f9b46f5b4604555d 240b02b6c94ae9bb4ea2ebf1ef3183bb41da1809
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# How to keep geometry edges always vertical?
Hi,
I would like to know if we can choose to keep the edges of a geometry vertically.
I’m trying to create a ramp by means of 3 curves but the edges are slightly sloped.
I used the surface.thicken node.
Thank you in advance.
make some more geometry and use geometry.split, cutting your geometry?
Surface.Thicken extrudes a surface along the normal(s), so it’s likely that you’d get non-vertical edges on a curving ramp.
Instead take the original closed curve (before it’s a surface), and translate it along the Z axis by the distance you want to thicken it by, and then use a Solid.ByLoft to generate a solid which maintains the vertical edges.
Be careful on this one though, you could most likely end up with non uniform thickness. The reason why surface thicken uses the normal is to ensure uniform thickness across the whole domain of the original surface.
One method you could do is the cookie cutter method, which is to over size the original surface, then thicken, then use a another vertical surface profile that matches the desired profile to perform a boolean cut on the solid.
1 Like
Hi Jacob,
Thank you for your help.
For the moment I don’t have any closed curve. I have 3 curves.
I’m trying to offset every curve and then close them. After that I guess I can use a Solid.ByLoft.
But I don’t know how to close every parallel curve with its original curve.
Think I would need the full graph to see the data structure so I know how to close this one out. Can you post it and whatever may be needed to reduce the depth?
Certainly a good thing to keep in mind. I’ll spend some time trying to make an example to illustrate this.
1 Like
Hi Jacob,
Attached the rvt file and graph. I use 3 construction 2d curves(splines) from the rvt file.
Thank you.ramp_test.rvt (3.0 MB) ramp1.dyn (46.0 KB)
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## Saturday, February 23, 2019
### FREE MATH LESSON - “Common Core Math Warm Up Freebies”
by Tessa Maguire
Kindergarten - 1st Grade
This is a 4 page sample of my math warm up activities. Each page has 5 different skills. It is aligned to the Kindergarten and 1st grade Common Core standards.
The full pack's skills and standards included:
K.OA.1, & 1.OA.1 Creating number sentences from pictures
K.OA.2 Solve addition and subtraction word problems
K.OA.3 Making representations of numbers
K.OA.4 & 1.OA.4 & 1.OA.8 Find the missing number to solve the equation.
K.NBT.1 & 1.NBT.2 Understanding tens and ones
1.OA.3 Fact Families
1.OA.6 Adding and subtracting within 20 fluently.
1.OA.7 Determining if equations are true or false.
1.NBT.3 Comparing two digit numbers
1.NBT.5 & 1.NBT.6 Understanding place value
1.MD.3 Telling time to the hour
1.MD.4 Interpreting data from graphs
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#### Measurement and Measuring Instruments One Mark Questions
9th Standard
Reg.No. :
•
•
•
•
•
•
Science
Time : 01:00:00 Hrs
Total Marks : 30
10 x 1 = 10
1. Choose the correct one
(a)
mm< cm < m < km
(b)
mm > cm > m > km
(c)
km < m < cm < mm
(d)
mm > m> cm> km
2. Rulers, measuring tapes and metre scales are used to measure
(a)
Mass
(b)
Weight
(c)
Time
(d)
Length
3. 1 metric ton is equal to
(a)
100 quintals
(b)
10 quintals
(c)
1/10 quintals
(d)
1/100 quintals
4. Distance between Chennai and Kanyakumari can be found in
(a)
Kilometres
(b)
Metres
(c)
Centimetres
(d)
Millimetres
5. Which among the following is not a device to measure mass
(a)
Spring balance
(b)
Beam balance
(c)
Physical balance
(d)
Digital balance
6. ___________is the derived quantity.
(a)
Length
(b)
Volume
(c)
Mass
(d)
Time
7. _________is not the unit of length.
(a)
Muzham
(b)
Furlong
(c)
Mile
(d)
Hour
8. Unit of acceleration is________.
(a)
ms-2
(b)
ms-1
(c)
$\frac{m}{s}$
(d)
$\frac{s}{m}$
9. The diameters of spherical objects are measured with a_________scale.
(a)
pitch
(b)
meter
(c)
(d)
vernier
10. _________is the unit distance used to measure astronomical objects outside the solar system.
(a)
Parsec
(b)
km
(c)
cm
(d)
litre
11. 10 x 1 = 10
12. Metre is the unit of ________
()
length
13. 1 kg of rice is weighed by ______
()
beam balance
14. The thickness of a cricket ball is measured by _______
()
Vernier Caliper
15. The radius of a thin wire is measured by ________
()
Screw gauge
16. A physical balance measures small differences in mass up to ______
()
1 milligram
17. The SI unit of volume is________.
()
Cubic metre
18. Least count of screw gauge__________.
()
0.01 mm
19. Least count of Vernier caliper is___________.
()
0.01 cm
20. Length, mass and time are called__________.
()
Fundamental unit
21. 1 Angstrom (A0)=__________m.
()
10-10
22. 5 x 1 = 5
23. A physical balance is more sensitive than a beam balance as it can accurately measure even a very small mass, even milligram
(a) True
(b) False
24. One Celsius degree is an interval of 1K and zero degree Celsius is 273.15 K
(a) True
(b) False
25. Two pan balance is commonly used in provisions and grocery shops.
(a) True
(b) False
26. Weight is the fundamental quantity.
(a) True
(b) False
27. In screw gauge the plane surface of the screw and opposite plane are brought into contact. If the zero of the head scale lies above the pitch scale axis the zero error is positive.
(a) True
(b) False
28. 5 x 1 = 5
29. Temperature
30. (1)
Hooke's law
31. Mass
32. (2)
Matter
33. Least count of screw gauge
34. (3)
Thermometre
35. Spring balance
36. (4)
Beam balance
37. Mass
38. (5)
0.01 mm
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https://brainmass.com/physics/energy/electrostatic-potential-energy-of-a-distribution-of-charges-121849
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# Electrostatic potential energy of a distribution of charges
Three point charges are on the x axis: q1 at the origin, q2 at x=3 m, and q3 at x= 6 m. Find the electrostatic potential energy for (a) q1=q2=q3=2uC (u is the fancy u sign); (b) q1=q2=2uC, and q3 -2uC; and (c) q1=q3 = 2uC, and q2= -2uC.
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# Why do we use “Sum of Squared Errors” as loss function in linear regression? [duplicate]
What is a loss function? How can we relate the slope of Linear Regression with Sum of Squared Errors?
SSE is used in linear regression because it directly relates to the portion of the variance of outcome $$Y$$ that is not explained (cannot be contributed) to the difference is the values of (the) predictor(s) $$X$$. It is a measure of 'predicability' of the $$X$$'s for the value of $$Y$$.
The SSE directly relates to the slope of a linear regression model because it is the sum of the squared deviations of a given $$(X,Y)$$ from $$(X,\hat{Y})$$ where $$\hat{Y}$$ is the predicted value based on the model and the given $$X$$.
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11. Currying in Python
General Idea
In mathematics and computer science, currying is the technique of breaking down the evaluation of a function that takes multiple arguments into evaluating a sequence of single-argument functions. Currying is also considered to be a design pattern. Currying is also used in theoretical computer science, because it is often easier to transform multiple argument models into single argument models.
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Origin of the Name
The "Curry" in "Currying" has nothing to do with the spicy curry powder, even though it can make your Python code more spicy. The name is a reference to the logician and mathematician Haskell Brooks Curry, and he is quoted in "THE KLEENE SYMPOSIUM", (Proceedings of the Symposium held June 18-24, 1978 at Madison, Wisconsin. U.S.A.):
Some contemporary logicians call this way of looking at a function “currying”, because I made extensive use of it; but Schonfinkel had the idea some 6 years before I did.
This is why there is also - though seldomly - the name "Schönfinkelisation" being used. There are even further roots of the concept going back to the end of the 19th century and the mathematician Gottlob Frege.
