content stringlengths 86 994k | meta stringlengths 288 619 |
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This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version.
This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the
License, or (at your option) any later version. | {"url":"https://doc.wikimedia.org/mediawiki-core/REL1_39/php/dir_a73ad9079d04c4767e04ca7acf69ed7a.html","timestamp":"2024-11-10T12:58:26Z","content_type":"application/xhtml+xml","content_length":"21827","record_id":"<urn:uuid:1b59071f-7e9c-4efc-97ae-cba12402e9e0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00494.warc.gz"} |
How to Calculate Average Acceleration.
What is Acceleration?
Prior to discussing about how to calculate average acceleration, let’s define it. Most people tend to relate high speed to acceleration and that’s clearly not true as acceleration only exists when
there is change in velocity.
For instance, if an aeroplane is traveling in a straight line at a constant speed of 1,000 K mph and a car is also traveling in a straight line at the speed of 80 K mph, there is no acceleration here
as the two are neither increasing or decreasing speed or changing direction.
Both acceleration and speed or velocity are vector quantities.
Since velocity is the rate of change in displacement, hence, acceleration is the rate of change of the velocity of an object with respect to time.
We say a body is in a positive acceleration if there is an increase in velocity with time while a negative acceleration means that the speed reduces with time. Sometimes we say a body is in zero
acceleration to mean that its it at rest.
What is Average Acceleration?
Average Velocity is the rate of change in velocity over a time period.
Therefore to calculate Average Velocity we divide the change in velocity by the change in time.
The SI unit used to measure acceleration is m/s².
Formula to calculate average acceleration.
Change in velocity is the difference between the initial velocity and the final velocity.
Change in time mostly means, the difference in time from time 0 to the final time recorded.
Example 1:
Loise just bought a new car which goes from 0 to 50 m/s in just 5 seconds. Calculate the acceleration of the car.
Therefore, the average acceleration of the car is 10m/s².
Example 2:
A car is driving down the road at 20 meters per second when the driver notices a stop sign up ahead. The driver applies the brakes and comes to a stop in 5 seconds. What was the average acceleration
when the brakes were applied.
In this word problem we are dealing with negative acceleration as the car’s initial velocity was 20m/s before it stopped.
Therefore, the average acceleration or rather, the average deceleration of the car is -4m/s². | {"url":"https://www.learntocalculate.com/calculate-average-acceleration/","timestamp":"2024-11-14T08:26:31Z","content_type":"text/html","content_length":"59715","record_id":"<urn:uuid:0f44ba83-4856-4cfb-a29f-6a48648a6aa5>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00318.warc.gz"} |
Borel construction
I've done a very slight edit including the alternate name
homotopy quotient
but that leads to something in which homotopy quotient appears only very weakly.
That is very useful and very ’timely’ as I was looking for a good reference for the Borel construction and orbifolds. :-)
started an entry on the Borel construction, indicating its relation to the nerve of the action groupoid.
added a subsection called
As a homotopy colimit over the category associated to $G$
This entry was lacking a decent reference. I have taken the liberty now of pointing to
• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Prop. 3.73 in: Principal $\infty$-bundles – Presentations, Journal of Homotopy and Related Structures, Volume 10, Issue 3 (2015), pages 565-622 (
arXiv:1207.0249, doi:10.1007/s40062-014-0077-4)
Please feel invited to add you favorite classical textbook account on the Borel construction, instead (which is?)
diff, v11, current
added mentioning of the simplicial version and some lines (here) relating to the model structure on simplicial group actions
diff, v13, current
I have written out a proof (here) that the topological Borel construction of a well-pointed topological group action sits in the evident homotopy fiber sequence
diff, v15, current
I have added statement and proof (here) that the topological Borel construction of a free action is weakly equivalent – under some sufficient conditions – to the plain quotient.
diff, v17, current
have removed, in that Prop, the assumption that the fundamental group is abelian, and instead added the remark to the proof that the five-lemma still applies.
Of course it does. I have tried to make up for being silly here, previously, by expanding a little at five lemma on the case of homological categories.
diff, v19, current | {"url":"https://nforum.ncatlab.org/discussion/3705/","timestamp":"2024-11-05T00:41:57Z","content_type":"application/xhtml+xml","content_length":"51511","record_id":"<urn:uuid:82ae82f2-4d7e-49da-8309-4144faec6e6f>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00046.warc.gz"} |
How to Improve Prediction With Keras And Tensorflow?
To improve predictions with Keras and TensorFlow, there are several techniques that can be applied. One of the most important methods is to carefully preprocess the data before feeding it into the
neural network. This includes normalization, scaling, and handling missing values.
Another key aspect is to choose the appropriate neural network architecture for the specific problem at hand. This involves selecting the number of layers, the type of activation functions, and the
optimizer to use during training.
Hyperparameter tuning is also crucial for improving prediction accuracy. This involves adjusting parameters such as learning rate, batch size, and dropout rates to find the optimal configuration for
the model.
Regularization techniques, such as L1 and L2 regularization, can also help to prevent overfitting and improve the generalization of the model.
Finally, monitoring the training process with tools like TensorBoard and early stopping can help to identify potential issues early on and improve the overall performance of the model.
What is the rationale behind using different activation functions in hidden layers of a neural network in Keras?
The rationale behind using different activation functions in hidden layers of a neural network in Keras is to introduce non-linearity in the model. Without non-linear activation functions, the neural
network would essentially be a series of linear transformations, which would limit its ability to model complex, non-linear relationships in the data.
Different activation functions have different properties, such as different ranges of output values, different gradients, and different levels of computational efficiency. By using a variety of
activation functions in different hidden layers, we can introduce diversity and flexibility in the model, allowing it to learn complex patterns and relationships in the data more effectively.
For example, popular activation functions like ReLU (Rectified Linear Unit) are commonly used in hidden layers of deep neural networks due to its simplicity and computational efficiency, while
activation functions like Sigmoid or Tanh are used in the output layer for binary or multi-class classification tasks. By experimenting with different activation functions, we can find the
combination that helps the neural network converge faster, generalize better, and achieve higher performance on the given task.
What is the impact of batch size on prediction accuracy in Keras and TensorFlow?
The impact of batch size on prediction accuracy in Keras and TensorFlow can vary depending on the specific dataset and model being used.
In general, a larger batch size can lead to faster training times as more data is processed in each iteration. However, larger batch sizes can also result in lower prediction accuracy as the model
may not be able to generalize as well to unseen data. This is because larger batch sizes can lead to a loss of diversity in the data samples being processed, potentially causing the model to converge
to a suboptimal solution.
On the other hand, smaller batch sizes can lead to slower training times but may result in better prediction accuracy as the model is exposed to a greater variety of data samples in each iteration.
Smaller batch sizes can also help the model generalize better to unseen data by preventing overfitting.
Overall, it is important to experiment with different batch sizes and monitor the resulting prediction accuracy to determine the optimal batch size for a given dataset and model.
What is the importance of cross-validation in evaluating the generalization performance of your neural network model in Keras?
Cross-validation is important in evaluating the generalization performance of a neural network model in Keras because it helps to prevent overfitting and provides a more accurate estimate of how well
the model will perform on unseen data.
By splitting the dataset into multiple subsets and training the model on different combinations of training and validation sets, cross-validation allows for a more robust assessment of the model's
performance across various data subsets. This helps to ensure that the model is not just memorizing the training data, but actually learning to generalize to new, unseen data.
Additionally, cross-validation helps to identify any potential issues with the model, such as high variance or bias, by providing a more comprehensive evaluation of the model's performance. This can
help to guide further optimization and fine-tuning of the model to improve its overall generalization performance.
Overall, cross-validation is an important tool in evaluating the generalization performance of a neural network model in Keras, as it provides a more accurate and reliable assessment of the model's
ability to generalize to new, unseen data.
How to evaluate the performance of your neural network model in Keras and TensorFlow?
To evaluate the performance of a neural network model in Keras and TensorFlow, you can use the model.evaluate() function which calculates the loss and any metrics specified during the model
Here's an example of how to evaluate the performance of a neural network model:
1. Compile your model:
1 model.compile(optimizer='adam', loss='categorical_crossentropy', metrics=['accuracy'])
1. Train your model on training data:
1 model.fit(X_train, y_train, epochs=10, batch_size=32, validation_split=0.2)
1. Evaluate the model on the test data:
1 loss, accuracy = model.evaluate(X_test, y_test)
2 print("Test loss:", loss)
3 print("Test accuracy:", accuracy)
This will output the loss and accuracy of the model on the test data. You can also use other metrics specified during model compilation by passing them to the metrics parameter of the evaluate()
Additionally, you can use confusion matrix, precision, recall, F1 score, and other evaluation metrics to further evaluate the performance of the model.
What is transfer learning and how can it improve prediction accuracy with Keras?
Transfer learning is a machine learning technique where a model trained on one task is reused or adapted for a different task. In the context of deep learning, transfer learning involves leveraging
pre-trained neural network models and fine-tuning them for a specific task.
In Keras, transfer learning can be implemented by loading a pre-trained model, removing the output layer, and adding a new output layer tailored to the specific task at hand. The pre-trained model's
weights are frozen, and only the weights of the new output layer are updated during training. This allows for faster and more efficient training, as the model has already learned general features
from a large dataset.
By using transfer learning with Keras, prediction accuracy can be improved for tasks with limited training data or when working with complex data. It allows for leveraging the knowledge learned by
the pre-trained model, which can lead to better performance on the target task. Additionally, transfer learning can help reduce the risk of overfitting and improve generalization capabilities.
What is the role of optimizer functions in improving prediction accuracy with Keras and TensorFlow?
Optimizer functions are essential in improving prediction accuracy in Keras and TensorFlow by adjusting the weights of the neural network during the training process. The optimizer helps in
minimizing the error by updating the weights based on the gradients of the loss function with respect to those weights.
The role of the optimizer functions include the following:
1. Update weights: Optimizer functions update the weights of the neural network by minimizing the loss function. This process helps in improving the prediction accuracy of the model.
2. Speed up convergence: Optimizer functions help in speeding up the convergence of the training process by efficiently adjusting the weights. This leads to faster training and better prediction
3. Prevent overfitting: Some optimizer functions, such as Adam and RMSprop, have built-in mechanisms to prevent overfitting by adjusting learning rates dynamically during training. This helps in
improving the generalization of the model and preventing it from memorizing the training data.
4. Increase stability: Optimizer functions can also help in increasing the stability of the training process by reducing the likelihood of sudden changes in weight values. This leads to better
prediction accuracy and smoother training.
Overall, optimizer functions play a crucial role in improving prediction accuracy by efficiently updating the weights of the neural network during the training process. By choosing the right
optimizer function and tuning its parameters, you can enhance the performance of your model and achieve better prediction accuracy. | {"url":"https://article-blog.kdits.ca/blog/how-to-improve-prediction-with-keras-and-tensorflow","timestamp":"2024-11-01T19:49:34Z","content_type":"text/html","content_length":"162058","record_id":"<urn:uuid:f507bf5d-3f39-41e6-a1df-2b4233d75095>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00735.warc.gz"} |
AECOM (ACM) Stock Forecast for 2026. Acm Prediction
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AECOM (ACM) Stock Forecast for 2026. Acm Shares Prediction
Updated: November 13, 2024 (01:56)
Sector: Industrials
The share price of AECOM (ACM) now
What analysts predict: $113.56
52-week High/Low: $115.74 / $82.05
50/200 Day Moving Average: $103.3 / $94.24
This figure corresponds to the Average Price over the previous 50/200 days. For AECOM stocks, the 50-day moving average is the resistance level today.
For AECOM stocks, the 200-day moving average is the resistance level today.
Are you interested in AECOM stocks and want to buy them, or are they already in your portfolio? If yes, then on this page you will find useful information about the dynamics of the AECOM stock price
in 2026. How much will one AECOM share be worth in 2026? Is it worth taking profit / loss on ACM stock now or waiting? What are analysts' forecasts for AECOM stock?
We forecast AECOM stock performance using neural networks based on historical data on ACM stocks. Also, when forecasting, technical analysis tools are used, world geopolitical and news factors are
taken into account. This corporation stock prediction results are shown below and presented as a graph, table and text information.
AECOM stock forecasts are adjusted once a day based on the closing price of the previous trading day.
The minimum target price for AECOM analysts is $113.56. Today 200 Day Moving Average is the support level (94.24 $). 50 Day Moving Average is the support level (103.3 $).
Historical and forecast chart of AECOM stock
The chart below shows the historical price of AECOM stock and a prediction chart for the next year. For convenience, prices are divided by color. Forecast prices include: Optimistic Forecast,
Pessimistic Forecast, and Weighted Average Best Forecast. Detailed values for the ACM stock price can be found in the table below.
AECOM (ACM) Forecast for 2026
Month Target Pes. Opt. Vol., %
Jan 110.74 104.45 115.79 9.79 %
Feb 108.26 106.26 111.11 4.36 %
Mar 104.18 101.68 109.10 6.80 %
Apr 103.52 98.30 108.90 9.73 %
May 104.35 99.92 106.68 6.34 %
Jun 109.10 102.30 113.73 10.05 %
Jul 105.00 98.62 107.44 8.21 %
Aug 105.25 101.55 108.45 6.37 %
Sep 102.73 100.26 108.73 7.79 %
Oct 95.41 93.43 99.15 5.77 %
Nov 99.46 96.12 101.37 5.18 %
Dec 100.97 95.80 105.58 9.26 %
AECOM information and performance
AECOM Address
1999 AVENUE OF THE STARS, SUITE 2600, LOS ANGELES, CA, US
Market Capitalization: 14 959 196 000 $
Market capitalization of the AECOM is the total market value of all issued shares of a company. It is calculated by the formula multiplying the number of ACM shares in the company outstanding by the
market price of one share.
EBITDA: 1 066 792 000 $
EBITDA of AECOM is earnings before interest, income tax and depreciation of assets.
PE Ratio: 41.17
P/E ratio (price to earnings) - shows the ratio between the price of a share and the company's profit
PEG Ratio: 0.362
Price/earnings to growth
DPS: 0.84
Dividend Per Share is a financial indicator equal to the ratio of the company's net profit available for distribution to the annual average of ordinary shares.
DY: 0.0079
Dividend yield is a ratio that shows how much a company pays in dividends each year at the stock price.
EPS: 2.71
EPS shows how much of the net profit is accounted for by the common share.
Quarterly Earnings Growth YOY: -0.994
Quarterly Revenue Growth YOY: 0.133
Trailing PE: 41.17
Trailing P/E depends on what has already been done. It uses the current share price and divides it by the total earnings per share for the last 12 months.
Forward PE: 22.03
Forward P/E uses projections of future earnings instead of final numbers.
EV To Revenue: 1.031
Enterprise Value (EV) /Revenue
EV To EBITDA: 17.8
The EV / EBITDA ratio shows the ratio of the cost (EV) to its profit before tax, interest and amortization (EBITDA).
Shares Outstanding: 134067000
Number of issued ordinary shares
Shares Float: N/A
Number of freely tradable shares
Shares Short Prior Month: N/A
Shares Short Prior Month - the number of shares in short positions in the last month.
Percent Institutions: N/A
AECOM price target for 2026 by month
Target values for the price of one AECOM share for Jan 2026.
The weighted average target price per AECOM share in Jan 2026 is:
In Jan, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Feb 2026.
The weighted average target price per AECOM share in Feb 2026 is:
In Feb, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Mar 2026.
The weighted average target price per AECOM share in Mar 2026 is:
In Mar, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Apr 2026.
The weighted average target price per AECOM share in Apr 2026 is:
In Apr, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for May 2026.
The weighted average target price per AECOM share in May 2026 is:
In May, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Jun 2026.
The weighted average target price per AECOM share in Jun 2026 is:
In Jun, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Jul 2026.
The weighted average target price per AECOM share in Jul 2026 is:
In Jul, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Aug 2026.
The weighted average target price per AECOM share in Aug 2026 is:
In Aug, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Sep 2026.
The weighted average target price per AECOM share in Sep 2026 is:
In Sep, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Oct 2026.
The weighted average target price per AECOM share in Oct 2026 is:
In Oct, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Nov 2026.
The weighted average target price per AECOM share in Nov 2026 is:
In Nov, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
Target values for the price of one AECOM share for Dec 2026.
The weighted average target price per AECOM share in Dec 2026 is:
In Dec, the
dynamics for AECOM shares will prevail with possible monthly volatility of
volatility is expected.
Pessimistic target level:
Optimistic target level:
AECOM (ACM) stock dividend
AECOM last paid dividends on 10/02/2024. The next scheduled payment will be on 10/18/2024. The amount of dividends is $0.84 per share. If the date of the next dividend payment has not been updated,
it means that the issuer has not yet announced the exact payment. As soon as information becomes available, we will immediately update the data. Bookmark our portal to stay updated.
Last Split Date: 01/01/1970
Splitting of shares is an increase in the number of securities of the issuing company circulating on the market due to a decrease in their value at constant capitalization.
For example, a 5: 1 ratio means that the value of one share will decrease 5 times, the total amount will increase 5 times. It is important to understand that this procedure does not change the
capitalization of the company, as well as the total value of assets held in private hands. | {"url":"https://pandaforecast.com/stock_forecasts/forecast_acm/for2026/","timestamp":"2024-11-14T11:54:10Z","content_type":"text/html","content_length":"171147","record_id":"<urn:uuid:74ecbfd9-4fe2-4648-bc71-06d7896e2a33>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00837.warc.gz"} |
GENPROCRUSTES procedure • Genstat v21
Performs a generalized Procrustes analysis (G.M. Arnold & R.W. Payne).
PRINT = string Printed output required (analysis, centroid, column, individual, monitoring); default anal, cent
SCALING = Type of scaling to use (none, isotropic, separate); default none
string token
METHOD = string Method to be used (Gower, TenBerge); default Gowe
NROOTS = scalar Number of roots (i.e. dimensions) to print for the output configurations, consensus and rotation matrices, and number of dimensions to save with the XOUTPUT, CONSENSUS and ROTATIONS
paramaters if their matrices have alread not been defined; default is to print and save all the dimensions
PLOT = string Controls which graphs to display (consensus, individuals, projections); default * i.e. none
NDROOTS = Number of dimensions to display in the consensus and individuals plots; default 3
TOLERANCE = The algorithm is assumed to have converged when (last residual sum of squares) – (current residual sum of squares) < TOLERANCE × (number of configurations); default 0.00001
MAXCYCLE = Limit on number of iterations; default 50
XINPUT = pointers Each pointer points to a set of matrices holding the original input configurations
XOUTPUT = pointers Each pointer points to a set of matrices to store a set of final (output) configurations
CONSENSUS = matrices Stores the final consensus configuration from each analysis
ROTATIONS = pointers Each pointer points to a set of matrices to store the rotations required to transform each set of XINPUT configurations to their final (scaled) XOUTPUT configurations
RESIDUALS = pointers Each pointer points to a set of matrices to store the distances of a set of scaled XINPUT configurations from its consensus
RSS = scalars Stores the residual sum of squares from each analysis
ROOTS = diagonal matrices Stores the latent roots from referring the centroid configuration to its principal axis form (consensus) for each analysis
WSS = scalars Stores the initial within-configuration sum of squares from each analysis
SCALINGFACTOR = variates Stores the isotropic scaling factors for configurations from each analysis
PROJECTIONS = pointers Each pointer points to a set of matrices to store a set of projection matrices
An N × V matrix represents a configuration of N points in V dimensions. Given a set of M such matrices (XINPUT), a generalized Procrustes analysis iteratively matches them to a common centroid
configuration by the operations of translation to a common origin, rotation/reflection of axes and possibly also scale changes. This matching seeks to minimise the sum of the squared distances
between the centroid and each individual configuration summed over all points (the Procrustes statistic for each configuration and the centroid, summed over all configurations). The final centroid is
referred to principal axes to give a unique consensus configuration. Two methods of scaling are available (controlled by the SCALING option). Isotropic scaling, which scales the all the dimensions of
each configuration by an equal amount, takes place during the Procrustes analysis. The alternative is to scale each configuration prior to the analysis so that the trace of each matrix is one (see
Arnold 1992). If this latter method is used, the subsequent residuals represent pure lack-of-fit and the scaling factors given in the results represent differences in relative size/spread of the
original (centred) configurations, whereas for overall isotropic scaling the scaling factor contains components of both size and lack-of-fit.
Procedure GENPROCRUSTES carries out a generalized Procrustes analysis and has parameters for saving various results for future use (XOUTPUT, CONSENSUS, ROTATIONS, RESIDUALS, RSS, ROOTS, WSS,
SCALINGFACTOR, PROJECTIONS). There are options for different methods to use for the matching (SCALING, METHOD), control of convergence (TOLERANCE, MAXCYCLE) and printing and plotting of results
(PRINT, PLOT, NROOTS and NDROOTS).
Note that the special case of M=2 corresponds to the classical pairwise Procrustes matching (ROTATE directive) except that by fitting each configuration to a common centroid the requirement to regard
one of the initial configurations as fixed is obviated.
Options: PRINT, SCALING, METHOD, NROOTS, PLOT, NDROOTS, TOLERANCE, MAXCYCLE.
Parameters: XINPUT, XOUTPUT, CONSENSUS, ROTATIONS, RESIDUALS, RSS, ROOTS, WSS, SCALINGFACTOR, PROJECTIONS.
The default method used for generalized Procrustes analysis in GENPROCRUSTES is that described by Gower (1975). Each input configuration (XINPUT – referred to henceforth as X[i], i=1…M) is initially
column-centred, with the individual column means for each configuration optionally printed (by including the column setting with the PRINT option). If separate scaling is requested (option SCALING=
separate), the matrices are also scaled to have trace one (see Arnold 1992). A constraint is required on the overall sum of squares to prevent the trivial solution of matching by all configurations
collapsing to the origin. In this procedure the constraint used is
∑ ( trace ( X[i]′ X[i] ) ) = M.
An initial estimate of the centroid is found from these centred and scaled configurations; firstly X[2] is rotated to X[1], with the rotated X[2] saved as the new X[2] and the centroid computed as
the mean of X[1] and the new X[2]; X[3] is rotated to this centroid which is then recalculated as the mean of the three current configurations; and so on until all configurations X[i] (i=1…M) have
been included. The centroid thus found is taken as the initial centroid estimate Y, with the rotated values as the new X[i]. The initial residual sum of squares S[r] is calculated as
Sr = M × ( 1 – trace ( Y′ Y )).
Each of the current configurations X[i] is then rotated to Y and the rotated position saved as the new X[i]. The updated estimate of the centroid Y[n] is calculated as the mean of the new X[i] (i=
1…M) and the new residual sum of squares calculated as
Sr[n] = M × ( 1 – trace ( Y[n]′ Y[n] )).
If isotropic scaling has been requested (option SCALING=isotropic) new estimates ro[i]′ of the individual scaling factors ro[i] (originally set to 1) are now found by
ro[i]′/ro[i]= √( trace( X[i]′Y[n] )/( trace( X[i]′X[i] ) × trace( Y[n]′Y[n] )))
and each X[i] is updated by a factor of ro[i]′/ro[i]. The centroid is then recalculated as the mean of the new X[i] and the new residual sum of squares calculated in a similar manner to before. If
the change in residual Sr is less than a preset tolerance (controlled by option TOLERANCE) the algorithm is taken to have converged. If not, the process is repeated until the tolerance is reached, up
to a maximum number of iterations as set by the option MAXCYCLE (default 50) after which a message of non-convergence is printed and the procedure terminated. Monitoring information about convergence
can be printed by including the monitoring setting with the PRINT option.
After convergence a unique consensus configuration is found by referring the final centroid to principal axes; the corresponding latent roots may be saved using the ROOTS parameter. Final results for
the consensus and individual configurations (referred to the same principal axes) may be printed using the centroid and individual settings of the PRINT option, and/or saved using the parameters
XOUTPUT, CONSENSUS and ROTATIONS. By default, results are presented and saved for the maximum available dimensionality but the option NROOTS allows a reduced number of dimensions to be set. Analysis
of variation for the M configurations (including the individual scaling factors) and for the N points, along with the initial within and between configurations sums of squares (WSS and BSS), the
final residual sum of squares (RSS) and number of steps in the iteration process may be printed using the analysis setting of the PRINT option. The initial within-configuration sum of squares, final
residual sum of squares and individual isotropic scaling factors may also be saved using, respectively, the WSS, RSS and SCALINGFAC parameters. (Note that the final results are still scaled by the
original factor from the initial overall constraint; to return to the original scale all sums of squares need adjustment by a factor of WSS/M and configurations by the square root of that factor).
Independently of the choice of dimensionality for printing and saving, the NDROOTS option controls the dimensionality of the graphical output requested using the PLOT option (default 3). The
consensus setting plots the consensus solution in the chosen dimensionalty, and the individual setting gives the individual final configurations as well as the consensus. The projection setting
displays the projections (calculated from the individual rotation matrices scaled by the singular values from the consensus solution in principal axis form) as vectors labelled by configuration
number and colour-coded for order of column. This projection plot can be particularly helpful in comparing the use of terms/attributes (columns of the configurations) by individual assessors in
sensory analysis, both in conventional and free-choice profiling; see Arnold & Collins (1993) for further details.
Modifications to the method described above are given in TenBerge (1975), and may be invoked by the TenBerge setting of the METHOD option. This may give considerable savings in the time to reach
convergence (Arnold 1988).
Arnold, G.M. (1988). Comparisons of algorithms for generalized Procrustes analyses. Genstat Newsletter, 22, 7-11.
Arnold, G.M. (1992). Scaling factors in generalized Procrustes analysis. Computational Statistics, Volume 1, Proceedings of the 10th Symposium on Computational Statistics, COMPSTAT, Neuchatel,
Switzerland, August 1992, 61-66.
Arnold, G.M. & Collins, A.J. (1993). Interpretation of transformed axes in multivariate analysis. Applied Statistics, 42, 381-400.
Gower, J.C. (1975). Generalized Procrustes analysis. Psychometrika, 40, 33-51.
TenBerge, J.M.F. (1977). Orthogonal Procrustes rotation for two or more matrices. Psychometrika, 42, 267-276.
See also
Directives: ROTATE. FACROTATE.
Procedures: PCOPROCRUSTES, SAGRAPES.
Commands for: Multivariate and cluster analysis.
CAPTION 'GENPROCRUSTES example',!t('Data from',\
'Gower (1975), Psychometrika, 40, pages 33-51.',\
'Note, however, that in Table 3 the scaling factors printed',\
'were SQRT(ro[i]) instead of ro[i], and in Table 4 the',\
'Between and Within Judges sums of squares were transposed.');\
MATRIX [ROWS=9; COLUMNS=7] X[1...3]
READ [SERIAL=yes] X[]
71 70 34 72 72 71 35 :
55 63 53 77 79 57 49 :
27 78 85 89 92 81 41 :
GENPROCRUSTES [PRINT=analysis,centroid,column,individual,monitoring;\
SCALING=isotropic] XINPUT=X | {"url":"https://genstat21.kb.vsni.co.uk/knowledge-base/genprocr/","timestamp":"2024-11-14T12:06:37Z","content_type":"text/html","content_length":"51391","record_id":"<urn:uuid:5f21e913-b2c8-44be-9902-2eee9c404e54>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00729.warc.gz"} |
Understanding platykurtic in everyday terms
Statistics gives us powerful tools to comprehend and explain data, but sometimes, its vocabulary sounds like an alien language.
An example of such a word is kurtosis – it denotes the shape of a statistical distribution. It basically tells us how much data bunches around the mean in relation to the tails of a distribution.
Within this domain of kurtosis exists a kind called platykurtic distributions.
So what exactly does platykurtic mean, and why does it matter in the context of statistical analysis?
Understanding Kurtosis
Kurtosis is a statistical measure that describes the extent of scatter in the distribution of observations. It shows how frequent outliers are.
There are three categories of kurtosis: Mesokurtic, Leptokurtic and Platykurtic.
• Mesokurtic: Distributions with medium kurtosis (medium tails). Normal distributions fall into this category, with a kurtosis of 3.
• Platykurtic: Distributions with low kurtosis (thin tails). These distributions have fewer tail data, pushing the tails of the bell curve away from the mean.
• Leptokurtic: Distributions with high kurtosis (fat tails). These distributions have more tail data, bringing the tails in towards the mean.
The tails of a distribution tell us how probable or frequent it is to have values that are exceedingly high or low compared to the average. Conversely, they represent how often the outliers occur.
Kurtosis is sometimes mistaken for a measure of peakedness in a distribution. Yet, it measures how its tails compare with the overall distribution shape.
For example, a sharply peaked distribution can have low kurtosis and, conversely, an even lower peak with high kurtosis. Therefore, it measures “tailedness” rather than “peakedness”.
What is Platykurtic?
Platykurtic is a statistical distribution or data set that exhibits a flat shape, unlike other distributions with peakedness or elongated tails. This is usually used to refer to the form of
platykurtic curve that has less height and a shorter side than the normal distribution.
Simply put, when we talk about platykurtic distribution, it means that the number of extreme values in such a case will be reduced as well as resulting in more spread out data points compared to
normal distribution.
Understanding platykurtic distributions is vital in different fields as they enable data analysts to interpret patterns in data and then use them for decision-making processes based on how far apart
or near together these data points are.
Example of Platykurtic Distributions
An example of a platykurtic distribution in finance can be observed in the distribution of daily returns of certain stocks or assets. Platykurtic distributions in finance often indicate a situation
where the data points are spread out with fewer extreme values compared to a normal distribution.
For instance, consider a stock with a platykurtic distribution of daily returns. In this scenario, the daily returns exhibit a flatter peak and thinner tails compared to a normal distribution. This
suggests that the stock’s returns are less volatile, with fewer occurrences of extreme gains or losses.
Such a distribution might be observed in stable, mature companies with steady growth patterns and relatively low volatility in their stock prices. Investors might interpret this as a lower-risk
investment compared to stocks with leptokurtic distributions, which have fatter tails and higher volatility.
Advantages of Platykurtic Distributions
Platykurtic distributions offer both benefits and drawbacks worth exploring. Let’s start with the advantages of platykurtic distributions:
1. Enhanced representation of homogeneity
One of the primary advantages of platykurtic distributions is their ability to represent a more homogeneous set of data. This means that the values within the dataset are more evenly spread out,
indicating a level of consistency and uniformity in the data points.
For researchers and analysts, this homogeneity can simplify the interpretation of data, as it suggests less deviation from the mean and fewer outliers.
This characteristic is particularly beneficial in fields where data uniformity is desirable, such as quality control in manufacturing processes.
2. Lower susceptibility to outliers
Platykurtic distributions are generally less affected by outliers compared to leptokurtic distributions, which have sharper peaks and heavier tails.
The flatter peak and lighter tails of platykurtic distributions mean that extreme values have a lesser impact on the overall distribution.
This is advantageous in data analysis, as it can lead to more stable estimates of central tendency and variability, making statistical conclusions more reliable, especially in environments where
outliers are expected to be minimal or non-influential.
3. Improved predictability in certain contexts
In scenarios where data points are expected to be more uniformly distributed without significant anomalies, platykurtic distributions can offer improved predictability.
This predictability comes from the distribution’s characteristic of having data spread more evenly across the range, reducing the impact of extreme variations.
For instance, in social sciences, where extreme outliers are less common, a platykurtic distribution could indicate that the population being studied behaves more uniformly, allowing for more
accurate predictions of social trends.
Limitations of Platykurtic Distributions
1. Reduced peakedness and tails
Platykurtic distributions have a lower peak around the mean and lighter tails compared to a normal distribution.
This means that data points tend to be more spread out, leading to a broader range of values but with fewer extreme values (outliers).
This characteristic can make it challenging to identify significant deviations or outliers in the data set, as the distribution does not emphasise the tails where these values typically lie.
2. Misinterpretation of data
Due to their flatter nature, platykurtic distributions might lead to misinterpretations of the variability or dispersion of the data.
Analysts or researchers might underestimate the spread of the data because the flatter peak suggests a less dramatic variance than might actually be present.
This can affect decision-making processes, especially in fields where understanding the distribution of data is crucial, such as finance and risk management.
3. Analytical limitations
Statistical models often assume normality or specific kurtosis characteristics that do not align with platykurtic distributions.
This misalignment can lead to complications or inaccuracies when applying certain statistical tests or models that assume data follows a more normal distribution.
Consequently, analysts may need to use alternative methods or adjustments to analyse the data accurately, potentially complicating the analysis process.
So, understanding what platykurtic distributions mean helps you make better sense of data. It’s like having a special lens to see patterns more clearly in finance, biology, and more. Knowing these
statistical terms can really open up new ways of understanding the world around you.
If you want to learn more about stats and data analysis, check out StockGro blogs.
How can I recognise a platykurtic distribution?
Look for a flatter peak and thinner tails in the graph, indicating less data clustering around the mean.
What causes a distribution to be platykurtic?
Factors like diverse or widely dispersed data points contribute to a platykurtic shape, spreading the data out more evenly.
Is Platykurtic good or bad?
It depends on the context. Platykurtic distributions may indicate less risk in some situations but can make predictions less precise in others.
Can platykurtic data be reliable for decision-making?
Yes, but it’s essential to understand the implications. While platykurtic distributions offer insights, they may require additional analysis for accurate interpretation.
How does platykurtic differ from other distributions like leptokurtic?