Currying Functions
Currying is the technique of breaking down the evaluation of a function that takes multiple arguments into evaluating a sequence of single-argument functions.
In mathematical notation it looks like this: If we have a function $f$ which takes $n$ arguments, we can 'replace' it by a composition of $n$ functions $f_1, f_2, \ldots f_n$ where each takes only one argument:
$$let x = f(a_1, a_2, a_3)$$
we will get the same value x as if we call:
$$f_2 = f_1(a_1)$$
$$f_3 = f_2(a_2)$$
and
$$x = f_3(a_3)$$
An example implemented in Python could look like this:
def f(a1, a2, a3):
return a1 * a2 * a3
def f1(a1):
def f2(a2):
def f3(a3):
return f(a1, a2, a3)
return f3
return f2
for i in range(1, 10):
print(f(i, i+1, i+2), f1(i)(i+1)(i+2))
OUTPUT:
6 6
24 24
60 60
120 120
210 210
336 336
504 504
720 720
990 990
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Example BMI
We illustrate this with the function BMI, which we had used in our chapter on functions. We write a curried version of BMI:
def BMI(weight, height):
return weight / height**2
def bmi_curried(height):
def bmi_weight(weight):
return BMI(weight, height)
return bmi_weight
bmi_curried(1.76)(72)
OUTPUT:
23.243801652892564
Using partial
The function partial of the module functools of Python allows you to create partially applied functions. A partially applied function is a new function that is derived from an existing function by fixing a certain number of arguments in advance. The result is a function that can be called with the remaining arguments. We can use it for the commposition of functions.
We demonstrate the way of working in the following examples:
from functools import partial
def f(a1, a2):
return a1 * a2
partial(f, 3)(4)
OUTPUT:
12
from functools import partial
def f(a1, a2, a3):
return a1 * a2 * a3
partial(f, 3, 2)(4)
OUTPUT:
24
partial(f, 3)(2, 4)
OUTPUT:
24
partial(partial(f, 3), 2)(4)
OUTPUT:
24
partial(f, 3)(2, 4)
OUTPUT:
24
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Using partial
The function partial of the module functools of Python allows you to create partially applied functions. A partially applied function is a new function that is derived from an existing function by fixing a certain number of arguments in advance. The result is a function that can be called with the remaining arguments. We can use it for the commposition of functions.
We demonstrate the way of working in the following examples:
Decorator for currying
def f(x, y, z):
return x * y * z
The following Python function curry is a higher-order function that transforms a regular function into a curried function. The curried function returns a new function for each argument provided, and you can apply these functions one at a time, which can be useful in various scenarios. Inside the curry function, a nested function curried is defined. It accepts a variable number of arguments using *args. This allows the curried function to accept any number of arguments.
print(f'{args}'): simply prints the current arguments to the console. It's not necessary for the functionality of course, it's included for debugging and illustration purposes.
if len(args) == func.__code__.co_argcount: This condition checks whether the number of arguments provided (len(args)) matches the number of arguments that the original function func expects. The co_argcount attribute is used to retrieve the number of arguments of the func function.
return func(*args): If the number of arguments matches, the original function func is called with the provided arguments, and the result is returned.
else: return lambda x: curried(*(args + (x,))) If the number of arguments doesn't match, a new lambda function is returned. This lambda function takes a single argument x, and it calls the curried function with the existing args and the new argument x appended. This effectively allows you to build up (accumulate) the arguments one by one and create a chain of unary functions.
The co_argcount sub-attribute returns the number of positional arguments (including arguments with default values). To learn more about the __code__-attribute you can consult our page Argument Count for Advanced Python Programmers
def curry(func):
def curried(*args):
print(f'{args}')
if len(args) == func.__code__.co_argcount:
return func(*args)
else:
return lambda x: curried(*(args + (x,)))
return curried
@curry
def prod3(x, y, z):
return x + y + z
prod3(3)(4)(5)
OUTPUT:
(3,)
(3, 4)
(3, 4, 5)
12
Another exaple for using our curry function. We first repeat the curry without the debugging print:
def curry(func):
def curried(*args):
if len(args) == func.__code__.co_argcount:
return func(*args)
else:
return lambda x: curried(*(args + (x,)))
return curried
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Another Example for Currying
Let's consider a function for calculating the total cost of a shopping cart, with the ability to apply various discounts at different stages. This example will use multiple curried functions to apply item prices, quantities, and discounts incrementally.
First, we define a curried function for calculating the cost of individual items:
@curry
def calculate_item_cost(price, quantity):
return price * quantity
Next, we create a function to apply a percentage discount to the cost. This function takes a percentage and returns a curried function:
@curry
def subtract_percentage(discount_percentage, cost):
#print(f"{cost=}, {discount_percentage=}")
discount_amount = (cost * discount_percentage) / 100
return cost - discount_amount
For the following, it might be useful to be familiar with to simple forms of price reduction in the business world:
Rebates and discounts are two distinct pricing strategies that businesses use to incentivize customers. The main distinction between discounts and rebates lies in the timing of the price reduction. Discounts provide immediate cost savings at the point of sale, while rebates offer savings after a purchase has been made, contingent on meeting specific criteria. Both strategies can be effective in influencing customer behavior and boosting sales, but they are suited to different business scenarios and objectives.
Now, let's use these curried functions to calculate the total cost of a customers order list in the end:
# Calculate the cost of individual items
item1 = calculate_item_cost(10)(2) # Item 1 costs € 10 with a quantity of 2
item2 = calculate_item_cost(5)(3) # Item 2 costs € 5 with a quantity of 3
print(f"{item1=}, {item2=}")
# Apply percentage discounts
calculate_discounted_price = subtract_percentage(12)
calculate_rebated_price = subtract_percentage(3)
# Calculate the total cost with discounts
total_cost = calculate_rebated_price(calculate_discounted_price(item1 + item2))
print(f"Total cost: €{total_cost:.2f}")
OUTPUT:
item1=20, item2=15
Total cost: €29.88
We could have done the calculation directly on each item:
subtract_percentage(3)(subtract_percentage(12)(calculate_item_cost(10)(2)))
OUTPUT:
17.072000000000003
In the previous example we used the same rebate and discount on each item. Usually, they differ from product to product.
Let's assume that we have the following dictionary as a shopping list for items. Each item consists of a tuple with (quantity, price, discount, rebate). We can do the summation in a clean and readable way:
shopping_list = [(2, 10, 12, 3), (3, 5, 12, 3)]
total = 0
sub_perc = subtract_percentage # just a shorter name
for quantity, price, discount, rebate in shopping_list:
subtotal = sub_perc(rebate)(sub_perc(discount)((calculate_item_cost(price)(quantity))))
#print(subtotal)
total += subtotal
print(f"Total cost: €{total:.2f}")
OUTPUT:
Total cost: €29.88
Currying Function with an Arbitrary Number of Arguments
One interesting question remains: How to curry a function across an arbitrary and unknown number of parameters?