Unlike leptokurtic distributions with taller peaks and fatter tails, platykurtic distributions have flatter peaks and thinner tails, indicating less concentration of data around the mean. | {"url":"https://www.stockgro.club/learn/share-market/understanding-platykurtic/","timestamp":"2024-11-06T21:13:56Z","content_type":"text/html","content_length":"112337","record_id":"<urn:uuid:c4419ee1-0a14-42a8-8880-b22e8b9b78e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00618.warc.gz"} |
Primordial Universe and Inflation
Looking back in time observing the first light ever emitted, the Cosmic Microwave Background, we can investigate and understand how the Universe was in its first instants, how it evolved up to what
we observe today and how structures as galaxies and galaxy clusters originated. The classical Big Bang theory states that the Universe was generated from a hot dense state and subsequently expanded
and cooled allowing the formation of particles, matter and cosmic structures.
Thanks to cosmological observations, the classical theory has been integrated, and became the standard cosmological model, with the introduction of the dark sectors, like the dark energy and dark
matter to explain the recent acceleration of the Universe expansion and the large-scale structures formation and dynamics. The addition of an inflationary phase allowed solving some remaining issues
of the model.
Inflation is defined as a phase of accelerated expansion. During this period the energetic content of the Universe is dominated by a component which exerts a negative pressure (with an equation of
state parameter smaller than -1/3) allowing a quasi-exponential expansion, the simplest candidate which satisfies this condition being a scalar field. In the simplest inflationary models, a scalar
field, called inflaton, is the only responsible of inflation but there are also several models predicting the existence of multiple fields with different natures, like scalars, pseudo-scalars,
vectors etc., involved in the inflationary phase. An inflationary phase taking place before the hot Big Bang would explain the size of our observable Universe (which must be at least the CMB largest
observable scale) and its almost perfect Euclidean geometry (flatness) together with the abundances of topological relicts, which have been the three historical problems affecting the original Big
Bang theory. Inflation also provides a natural mechanism to generate the primordial fluctuations (through quantum effects) which after inflation grew up through gravitational instability generating
galaxies and galaxy clusters.
Inflation is supported both on theoretical and observational grounds (the spectral index of scalar perturbations being different from scale invariance in first row) but is still lacking the
definitive proof: the presence of a B-mode polarization signal from primordial gravitational waves, an unavoidable prediction of inflation. This signal is the primary target of the LiteBIRD mission
and its detection will provide the energetic scale at which primordial perturbations were created. The measurement of the primordial B-mode angular power spectrum, supported also by non-Gaussianity
measures also provided by LiteBIRD, will disclose the dynamics of the early Universe and select which model of inflation among the several proposed occurred. On the other hand, it may even point
towards inflation alternatives like bouncing models or Ekpyrotic Universes, carrying in all the cases profound consequences for the fundamental physics we know. In fact, the early Universe, and as a
consequence its observables like the CMB, represents the best high energy laboratory that we have to test and investigate fundamental and particle physics allowing to test theories at energies not
ever reachable by any Earth bound experiment.
LiteBIRD might also provide information on the relic neutrinos, that weak processes occurring in the early Universe brought in thermal equilibrium with the rest of the cosmological plasma. Flavour
oscillation experiments have shown that neutrinos have a mass. However, oscillation experiments only measure mass differences between the three neutrino mass eigenstates. Thus, we do not know how
massive they are – we only know that they are much lighter than the other known fermions in the Standard Model (SM) of particle physics, like the electron. In fact, laboratory experiments looking at
the beta decay of 3H have shown that the electron neutrino should be lighter than ~2 eV, while future experiments using the same technique could improve this limit by one order of magnitude.
Moreover, we do not know which of the two possibilities for the neutrino mass ordering, i.e the so-called normal (the two lighter neutrinos are closer in mass) or inverted (the two heavier neutrinos
are closer in mass) ordering, is actually realized in nature. The smallness of neutrino masses is a puzzling fact by itself, and might point to the fact that neutrinos do not acquire mass exclusively
through their coupling to the Higgs boson, but that instead the mass generation mechanism is related to some high-energy scale, far higher than the electroweak scale probed in accelerators on Earth.
Cosmological observations are a powerful probe of neutrino masses, thanks to the peculiar effect that such light particles have on structure formation, slightly hindering the clustering of matter at
small scales due to their large thermal velocities. In fact, cosmology currently provides the strongest constraints on neutrino masses: the 2018 Planck data, together with measurements of baryon
acoustic oscillations (BAO), constrain the sum of neutrino masses m to be below 0.12 eV (95% CL). This value is very close to 0.1 eV, the value that could allow to discriminate between the two
possibilities for neutrino mass ordering. An accurate measurement of the small-scale CMB lensing, that probes the distribution of matter between us and the last scattering surface, together with
baryon acoustic oscillation data or galaxy lensing/clustering data would allow to reach a higher sensitivity on neutrino masses, with an uncertainty ( m) = 0.02 eV or better. Given that the results
of oscillation experiments imply m> 0.06 eV, future-generation CMB experiments might be able to finally provide a statistically significant (>3) measurement of neutrino masses, as well as evidence
for the normal mass ordering if the sum of the masses is close to 0.06 eV.
LiteBIRD will not measure the small-scale pattern of CMB anisotropies necessary to probe the CMB lensing effect, that will instead be probed by ground-based experiments like the future CMB-S4
experiment. However, a precise (cosmic-variance limited) measurement of the optical depth to reionization like that provided by LiteBIRD is of paramount importance to reach the sensitivity to m
quoted above. In fact, the suppression of anisotropy power caused by reionization can somehow “confuse” the signature of massive neutrinos. Observing the large-scale polarization anisotropies,
unaccessible from the ground, as LiteBIRD is specifically designed to do, would provide a measurement of the reionization peak and an independent, cosmic-variance limited (() = 0.002) estimate of ,
minimizing the “confusion” effect described above.
1. A. Starobinsky, “Spectrum of relict gravitational radiation and the early state of the universe”, JETP Lett. 30, 682 (1979)
2. A. Starobinsky, “A new type of isotropic cosmological models without singularity” Phys. Lett. B91, 99 (1980).
3. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23 (1981) 34.
4. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,” Phys. Lett. B 108 (1982) 389.
5. F. Mukhanov and G. Chibisov, “Quantum fluctuations and a nonsingular universe” JETP Lett. 33, 532 (1981)
6. F. Mukhanov and G. Chibisov, “Vacuum energy and large-scale structure sf the Universe”, Sov.Phys.JETP 56, 258 (1982)
7. A. Starobinsky, “Dynamics of phase transition in the new inflationary universe scenario and generation of perturbations” Phys. Lett. B117, 175 (1982)
8. M. Maldacena, “Non-Gaussian features of primordial fluctuations in single field inflationary models,” JHEP 0305 (2003) 013.
Planck Collaboration “Planck 2018 Results X: Constraints on Inflation” arXiv:1807.06211 (2018).
R.H. Brandenberger “Introduction to Early Universe Cosmology” PoS (ICFI 2010) 001
1. L. Lehners “Ekpyrotic and cyclic cosmology” Physics Reports Volume 465, Issue 6, September 2008, Pages 223-263
“Neutrino Masses, Mixing, and Oscillations” in M. Tanabashi et al. (Particle Data Group), “Review of particle physics” Phys. Rev. D. 98, 03001 (2018).
A.D. Dolgov. “Neutrinos in cosmology”. Phys Rept. (2002) 370:333–535.
79.
Lesgourgues, S. Pastor, “Massive neutrinos and cosmology” Phys Rept. (2006) 429:307–79.
80. Gerbino and M. Lattanzi, “Status of Neutrino Properties and Future Prospects—Cosmological and Astrophysical Constraints”, Front. Phys. 5:70. (2018).
“Neutrinos in Cosmology” in M. Tanabashi et al. (Particle Data Group), “Review of particle physics” Phys. Rev. D. 98, 03001 (2018).
Planck Collaboration, “Planck 2018 results. VI. Cosmological parameters”, arXiv:1807.06209 [astro-ph.CO].
1907. Abazajian et al., “CMB-S4 Science Case, Reference Design, and Project Plan”, arXiv:1907.04473 [astro-ph.IM]. | {"url":"https://www.litebird-europe.eu/litebird-europe-draft-web-page/primordial-universe-and-inflation","timestamp":"2024-11-15T04:33:38Z","content_type":"text/html","content_length":"22283","record_id":"<urn:uuid:707f86af-4318-44a4-a767-22e1be97307d>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00336.warc.gz"} |
Checkbox condition in a child row, conditional formatting of parent?
I have an uber parent row that is the mother of a group of related tests for software QA. In addition, some of the children of that mother have their own children.
I would like any parent row to turn red if any of its children fail a test, as noted by a "Fail" checkbox column, and bonus points for conversely turning green when all of the children have passed
(as denoted by a "Pass" checkbox column).
• I am looking for a similar solution, in my post too. Thanks for posting this.
• Using a formula in your parent row in that column would make it auto check when at least one child row is checked.
=IF(COUNTIF(CHILDREN(), 1) > 0, 1, 0)
Try something like that out.
• Nice, totally works—the auto check is a great solution, thanks!
I do wish I understood how it works, exactly. Like in common language, what is being said? I'm trying to wrap my head around how to understand and construct these formulas, I have no Excel
background in such things, alas.
This is how I'm trying to break it down, seems like what is in bold is saying what I've noted below:
=IF(COUNTIF(CHILDREN(), 1) > 0, 1, 0)
"if this row has children with a checkbox in this column and it is checked"
...but the opening IF, the "greater than" sign and the 0, 1, 0 (false, true, false)....just not quite getting how that is resulting in the parent being auto-checked :-/
I also can foresee issues if a row that is bot a parent and a child has the checkbox manually messed with, as that seems to delete the formula from the cell. Wish there was a way to lock
Anyway, THANKS AGAIN!
• Hi,
=IF(COUNTIF(CHILDREN(), 1) > 0, 1, 0)
If the number of children with a checkbox checked is more than zero > then check the box and if none are checked > then do nothing.
Hope that helps!
Have a fantastic week!
Andrée Starå
Workflow Consultant @ Get Done Consulting
SMARTSHEET EXPERT CONSULTANT & PARTNER
Andrée Starå | Workflow Consultant / CEO @ WORK BOLD
W: www.workbold.com | E:andree@workbold.com | P: +46 (0) - 72 - 510 99 35
Feel free to contact me for help with Smartsheet, integrations, general workflow advice, or anything else.
• Thanks for your reply, Andrée!
So basically this:
IF = If
(COUNTIF(CHILDREN(), = number of children with a checkbox
1) = checked
> 0,1, = is more than zero check the box (in this cell)
0) = else do nothing
Is that right? Doesn't seem quite right :-/
• Suzanne,
That is correct. This would go in the parent row of the Fail column to basically say that if at least one child row fails, the parent row will be a fail.
To mark the parent row as Pass if ALL children pass, you would use something like this...
=IF(COUNTIFS(CHILDREN(), 1) = COUNT(CHILDREN(Task@row)), 1)
These formulas would go only in the parent rows.
• Happy to help!
That is correct! What doesn't seem right?
SMARTSHEET EXPERT CONSULTANT & PARTNER
Andrée Starå | Workflow Consultant / CEO @ WORK BOLD
W: www.workbold.com | E:andree@workbold.com | P: +46 (0) - 72 - 510 99 35
Feel free to contact me for help with Smartsheet, integrations, general workflow advice, or anything else.
• Hi Andrée-
It works fine, I'm just trying to understand how the formulas are built...learn the language, if you will. Thank you for confirming :-)
• Thanks, Paul. This solution is working for me, with the caveats that 1) when/if someone manually fusses with the checkbox to which the formula has been added, the formula apparently deletes
itself; and 2) I am applying the formula to all of the checkboxes in the column to allow for the flexibility that a child could become a parent at any moment (just like in real life, alas). This
isn't causing a problem (yet).
My goal is that this be a bullet-proofQA tool that can be easily used by others who are not very familiar with Smartsheet. I've used it quite a bit, but am ashamed to say I haven't invested the
time to really understand how to string together formulas, and instead just tend to mimic stuff I find online. I really want to gain a better understanding!
• Unfortunately there is no way to preserve the formula if someone manually alters a checkbox.
This will also be true when/if a child becomes a parent row. If a child row has already been manually changed and the formula overwritten, the formula would need to be re-added once the row
becomes a parent row.
One option to help with this would be to lock all of the parent rows and make it so that people can only edit child rows.
• Excellent!
Happy to help!
Here's an excellent resource for more information about the different functions and their structure: https://help.smartsheet.com/functions
Hope that helps!
SMARTSHEET EXPERT CONSULTANT & PARTNER
Andrée Starå | Workflow Consultant / CEO @ WORK BOLD
W: www.workbold.com | E:andree@workbold.com | P: +46 (0) - 72 - 510 99 35
Feel free to contact me for help with Smartsheet, integrations, general workflow advice, or anything else.
• There is also a template in the Solution Center that is called "Smartsheet Formula Examples". It is an interactive sheet with sample data in it. It has an example of every function that you can
experiment with. If you happen to accidentally save it after messing something up, you can delete the sheet and download a fresh one from the same template. The only catch with this one is that
you will need to occasionally download a fresh copy to ensure your examples sheet is completely up to date.
• Thanks, again Andrée! Really appreciate the swift assistance.
• Hey Paul, thanks very much. Now that you mention it, I remember that resource, and will seek it out again for sure. I've been away from Smartsheet for a couple years and am hoping to sell it in
to my new org...just gotta get my ducks in a row first!
Help Article Resources | {"url":"https://community.smartsheet.com/discussion/53031/checkbox-condition-in-a-child-row-conditional-formatting-of-parent","timestamp":"2024-11-07T08:50:45Z","content_type":"text/html","content_length":"446883","record_id":"<urn:uuid:a06aa1c2-65bd-4333-bd49-9308dc16c7c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00484.warc.gz"} |
Determine which of the following functions are one to one
and explain your reasoning.
1. f(x)=...
Determine which of the following functions are one to one and explain your reasoning. 1. f(x)=...
[Determine which of the following functions are one to one and explain your reasoning.]
1. f(x)= x^2-3x-10
2. g(x)= -2x^3+1 | {"url":"https://justaaa.com/math/1300747-determine-which-of-the-following-functions-are","timestamp":"2024-11-11T05:00:42Z","content_type":"text/html","content_length":"38560","record_id":"<urn:uuid:c409fbb7-8197-475e-9505-c55c64e286ec>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00239.warc.gz"} |
Understanding Man-Hours: The Intersection of Labor and Time
Written on
Chapter 1: The Concept of Man-Hours
In previous discussions, we explored the multiplication of different items—like apples with oranges—to derive a total quantity. Now, we shift our focus to multiplying individuals by time to generate
what we call man-hours, a term that may initially seem peculiar.
After examining how to multiply various quantities (like apples and oranges), we will now consider how to multiply people (men) by a measurable unit (time).
Understanding the Term "Man-Hour"
What exactly is a man-hour? It is not simply a rate defined as “men per hour” nor a measure of “work performed per person each hour.” Instead, a man-hour serves as a unit that compares the volume of
work performed by one person over a single hour against other tasks.
The formula can be expressed as:
Number of people × Number of hours = Man-hours
While related to the rate of work done per individual in an hour, the man-hour is a distinct unit. Its utility is prevalent in project management, where it assists in estimating both time and
financial resources needed for various tasks.
What kind of work does it measure? This often varies by context. For instance, in construction, it might be quantified as “blocks laid” or “ditches dug.”
Assuming an average of 50 blocks laid per person per hour, we can express this as:
50 blocks / 1 man / 1 hour
Reorganizing this, we arrive at:
50 ÷ 1 ÷ 1 blocks / man / hour = 50 blocks / man-hour
This transition from “blocks / man / hour” to “blocks / man-hour” may seem confusing if unfamiliar, but it represents a new unit—combining a multitude (man) and a magnitude (time).
Visualizing Man-Hours
To comprehend the concept of man-hours, it’s important to clarify what it is not. A man-hour is not a rate like “men per hour.” For example, if 72 individuals traverse a bridge in one day, we could
express this as:
72 men / 24 hours = 3 men / hour
Yet, again, a man-hour is not a simple rate; it is a compound unit that combines both the number of individuals and the time they spend working.
One might visualize the man-hour as a combination of a stick figure and an hourglass or perhaps a figure integrated within a clock face. The essence is to convey that the man-hour combines two
separate entities into a single measure.
In practical terms, man-hours are utilized to gauge the time required for completing tasks, and since “time is money,” they can also serve to estimate the cost involved.
Multiply time increments, multiply measures (personal video) - YouTube: This video delves into how to multiply time increments, aiding in understanding the essence of man-hours and their
Applying Man-Hours in Project Management
Now, let’s consider how to apply the concept of man-hours effectively. For instance, if a foreman understands that an average worker can lay 50 blocks in one hour, they can use this information to
estimate the time needed for larger tasks.
Suppose a project requires laying 200 blocks. The comparison would be:
50 blocks / 1 man-hour : 200 blocks / ? man-hours
By analyzing these figures, we find that 200 blocks is four times the amount of 50 blocks:
200 ÷ 50 = 4
Thus, the foreman concludes that four man-hours are necessary. This could be achieved by having one worker lay blocks for four hours, or four workers could each work for just one hour.
The Mythical Woman-Month
After discussing man-hours, it is worth considering an analogous unit: the “woman-month.” This term could describe the work done by one woman over a month.
Using pregnancy as an example, it typically takes one woman about nine months to complete a pregnancy. This can be represented mathematically as:
1 pregnancy / 1 woman / 9 months
If we desire to accelerate the process and finish a pregnancy within one month, we could think about either increasing the number of women or extending the time. However, it becomes absurd to suggest
that nine women could complete a pregnancy in one month, as this ignores the biological reality.
This discussion highlights the risk of relying too heavily on arithmetic when evaluating complex situations. The famed work, "The Mythical Man-Month" by Fred Brooks, illustrates this point well,
emphasizing that adding more individuals to a project often complicates rather than simplifies the process.
Multiplication of Hours, Minutes & Seconds | Multiplying Time - YouTube: This video further explores the multiplication of time units, reinforcing the principles discussed in this article.
In conclusion, while units like man-hours and woman-months can be useful for measuring work and time, it is crucial to remain aware of their limitations and the contexts in which they apply. When
applied thoughtfully, these concepts can greatly aid in project management and resource allocation. | {"url":"https://provocationofmind.com/understanding-man-hours-labor-time.html","timestamp":"2024-11-09T19:42:18Z","content_type":"text/html","content_length":"12713","record_id":"<urn:uuid:09de95f5-b939-475f-93df-aa364b21ebb3>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00786.warc.gz"} |
Ghost Ratings
There are two formulas used to calculate your GR, one for 1v1 games and one for non-1v1 games. Both of these formulas are based on the ELO rating system. However, they have different ways of
computing the score that you are expected to attain given the ratings of the competition you are playing against and the actual score that you achieve in a given game.
In both 1v1 and non-1v1 games, your rating is calculated using the following general formula:
ratingAdjustment = x * (expectedScore - actualScore)
Classic Formula
In the non-1v1 formula, "x" represents a scaling modifier that changes based on the settings of the game, derived by the following equation:
x = sumOfEachPlayersGR / (17.5 * pressWeightModifier * variantWeightModifier)
The press weight modifiers and variant weight modifiers for each category is listed with the category details.
We calculate your expected score in a non-1v1 game with the following equation:
expectedScore = yourGR / sumOfEachPlayersGR
If you draw in a draw-size scoring (DSS/WTA) game, we calculate your actual score with the following equation:
actualScore = 1 / numberOfPlayersInTheDraw
If you draw in a sum-of-squares scoring (SoS) game, we calculate your actual score with the following equation:
actualScore = yourSupplyCenters ^ 2 / allSupplyCenters ^ 2
Your actual score in a non-1v1 game is always 1 if you solo and 0 if you are defeated.
1v1 Formula
In a 1v1 formula, "x" is always 32. This value is constant, and therefore never changes.
We calculate your expected score in a 1v1 game with the following equation:
expectedScore = (10 ^ (yourGR / 400)) / ((10 ^ (yourGR / 400)) + (10 ^ (opponentGR / 400)))
Your actual score in a 1v1 game is 1 if you solo, 0.5 if you draw, and 0 if you are defeated. | {"url":"http://kaffelport.net/ghostRatings.php","timestamp":"2024-11-06T16:41:58Z","content_type":"application/xhtml+xml","content_length":"25208","record_id":"<urn:uuid:f3f66ef1-ee12-47de-a8d5-e5b983afedbd>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00282.warc.gz"} |
Bootstrapped Association Rule Mining in R
Association rule mining is a machine learning technique designed to uncover associations between categorical variables in data. It produces association rules which reflect how frequently categories
are found together. For instance, a common application of association rule mining is the “Frequently Bought Together” section of the checkout page from an online store. Through mining across
transactions from that store, items that are frequently bought together have been identified (e.g., shaving cream and razors).
Association rules are structured as “if-then” statements. For instance, if a customer bought shaving cream, then they also bought razors. The “if-then” portions of association rules are often
referred to as the antecedent (“if”) and the consequent (“then”). It is worth noting that association rules merely denote co-occurrence, not causal patterns. Based on the results of association rule
mining, we cannot determine the cause of buying razors, merely that it is associated with purchasing shaving cream.
There are several algorithms available for association rule mining. We will focus on the Apriori algorithm for this article as it is one of the most common. The Apriori algorithm will search the data
for the most frequently occurring sets of items. Typically, the antecedent can contain any number of items while the consequent will contain only one item. To evaluate the importance of each rule,
several common metrics can be evaluated. For this article we will focus on only three: support, confidence, and lift. Support is defined as how often this item set appears in a dataset. Confidence
represents the proportion of rules containing both the antecedent and the consequent. Lift represents how much more likely the consequent is to occur when the antecent is present as compared to when
it is absent; it is the ratio of the confidence of a rule to the frequency of the consequent in the whole dataset.
Association rule mining in R
To conduct association rule mining in R (v4.4.0; R Core Team 2023), we will use the apriori() function from the arules package (v1.7-7; Hahsler et al., 2023). Within the arules package there is a
dataset called Groceries. This dataset represents transactions at a grocery store for 1 month (30 days) from a real world grocery store. There are 9,835 transactions across 169 categories. First, we
will load the arules package. Then, we will load in the Groceries dataset. If you don’t already have the arules package, you can install it by running install.packages(“arules”) in the console.
library(arules) # Load the arules package
data(Groceries) # Read Groceries into Global Environment
Let’s take a look at the Groceries dataset. To inspect the dataset, let’s first call the class() function to show us the class of the object.
class(Groceries) # Inspect class of Groceries dataset
[1] "transactions"
[1] "arules"
The Groceries dataset is of class “transactions”. This class of object is specific to the arules package. The functions in the arules package used to conduct association rule mining specifically take
objects of class “transactions”. To convert an object to the class “transactions”, the transactions() function can be used. Objects of this class have a unique structure. First, these objects are S4
objects, meaning that to call anything contained within Groceries we need to use the “@” symbol. In order to look at the items contained in Groceries, we can call the itemInfo object by running
Groceries@itemInfo. This would print a data frame of all the items and their information contained in the Groceries object. Since it’s a data frame, we can wrap this object in head() to look at the
first six rows.
labels level2 level1
1 frankfurter sausage meat and sausage
2 sausage sausage meat and sausage
3 liver loaf sausage meat and sausage
4 ham sausage meat and sausage
5 meat sausage meat and sausage
6 finished products sausage meat and sausage
This shows us that there are three variables called labels, level2, and level1. The labels variable contains all of the items; level2 shows what category those items are in, and level1 shows the
category that level2 is in.
The apriori() function in the arules package will allow us to apply the Apriori algorithm to these data. This function defaults to mining only rules with a minimum support value of 0.10, a minimum
confidence value of 0.80, and a maximum of 10 items in the transactions. In order to prevent the function from running for too long, it will also time out checking for subsets after 5 seconds.
Transaction lengths refer to the entire itemset, including both the antecedent and the consequent. This function defaults to a minimum transaction or itemset length of 1, meaning that a rule could be
returned containing only the antecedent. Let’s run the first model using these preset values and save the model as an object called rules_1.
rules_1 <- apriori(Groceries)
Parameter specification:
confidence minval smax arem aval originalSupport maxtime support minlen
0.8 0.1 1 none FALSE TRUE 5 0.1 1
maxlen target ext
10 rules TRUE
Algorithmic control:
filter tree heap memopt load sort verbose
0.1 TRUE TRUE FALSE TRUE 2 TRUE
Absolute minimum support count: 983
set item appearances ...[0 item(s)] done [0.00s].
set transactions ...[169 item(s), 9835 transaction(s)] done [0.00s].
sorting and recoding items ... [8 item(s)] done [0.00s].
creating transaction tree ... done [0.00s].
checking subsets of size 1 2 done [0.00s].
writing ... [0 rule(s)] done [0.00s].
creating S4 object ... done [0.00s].
Using the default setting, this returns 0 relevant rules. Let’s adjust some of the default settings and see how this affects the model. Within the apriori() function we will pass a named list to the
argument called parameter. In this list, we will adjust minimum support values with an argument called supp, minimum confidence values with an argument called conf, and the maximum length of rules
with an argument called maxlen. We will save this model to an object called rules_2.
rules_2 <- apriori(Groceries, parameter = list(supp = .01, # Minimum Support value
conf = .5, # Minimum Confidence value
maxlen = 20)) # Maximum rule length
Parameter specification:
confidence minval smax arem aval originalSupport maxtime support minlen
0.5 0.1 1 none FALSE TRUE 5 0.01 1
maxlen target ext
20 rules TRUE
Algorithmic control:
filter tree heap memopt load sort verbose
0.1 TRUE TRUE FALSE TRUE 2 TRUE
Absolute minimum support count: 98
set item appearances ...[0 item(s)] done [0.00s].
set transactions ...[169 item(s), 9835 transaction(s)] done [0.00s].
sorting and recoding items ... [88 item(s)] done [0.00s].
creating transaction tree ... done [0.00s].
checking subsets of size 1 2 3 4 done [0.00s].
writing ... [15 rule(s)] done [0.00s].
creating S4 object ... done [0.00s].
This returns a set of 15 rules. We can use the inspect() function to return the rules listed in this object.
rules_2_summary <- head(inspect(rules_2))
lhs rhs support
[1] {curd, yogurt} => {whole milk} 0.01006609
[2] {other vegetables, butter} => {whole milk} 0.01148958
[3] {other vegetables, domestic eggs} => {whole milk} 0.01230300
[4] {yogurt, whipped/sour cream} => {whole milk} 0.01087951
[5] {other vegetables, whipped/sour cream} => {whole milk} 0.01464159
[6] {pip fruit, other vegetables} => {whole milk} 0.01352313
confidence coverage lift count
[1] 0.5823529 0.01728521 2.279125 99
[2] 0.5736041 0.02003050 2.244885 113
[3] 0.5525114 0.02226741 2.162336 121
[4] 0.5245098 0.02074225 2.052747 107
[5] 0.5070423 0.02887646 1.984385 144
[6] 0.5175097 0.02613116 2.025351 133
In the output of rules_2 there are columns labeled rhs (right hand side) and lhs (left hand side). These refer to the antecedent (lhs) and consequent (rhs). The first rule can be read as “if curd and
yogurt were purchased, then so was whole milk.”
The way we specified the apriori() function for rules_2 allowed it to search through all possible antecedent and consequent combinations. If we want to, we can also define which items we would like
to have in the antecedent and which items should be in the consequent. A list of possible values can be given for consideration in the antecedent. The algorithm will search through possible
combinations, ranging from no items to all items. A list of possible values will also be given for consideration in the consequent, however only one item can be present in the consequent within a
single rule. The algorithm will produce a set of rules in which the consequents will contain one of the items from the list provided and the antecedents will contain any combination of items from the
list provided. In the rules_3 object, we will use the appearance argument of the apriori() function to set these values. The appearance argument takes a list of rhs and lhs values.
rules_3 <- apriori(Groceries,
# Consequent
appearance = list(rhs = c("whole milk", "other vegetables"),
# Antecedent
lhs = c("yogurt", "whipped/sour cream", "tropical fruit", "root vegetables",
"citrus fruit", "domestic eggs", "pip fruit", "butter", "curd", "rolls/buns")),
parameter = list(supp = .01, # Minimum Support value
conf = .2, # Minimum Confidence value
maxlen = 20)) # Maximum rule length
Parameter specification:
confidence minval smax arem aval originalSupport maxtime support minlen
0.2 0.1 1 none FALSE TRUE 5 0.01 1
maxlen target ext
20 rules TRUE
Algorithmic control:
filter tree heap memopt load sort verbose
0.1 TRUE TRUE FALSE TRUE 2 TRUE
Absolute minimum support count: 98
set item appearances ...[12 item(s)] done [0.00s].
set transactions ...[12 item(s), 9835 transaction(s)] done [0.00s].
sorting and recoding items ... [12 item(s)] done [0.00s].
creating transaction tree ... done [0.00s].
checking subsets of size 1 2 3 done [0.00s].
writing ... [37 rule(s)] done [0.00s].
creating S4 object ... done [0.00s].
rules_3_summary <- head(inspect(rules_3))
lhs rhs support confidence coverage
[1] {} => {whole milk} 0.25551601 0.2555160 1.00000000
[2] {curd} => {other vegetables} 0.01718353 0.3225191 0.05327911
[3] {curd} => {whole milk} 0.02613116 0.4904580 0.05327911
[4] {butter} => {other vegetables} 0.02003050 0.3614679 0.05541434
[5] {butter} => {whole milk} 0.02755465 0.4972477 0.05541434
[6] {domestic eggs} => {other vegetables} 0.02226741 0.3509615 0.06344687
lift count
[1] 1.000000 2513
[2] 1.666829 169
[3] 1.919481 257
[4] 1.868122 197
[5] 1.946053 271
[6] 1.813824 219
It is possible for redundant rules to be produced. To find all such rules, we can use the is.subset() function from the arules package.
rules_3.sorted <- sort(rules_3, by = "lift") # Sort by lift
# Identify redundant rules
subset.matrix <- is.subset(rules_3.sorted,rules_3.sorted)
subset.matrix[lower.tri(subset.matrix,diag=T)] <- F
redundant <- colSums(subset.matrix) >= 1
# Which rules are redundant?
# Remove redundant rules
rules_3_pruned <- rules_3.sorted[!redundant] # Remove redundant rules
No rules were returned as redundant.
Applying association rule mining to a sample provides insight into the association rules and pairings present in a particular sample. However, in many analyses, we are more interested in generalizing
to other samples. We can apply a bootstrapping approach to assess the reproducibility of the rules we uncovered. For this example, we will generate 1,000 bootstrapped samples by randomly sampling
(with replacement) from the original dataset. Then, on each bootstrapped sample we will apply the Apriori algorithm. We investigate the association rules found in each sample and retain only those
rules that appeared in 90% or more of the bootstrapped samples. Association rule strength metrics (confidence, lift, and support) can then be computed on each bootstrapped sample. From this, we can
calculate the mean and 95% confidence interval for these metrics.
Bootstrapping the apriori() function
To bootstrap the apriori() function, we can set up a function called boot_apriori(). This function will accept seven arguments: data (dataset), rhs (consequent), lhs (antecedent), confidence (minimum
confidence), minlen (minimum rule length), maxlen (maximum rule length), and supp (minimum support value). First, the function will produce a bootstrapped sample using an embedded function called
bootstrap_sample(). This function will sample with replacement and create a new sample with only those cases. Using the data produced by resampling with replacement, the apriori() function is then
applied. The rules produced by the Apriori algorithm will be assessed for redundancy and the final pruned set of rules will be returned.
boot_apriori <- function(data, rhs, lhs, confidence, minlen, maxlen, supp){
# Create bootstrap samples using embedded function
bootstrap_sample <- function(data) { # This function only uses the given data
sampled_ids <- sample(seq_along(data), replace = TRUE) # Sample with replacement
sampled_transactions <- data[sampled_ids] # Create new data with only those cases
# Generate a bootstrap sample of the data
boot_data <- bootstrap_sample(data = data)
# Adjusting parameters
boot_rules_eh <- apriori(boot_data,
parameter = list(confidence = confidence, minlen=minlen,
maxlen = maxlen, supp = supp),
appearance = list(lhs=lhs, # antecedent (if)
rhs=rhs)) # consequent (then)
## find redundant rules
boot_rules_eh.sorted <- sort(boot_rules_eh, by = "lift")
if(length(boot_subset.matrix) > 0){
} else {
boot_rules_eh.pruned <- boot_rules_eh
We could apply this function 1,000 times using a process such as a for() loop or the replicate() function. However, this could take a very long time to run. Let’s see how long one iteration takes to
run using the tic() and toc() functions from the tictoc package (v1.2.1; Izrailev, 2023).