We can use a nested function to make it possible to "curry" (accumulate) the arguments. We will need a way to tell the function calculate and return the value. If the funtions is called with arguments, these will be curried, as we have said. What if we call the function without any arguments? Right, this is a fantastic way to tell the function that we finally want to the the result. We can also clean the lists with the accumulated values:
def arimean(*args):
return sum(args) / len(args)
def curry(func):
# to keep the name of the curried function:
curry.__curried_func_name__ = func.__name__
f_args, f_kwargs = [], {}
def f(*args, **kwargs):
nonlocal f_args, f_kwargs
if args or kwargs:
f_args += args
f_kwargs.update(kwargs)
return f
else:
result = func(*f_args, *f_kwargs)
f_args, f_kwargs = [], {}
return result
return f
curried_arimean = curry(arimean)
curried_arimean(2)(5)(9)(5)
OUTPUT:
<function __main__.curry.<locals>.f(*args, **kwargs)>
We have to call the function again with an empty argument, because otherwise it doesn't know that it's at the end. this is not "proper" currying, but it works:
curried_arimean(2)(5)(9)(5)()
OUTPUT:
5.25
arimean(2, 5, 9, 5)
OUTPUT:
5.25
Including some prints might help to understand what's going on:
def arimean(*args):
return sum(args) / len(args)
def curry(func):
# to keep the name of the curried function:
curry.__curried_func_name__ = func.__name__
f_args, f_kwargs = [], {}
def f(*args, **kwargs):
nonlocal f_args, f_kwargs
if args or kwargs:
print("Calling curried function with:")
print("args: ", args, "kwargs: ", kwargs)
f_args += args
f_kwargs.update(kwargs)
print("Currying the values:")
print("f_args: ", f_args)
print("f_kwargs:", f_kwargs)
return f
else:
print("Calling " + curry.__curried_func_name__ + " with:")
print(f_args, f_kwargs)
result = func(*f_args, *f_kwargs)
f_args, f_kwargs = [], {}
return result
return f
curried_arimean = curry(arimean)
curried_arimean(2)(5)(9)(7)
# it will keep on currying:
curried_arimean(5, 9)
print(curried_arimean())
OUTPUT:
Calling curried function with:
args: (2,) kwargs: {}
Currying the values:
f_args: [2]
f_kwargs: {}
Calling curried function with:
args: (5,) kwargs: {}
Currying the values:
f_args: [2, 5]
f_kwargs: {}
Calling curried function with:
args: (9,) kwargs: {}
Currying the values:
f_args: [2, 5, 9]
f_kwargs: {}
Calling curried function with:
args: (7,) kwargs: {}
Currying the values:
f_args: [2, 5, 9, 7]
f_kwargs: {}
Calling curried function with:
args: (5, 9) kwargs: {}
Currying the values:
f_args: [2, 5, 9, 7, 5, 9]
f_kwargs: {}
Calling arimean with:
[2, 5, 9, 7, 5, 9] {}
6.166666666666667
def arimean(*args):
return sum(args) / len(args)
def curry(func):
f_args, f_kwargs = [], {}
def f(*args, **kwargs):
nonlocal f_args, f_kwargs
if args or kwargs:
f_args += args
f_kwargs.update(kwargs)
return f
else:
result = func(*f_args, **f_kwargs)
f_args, f_kwargs = [], {}
return result
return f
curried_arimean = curry(arimean)
result = curried_arimean(2)(5)(9)(7)(5, 9)()
print("Result:", result)
We can write it also as a class, which uses the __call__ method:
def arimean(*args):
return sum(args) / len(args)
class Curry:
def __init__(self, func, *args):
self.func = func
self.args = args
def __call__(self, *args):
print(args, self.args)
if not args:
return self.func(*self.args)
return Curry(self.func, *(self.args + args))
curried_arimean = Curry(arimean)
result = curried_arimean(2)(5)(9)(7)(5, 9)
print("Result:", result, type(result))
OUTPUT:
(2,) ()
(5,) (2,)
(9,) (2, 5)
(7,) (2, 5, 9)
(5, 9) (2, 5, 9, 7)
Result: <__main__.Curry object at 0x7fdfe3c103d0> <class '__main__.Curry'>
c = curried_arimean(2)(5)(9)(7)(5)(9)()
c
OUTPUT:
(2,) ()
(5,) (2,)
(9,) (2, 5)
(7,) (2, 5, 9)
(5,) (2, 5, 9, 7)
(9,) (2, 5, 9, 7, 5)
() (2, 5, 9, 7, 5, 9)
6.166666666666667
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SEARCH HOME
Math Central Quandaries & Queries
Question from Cathy, a parent: This question was on my daughter's geometry assignment. Write a general rule of formula for finding the number of segments that can be named by a given number of points on a line. For example, 2 points on a line = 1 line segment; 3 points on a line = 3 segments; 5 points on a line = 10 segments.
Cathy,
Suppose you look at how many ways you can pick two distinct points. First note that if you pick point a and then point b they determine the same line segment as picking point b first and then point a. If you had 5 points you can pick one of them in 5 ways and then a second point in 4 ways - 20 ways to pick them consecutively, however, from what we've just said, they only determine 10, not 20 segments. If you had 6 points it would lead to (6x5)/2 =15 segments, etc.
Penny
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.
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# math
posted by .
the triangular sail on a sailboat has a base of 4 feet and height of 6 feet. aubrey will paint the sail using a special white paint. a can of this paint covers 10 square feet. how many cans of white paint will she need
• math -
See later post and divide by 10.
• math -
I do not understand
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# America Has Two Feet. It’s About to Lose One of Them
## America Has Two Feet. It’s About to Lose One of Them
### RE: America Has Two Feet. It’s About to Lose One of Them
We should lose both of them, but that's a whole 'nuther discussion.
============
"Is it the only lesson of history that mankind is unteachable?"
--Winston S. Churchill
### RE: America Has Two Feet. It’s About to Lose One of Them
America is much more than two feet, it is a centipede.
### RE: America Has Two Feet. It’s About to Lose One of Them
may be it's time for metric units
### RE: America Has Two Feet. It’s About to Lose One of Them
Actually the 'National Institute of Standards and Technology' units used in this country includes only the 'yard' and the 'pound'. All other units for lengths and mass are derived from these TWO STANDARDS. And since 1893, as a result of the 'Mendenhall Order' these standards are defined using SI or Metric units. The yard was defined as being equal to 3600⁄3937 meter, and the pound was defined as 0.4535924277 kilogram. This was later revised in 1959 to where the yard was now defined as 0.9144 meter, and the pound as 0.45359237 kilogram. Part of the problem in the US is that we really have TWO different standards for the foot, which is what led to this issue being discussed here now. We have the normal everyday 'foot' and then we have the 'survey foot'. That's the issue which is being reconciled.
That being said, very few people are aware that the United States officially adopted the Metric System in 1866. It's just that the 'Metric Act of 1866', while it officially put the US on the Metric System, it did NOT require that anyone actually use the Metric units for industry or commerce. And while it did not require the use of Metric units, the biggest thing that it did accomplish was that it protected companies from being held liable if they did decide to use Metric rather then Imperial or 'traditional' units of measure. Therefore the act merely made it LEGAL to use Metric units, but it did not make them mandatory.
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
### RE: America Has Two Feet. It’s About to Lose One of Them
Feet and pounds and inchies are still a sign of English colonization. By the way you still drive like in England?
regards
luis
### RE: America Has Two Feet. It’s About to Lose One of Them
NIST has the document from 1959 that defines the yard as 0.9144 m, and the survey foot as 1200/3937 m
https://www.nist.gov/system/files/documents/2017/0...
TTFN (ta ta for now)
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### RE: America Has Two Feet. It’s About to Lose One of Them
Interesting that in the article, the point is made that the foot/inch/etc are defined in terms of the metre, as the US was one of the original signatories to the metric system.
### RE: America Has Two Feet. It’s About to Lose One of Them
(OP)
#### Quote (0707)
Feet and pounds and inchies are still a sign of English colonization. By the way you still drive like in England?
They drive like they are in Canada.
--
JHG
### RE: America Has Two Feet. It’s About to Lose One of Them
Sometimes when "we" drive we shoot each other and run each other off the road (we don't care which side).
### RE: America Has Two Feet. It’s About to Lose One of Them
That is right monkeydog! but that situation is universal (we don't care which side) since we didn´t jump too far and, we will keep alive, being in condition to sign the friendly assurance car book...
luis
### RE: America Has Two Feet. It’s About to Lose One of Them
Where I live, New Brunswick Canada we (not me, obviously) drove on the left, until 1921 or so, when it was changed due to the proximity of the US and the tourists coming in, it was felt it made more sense.
### RE: America Has Two Feet. It’s About to Lose One of Them
I can remember when Sweden switched to right-hand drive. I wasn't there I just remember reading about it. They picked a Sunday figuring that there would be the least amount of traffic, but the problem was that everyone showed-up so as to experience that first day and it created a real mess. Sort of like in China when you drive from Hong Kong into China proper. Hong Kong is Left-hand drive and China is Right-hand drive, so you have to switch at the border, after passing through customs and immigration. While I've never had to do that as a driver, I have been a front-seat passenger in cars that did make that transition, and it's feels very weird when you're where the car's set=up doesn't match where you're driving. Now, I've driven in Left-hand drive countries, but always with a car designed for that situation.