# Load the tictoc package
tic() # Start the timer
set.seed(15) # Set seed
test <- boot_apriori(Groceries,
# Consequent
rhs = c("whole milk", "other vegetables"),
# Antecedent
lhs = c("yogurt", "whipped/sour cream", "tropical fruit", "root vegetables",
"citrus fruit", "domestic eggs", "pip fruit", "butter", "curd", "rolls/buns"),
supp = .01, # Minimum Support value
conf = .2, # Minimum Confidence value
minlen = 1, # Minimum rule length
maxlen = 20) # maximum rule length
Parameter specification:
confidence minval smax arem aval originalSupport maxtime support minlen
0.2 0.1 1 none FALSE TRUE 5 0.01 1
maxlen target ext
20 rules TRUE
Algorithmic control:
filter tree heap memopt load sort verbose
0.1 TRUE TRUE FALSE TRUE 2 TRUE
Absolute minimum support count: 98
set item appearances ...[12 item(s)] done [0.00s].
set transactions ...[12 item(s), 9835 transaction(s)] done [0.00s].
sorting and recoding items ... [12 item(s)] done [0.00s].
creating transaction tree ... done [0.00s].
checking subsets of size 1 2 3 done [0.00s].
writing ... [37 rule(s)] done [0.00s].
creating S4 object ... done [0.00s].
toc() # Stop current timer
test_summary <- head(inspect(test))
lhs rhs support
[1] {tropical fruit, root vegetables} => {other vegetables} 0.01148958
[2] {yogurt, whipped/sour cream} => {other vegetables} 0.01057448
[3] {root vegetables, yogurt} => {other vegetables} 0.01220132
[4] {butter, yogurt} => {whole milk} 0.01016777
[5] {root vegetables, rolls/buns} => {other vegetables} 0.01016777
[6] {tropical fruit, yogurt} => {other vegetables} 0.01443823
confidence coverage lift count
[1] 0.5765306 0.01992883 3.000094 113
[2] 0.5200000 0.02033554 2.705926 104
[3] 0.5150215 0.02369090 2.680019 120
[4] 0.6451613 0.01576004 2.490252 100
[5] 0.4784689 0.02125064 2.489810 100
[6] 0.4565916 0.03162176 2.375968 142
In this test we found 37 rules. This process did not take very long; however, to do this 1,000 times would take much longer. To speed this process up, let’s run this function using parallel
Parallel processing
Parallel processing allows for a large task to be split into smaller tasks efficiently distributed throughout the CPUs of a machine. The parallel package is included with R and contains a set of
functions that can easily parallelize most processes. To parallelize this process, we use the clusterExport() function. In order to use this function, we need to save all information that will be
necessary for the boot_apriori() function into the Global Environment. This includes all arguments being passed to boot_apriori(). Then, we will define the number of clusters to use with the
makeCluster() and the detectCores() functions. Finally, we can use the parLapply() function to run the boot_apriori() function 1,000 times and produce a list called models_1.
# Load the parallel package
nreps <- 1000 # Repeat the process 1000 times
# Save all arguments as objects
rhs = c("whole milk", "other vegetables")
lhs = c("yogurt", "whipped/sour cream", "tropical fruit", "root vegetables",
"citrus fruit", "domestic eggs", "pip fruit", "butter", "curd", "rolls/buns")
confidence = .05
minlen = 1
maxlen = 20
supp = .01
# Define the number of clusters
cl <- makeCluster(detectCores() - 1)
# Export objects
clusterExport(cl, varlist = c("Groceries", "rhs", "lhs", "supp","boot_apriori","nreps",
models_1 <- parLapply(cl, 1:nreps, function(x)
boot_apriori(Groceries, rhs, lhs, confidence, minlen, maxlen, supp))
# Stop the cluster
By bootstrapping association rule mining, we can gain insight into which rules would replicate over and over again if we were to keep resampling from the population. This gives a better idea of
whether or not a rule is an association that happens to appear in the given data, or is one that actually exists in the population. To investigate that in models_1, we can look to see which rules
exist in 90% or more of the bootstrapped samples.
# Organize the list of models produced by boost_apriori into a data.frame
rules <- lapply(models_1, inspect)
# Organize into data.frame
results <- do.call(rbind, rules)
# Count how many samples a rule appeared in
results_count <- results %>%
select(lhs,rhs) %>%
group_by(lhs, rhs) %>%
summarise(boot = n()) %>%
results <- merge(results, results_count)
# Remove bootstrapped samples with less than 90% of the rules reproduced
results_final <- results[results$boot > .9*nreps,]
table(results_final$lhs, results_final$rhs)
{other vegetables} {whole milk}
{} 921 921
{butter} 921 921
{citrus fruit} 921 921
{curd} 921 921
{domestic eggs} 921 921
{pip fruit} 921 921
{rolls/buns} 921 921
{root vegetables, rolls/buns} 0 913
{root vegetables, yogurt} 911 921
{root vegetables} 921 921
{tropical fruit, root vegetables} 903 0
{tropical fruit, yogurt} 902 921
{tropical fruit} 921 921
{whipped/sour cream} 921 921
{yogurt, rolls/buns} 0 910
{yogurt} 921 921
These are the rules that appeared in more than 90% of the bootstrapped samples. The first row indicates two rules. Since the antecedent is an empty set, we can interpret this as “no matter what other
items were bought, other vegetables were bought” and “no matter what other items were bought, whole milk was bought.” To avoid getting rules with an empty antecedent set, simply set the minimum rule
length to 2. For the next row, two more rules are given: “if butter was bought, then other vegetables were bought” and “if butter was bought, then whole milk was bought.”
Since we have bootstrapped samples, we can also calculate the mean and 95% confidence intervals for any metric of interest. In the below example, we will do this for confidence, but the variable can
be changed to look at the same parameters for lift or support.
library(tidyr) # unnest.wider()
library(Hmisc) # for smean.cl.normal()
results_final %>%
group_by(rhs, lhs) %>%
N = n(),
ci = list(smean.cl.normal(confidence))
) %>%
# A tibble: 29 × 6
# Groups: rhs [2]
rhs lhs N Mean Lower Upper
<chr> <chr> <int> <dbl> <dbl> <dbl>
1 {other vegetables} {butter} 921 0.361 0.360 0.362
2 {other vegetables} {citrus fruit} 921 0.349 0.348 0.350
3 {other vegetables} {curd} 921 0.322 0.321 0.324
4 {other vegetables} {domestic eggs} 921 0.351 0.349 0.352
5 {other vegetables} {pip fruit} 921 0.345 0.344 0.346
6 {other vegetables} {rolls/buns} 921 0.232 0.231 0.232
7 {other vegetables} {root vegetables, rolls/buns} 904 0.502 0.500 0.505
8 {other vegetables} {root vegetables, yogurt} 914 0.500 0.498 0.502
9 {other vegetables} {root vegetables} 921 0.434 0.433 0.435
10 {other vegetables} {tropical fruit, yogurt} 904 0.420 0.418 0.421
# ℹ 19 more rows
Here only the first 10 rows are displayed. To see all rows simply add %>% print(n = Inf). For the first rule, “if butter was bought, then so were other vegetables,” we can interpret the mean
confidence value as the percentage of transactions that supported this rule. The confidence intervals can be interpreted as follows: We are 95% confident that the true percentage of transactions
supporting this rule falls between 36.0% and 36.2%.
• Hahsler, M., Buchta, C., Gruen, B., & Hornik, K. (2023). arules: Mining Association Rules and Frequent Itemsets. R package version 1.7-6, https://CRAN.R-project.org/package=arules
• Izrailev, S. (2023). tictoc: Functions for Timing R Scripts, as Well as Implementations of “Stack” and “StackList” Structures. R package version 1.2. https://CRAN.R-project.org/package=tictoc
• R Core Team (2023). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/
Laura Jamison
StatLab Associate
University of Virginia Library
May 20, 2024
Categories related to this article: | {"url":"https://library.virginia.edu/data/articles/bootstrapped-association-rule-mining-r","timestamp":"2024-11-13T06:49:23Z","content_type":"text/html","content_length":"85665","record_id":"<urn:uuid:6d1163ef-ce1c-4eba-8ce7-af037bde4931>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00338.warc.gz"} |
Multiplication Coloring Worksheets
Math, especially multiplication, develops the foundation of countless academic self-controls and real-world applications. Yet, for numerous learners, mastering multiplication can position a
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Introduction to Math Multiplication Coloring Worksheets
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Coloring the fact picture builds memory association between the fact picture Individualize practice by having kids work on their frequently missed facts Create booklets of the coloring pages as an
ongoing memory tool QUICK DOWNLOAD Click on these links to download the coloring pages in batches Number Pictures 2x2 2x6 2x7 3x7 3x8
These math worksheets also help in improving the fine motor skills of students by making sure the coloring is done within the lines Students also tend to develop good hand and eye coordination while
learning math Multiplication coloring worksheets provide a sense of pride and achievement to younger children when they feel they have created
Value of Multiplication Technique Comprehending multiplication is critical, laying a strong foundation for advanced mathematical ideas. Math Multiplication Coloring Worksheets provide structured and
targeted method, cultivating a much deeper understanding of this basic math operation.
Advancement of Math Multiplication Coloring Worksheets
27 Free Multiplication Coloring Worksheets Motorolai425softwareun39110
27 Free Multiplication Coloring Worksheets Motorolai425softwareun39110
Practice Multiplication Facts with these Color by Number Worksheets Third grade and fourth grade students who are learning their multiplication facts will have a great time completing these fun
coloring pages They also make for a fun art activity for students in later grades This collection of worksheets is growing and I ll continue adding
Provide an worksheet consisting of an image of a flower for kids to color They need to multiply the given numbers and color the region where the correct answer is mentioned for example 2 x 8 16 Check
out the flower multiplication coloring worksheet given below To color the flower in this worksheet first solve the multiplication problem
From typical pen-and-paper workouts to digitized interactive formats, Math Multiplication Coloring Worksheets have advanced, accommodating varied discovering designs and choices.
Kinds Of Math Multiplication Coloring Worksheets
Standard Multiplication Sheets Basic exercises concentrating on multiplication tables, aiding students build a strong arithmetic base.
Word Issue Worksheets
Real-life scenarios integrated into troubles, enhancing important reasoning and application skills.
Timed Multiplication Drills Examinations developed to boost speed and precision, helping in fast mental math.
Benefits of Using Math Multiplication Coloring Worksheets
10 Best Free Printable Multiplication Coloring Worksheets PDF For Free At Printablee
10 Best Free Printable Multiplication Coloring Worksheets PDF For Free At Printablee
8 Multiplication Coloring Page Free Printable January 22 2023 These multiplication coloring page worksheets will help to visualize and understand multiplication as well as develop the color sense of
3rd and 4th grade students Students will learn basic multiplication methods and can improve their basic math and coloring skills with our
This color by multiplication worksheet includes a key for the colors to use for each square The color to use depends on the answer to the equation For example 3x3 9 and will be colored light blue The
answers can range from 0 70 with 7 different colors needed to complete this math multiplication mosaic
Boosted Mathematical Abilities
Consistent technique hones multiplication effectiveness, boosting general mathematics capacities.
Enhanced Problem-Solving Talents
Word issues in worksheets develop logical thinking and method application.
Self-Paced Learning Advantages
Worksheets suit private learning speeds, promoting a comfy and adaptable discovering atmosphere.
How to Produce Engaging Math Multiplication Coloring Worksheets
Including Visuals and Shades Dynamic visuals and shades capture attention, making worksheets visually appealing and involving.
Including Real-Life Scenarios
Relating multiplication to daily situations adds relevance and practicality to workouts.
Customizing Worksheets to Various Ability Levels Customizing worksheets based upon varying proficiency levels makes sure inclusive learning. Interactive and Online Multiplication Resources Digital
Multiplication Devices and Games Technology-based sources use interactive discovering experiences, making multiplication appealing and pleasurable. Interactive Websites and Apps On-line systems
provide diverse and easily accessible multiplication method, supplementing traditional worksheets. Personalizing Worksheets for Different Knowing Styles Aesthetic Learners Aesthetic help and layouts
help understanding for learners inclined toward visual learning. Auditory Learners Spoken multiplication troubles or mnemonics satisfy learners that understand ideas through acoustic methods.
Kinesthetic Learners Hands-on activities and manipulatives sustain kinesthetic learners in comprehending multiplication. Tips for Effective Implementation in Learning Uniformity in Practice Normal
technique reinforces multiplication skills, promoting retention and fluency. Balancing Repeating and Range A mix of repetitive workouts and diverse problem layouts keeps rate of interest and
comprehension. Providing Useful Feedback Feedback help in identifying areas of enhancement, encouraging continued progress. Challenges in Multiplication Technique and Solutions Inspiration and
Engagement Difficulties Tedious drills can bring about uninterest; innovative strategies can reignite motivation. Getting Rid Of Worry of Mathematics Adverse perceptions around mathematics can hinder
progress; developing a favorable knowing setting is crucial. Impact of Math Multiplication Coloring Worksheets on Academic Performance Research Studies and Study Findings Research study suggests a
positive relationship in between consistent worksheet use and boosted mathematics performance.
Math Multiplication Coloring Worksheets emerge as flexible devices, fostering mathematical proficiency in learners while fitting varied knowing styles. From fundamental drills to interactive on the
internet sources, these worksheets not only improve multiplication skills however additionally promote critical thinking and analytical abilities.
Free Printable Color By Number Multiplication Worksheets Free Printable
Multiplication Coloring Worksheets Multiplication Color worksheets Free multiplication
Check more of Math Multiplication Coloring Worksheets below
15 Best Halloween Multiplication Coloring Printables Printablee
10 Best Free Printable Multiplication Coloring Worksheets Printablee
Free Printable Math Multiplication Coloring Worksheets FREE PRINTABLE
Multiplication Coloring Pages At GetDrawings Free Download
Free Printable Multiplication Coloring Worksheets
Multiplication Coloring Pages At GetDrawings Free Download
Multiplication Coloring Worksheets Free Online PDFs Cuemath
These math worksheets also help in improving the fine motor skills of students by making sure the coloring is done within the lines Students also tend to develop good hand and eye coordination while
learning math Multiplication coloring worksheets provide a sense of pride and achievement to younger children when they feel they have created
Multiplication Coloring Coloring Squared
Multiplication Coloring We hope you like these multiplication worksheets If you enjoy them check out Coloring Squared Multiplication and Division It collects our basic and advanced multiplication and
division pages into an awesome coloring book Super Multiplication and Division 50 puzzles 14 95
These math worksheets also help in improving the fine motor skills of students by making sure the coloring is done within the lines Students also tend to develop good hand and eye coordination while
learning math Multiplication coloring worksheets provide a sense of pride and achievement to younger children when they feel they have created
Multiplication Coloring We hope you like these multiplication worksheets If you enjoy them check out Coloring Squared Multiplication and Division It collects our basic and advanced multiplication and
division pages into an awesome coloring book Super Multiplication and Division 50 puzzles 14 95
Multiplication Coloring Pages At GetDrawings Free Download
10 Best Free Printable Multiplication Coloring Worksheets Printablee
Free Printable Multiplication Coloring Worksheets
Multiplication Coloring Pages At GetDrawings Free Download
Free Printable Math Coloring Pages For Kids Cool2bKids
Math Coloring Pages Multiplication Coloring Home
Math Coloring Pages Multiplication Coloring Home
10 Best Free Printable Multiplication Coloring Worksheets PDF For Free At Printablee
Frequently Asked Questions (Frequently Asked Questions).
Are Math Multiplication Coloring Worksheets suitable for every age groups?
Yes, worksheets can be customized to different age and skill levels, making them versatile for numerous students.
Just how frequently should students practice utilizing Math Multiplication Coloring Worksheets?
Constant practice is crucial. Regular sessions, preferably a couple of times a week, can yield substantial renovation.
Can worksheets alone improve mathematics skills?
Worksheets are an important device however needs to be supplemented with varied understanding methods for thorough ability advancement.
Are there on the internet platforms providing complimentary Math Multiplication Coloring Worksheets?
Yes, numerous instructional websites use open door to a wide range of Math Multiplication Coloring Worksheets.
How can parents sustain their children's multiplication method in the house?
Encouraging regular practice, supplying aid, and developing a favorable understanding setting are beneficial actions. | {"url":"https://crown-darts.com/en/math-multiplication-coloring-worksheets.html","timestamp":"2024-11-05T06:28:01Z","content_type":"text/html","content_length":"29051","record_id":"<urn:uuid:ac20132f-0e18-4bcd-873b-b892bc4f7d7a>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00260.warc.gz"} |
Marty and Ethan both wrote a function, but in different ways. Marty y plus 3 equals StartFraction 1 Over 3 EndFraction lef - DocumenTVMarty and Ethan both wrote a function, but in different ways. Marty y plus 3 equals StartFraction 1 Over 3 EndFraction lef
Marty and Ethan both wrote a function, but in different ways. Marty y plus 3 equals StartFraction 1 Over 3 EndFraction lef
Marty and Ethan both wrote a function, but in different ways.
y plus 3 equals StartFraction 1 Over 3 EndFraction left-parenthesis x plus 9 right-parenthesis.
A two column table with 5 rows. The first column, x, has the entries, negative 4, negative 2, 0, 2. The second column, y, has the entries, 9.2, 9.6, 10, 10.4.
Whose function has the larger slope?
Marty’s with a slope of 2/3
Ethan’s with a slope of 2/5
Marty’s with a slope of 1/3
Ethan’s with a slope of 1/5
in progress 0
Mathematics 3 years 2021-08-24T15:01:09+00:00 2021-08-24T15:01:09+00:00 1 Answers 99 views 0 | {"url":"https://documen.tv/question/marty-and-ethan-both-wrote-a-function-but-in-different-ways-marty-y-plus-3-equals-startfraction-24114593-95/","timestamp":"2024-11-09T08:07:01Z","content_type":"text/html","content_length":"84604","record_id":"<urn:uuid:e5bbc9bc-e3f2-40a0-9d36-fc57e62a882b>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00257.warc.gz"} |
Gary Miller (computer scientist)
Gary Lee Miller is a professor of Computer Science at Carnegie Mellon University, Pittsburgh, United States. In 2003 he won the ACM Paris Kanellakis Award (with three others) for the Miller–Rabin
primality test. He was made an ACM Fellow in 2002^[1] and won the Knuth Prize in 2013.^[2]
Miller received his Ph.D. from the University of California, Berkeley in 1975 under the direction of Manuel Blum. His Ph.D. thesis was titled Riemann's Hypothesis and Tests for Primality.
Apart from computational number theory and primality testing, he has worked in the areas of computational geometry, scientific computing, parallel algorithms and randomized algorithms. Among his
Ph.D. students are Susan Landau, F. Thomson Leighton, Shang-Hua Teng, and Jonathan Shewchuk.
1. ↑ "ACM Awards Knuth Prize to Creator of Problem-Solving Theory and Algorithms" (Press release). Association for Computing Machinery. Retrieved 31 October 2013.
External links
This article is issued from
- version of the 4/13/2016. The text is available under the
Creative Commons Attribution/Share Alike
but additional terms may apply for the media files. | {"url":"https://ipfs.io/ipfs/QmXoypizjW3WknFiJnKLwHCnL72vedxjQkDDP1mXWo6uco/wiki/Gary_Miller_(computer_scientist).html","timestamp":"2024-11-05T05:48:47Z","content_type":"text/html","content_length":"15787","record_id":"<urn:uuid:75d24b2b-e1f9-4a33-ab3c-c47b895b7fdb>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00476.warc.gz"} |
Modeling an approximation of a converging gaussian beam in non-sequential mode | Zemax Community
I would like to model an approximation of a converging Gaussian beam source in the non-sequential mode. I first tried to model the source with Source DLL, (Guassianbeam.dll) and use a negative
position (Position in the object editor), so that the source is behind the waist, but it doesn’t look like the beams are converging on the waist. Of course, I understand why the divergence angle
should always be positive.
Basically, my model should be relatively simple: a converging gaussian beam hits a reflective object.The Gaussian beam waist is located inside the object. The detector’s distance from the beam waist
is also negative (~-40mm).
Perhaps I could use some lens objective to recreate an approximation of what I need, but it would be nice to have the appropriate source directly. I am using non-sequential mode, because there are
presumably some multiple reflections occurring inside the object.
Thank you, | {"url":"https://community.zemax.com/got-a-question-7/modeling-an-approximation-of-a-converging-gaussian-beam-in-non-sequential-mode-572","timestamp":"2024-11-12T12:32:10Z","content_type":"text/html","content_length":"200047","record_id":"<urn:uuid:cefa558f-b5c5-4ae2-a196-08f68bb8e8b3>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00399.warc.gz"} |
Understanding Betting Odds
Understanding Betting Odds
Odds are an important facet of sports betting. Understanding them and how to use them is crucial if you want to turn into a successful sports bettor. It’s likely that used to calculate how much money
you get back from winning wagers, but that’ s not every.
What you might not have known is that there are many different ways of expressing probabilities, or that odds are tightly linked to the probability of a bet winning.
They also dictate whether or not any particular wager represents good value or perhaps not, and value is usually something that you should always consider when ever deciding what bets to put. Odds
play an built-in role in how bookmakers make money too.
We cover everything you need to find out about odds on this page. We urge you to check out read through all this information, especially if you are relatively new to gambling.
However , if you want a visual overview of everything we all cover on this page, make sure to view our infographic in the this subject.
The Basics of Odds
As we’ ve already stated, odds are accustomed to determine the amounts released on winning bets. This is exactly why they are often referred to as the “ price” of a wager. A wager can have a price
that’ s either odds about or odds against.
Odds On – The potential amount you can earn will be less than the amount staked.
Odds Against – The potential amount you can win will be greater than the total amount staked.
You’ ll still make a profit via winning an odds about bet, as your initial risk is returned too, but you have to risk an amount that’ s higher than you stand to gain. Big favorites are usually odds
on, as they are very likely to win. When wagers may lose than win, they may typically be odds against.
Odds can be even money. A winning sometimes money bet will go back exactly the amount staked in profit, plus the original stake. So you basically double your dollars.
Different Possibilities Formats
Underneath are the three main formats used for expressing betting odds.
Moneyline (or American)
Most likely, you’ ll run into all of these formats when playing online. Some sites enable you to choose your format, sometimes don’ t. This is why knowing all of them is extremely beneficial.
This is the format most commonly used by simply betting sites, with the possible exception of sites which have a predominantly American customer base. This is probably because it is the simplest in
the three formats. Decimal probabilities, which are usually displayed applying two decimal places, present exactly how much a winning wager will certainly return per unit secured.
Here are some examples. Bear in mind, the total return includes the first stake.
Samples of Winning Wagers Returned Every Unit Staked
The calculation required to see the potential return when using fracci?n odds is very simple.
Stake x Odds = Potential Returns
In order to work out the potential revenue just subtract one from your odds.
Stake x (Odds – 1) = Potential Profit
Using the decimal format is as easy as that, which is why most betting sites stick with it. Note that 2 . 00 is the equivalent of even money. Anything higher than 2 . 00 is odds against, and anything
lower is odds on.
Moneyline odds, also known as American chances, are used primarily in the United States. Certainly, the United States always has to be diverse. Surprise, surprise. This file format of odds is a
little more difficult to understand, but you’ lmost all catch on in no time.
Moneyline odds could be either positive (the relevant number will be preceded with a + sign) or negative (the relevant number will be preceded by a – sign).
Positive moneyline odds show how much profit a winning bet of $100 would make. So if you saw likelihood of +150 you would know that a $100 wager could win you $150. In addition to that, you’ d also
get your stake back, for a total go back of $250. Here are some extra examples, showing the total potential return.
Sort of Total Potential Return 1
Negative moneyline odds show how much you should bet to make a $100 profit. So if you saw odds of -120 you would know that a wager of $120 could gain you $100. Again you will get your stake back, for
your total return of $220. To further clarify this concept, check out these additional examples.
Example of Total Potential Return 2
The easiest way to calculate potential profits from moneyline odds is to use the following formula when they are positive.
Stake x (Odds/100) = Potential Profit
If you want to find out the total potential return, easily add your stake to the result.
Pertaining to negative moneyline odds, this formula is required.
Stake / (Odds/100) = Potential Profit
Again, simply add your stake to the result intended for the total potential return.
Note: the equivalent of actually money in this format is bettingday.xyz definitely +100. When a wager is odds against, positive numbers are used. When a wager is definitely odds on, negative
statistics are used.
Fragmentary; sectional
Fractional it’s likely that most commonly used in the United Kingdom, where they are simply used by bookmaking shops and course bookies at equine racing tracks. This formatting is slowly being
changed by the decimal format even though.
Here are some basic examples of fractional odds.
2/1 (which has been said to as two to one)
10/1 (ten to one)
10/1 (ten to one)
And now some slightly more complicated good examples.
7/4 (seven to four)
5/2 (five to two)
15/8 (fifteen to eight)
These examples are all possibilities against. The following are some examples of odds on.
1/2 (two to one on)
10/11 (eleven to ten on)
4/6 (six to four on)
Note that even money is technically expressed as 1/1, but is typically referred to basically as “ evens. ”
Working out profits can be overwhelming at first, although don’ t worry. You can expect to master this process with enough practice. Each fraction displays how much profit you stand to make on a
winning guess, but it’ s up to you to add in your initial position.
The following calculations is used, where “ a” is the first number inside the fraction and “ b” is the second.
Stake x (a/b) = Potential Profit
Some people prefer to convert fractional odds into decimal chances before calculating payouts. To do this you just divide the first number by the second number and add one. So 5/2 in decimal odds
would be a few. 5, 6/1 would be several. 0 and so on.
Odds, Probability & Intended Probability
To create money out of wagering, you really have to recognize the difference among odds and probability. Even though the two are fundamentally linked, odds aren’ t actually a direct reflection of the
probability of something happening or certainly not happening.
Probability in sports betting is very subjective, plain and simple. Both bettors and bookmakers alike are going to have a difference of opinion when it comes to guessing the likely outcome of any
Possibilities typically vary by five per cent to 10%: sometimes less, sometimes more. Successful gambling is largely about making accurate assessments about the likelihood of an outcome, and then
deciding if the odds of that result make a wager worth it.
To make that determination, we need to understand meant probability.
In the context of wagering, implied probability is what the odds suggest the chances of any given result happening are. It can help us to calculate the bookmaker’ s advantage in a bets market. More
importantly, implied probability is something that can really help us determine whether or not a bet offers us value.
A great rule of thumb to have by is this; only ever before place a wager when there’ s value. Value prevails whenever the odds are placed higher than you think they should be. Implied probability
tells us whether or not this is the case.
To explain implied probability more obviously, let’ s look at this hypothetical tennis match. Imagine there’ s a match between two players of an the same standard. A bookmaker provides both players
the exact same probability of winning, and so prices the odds at 2 . 00 (in decimal format) for each participant.
In practice a bookmaker would never set chances at 2 . 00 about both players, for factors we explain a little after. For the sake of this example, even though, we will assume it’s this that they did.
What these odds are telling all of us is that the match is essentially just like a coin flip. There are two possible outcomes and each one is just as likely because the other. In theory, every single
player has a 50% probability of winning the match.
This 50% certainly is the implied probability. It’ t easy to work out in such a basic example as this one although that’ s not always the truth. Luckily, there’ s a formula for converting quebrado
odds into implied likelihood.
Implied Possibility = 1 / quebrado odds
This will give you a number of between actually zero and one, which is how probability should be expressed. It’ s easier to think of probability as a percentage though, and this can be calculated by
multiplying the consequence of the above formula by 90.
The odds in our tennis match example happen to be 2 . 00 as we’ ve already stated. So 1 / 2 . 00 is. 50, which multiplied by 100 gives us 50%.
If each player truly have have a 50% possibility of winning this match, after that there would be no point in placing wager on either one. You’ ve got a 50 percent chance of doubling your money, and
a 50% chance of getting rid of your stake. Your expectancy is neutral.
However , you might think that one person is more likely to win. Perhaps you have been following their contact form closely, and you believe that one of many players actually has a 60 per cent chance
of beating his challenger.
In this case, benefit would exist when betting on your preferred player. In case your opinion is accurate, you’ ve got a 60 per cent chance of doubling your money in support of a 40% chance of
shedding your stake. Your expectation is now positive.
We’ ve really simplified things here, as the objective of this page is just to explain each of the ways in which odds are relevant once betting on sports. We’ ve written another article which
explains implied possibility and value in considerably more detail.
For now, you should just understand that chances can tell us the intended probability of a particular final result happening. If our perspective is that the actual probability is certainly higher
than the implied likelihood, then we’ ve discovered some value.
Finding value is a crucial skill in sports betting, and one that you should try to master if you want to be successful.
Well balanced Books & The Overround
How do bookmakers make money? It is simple seriously; they try to take more money in losing wagers than they pay out in winning wagers. In reality, though, it isn’ t quite that easy.
If they offered completely fair chances on an event then they will not be guaranteed a profit and would be potentially exposed to risk. Bookmakers do NOT expose themselves to risk. Their objective is
to make a profit on every function they take bets on. This is where a balanced book and the overround come in play.
As we mentioned in the playing example above, in practice you wouldn’ t actually see two equally likely outcomes both priced at 2 . 00 by a bookmaker. Although this would technically represent fair
odds, this is NOT how bookmakers work.
For every celebration that they take bets in, a bookmaker will always check out build in an overround. They’ ll also try to make certain that they have balanced books.
When a terme conseill? has a balanced book for your event it means that they stand to pay out roughly the same amount involving regardless of the outcome. Let’ ersus again use the example of the
tennis match with odds of 2 . 00 of each player. If a bookmaker took $10, 500 worth of action on each of your player, then they would have a well-balanced book. Regardless of which participant wins,
they have to pay out an overall total of $20, 000.
Of course , a terme conseill? wouldn’ t make anything in the above scenario. They have taken a total of 20 dollars, 000 in wagers and paid the same amount out. Their goal is to be in a situation
exactly where they pay out less than they get in.
That is why, in addition to having a balanced reserve, they also build in the overround.
The overround is also known as vig, or juice, or margin. It’ s effectively a commission that bookmakers impose their customers every time they create a wager. They don’ testosterone levels directly
charge a fee though; they just reduce the possibilities from their true probability. So the odds that you would see on a tennis match exactly where both players were evenly likely to win would be
regarding 1 . 91 on each participant.
If you again assumed that they took $, 000 on each player, they would now be guaranteed a profit whichever player wins. All their total pay-out would be $19, 100 in winning bets against the total of
20 dollars, 000 they have taken. The $900 difference is the overround, which is usually expressed to be a percentage of the total book.
This above scenario is an ideal situation meant for my bookmaker. The volume of bets a bookmaker consumes is so important to them, mainly because their goal is to generate income. The more money they
take, the more likely they are to be able to create a well-balanced book.
The overround and the need for a balanced book is also why you can often see the odds meant for sports events changing. If the bookmaker is taking excessively on a particular outcome, they will
probably reduce the odds to discourage any further action.
Also, they might enhance the odds on the other possible final result, or outcomes, to inspire action against the outcome they have taken too many wagers about.
Be aware; bookies are not always successful in creating a balanced book, and do sometimes lose money by using an event. In fact , bookmakers losing money on an event isn’ capital t uncommon by any
means, BUT they do generally get close to becoming balanced far more often than not.
Remember though, just because the bookmakers be sure they turn a profit in the long run doesn’ t mean you can’ t beat them. You don’ t have to cause them to become lose money overall, you just have
to concentrate on making more money from your earning wagers than you lose with your losing wagers.
This may sound complicated, however it isn’ t. As long as you own a basic understanding of how bookies use overrounds and well balanced books and as long as you have a general understanding of how
odds are utilised in betting, then you have what you should be successful. | {"url":"https://alchemist-corp.com/understanding-betting-odds-146-2/","timestamp":"2024-11-08T04:43:08Z","content_type":"text/html","content_length":"73987","record_id":"<urn:uuid:07c2313c-344d-43b6-95fb-aa57fa1b0ae9>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00378.warc.gz"} |
Computing the p-Spectral Radii of Uniform Hypergraphs with Applications
The p-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of
hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the p-spectral radius of a uniform hypergraph
(CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral
radius of the adjacency tensor of a uniform hypergraph, CSRH outperforms existing approaches. Furthermore, CSRH is competent to calculate the p-spectral radius of a hypergraph with millions of
vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the p-spectral radius
Scopus Subject Areas
• Software
• Theoretical Computer Science
• Numerical Analysis
• Engineering(all)
• Computational Theory and Mathematics
• Computational Mathematics
• Applied Mathematics
User-Defined Keywords
• Eigenvalue
• Hypergraph
• Large scale tensor
• Network analysis
• p-spectral radius
• Pagerank
Dive into the research topics of 'Computing the p-Spectral Radii of Uniform Hypergraphs with Applications'. Together they form a unique fingerprint. | {"url":"https://scholars.hkbu.edu.hk/en/publications/computing-the-p-spectral-radii-of-uniform-hypergraphs-with-applic","timestamp":"2024-11-13T02:10:12Z","content_type":"text/html","content_length":"55632","record_id":"<urn:uuid:50972bd1-3d2b-4c92-a36f-66af336ecfae>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00061.warc.gz"} |
28 7th Grade Math Problems With Answers And Worked Examples
7th grade math problems build on students’ reasoning skills and understanding of key mathematical concepts.
We’ve created 28 7th grade math problems for your classroom, suitable for use as additional practice before an assessment, as a Do Now activity, or for an Exit Slip to check for understanding after
a lesson. Math teachers designed these problems to build on students’ math skills and help to prepare them for the rigor expected ahead in 8th grade and in high school courses like Algebra,
Geometry, and Statistics.
What are 7th grade math problems?
7th grade math problems combine 7th grade math concepts with grade-level appropriate reasoning and problem solving skills.
7th grade math content includes computing with all rational numbers in all four operations, using the correct order of operations to simplify expressions, solving problems involving percentages,
using proportional reasoning, solving 1 and 2 step equations, multi-step inequalities and extending knowledge of geometry.
Why focus on problem solving in 7th grade?