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
### RE: America Has Two Feet. It’s About to Lose One of Them
Its like coming off the cross channel ferry into the UK .
Signs for the first 3 miles saying "Drive on the left", in three different languages. And signs giving distances in miles, with kilometers in parentheses underneath.
B.E.
You are judged not by what you know, but by what you can do.
### RE: America Has Two Feet. It’s About to Lose One of Them
I'm still not sure if it's a joke.
### RE: America Has Two Feet. It’s About to Lose One of Them
#### Quote (JohnRBaker)
Now, I've driven in Left-hand drive countries, but always with a car designed for that situation.
I've done a bit of all four permutations (including all in the space of 24 hours once - but at least that time, one of the vehicles was big enough that what side of the road I was driving on was everybody else's problem). Psychologically, it's not actually as difficult as you might expect. The bit I don't like though is how much harder it is to see what's happening up ahead on the other side of the road.
A.
### RE: America Has Two Feet. It’s About to Lose One of Them
While I've never done it, I've concluded that it might harder to drive a motorcycle in a country where the drive-lane is the opposite of what you're used to. At least in a car, you're constantly aware that things are the other way round, but on a bike, there would be no clues other than the driving environment itself, which on regular two-lane roads, could prove problematic. And for the record, I've owned three motorcycles, although it's been years since I driven one.
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
### RE: America Has Two Feet. It’s About to Lose One of Them
Pirates keep going on having weight in our society.
luis
### RE: America Has Two Feet. It’s About to Lose One of Them
I rode a motorcycle in Australia last year, first experience with driving/riding on the left. Only times I made a mistake was when there was no one else around due to not having a visual cue, although that also meant that it wasn't a problem. The rental car that I drove first had a bigger nuisance...the mirrored stalks! Got wipers when I wanted turn signals many times.
### RE: America Has Two Feet. It’s About to Lose One of Them
Feet of clay. Two of them.
Cf Daniel 2:41-43
"Everyone is entitled to their own opinions, but they are not entitled to their own facts."
### RE: America Has Two Feet. It’s About to Lose One of Them
I've heard a lot of folks ask when the US will switch to metric. My standard answer is that 25.4 isnt metric.
### RE: America Has Two Feet. It’s About to Lose One of Them
As I've noted before, officially we're already on the metric system, it's just voluntary. Besides, right now I'm afraid that if we tried to get people to use it more widely, we'd be accused of some sort of anti-American plot. And I fear it would fall along the same political lines as certain issues are now dividing us, like ignoring science and pushing conspiracy theories.
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
### RE: America Has Two Feet. It’s About to Lose One of Them
I doubt the US will ever really convert to metric. They were supposed to in 1975 I think. I graduated high school that year. They never will because it would require way too much expense and headaches. If the government (and thereby the tax-payer) would very heavily subsidize businesses it may occur but even then I doubt it. the US has too much to convert that is not easily converted.
They converted rebar so far, that is not hard to do. They converted nuts and bolts on vehicles, that is not hard to do.
Look at plumbing, can't reduce the diameter of a pipe in the direction of flow. How do you replace a section of damaged 8" pipe?
While lumber itself is easy to convert, the use of lumber is not easy. Try selling metric lumber when all remodels will require the old system. If you open up the first metric lumber mill, you better have deep pockets. Cabinets would no longer be 24"x36" as a standard.
Luckily, I am pretty good at converting parts per million and roof slopes to their metric equivalents.
### RE: America Has Two Feet. It’s About to Lose One of Them
I had to buy a couple of sheets of furniture grade plywood at Home Depot a couple of months ago and I noticed that while the 48 x 96 inch size was unchanged, that it came in thicknesses of 5.2mm, 12mm and 18mm. Not a 100% conversion, but it's a move in that direction.
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
### RE: America Has Two Feet. It’s About to Lose One of Them
#### Quote (Ron247)
Look at plumbing, can't reduce the diameter of a pipe in the direction of flow. How do you replace a section of damaged 8" pipe?
While lumber itself is easy to convert, the use of lumber is not easy. Try selling metric lumber when all remodels will require the old system. If you open up the first metric lumber mill, you better have deep pockets. Cabinets would no longer be 24"x36" as a standard.
Canada has been trying to wade through those conversions since 1975 - not sure where they are at in the process now.
### RE: America Has Two Feet. It’s About to Lose One of Them
(OP)
dauwerda,
You do realise that a three inch pipe is not three inches in diameter? If we were to re-designate what we now call 3" pipe as 80mm pipe, we would be more accurate.
I designed and built back steps for my house mostly out of 2×6s, relying on the lumber tables in my first semester mechanics of materials textbook. This claims that the size is 1‑5/8 by 5‑5/8". You know you are getting old when...
--
JHG
### RE: America Has Two Feet. It’s About to Lose One of Them
Of course I realize that - it wasn't my example. I was commenting on the fact that Canada seems to be figuring out how to deal with the "standard size" issue, so the US should be able to figure it out as well.
Perhaps Ron247's use of pipe is not a good example of the issue, the standard cabinet size may be better.
I believe the point being made is that it is nice to use even increments for standard sizes, so new standard sizes would need to be made using even increments of the metric dimensions, which would not align perfectly with the current even increments that are used with imperial dimensions.
The example that comes to mind for me is the standard plywood size - in the US it is 4'x8' which is approximately 1220mmx2440mm in metric - not nice even increments. The metric "equivalent" would be 1200x2400 I believe. Both sizes would have to be made available during the transition period - which would have to last years to ensure that existing items could still be repaired or added to.
### RE: America Has Two Feet. It’s About to Lose One of Them
So how would someone go about getting full size lumber? I ran into that problem years ago. The 2 X 4 of today was too small.
I already need two sets of tools to work on my car, for the small part remaining that I can work on. Had to buy a special socket to replace the oxygen sensor.
I believe the biggest thing would be land sizes and recording of old deeds.
Lets see, in some parts of Texas, the land size was measured by how far someone go while riding a mule while smoking two cigars (or something like that).
Standardize that.
Also I noticed that tires are in inch diameters, and mm width.
### RE: America Has Two Feet. It’s About to Lose One of Them
You have to ask for what they call 'dimensional' lumber. I bought some 1 x 4 last year for another project where I actually needed it to be 1 x 4, and I had no problem finding it at Home Depot.
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
### RE: America Has Two Feet. It’s About to Lose One of Them
So what you bought was probably a 4/4 x 16/4 board, not a 1 x 4. That fact that no nominal size of anything I'm familiar with is the actual size makes that whole transition much easier. The same thing can have two nominal dimensions, one in customary units and one in SI units, but the thing actually has one size. The actual metrification would occur when you start cutting it to length in SI units rather than in customary units.
### RE: America Has Two Feet. It’s About to Lose One of Them
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
### RE: America Has Two Feet. It’s About to Lose One of Them
The pipe example I used is a real problem with metric conversion. Regardless of the actual diameter of an 8" pipe, there is no "standard" metric pipe of equivalent diameter. So if you splice an 8" line with a standard metric size pipe (not some custom diameter that is really close to 8"), you are decreasing the diameter on one end or the other of a splice. This is more a sewage application than pure fluid flow.
while we can physically make metric lumber, a standard metric wall stud would not be 1.5"x3.5"x92.625". It would be some even number of millimeters that may be somewhat close to those dimensions but not close enough. They also space framing at 60 cm on center, not 24" on center. Therefore, real metric sheet material is not 8' long. The point I am making, it without heavy \$ subsidies, no one wants to convert their plant operation and an oppressive law that forced the issue would not go over well.
Standardization has to do with nice round numbers when possible not just labeling something two different ways. But give them 25.4 centimeters and they will take a 1.6 kilometers, I always say. Did I convert that right? Let me check with a Denver Bronco fan to see.
### RE: America Has Two Feet. It’s About to Lose One of Them
With regard to the metric system in the USA, in 1977 my older son came home with math homework that was entirely metric to metric conversions. I asked him if his teacher ever gave them conversions that included English units and he replied, "No!"
I made an appointment with Miss Jeski, his sixth grade math teacher, and in our session, she informed me that next year the United States would be converting all units of measure from English to Metric. We had a little discussion about how absurd that assumption was and how ill prepared her students would be if her current approach were to be followed.
Later I assumed that she must have seen some distorted reporting [say it ain't so!] in the news media about...
The Metric Conversion Act is an Act of Congress that U.S. President Gerald Ford signed into law on December 23, 1975.
https://en.wikipedia.org/wiki/Metric_Conversion_Ac...
...or maybe she jumped to a conclusion?
Skip,
Just traded in my OLD subtlety...
for a NUance!
### RE: America Has Two Feet. It’s About to Lose One of Them
(OP)
#### Quote (dauwerda)
Of course I realize that - it wasn't my example. I was commenting on the fact that Canada seems to be figuring out how to deal with the "standard size" issue, so the US should be able to figure it out as well.
I am Canadian, based in Toronto. A company I used to work for had a new building custom designed. I saw the drawings. The architects worked in metric — metres and millimetres. Everything was converted to feet and inches for the builders.
--
JHG
### RE: America Has Two Feet. It’s About to Lose One of Them
To convey a measurement, the reader needs to be able have a feel of what that is. Most people in the US have a washing machine, and have a good idea what size that is.
An inch is about the size of the length between two of my finger joints. A millimeter is how much?
A yard is about one large step. A foot is the outside size of a size 10 or 11 shoe.
Never learned that stuff about the metric system, but really learned the English system in shop, where I had to use the measurements.
### RE: America Has Two Feet. It’s About to Lose One of Them
I have a tape measure marked off in washing machines on one side and refrigerators on the other side. It is real handy for laying out kitchens.
### RE: America Has Two Feet. It’s About to Lose One of Them
I used to work in a shop that made exhaust pipes for yachts.
We shipped all over the world, and worked in metric or US units as necessary.
Then one of our best fabricators got promoted to a designer.
For way too long, he used 25 mm = 1 inch for his conversions.
We discovered it when some beautiful polished pipes we shipped to the other side of the world didn't fit right.
That was a fairly expensive lesson.
Mike Halloran
Corinth, NY, USA
### RE: America Has Two Feet. It’s About to Lose One of Them
I have a measuring tape that's marked off in washing machines on one side, and black diamonds on the other.
Best of both worlds.
### RE: America Has Two Feet. It’s About to Lose One of Them
I used to work for a company in Pennsylvania that designed their equipment in metric dimensions. We used metric equivalent dimensions for material thickness of the steel because buying metric plate was about double the cost. All lengths were in metric dimensions.
When I started there, they put trailing zeros on dimensions for tolerance value place holders. I got in some heated discussions, but finally won out, when our CAD system no longer allowed values like 80.00 to be displayed for a length and was now just 80. The engineers said that all of their stackup tolerances had been changed. It lead them to starts tolerancing individual dimensions if they wanted a tighter tolerance than we had for no decimal places in the titleblock.
"Wildfires are dangerous, hard to control, and economically catastrophic."
Ben Loosli
### RE: America Has Two Feet. It’s About to Lose One of Them
Please let an engineer living in Scandinavia comment.
Timber!! Wood material measurements were the last to be accepted converted here. In practice it is easier to handle and understand the metric system once you are used to it. It will also give a more accurate description of tolerences, often given by (say) +/- 1 mm thickness of a plank from different sawmills.
Valves and piping: Once, at an international exhibition in Germany, I asked a Chinese representative of a Chinese valve company, competing with mine, if his factory could supply ANSI and European flanges. This was some years ago, and Offshore North Sea used (at that time) mostly ANSI. The answer: 'We can supply both, but we wish mainly to supply European. We see the largest potential here as they have a smaller market now, and we want to grow and compete. (No comment!)
It is not seen as a large problem in Europe to connect ANSI and European dimensions by suitable connecting pieces and/or conerting flanges.
On driving: I have been driving on the left (wrong) side in Sweden, UK and Australia. The problem is to use a system where you have learned instinctively to turn to nearest ditch to avoid accidents. If you converse from right to left, or opposite, this will then lead to front collisions. Also driving a car with the wheel 'on the wrong side' is aproblem.
And lastly, on different feet: To me this seems to be a 'microscopic problem' compared to the other topics discussed.
### RE: America Has Two Feet. It’s About to Lose One of Them
#### Quote (gerhardl)
To me this seems to be a 'microscopic problem' compared to the other topics discussed.
Well, since this first became an issue with respect to land surveying, even a 'microscopic problem' could prove a disaster if not properly resolved.
BTW, I thought Sweden moved over from driving on the Left to driving on the Right years ago, like back in the 60's.
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
### RE: America Has Two Feet. It’s About to Lose One of Them
Not english or metric, but years ago I talked to someone from Pueblo, a city in Colorado, who said they were having problems finding water piping with a Pueblo thread.
I thought it odd, why not just use a common thread. After a little looking, I found out that at the time Pueblo was developing there water system, no one had set a common thread. So the city developed their own standard. Now the rest of the world has moved on, but they still have their own standard, and everything is in that standard.
Sort of like the US is invested in the english system, and it is so hard to change.
### RE: America Has Two Feet. It’s About to Lose One of Them
#### Quote (John R. Baker, P.E.)
BTW, I thought Sweden moved over from driving on the Left to driving on the Right years ago, like back in the 60's
That's correct, IIRC it was in 1969 or 1970. There could not yet have existed an expressway system with ramps that would probably not be reversible. Apparently it went off without a significant hitch.
(Could you imagine attempting something like that in today's climate of conspiracy theories and fake news?)
"Everyone is entitled to their own opinions, but they are not entitled to their own facts."
### RE: America Has Two Feet. It’s About to Lose One of Them
I thought it was 1968. Mid-day, on a Sunday, I think. Would have been disruptive, but that eliminated a bunch of other problems. Leave the "wrong" side drivers to a bunch of islands (of wildly varying size) or a sub-continent. Everywhere else they drive on the right side. Sweden, pre-change, just didn't fit the rest of the pattern.
I’ll see your silver lining and raise you two black clouds. - Protection Operations
### RE: America Has Two Feet. It’s About to Lose One of Them
And I understand that virtually all of the vehicles produced in Sweden up to that time, mostly Volvo's and Saab's, were designed for driving on the right side of the road, like the rest of Europe, sans the UK of course. Primarily because most of their auto and truck production was for export.
Note that the change took place on Sunday, September 3, 1967:
http://realscandinavia.com/this-day-in-history-swe...
John R. Baker, P.E. (ret)
EX-Product 'Evangelist'
Irvine, CA
Siemens PLM:
UG/NX Museum:
The secret of life is not finding someone to live with
It's finding someone you can't live without
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# Solar power for 1 amp relay board
Hello , I want to power “XXX-Channel 1-Amp SPDT Signal Relay Shield” with photon, with solar power/battery, any project I can use as a guide?
thanks
Hi Sergio,
That board will draw about 46 mA at 12VDC continuously if it is to maintain communication with the Particle Cloud. When the relay is energized there will be an additional current draw of approximately 12.5 mA at 12VDC.
Thanks Travis, a question, 12.5 mA is per relay?
That is correct. Each energized relay will draw 12.5mA when energized.
What are you powering with the Relays? The 12.5 mA per relay is in addition to the actual load.
If you share your load and estimated duty cycle, a simplified power budget can be calculated with the info Travis posted.
That will point you in the right direction for Solar Panel and battery size.
1 Like
I thought about tagging you on this one @rfontaine since you have some good recent experience with Solar projects. Didn’t want to seem pushy though
@rfontaine, the relays will turn off/on a already powered 12v electric fence.
10-4
The Quick Version:
Calc’s will assume 12.0V, even though your battery will hopefully stay well above that.