Excelling at computation is great; excelling at computation but not understanding application is pointless. The purpose of mathematics is to learn to reason, find patterns, and use logic, so solving
math problems that are rich, deep, and complex is necessary to meet those purposes.
Mathematics today has advanced beyond just computation and evolved into a marriage of numbers and words – not only do students need to get to a numerical answer they also need to be able to
communicate their thinking and processes along the way. They need repeated practice in reasoning, perseverance, and rational thinking to solve problems throughout their lives.
7th grade is a perfect time to extend problem solving activities in the classroom. The curriculum moves away from much of the memorization required in elementary school of basic addition facts and
times tables, and builds on 6th grade content to focus on making connections between mathematical reasoning, logic, and numbers.
28 7th Grade Math Problems
28 7th grade math questions targeting the skills and knowledge needed for the four operations, order of operations, percentages, proportional relationships, one-step and two-step equations, geometry
and word problems. Includes answers and explanations.
Download Free Now!
The 7th grade math curriculum
This article focuses on problems for 7th grade students, especially concepts taught in pre-algebra. Pre-algebra includes concepts of proportional relationships, algebraic expressions, equivalent
expressions, equations, rational numbers, graphs, and extension of knowledge of the number system, including whole numbers and mixed numbers. The problems align with common core standards.
Seventh grade math can be considered a turning point in mathematics. It bridges the concepts taught in elementary school, including the foundations of arithmetic and number sense, with the rigor
expected ahead in high school courses like Algebra, Geometry, and Statistics.
7th grade math problems
We’ve designed 28 7th grade math problems to use as a whole class or in a Tier 2 or Tier 3 intervention to help develop and secure students’ mathematical knowledge.
7th grade math problems: Four operations
1. Solve -4 + 10. Use the number line.
Solution: 6
2. Solve: -8 – 12. Use the number line.
Solution: -20
3. Solve: 4(-3)(-2)
Solution: 24
4. Bill said the answer to -3 – 12 is 9. What mistake did he make? What is the correct answer?
Solution: Bill knows that the opposite of subtraction is addition, but he forgot to take the opposite of 12, so he re-wrote the problem as -3 + 12. Since we are subtracting 12 from -3, the answer is
the same as -3+-12, which is -15.
7th grade math problems: Order of operations
5. Solve: 2(10-8) Ă· 2 + 4
Solution: 2(10-8) Ă· 2 + 4
2(2) Ă· 2 + 4
4 Ă· 2 + 4
2 + 4
6. Solve: (3 + 10 Ă· 2 – 6) x 6
Solution: (3 + 10 Ă· 2 – 6) x 6
(3 + 5 – 6) x 6
(8 – 6) x 6
2 x 6
7. Solve: -5(8) Ă· 2 + 6
Solution: -5(8) Ă· 2 + 6
              -40 ÷ 2 + 6
-20 + 6
                    -14
8. Solve: (-2)3 – 2 + 6 Ă· 3
Solution: (-2)3 – 2 + 6 Ă· 3
-8 – 2 + 6 Ă· 3
-8 – 2 + 2
-10 + 2
7th grade math problems: Percentages
9. Isabella got 16 out of 40 questions wrong on her quiz. What percent did she get correct?
Solution: \frac{16}{40} can be simplified to \frac{2}{5}, which is equivalent to \frac{40}{100} or 40%. If Isabella got 40% incorrect, she got 60% correct (100-40=60).
10. Without doing any computation, explain whether \frac{38}{72} is greater than or less than 50%.
Solution: \frac{38}{72} is greater than 50%. \frac{36}{72} is equivalent to \frac{1}{2}, which is equivalent to 50%. Since 38 is a little greater than 36, \frac{38}{72} is a little greater than 50%.
11. Put the following in order from least to greatest: \frac{3}{4}, 76%, 0.68, \frac{3}{5}, \frac{35}{50}, 0.702
Solution: \frac{3}{5}, 0.68, \frac{35}{50}, 0.702, \frac{3}{4}, 76%
12. A store marked all shoes on sale for 30% off. What percent will Sam pay for shoes? Explain your answer.
Solution: Sam will pay 70% for shoes. The full price is 100%, so if 30% is saved, the remaining 70% will be the sales price.
7th grade math problems: Proportional relationships
13. \frac{5}{6} = \frac{x+2}{15}
Solution: 6(x+2) = 5(15)
6x + 12 = 75
-12 -12
\frac{6x}{6}= \frac{63}{6}
x = 10.5
14. Three out of every five students are wearing jeans. If there are 20 students in total, how many are wearing jeans?
Solution: \frac{3}{5} = \frac{x}{20}
3 (20) = 5x
60 = 5x
12 = x
15. Three out of every five students are wearing jeans. If there are 20 students in all, how many are not wearing jeans?
Solution: From the last problem, we saw that \frac{3}{5} is the same as the 12 students wearing jeans. If there are 20 students total, we can subtract the 12 wearing jeans from the 20 total to find
that 8 are not wearing jeans. We could also set up this proportion and solve to get 8.
\frac{2}{5} =\frac{x}{20}
16. A museum requires 12 chaperones for the 60 students attending the field trip. How many students are assigned to each chaperone?
\frac{12}{60} = \frac{1}{x}
12x = 60(1)
12x = 60
x = 5
Each chaperone will have a group of 5 students.
7th grade math problems: One-step equations and two-step equations
17. Solve: x + 7.1 = 15.9
Solution: x + 7.1 = 15.9
-7.1 -7.1
x = 8.8
18. Solve: x – 63 = 106.75
Solution: x – 63 = 106.75
+63 +63
x = 169.75
19. Solve: 6(x + 3) = -6
Solution: 6x + 18 = -6
– 18 -18
\frac{6x}{6} = \frac{-24}{6}
                        x = -4
20. Solve: 0.5x + 10 = 36
Solution: 0.5x + 10 = 36
-10 -10
0.5x = 26
x = 52
7th grade math problems: Geometry
21. Madison measured this angle with her protractor and said “It is 60°.”
Without measuring the angle, Bella said she could tell Madison’s answer was incorrect. How did Bella know this?
Solution: Bella knew this angle could not be 60° because this angle is obtuse
but a 60° angle is acute.
22. Find the circumference of the circle.
Solution: C = πd
C = 15Ď€
23. Use the figure to fill in the blanks:
Angles A and B are _________________ angles so their measures are _______________________________.
Solution: Angles A and B are vertical angles so their measures are equal.
24. Find the value of x.
Solution: 5x + 4x =90
9x = 90
x = 10
7th grade math problems: Math word problems
25. The 7th Graders at Marxville Middle School voted for their student council representatives. There were 200 votes cast in all. How many votes did the winner get?
Solution: Alexandra won the election with 30% of the votes. To find 30% of the 200 total votes, we can multiply 0.3 (200) to discover that she got 60 votes in all.
26. Brian runs every 12 days and Stella every 8 days. Both Brian and Stella ran today. How many days will it be before they both run on the same day again?
Solution: This is a Least Common Multiple problem. Brian runs on days 12, 24, 36, 48… and Stella runs on days 8, 16, 24, 32…, so they will both run again on Day 24.
27. Mr. Orlando is planting his vegetable garden this summer. He plants \frac{3}{4} of the garden with peppers and \frac{1}{4} with tomatoes. Of the peppers, \frac{1}{3} are red peppers. What
fraction of the entire garden will be red peppers?
Solution: Red peppers will make up \frac{1}{3} of \frac{3}{4} (the pepper section) of the garden. \frac{1}{3} x \frac{3}{4} = \frac{1}{4}, so \frac{1}{4} of the entire garden will consist of red
28. Will the product of -45(96) be positive or negative? Without solving, how do you know?
Solution: The answer will be negative. Multiplying a negative number by a positive one always leads to a negative product.
Challenges in teaching 7th grade mathematics
Let’s face it, most 7th graders don’t wake up every morning excited to get to math class that day to expand on their mathematical knowledge and thinking! Often, there are much more pressing
matters in the mind of a 12 year old. However, with the right mindset and classroom climate, we can support our students to help them put forth their best effort every day.
1. Praise effort: Developing a growth mindset in the classroom is key. Even if a student struggles to come to an answer, it is critical to praise the effort. We must make the thinking part of
mathematics something to treasure rather than solely focus on correct numerical answers.
2. Include low threshold high ceiling activities: These go a long way in piquing the interest level of most students.
3. Have fun: Don’t be afraid to have some fun math activities and silliness in your lessons, whether that’s through funny estimation activities or videos to get students interested, engaged and
willing to try.
How can Third Space Learning help with 7th grade math?
STEM-specialist tutors help close learning gaps and address misconceptions for struggling 7th grade math students. One-on-one online math tutoring sessions help students deepen their understanding of
the math curriculum and keep up with difficult math concepts.
Each student works with a private tutor who adapts instruction and math lesson content in real-time according to the student’s needs to accelerate learning.
7th grade math tutoring session on proportional relationships.
4 top tips for teaching 7th grade math problems
Here are some teaching tips to overcome common challenges and support problem solving in your classroom:
• Focus on effort, not accuracy: Students come into 7th grade with many ideas about math. Many of these ideas are likely negative and may result in some math anxiety. They’ve heard math is hard,
you’re never going to use it, it’s confusing.
Be excited about what you’re teaching and learning! Praise kids for trying their best, not for always being correct. Consider establishing effort-based reward systems, such as publicly
nominating students for Mathematician of the Month.
• Check for understanding: Be sure students understand questions before they attempt to answer them. Have them rephrase the question or explain what they’re looking for to solve the problem.
• Ask for wrong answers: Students may feel a lack of confidence in math but if you ask them for a wrong answer, they may feel more inclined to answer. It can also prompt students to engage with and
make sense of the information given in the problem in a less pressured environment.
Alternatively, as teachers, you can give some suggestions. For example, “3/10 of students got an A on the last test. There are 20 students in the class. How many got an A?” Could 20 of the
students have gotten an A? No? Why not? Could 8.5 of the students have gotten an A? Why not? Then get more specific. Did more or less than 10 of the students get an A? How do you know?
• Use relatable, real-world problems: Sometimes, problem solving uses questions about gas mileage or building fences that kids either don’t relate to or don’t care about. Find out what your
students’ interests are and incorporate that into your math instruction. Alternatively, use seasonal and relatable contexts such as Thanksgiving math activities, summer math or mardi gras math.
7th grade math worksheets
Looking for more resources? Please see our selection of seventh grade math worksheets covering 7th grade key math topics and more. Each includes printable resources and step-by-step answer keys:
7th grade math problems FAQ
What math should a 7th grader know?
A 7th grader should be able to compute using all four operations with positive and negative rational numbers. They should be able to simplify expressions, solve one- and two- step equations, and
solve proportions. Additionally, 7th graders can work with probability of simple and compound events, as well as study concepts of geometry, including finding angle measurements and solving for area
and perimeter of regular and irregular polygons and circles.
What does 7th grade math focus on?
Seventh grade math focuses on working with positive and negative rational numbers in simplifying expressions and equations as well as solving one- and two-step equations. Additionally, extensive work
is done in solving proportions and ratios.
Is 7th grade math hard?
Seventh grade math is always achievable! Students who find it hard usually come into the grade without having mastered basic concepts and facts. Be sure students are automatic with facts across
addition, subtraction, multiplication, and division and have the ability to work with fractions and decimals. It will make the new concepts in 7th grade easier to learn. | {"url":"https://thirdspacelearning.com/us/blog/7th-grade-math-problems/","timestamp":"2024-11-13T21:58:47Z","content_type":"text/html","content_length":"154509","record_id":"<urn:uuid:3d64d57e-ad83-438f-aae8-fdf11bf713e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00493.warc.gz"} |
Scheduling to minimize average completion time revisited: Deterministic on-line algorithms
We consider the scheduling problem of minimizing the average weighted completion time on identical parallel machines when jobs are arriving over time. For both the preemptive and the nonpreemptive
setting, we show that straightforward extensions of Smith's ratio rule yield smaller competitive ratios compared to the previously best-known deterministic on-line algorithms, which are (4 + ε)
-competitive in either case. Our preemptive algorithm is 2-competitive, which actually meets the competitive ratio of the currently best randomized on-line algorithm for this scenario. Our
nonpreemptive algorithm has a competitive ratio of 3.28. Both results are characterized by a surprisingly simple analysis; moreover, the preemptive algorithm also works in the less clairvoyant
environment in which only the ratio of weight to processing time of a job becomes known at its release date, but neither its actual weight nor its processing time. In the corresponding nonpreemptive
situation, every on-line algorithm has an unbounded competitive ratio.
Original language English
Title of host publication Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Editors Klaus Jansen, Roberto Solis-Oba
Publisher Springer Verlag
Pages 227-234
Number of pages 8
ISBN (Electronic) 3540210792, 9783540210795
State Published - 2004
Externally published Yes
Publication series
Name Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume 2909
ISSN (Print) 0302-9743
ISSN (Electronic) 1611-3349
Dive into the research topics of 'Scheduling to minimize average completion time revisited: Deterministic on-line algorithms'. Together they form a unique fingerprint. | {"url":"https://portal.fis.tum.de/en/publications/scheduling-to-minimize-average-completion-time-revisited-determin","timestamp":"2024-11-08T11:05:54Z","content_type":"text/html","content_length":"52687","record_id":"<urn:uuid:3d3d0ffb-4cce-46e1-8781-fd798241134c>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00679.warc.gz"} |
The Top 5 Applications of RREF Calculators in Engineering
Reduced Row Echelon Form (RREF) calculators are powerful tools that are commonly used in engineering applications. These calculators perform row operations on matrices, ultimately simplifying them to
a form that is easy to use for further calculations. Today, we will explore the top five applications of RREF calculators in engineering.
Solving Linear Equations
RREF calculators are commonly used to solve systems of linear equations. Engineers use systems of linear equations to model physical problems, and RREF calculators provide a quick and efficient
method for solving them. By converting the augmented matrix to RREF form, engineers can easily read off the solutions to the system of equations.
Finding the Rank of a Matrix
The rank of a matrix is an important characteristic of a matrix that indicates the number of linearly independent rows or columns. In engineering, the rank of a matrix is often used to determine
whether a system is solvable or not. RREF calculators provide a simple way to find the rank of a matrix by performing row operations on the matrix until it is in RREF form. The number of nonzero rows
in the RREF matrix is equal to the rank of the original matrix.
Calculating Eigenvalues and Eigenvectors
Eigenvalues and eigenvectors are important in engineering because they are used to analyze the stability and behavior of physical systems. RREF calculators can be used to calculate eigenvalues and
eigenvectors of a matrix by performing row operations on the matrix until it is in RREF form. Once in RREF form, the eigenvalues and eigenvectors can be easily read off.
Solving Optimization Problems
Optimization problems are common in engineering, and RREF calculators can be used to solve them. Optimization problems involve finding the maximum or minimum of a function subject to certain
constraints. RREF calculators can be used to convert the system of equations describing the optimization problem to RREF form, which can then be used to easily solve for the maximum or minimum.
Analyzing Circuit Networks
In electrical engineering, RREF calculators can be used to analyze circuit networks. Circuit networks can be represented as a system of equations, which can be solved using RREF calculators. This
provides a quick and efficient way to analyze the behavior of electrical systems and determine their characteristics.
Additionally, RREF calculators can be used in various other engineering applications, such as in the design of mechanical systems or the analysis of fluid dynamics. By simplifying complex matrices to
their reduced row echelon form, engineers can quickly and efficiently perform calculations and make important decisions regarding the design and operation of physical systems.
It is important for engineers to be proficient in using RREF calculators and to understand the many applications in which they can be used. As technology continues to evolve, RREF calculators will
likely play an even larger role in engineering and other fields that rely heavily on matrix calculations.
RREF calculators are incredibly useful tools in engineering. They can be used to solve linear equations, find the rank of a matrix, calculate eigenvalues and eigenvectors, solve optimization
problems, and analyze circuit networks. RREF calculators simplify complex problems and allow engineers to quickly and easily analyze physical systems. They are an essential tool for any engineer
working with matrices and systems of equations. With the continued advancement of technology, RREF calculators will only become more powerful and useful in engineering applications. | {"url":"https://scatty.com/the-top-5-applications-of-rref-calculators-in-engineering/","timestamp":"2024-11-06T05:35:02Z","content_type":"text/html","content_length":"36769","record_id":"<urn:uuid:f77e0c7c-d3b0-4e3b-9d46-2cfdbb50c014>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00820.warc.gz"} |
Dealing with multiple integer libraries in Coq? on coq,coq-tactic
Dealing with multiple integer libraries in Coq?
111 views Asked by Siddharth Bhat At
I often get proof terms of the form:
Lemma of_nat_gt_0: forall (n: nat),
(Z.of_nat n >=? Int32.unsigned (Int32.repr 0)) = true.
The theorem is obviously true (Z of a natural will always be >= 0. Similarly, unsigned of a repr of a 0 will yield 0.