Also assume 1 relay powered 24/7.
Per Travis’ measurements = 46+ 12.5mA = 58.5 mA = 0.0585 Amps
Watt = Amps * Volts
Watts = 0.0585 * 12.0 = 0.7 watts.
For 1 day = 0.7 watts * 24 hours = 17 watt hours.
In a perfect world, a 17 Watt Solar Panel would recharge your battery with 1 hour of full sunlight.
The quick approach is to de-rate any Solar Panel 50% (I’d be happy to explain in more detail if ya care to read it).
For Instance, if you want to use a 12V 10 watt Solar Panel, go ahead and de-rate it to 5 watts.
We calculated that you need 17 watt hours per day, so you hope for 3.5 hours of Sunlight (17/5) each day to maintain the battery. That’s pretty optimistic, but can be done if your battery has enough reserve capacity for Cloudy Days.
Battery Capacity:
I’ll use a 7 amp-hour SLA battery for an example.
To prolong the life of the battery, you don’t want to cycle it below 50% capacity.
So de-rate the Battery Capacity 50% (similar to Solar Panel). For the Simplified Calc’s, this battery now has a 3.5 amp-hour effective capacity.
3.5 amp-hours / 0.0585 Amps = 60 hours of runtime, without Recharging. That sounds like a LONG time, but that’s only 2 cloudy days.
A full Power Budget gets much more involved, but this shows you how to get a decent idea pretty quickly.
Decide the minimum # of hours of sunlight that you expect in the Winter to size your Solar Panel.
Decide the maximum # of Cloudy days in a row that you expect in the Winter to size your Battery Capacity.
1 Like
Perfecto! you are the best!
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# Speical relativity problem: How Tall Really Is Your Favorite Superhero?
1. Sep 11, 2008
### jlk
1. The problem statement, all variables and given/known data
You are at the top of the Empire State Building on the 102nd floor, which is located 373m above the ground, when your favorite superhero flies over the building parallel to the ground at 70.0% the speed of light.
You have never seen your favorite superhero in real life. Out of curiosity you calculate her height to be 1.43m . If the superhero landed next to you, how tall would she be when standing?
What is the height of the 102nd floor of the Empire State Building as measured by the superhero while flying above it?
I'm not sure how to start the problem, but i do know you will have to use that formula.
2. Relevant equations
L= L0 square root of (1- v2/c2)
3. The attempt at a solution
2. Sep 11, 2008
### Hootenanny
Staff Emeritus
Can you start by identifying what each of the variables in the equation are and what they mean?
3. Sep 12, 2008
### jlk
i figure it out :)
thanks though.
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Explore BrainMass
Related Rates Word Problem
Not what you're looking for? Search our solutions OR ask your own Custom question.
This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!
A highway patrol helicopter hovers 3/10 mile above a level, straight interstate highway which has a posted speed limit of 65 miles per hour. The helicopter pilot sees a car on the highway and determines with radar that at that particular instant, the distance between the helicopter and the car is 1/2 mile and is increasing at a rate of 64 miles per hour. Should the driver of the car be ticketed for speeding? Set up and solve a related rates problem to justify answer. Hint: Find the car's speed along the highway.
https://brainmass.com/math/calculus-and-analysis/related-rates-word-problem-2735
Solution Preview
See the attached file for the diagram.
The best way to start this problem is by making a sketch. I've attached a word document that shows the situation.
The first thing to see here is ...
Solution Summary
The solution shows how to set up and solve a related rates problem to determine if a car should be ticketed for speeding.
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# rbind() and cbind() functions in R
Data Manipulation in R Data transformation
The `rbind` function combines vectors, matrices or data frames by rows while the `cbind` function combines them by columns. In this tutorial you will learn how to use these functions in several use cases.
## Binding rows with `rbind`
If you want to combine two or more vectors of the same length by binding them by rows you can input them to the `rbind` function. The output will be a matrix with as many rows as vectors and as many columns as elements.
``````x <- c(1, 2, 3)
y <- c("G1", "G2", "G3")
# Bind two vectors by rows
rbind(x, y)``````
`````` [,1] [,2] [,3]
x "1" "2" "3"
y "G1" "G2" "G3"``````
If the length of the vectors is not equal you will get a warning message indicating that the ânumber of columns of result is not a multiple of vector length (arg 1)â. In this scenario the elements of the smaller vectors will be recycled.
However, you can bind a single value (numeric or character) with a vector and the value will be recycled to the length of the largest vector.
``````x <- c(1, 2, 3)
# Bind integer and vector
rbind(2, x)``````
`````` [,1] [,2] [,3]
2 2 2
x 1 2 3``````
### Add new rows to a data frame or matrix
Consider the following sample data frame with three rows and two columns:
``````df <- data.frame(x = 1:3, y = c("G1", "G2", "G3"))
df``````
`````` x y
1 1 G1
2 2 G2
3 3 G3``````
If you want to bind a new row to it you can input the data frame and the vector to `rbind`, as shown below.
``````df <- data.frame(x = 1:3, y = c("G1", "G2", "G3"))
z <- c(4, "G4")
df <- rbind(df, z)
df``````
`````` x y
1 1 G1
2 2 G2
3 3 G3
4 4 G4``````
Note that you can also input several vectors to the function to bind several rows to the data frame.
``````df <- data.frame(x = 1:3, y = c("G1", "G2", "G3"))
z <- c(4, "G4")
w <- c(5, "G5")
df <- rbind(df, z, w)
df``````
`````` x y
1 1 G1
2 2 G2
3 3 G3
4 4 G4
5 5 G5``````
### Bind data frames by row
You can also bind two or more data frames together with the `rbind` function as long as they have the same number of columns and column names.
``````df1 <- data.frame(x = 1:3, y = c("G1", "G2", "G3"))
df2 <- data.frame(x = 15:19, y = c("G15", "G16", "G17", "G18", "g19"))
# Bind two data frames
df <- rbind(df1, df2)
df``````
`````` x y
1 1 G1
2 2 G2
3 3 G3
4 15 G15
5 16 G16
6 17 G17
7 18 G18
8 19 g19``````
### Bind data frames with different column names
Notice that if the names of the columns of the data frames are not the same you wonât be able to merge them despite there is a common column name. An error will arise in this scenario.
``````df1 <- data.frame(x = 1:3, y = c("G1", "G2", "G3"))
df2 <- data.frame(x = 15:19, z = c("Z1", "Z2", "Z3", "Z4", "Z5"))
# Bind two data frames
df <- rbind(df1, df2)
df``````
``````Error in match.names(clabs, names(xi)) :
names do not match previous names``````
In case that you need to bind two data frames by rows with different column names or with only some common column names you will need to use the `bind_rows` function from `dplyr`.
``````df1 <- data.frame(x = 1:3, y = c("G1", "G2", "G3"))
df2 <- data.frame(x = 15:18, z = c("Z1", "Z2", "Z3", "Z4"))
# Bind two data frames with different column names
library(dplyr)
df <- dplyr::bind_rows(df1, df2)
df``````
`````` x y z
1 1 G1 <NA>
2 2 G2 <NA>
3 3 G3 <NA>
4 15 <NA> Z1
5 16 <NA> Z2
6 17 <NA> Z3
7 18 <NA> Z4``````
### Row labels when binding rows
The `rbind` function provides an argument named `deparse.level` which defaults to `1`. If you set this argument to `0` the output will have no row labels and when set to `2` the labels will be constructed from the argument names.
``````x <- c(1, 2, 3)
# Bind integer and vector
rbind(2, x, deparse.level = 0)``````
`````` [,1] [,2] [,3]
[1,] 2 2 2
[2,] 1 2 3``````
## Binding columns with `cbind`
The `cbind` function binds vectors, data frames or matrices by columns. You can combine several vectors of the same length by columns passing them as input to `cbind`. The function will return a matrix with as many columns as vectors and as many rows as elements of the vectors.
``````x <- c(1, 2, 3)
y <- c("G1", "G2", "G3")
# Bind two vectors by columns
cbind(x, y)``````
`````` x y
[1,] "1" "G1"
[2,] "2" "G2"
[3,] "3" "G3"``````
You can also bind a single numeric or character value with other object. In this scenario the value will be repeated to fit the length of the other object.
``````x <- c(1, 2, 3)
# Bind a character and a vector
cbind("A", x)``````
`````` x
[1,] "A" "1"
[2,] "A" "2"
[3,] "A" "3"``````
The function allows you to add columns to a data frame or matrix passing as input the data frames and the new columns.
``````df <- data.frame(a = c("A", "B", "C"), b = c(1, 2, 3))
x <- c("AA", "BB", "CC")
cbind(df, x)``````
`````` x y
[1,] "1" "G1"
[2,] "2" "G2"
[3,] "3" "G3"``````
### Combining data frames
In addition, you can combine several data frames as long as they have the same number of rows, otherwise the function will throw an error.
``````df1 <- data.frame(a = c("A", "B", "C"), b = c(1, 2, 3))
df2 <- data.frame(c = c("D", "E", "F"), d = c(4, 5, 6))
# Combine two data frames
cbind(df1, df2)``````
`````` a b c d
1 A 1 D 4
2 B 2 E 5
3 C 3 F 6``````
If the number of rows of the data frames is not the same you will get an error indicating that âarguments imply differing number of rowsâ.
### Column labels when binding columns
The `deparse.level` argument from `cbind` determines the column labels. By default (`1`) the function will use the column names of the objects, if any, if `0` the output wonât have column names and if `2` the function will always construct column names from the input values.
``````x <- c(1, 2, 3)
# Bind an integer and a vector
cbind(3, x, deparse.level = 0)``````
`````` [,1] [,2]
[1,] 3 1
[2,] 3 2
[3,] 3 3``````
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# sir please try to answer the question number 5 of chapter electrostatics
Vikas TU
14149 Points
6 years ago
Charge q = 10^-6 C
Mass(m) = 2 g
• a = 1m
• B = 10 m
• Q = 10^-3 C
F =( kq1kq2)/r^2
F = (9x10^9 * 10^-9)/100
F =0.09 N
Speed = Force x (distance)^2
= 0.09 x 100
= 9 m/sec
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http://mathhelpforum.com/advanced-algebra/97734-exponention-composite-field-element.html
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# Thread: Exponention in Composite Field Element
1. ## Exponention in Composite Field Element
Hi,
How do i perform exponentiation to a composite field element? say the element is of the field GF(((2^2)^2)^2) and constructed from the following polynomials:
x^2 + x + {1000}
x^2 + x + {10}
x^2 + x + 1
please provide me with the working as detail as possible as im very confused with composite field arithmetic
2. Deducing as well as I can from your unexplained notation from which I have removed the confusing braces.
Since 10 satisifies x^2+x+1=0 and the field has characterstic 2
we have 10^2+10+1=0
10^2 = 10 + 1 = 11
11^2 = (10 + 1)^2 = 10^2 +1^2 = 11 + 1 = 10 (no 2.10.1 term since characteristic is 2)
Now can you work out 100^2 etc in the same way?
3. Originally Posted by alunw
Deducing as well as I can from your unexplained notation from which I have removed the confusing braces.
Since 10 satisifies x^2+x+1=0 and the field has characterstic 2
we have 10^2+10+1=0
10^2 = 10 + 1 = 11
11^2 = (10 + 1)^2 = 10^2 +1^2 = 11 + 1 = 10 (no 2.10.1 term since characteristic is 2)
Now can you work out 100^2 etc in the same way?
Thanks for the reply. But honestly I still dont quite get it. Forgive me for my insufficient knowledge in composite filed arithmetic. Can you show me how to obtain $\alpha^4$ where by $\alpha = {01111010}_2$ is an element of composite field $((GF(2^2)^2)^2)$ which is constructed from the following field polynomials: $x^2 + x + {1000}_2$ , $x^2 + x + {10}_2$ and $x^2 + x + 1$
and i cant work out 100^2 as well T-T
4. I can't work out what alpha^4 is because I don't understand your notation properly, but whatever it is you should work it out by first finding alpha^2 and then squaring that again. In all that follows I'm assuming addition is meant to be done using a bitwise XOR, whereas it might be that you actually have the elements successive powers of some generator of the cyclic group of non-zero elements, so that addition needs to be done using a table.
Note that given that 10 is a root of x^2+x+1 then we had enough information to work out 10*10. 10+1 cannot equal 10,0 or 1 so it makes sense to denote it by 11, and it will turn out we can complete the multiplication and addition tables for the field of 4 elements. 10*11 = 10*10 + 10*1 = 11 + 10 = 1, 11*11 = 10*10+1 = 11+1 = 10
Also 11*11+11+1 = 10 + 11 + 1 = 0 since characteristic is 2. In other words once we found one root of x^2+x+1 the other followed.
Next we presumably discover that the polynomial x^2+x+10 has no roots in our first splitting field. So we introduce another element 100 which is a root of this
Then we have 100^2+100 +10 = 0 so
100^2 = 100 + 10.
100+10 is certainly not equal to any of 0,1,10,11 or 100, so we are justified in denoting it by 110, and we can reason the same way to allow us to define 101 and 111 as well,
and work out what they square to:
101^2 = 100^2+1 = 111
110^2 = 100^2+10^2 = 110 +11 = 101
111^2 = 100^2+11^2 = 110+10 = 100
Incidentally we see now that 100^16=110^8=101^4=111^2=100. That means that 100^15=1. but 100^3 = 110*100 = 100*100+10*100=110+10*100 which cannot be 1
and 100^5 = 100*100^4=100*110^2 = 100*101 = 110+100 = 10. This shows that
our field must contain at least 16 elements.
So far we have 8 elements, but in fact we want at least 16, and this time we have more work to do to work out a complete multiplication table. In particular we don't yet know what 10*100 or 11*100 is. Also the other root of x^2+x+10 must be 101 since that is the only factor that could give us x^2+x+10 when we multiply the factors out and indeed 100*101 = 100*100 + 100 = 10.
Now 10*100 is not 0, or 1, or 10, or 11 (since 10*10=11) or 100, or 110 (since 100*100=110). Can it be 101 or 111?
Note that (10*100)^2 = 10^2*100^2 = 11*110 = 11*100+11*10 = 10*100+100+1. Thus 10*100 is not 101 otherwise we would have (10*100)^2 = 0.
Now if we have 10*100 = 111 then we have
100=11*111 = 10*111+111 = 10*100 + 10*11 + 111 = 111 + 10*11 + 111 = 1
Since that is certainly not the case we see that 10*100 must be a 9th element, and we may as well denote it by 10000. And we have 10000^2 = (10*100)^2 = 11*110 = 10*110+110 = 10*100 + 11 + 110 = 10*100+101=10101.
That would be enough information to complete a multiplication table. However, I made a fairly arbitrarary decision in decreeing 10*100=10000, since clearly I might just as well have assigned that value to one of the other multiplications I could not complete without adding a new element to the eight I already had. I suppose 10*100 is the logical choice to assign the value 10*100 to, as it the first entry in the multiplication table that we got stuck on.
This is clearly all very laborious but we could now complete the multiplication table for a field of 16 elements. Then we would have to (and could) show that x^2+x+1000 did no have any root in our field of 16 elements, and posit 10000 as one of its two roots, at which point we would know that 10000^2 = 11000. Then the same kind of reasoning would presumably eventually lead us to realise that 10*10000 and 100*10000 and 1000*10000 were all tricky products to calculate. Probably the best way to proceed would be to calculate all the powers of 10000 find its order (which presumably is a factor of 255 and probably is 255 though I don't know for sure - there certainly is an element of order 255 somewhere in the multiplicative group of non-zero elements in that field and very likely some structure theorem tells us that this is one of them).
In practice I'm sure one would use GAP or some similar program. I think GAP can do calculations in any finite field of reasonable order. Really all I'm doing here is some kind of mathematical equivalent of a Sudoko puzzle.
5. thanks alot for the reply! im still tryin my best to digest these. im still on halfway
by the way, what is GAP?
6. Originally Posted by classic_phohe
thanks alot for the reply! im still tryin my best to digest these. im still on halfway
by the way, what is GAP?
Groups, Algebra and Programming - it is a computer algebra system, and it is free to use if you want to. See here for more details, or see Wiki.
It allows you to do stuff with groups and other algebraic structures, like see if two permutation groups are isomorphic (and much nicer things too).
7. actually i need to derive an isomorphic matrix that map the element from field GF(2^8) to GF(((2^2)^2)^2) using 3 field polynomial. Can it be done from GAP or any other software?
8. Originally Posted by alunw
This is clearly all very laborious but we could now complete the multiplication table for a field of 16 elements. Then we would have to (and could) show that x^2+x+1000 did no have any root in our field of 16 elements, and posit 10000 as one of its two roots, at which point we would know that 10000^2 = 11000. Then the same kind of reasoning would presumably eventually lead us to realise that 10*10000 and 100*10000 and 1000*10000 were all tricky products to calculate. Probably the best way to proceed would be to calculate all the powers of 10000 find its order (which presumably is a factor of 255 and probably is 255 though I don't know for sure - there certainly is an element of order 255 somewhere in the multiplicative group of non-zero elements in that field and very likely some structure theorem tells us that this is one of them).
how do we do for cube instead of square? say 10000^3??
i do find it very difficult to calculate 10*10000 and 100*10000 and 1000*1000. say if i would like to find 100000^2, how do i justify i shd go for (100*1000)^2 or (10*10000)^2??
and you mentioned "...calculate all the powers of 10000 find its order..." what does this order means??
thanks alot!
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Search a number
junction numbers
A number that can be written as x + sod(x) for at least two x. more
The first 600 junction numbers :
101, 103, 105, 107, 109, 111, 113, 115, 117, 202, 204, 206, 208, 210, 212, 214, 216, 218, 303, 305, 307, 309, 311, 313, 315, 317, 319, 404, 406, 408, 410, 412, 414, 416, 418, 420, 505, 507, 509, 511, 513, 515, 517, 519, 521, 606, 608, 610, 612, 614, 616, 618, 620, 622, 707, 709, 711, 713, 715, 717, 719, 721, 723, 808, 810, 812, 814, 816, 818, 820, 822, 824, 909, 911, 913, 915, 917, 919, 921, 923, 925, 1001, 1003, 1005, 1007, 1009, 1011, 1012, 1013, 1014, 1015, 1016, 1018, 1020, 1022, 1024, 1026, 1102, 1104, 1106, 1108, 1110, 1112, 1114, 1116, 1118, 1203, 1205, 1207, 1209, 1211, 1213, 1215, 1217, 1219, 1304, 1306, 1308, 1310, 1312, 1314, 1316, 1318, 1320, 1405, 1407, 1409, 1411, 1413, 1415, 1417, 1419, 1421, 1506, 1508, 1510, 1512, 1514, 1516, 1518, 1520, 1522, 1607, 1609, 1611, 1613, 1615, 1617, 1619, 1621, 1623, 1708, 1710, 1712, 1714, 1716, 1718, 1720, 1722, 1724, 1809, 1811, 1813, 1815, 1817, 1819, 1821, 1823, 1825, 1910, 1912, 1914, 1916, 1918, 1920, 1922, 1924, 1926, 2002, 2004, 2006, 2008, 2010, 2012, 2013, 2014, 2015, 2016, 2017, 2019, 2021, 2023, 2025, 2027, 2103, 2105, 2107, 2109, 2111, 2113, 2115, 2117, 2119, 2204, 2206, 2208, 2210, 2212, 2214, 2216, 2218, 2220, 2305, 2307, 2309, 2311, 2313, 2315, 2317, 2319, 2321, 2406, 2408, 2410, 2412, 2414, 2416, 2418, 2420, 2422, 2507, 2509, 2511, 2513, 2515, 2517, 2519, 2521, 2523, 2608, 2610, 2612, 2614, 2616, 2618, 2620, 2622, 2624, 2709, 2711, 2713, 2715, 2717, 2719, 2721, 2723, 2725, 2810, 2812, 2814, 2816, 2818, 2820, 2822, 2824, 2826, 2911, 2913, 2915, 2917, 2919, 2921, 2923, 2925, 2927, 3003, 3005, 3007, 3009, 3011, 3013, 3014, 3015, 3016, 3017, 3018, 3020, 3022, 3024, 3026, 3028, 3104, 3106, 3108, 3110, 3112, 3114, 3116, 3118, 3120, 3205, 3207, 3209, 3211, 3213, 3215, 3217, 3219, 3221, 3306, 3308, 3310, 3312, 3314, 3316, 3318, 3320, 3322, 3407, 3409, 3411, 3413, 3415, 3417, 3419, 3421, 3423, 3508, 3510, 3512, 3514, 3516, 3518, 3520, 3522, 3524, 3609, 3611, 3613, 3615, 3617, 3619, 3621, 3623, 3625, 3710, 3712, 3714, 3716, 3718, 3720, 3722, 3724, 3726, 3811, 3813, 3815, 3817, 3819, 3821, 3823, 3825, 3827, 3912, 3914, 3916, 3918, 3920, 3922, 3924, 3926, 3928, 4004, 4006, 4008, 4010, 4012, 4014, 4015, 4016, 4017, 4018, 4019, 4021, 4023, 4025, 4027, 4029, 4105, 4107, 4109, 4111, 4113, 4115, 4117, 4119, 4121, 4206, 4208, 4210, 4212, 4214, 4216, 4218, 4220, 4222, 4307, 4309, 4311, 4313, 4315, 4317, 4319, 4321, 4323, 4408, 4410, 4412, 4414, 4416, 4418, 4420, 4422, 4424, 4509, 4511, 4513, 4515, 4517, 4519, 4521, 4523, 4525, 4610, 4612, 4614, 4616, 4618, 4620, 4622, 4624, 4626, 4711, 4713, 4715, 4717, 4719, 4721, 4723, 4725, 4727, 4812, 4814, 4816, 4818, 4820, 4822, 4824, 4826, 4828, 4913, 4915, 4917, 4919, 4921, 4923, 4925, 4927, 4929, 5005, 5007, 5009, 5011, 5013, 5015, 5016, 5017, 5018, 5019, 5020, 5022, 5024, 5026, 5028, 5030, 5106, 5108, 5110, 5112, 5114, 5116, 5118, 5120, 5122, 5207, 5209, 5211, 5213, 5215, 5217, 5219, 5221, 5223, 5308, 5310, 5312, 5314, 5316, 5318, 5320, 5322, 5324, 5409, 5411, 5413, 5415, 5417, 5419, 5421, 5423, 5425, 5510, 5512, 5514, 5516, 5518, 5520, 5522, 5524, 5526, 5611, 5613, 5615, 5617, 5619, 5621, 5623, 5625, 5627, 5712, 5714, 5716, 5718, 5720, 5722, 5724, 5726, 5728, 5813, 5815, 5817, 5819, 5821, 5823, 5825, 5827, 5829, 5914, 5916, 5918, 5920, 5922, 5924, 5926, 5928, 5930, 6006, 6008, 6010, 6012, 6014, 6016, 6017, 6018, 6019, 6020, 6021, 6023, 6025, 6027, 6029, 6031, 6107, 6109, 6111, 6113, 6115, 6117, 6119, 6121, 6123, 6208, 6210, 6212, 6214, 6216, 6218, 6220, 6222, 6224.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 9777726 values, from 101 to 99999970).
n\r 0 1
248888634888863 2
3325924232592423259242 3
42444433244443024444302444433 4
519555451955545195554519555461955545 5
6162962116296071629607162962116296351629635 6
71396826139681313968131396826139681313968221396813 7
812222171222214122221612222161222216122221612222141222217 8
9108641410864141086414108641410864141086414108641410864141086414 9
10977773977773977773977773977772977772977772977772977773977773 10
11889064887966889064889066889064887966889064889176889060889060889176
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.
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