However, they are just straight up annoying, because I need to deal with
1. Int32 from the CompCert.Integers module
2. Z.of_nat conversions.
Often, I also have terms from Pos and N definitions.
Thse proofs involve multiple manual rewrites to juggle into some standard form, and then an omega call.
is there some way to "normalize" all of these into a single unified representation?
I understand that this implicitly involves transferring between different rings (eg, Int32 is Z/(2^32 - 1)). It would be nice if there's some way to deal with this, because these are the proofs that
get annoyingly long. | {"url":"https://techqa.club/v/q/dealing-with-multiple-integer-libraries-in-coq-51045942","timestamp":"2024-11-01T23:01:24Z","content_type":"text/html","content_length":"33621","record_id":"<urn:uuid:b0600c1b-5363-43c7-9359-0de897331738>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00050.warc.gz"} |
Hermitian positive definite linear systems · Krylov.jl
(x, stats) = cg(A, b::AbstractVector{FC};
M=I, ldiv::Bool=false, radius::T=zero(T),
linesearch::Bool=false, atol::T=√eps(T),
rtol::T=√eps(T), itmax::Int=0,
timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
callback=solver->false, iostream::IO=kstdout)
T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.
(x, stats) = cg(A, b, x0::AbstractVector; kwargs...)
CG can be warm-started from an initial guess x0 where kwargs are the same keyword arguments as above.
The conjugate gradient method to solve the Hermitian linear system Ax = b of size n.
The method does not abort if A is not definite. M also indicates the weighted norm in which residuals are measured.
Input arguments
• A: a linear operator that models a Hermitian positive definite matrix of dimension n;
• b: a vector of length n.
Optional argument
• x0: a vector of length n that represents an initial guess of the solution x.
Keyword arguments
• M: linear operator that models a Hermitian positive-definite matrix of size n used for centered preconditioning;
• ldiv: define whether the preconditioner uses ldiv! or mul!;
• radius: add the trust-region constraint ‖x‖ ≤ radius if radius > 0. Useful to compute a step in a trust-region method for optimization;
• linesearch: if true, indicate that the solution is to be used in an inexact Newton method with linesearch. If negative curvature is detected at iteration k > 0, the solution of iteration k-1 is
returned. If negative curvature is detected at iteration 0, the right-hand side is returned (i.e., the negative gradient);
• atol: absolute stopping tolerance based on the residual norm;
• rtol: relative stopping tolerance based on the residual norm;
• itmax: the maximum number of iterations. If itmax=0, the default number of iterations is set to 2n;
• timemax: the time limit in seconds;
• verbose: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations;
• history: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;
• callback: function or functor called as callback(solver) that returns true if the Krylov method should terminate, and false otherwise;
• iostream: stream to which output is logged.
Output arguments
• x: a dense vector of length n;
• stats: statistics collected on the run in a SimpleStats structure.
solver = cg!(solver::CgSolver, A, b; kwargs...)
solver = cg!(solver::CgSolver, A, b, x0; kwargs...)
where kwargs are keyword arguments of cg.
See CgSolver for more details about the solver.
(x, stats) = cr(A, b::AbstractVector{FC};
M=I, ldiv::Bool=false, radius::T=zero(T),
linesearch::Bool=false, γ::T=√eps(T),
atol::T=√eps(T), rtol::T=√eps(T), itmax::Int=0,
timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
callback=solver->false, iostream::IO=kstdout)
T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.
(x, stats) = cr(A, b, x0::AbstractVector; kwargs...)
CR can be warm-started from an initial guess x0 where kwargs are the same keyword arguments as above.
A truncated version of Stiefel’s Conjugate Residual method to solve the Hermitian linear system Ax = b of size n or the least-squares problem min ‖b - Ax‖ if A is singular. The matrix A must be
Hermitian semi-definite. M also indicates the weighted norm in which residuals are measured.
Input arguments
• A: a linear operator that models a Hermitian positive definite matrix of dimension n;
• b: a vector of length n.
Optional argument
• x0: a vector of length n that represents an initial guess of the solution x.
Keyword arguments
• M: linear operator that models a Hermitian positive-definite matrix of size n used for centered preconditioning;
• ldiv: define whether the preconditioner uses ldiv! or mul!;
• radius: add the trust-region constraint ‖x‖ ≤ radius if radius > 0. Useful to compute a step in a trust-region method for optimization;
• linesearch: if true, indicate that the solution is to be used in an inexact Newton method with linesearch. If negative curvature is detected at iteration k > 0, the solution of iteration k-1 is
returned. If negative curvature is detected at iteration 0, the right-hand side is returned (i.e., the negative gradient);
• γ: tolerance to determine that the curvature of the quadratic model is nonpositive;
• atol: absolute stopping tolerance based on the residual norm;
• rtol: relative stopping tolerance based on the residual norm;
• itmax: the maximum number of iterations. If itmax=0, the default number of iterations is set to 2n;
• timemax: the time limit in seconds;
• verbose: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations;
• history: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;
• callback: function or functor called as callback(solver) that returns true if the Krylov method should terminate, and false otherwise;
• iostream: stream to which output is logged.
Output arguments
• x: a dense vector of length n;
• stats: statistics collected on the run in a SimpleStats structure.
solver = cr!(solver::CrSolver, A, b; kwargs...)
solver = cr!(solver::CrSolver, A, b, x0; kwargs...)
where kwargs are keyword arguments of cr.
See CrSolver for more details about the solver.
(x, stats) = car(A, b::AbstractVector{FC};
M=I, ldiv::Bool=false,
atol::T=√eps(T), rtol::T=√eps(T),
itmax::Int=0, timemax::Float64=Inf,
verbose::Int=0, history::Bool=false,
callback=solver->false, iostream::IO=kstdout)
T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.
(x, stats) = car(A, b, x0::AbstractVector; kwargs...)
CAR can be warm-started from an initial guess x0 where kwargs are the same keyword arguments as above.
CAR solves the Hermitian and positive definite linear system Ax = b of size n. CAR minimizes ‖Arₖ‖₂ when M = Iₙ and ‖AMrₖ‖M otherwise. The estimates computed every iteration are ‖Mrₖ‖₂ and ‖AMrₖ‖M.
Input arguments
• A: a linear operator that models a Hermitian positive definite matrix of dimension n;
• b: a vector of length n.
Optional argument
• x0: a vector of length n that represents an initial guess of the solution x.
Keyword arguments
• M: linear operator that models a Hermitian positive-definite matrix of size n used for centered preconditioning;
• ldiv: define whether the preconditioner uses ldiv! or mul!;
• atol: absolute stopping tolerance based on the residual norm;
• rtol: relative stopping tolerance based on the residual norm;
• itmax: the maximum number of iterations. If itmax=0, the default number of iterations is set to 2n;
• timemax: the time limit in seconds;
• verbose: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations;
• history: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;
• callback: function or functor called as callback(solver) that returns true if the Krylov method should terminate, and false otherwise;
• iostream: stream to which output is logged.
Output arguments
• x: a dense vector of length n;
• stats: statistics collected on the run in a SimpleStats structure.
solver = car!(solver::CarSolver, A, b; kwargs...)
solver = car!(solver::CarSolver, A, b, x0; kwargs...)
where kwargs are keyword arguments of car.
See CarSolver for more details about the solver.
(x, stats) = cg_lanczos(A, b::AbstractVector{FC};
M=I, ldiv::Bool=false,
check_curvature::Bool=false, atol::T=√eps(T),
rtol::T=√eps(T), itmax::Int=0,
timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
callback=solver->false, iostream::IO=kstdout)
T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.
(x, stats) = cg_lanczos(A, b, x0::AbstractVector; kwargs...)
CG-LANCZOS can be warm-started from an initial guess x0 where kwargs are the same keyword arguments as above.
The Lanczos version of the conjugate gradient method to solve the Hermitian linear system Ax = b of size n.
The method does not abort if A is not definite.
Input arguments
• A: a linear operator that models a Hermitian matrix of dimension n;
• b: a vector of length n.
Optional argument
• x0: a vector of length n that represents an initial guess of the solution x.
Keyword arguments
• M: linear operator that models a Hermitian positive-definite matrix of size n used for centered preconditioning;
• ldiv: define whether the preconditioner uses ldiv! or mul!;
• check_curvature: if true, check that the curvature of the quadratic along the search direction is positive, and abort if not, unless linesearch is also true;
• atol: absolute stopping tolerance based on the residual norm;
• rtol: relative stopping tolerance based on the residual norm;
• itmax: the maximum number of iterations. If itmax=0, the default number of iterations is set to 2n;
• timemax: the time limit in seconds;
• verbose: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations;
• history: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;
• callback: function or functor called as callback(solver) that returns true if the Krylov method should terminate, and false otherwise;
• iostream: stream to which output is logged.
Output arguments
• x: a dense vector of length n;
• stats: statistics collected on the run in a LanczosStats structure.
• A. Frommer and P. Maass, Fast CG-Based Methods for Tikhonov-Phillips Regularization, SIAM Journal on Scientific Computing, 20(5), pp. 1831–1850, 1999.
• C. C. Paige and M. A. Saunders, Solution of Sparse Indefinite Systems of Linear Equations, SIAM Journal on Numerical Analysis, 12(4), pp. 617–629, 1975.
solver = cg_lanczos!(solver::CgLanczosSolver, A, b; kwargs...)
solver = cg_lanczos!(solver::CgLanczosSolver, A, b, x0; kwargs...)
where kwargs are keyword arguments of cg_lanczos.
See CgLanczosSolver for more details about the solver.
(x, stats) = cg_lanczos_shift(A, b::AbstractVector{FC}, shifts::AbstractVector{T};
M=I, ldiv::Bool=false,
check_curvature::Bool=false, atol::T=√eps(T),
rtol::T=√eps(T), itmax::Int=0,
timemax::Float64=Inf, verbose::Int=0, history::Bool=false,
callback=solver->false, iostream::IO=kstdout)
T is an AbstractFloat such as Float32, Float64 or BigFloat. FC is T or Complex{T}.
The Lanczos version of the conjugate gradient method to solve a family of shifted systems
(A + αI) x = b (α = α₁, ..., αₚ)
of size n. The method does not abort if A + αI is not definite.
Input arguments
• A: a linear operator that models a Hermitian matrix of dimension n;
• b: a vector of length n;
• shifts: a vector of length p.
Keyword arguments
• M: linear operator that models a Hermitian positive-definite matrix of size n used for centered preconditioning;
• ldiv: define whether the preconditioner uses ldiv! or mul!;
• check_curvature: if true, check that the curvature of the quadratic along the search direction is positive, and abort if not, unless linesearch is also true;
• atol: absolute stopping tolerance based on the residual norm;
• rtol: relative stopping tolerance based on the residual norm;
• itmax: the maximum number of iterations. If itmax=0, the default number of iterations is set to 2n;
• timemax: the time limit in seconds;
• verbose: additional details can be displayed if verbose mode is enabled (verbose > 0). Information will be displayed every verbose iterations;
• history: collect additional statistics on the run such as residual norms, or Aᴴ-residual norms;
• callback: function or functor called as callback(solver) that returns true if the Krylov method should terminate, and false otherwise;
• iostream: stream to which output is logged.
Output arguments
• x: a vector of p dense vectors, each one of length n;
• stats: statistics collected on the run in a LanczosShiftStats structure.
• A. Frommer and P. Maass, Fast CG-Based Methods for Tikhonov-Phillips Regularization, SIAM Journal on Scientific Computing, 20(5), pp. 1831–1850, 1999.
• C. C. Paige and M. A. Saunders, Solution of Sparse Indefinite Systems of Linear Equations, SIAM Journal on Numerical Analysis, 12(4), pp. 617–629, 1975. | {"url":"https://jso.dev/Krylov.jl/dev/solvers/spd/","timestamp":"2024-11-13T01:08:03Z","content_type":"text/html","content_length":"38935","record_id":"<urn:uuid:92a7ba58-f279-409f-98e5-5627e6ee8790>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00455.warc.gz"} |
Nominal interest rates formulas
In finance and economics, the nominal interest rate or nominal rate of interest is either of two rate before adjusting for inflation, and a real rate is a constant- prices rate. The Fisher equation
is used to convert between real and nominal rates. 29 Jan 2020 Unlike the nominal rate, the real interest rate takes the inflation rate into account. The equation that links nominal and real interest
rates can be
Real Interest Rate Formula (Table of Contents) Formula; Examples; Calculator; What is the Real Interest Rate Formula? The term “real interest rate” refers to the interest rate that has been adjusted
by removing the effect of inflation from the nominal interest rate.In other words, it is effectively the actual cost of debt for the borrower or actual yield for the lender. The real interest rate is
the nominal rate of interest minus inflation, which can be expressed approximately by the following formula: Real Interest Rate = Nominal Interest Rate – Inflation Rate = Growth of Purchasing Power.
For low rates of inflation, the above equation is fairly accurate. Inflation rate calculator solving for nominal interest rate given real interest rate and inflation Inflation Rate Equations
Calculator Finance - Real Interest Rates - Formulas. Solving for nominal interest rate. Inputs: real interest rate (r) inflation rate (i) Conversions: real interest rate (r) = 0 = 0. inflation rate
(i) = 0 = 0. Unlike simple interest Simple Interest Simple interest formula, definition and example. Simple interest is a calculation of interest that doesn't take into account the effect of
compounding. In many cases, interest compounds with each designated period of a loan, but in the case of simple interest, it does not. A nominal interest rate is The Fisher equation provides the link
between nominal and real interest rates. To convert from nominal interest rates to real interest rates, we use the following formula: real interest rate ≈ nominal interest rate − inflation rate. To
find the real interest rate, we take the nominal interest rate and subtract the inflation rate. Read on to learn how to use Excel’s EFFECT formula to calculate an effective interest rate (APY) from a
nominal interest rate (APR). Use Excel’s EFFECT Formula. Suppose you want to figure out the effective interest rate (APY) from a 12% nominal rate (APR) loan that has monthly compounding. • Some
problems may state only the nominal interest rate. • Remember: Always apply the Effective Interest Rate in solving problems. • Published interest tables, closed-form time value of money formula, and
spreadsheet function assume that only Effective interest is applied in the computations. EGR2302-Engineering Economics
4) any time the interest rate is an APR, must start with this equation to convert to an effective interest rate. Khan Academy: Annual Percentage Rate and Effective
10 Nov 2015 Formula = Interest rate - (Interest rate*tax rate). = 10-(10*30%) = 7. This means that the effective interest earned after tax falls to 7 percent. The nominal interest rate is calculated
using the above formula as, Nominal interest rate calculation = ln (1 +12%). Nominal interest rate= 11.3329%. Nominal interest rate refers to the interest rate before taking inflation into account.
Nominal can also refer to the advertised or stated interest rate on a loan, without taking into account any fees or compounding of interest. The nominal interest rate formula can be calculated as: r
= m × [ ( 1 + i) 1/m - 1 ]. Nominal interest rate formula = [(1 + Real interest rate) * (1 + Inflation rate)] – 1. Real Interest Rate is the interest rate that takes inflation, compounding effect and
other charges into account. Inflation is the most important factor that impacts the nominal interest rate.
2 Jul 2019 What Is the Formula for Nominal Interest Rates? Nominal Interest Rate vs. Real Interest Rate; Nominal Interest Rate vs. Effective Interest Rate.
11 Nov 2014 It is calculated with this formula where i = nominal interest rate P = initial investment T = final (future) value N = number of compounding periods 19 Sep 2019 The nominal interest rate
formula calculates the nominal rate (i) based on an effective interest rate (r), and a number of compounding periods in 22 Oct 2011 Effective interest rate formulas. In the context of compound
interest, the effective annual rate of interest can be determined using the formula 1 Apr 2019 Effective rate helps determine the correct maturity amount as it accounts If one uses the nominal rate
of 8% in the above formula, the maturity 5 Sep 2018 It's going to get a little complicated here, so feel free to skip this section if math bores you. Here's the effective interest rate formula: 1 +
Nominal Interest Rate Formula – Example #1. ICICI bank is providing real interest rate which includes inflation 7% on 5-year bond and that time inflation rate is 4%
19 Sep 2019 The nominal interest rate formula calculates the nominal rate (i) based on an effective interest rate (r), and a number of compounding periods in 22 Oct 2011 Effective interest rate
formulas. In the context of compound interest, the effective annual rate of interest can be determined using the formula 1 Apr 2019 Effective rate helps determine the correct maturity amount as it
accounts If one uses the nominal rate of 8% in the above formula, the maturity 5 Sep 2018 It's going to get a little complicated here, so feel free to skip this section if math bores you. Here's the
effective interest rate formula: 1 + (nominal 11 Oct 2018 Learn to use nominal interest rate formula Excel. Also learn the differences between nominal and effective rates. Calculaute both using my
Nominal Annual Interest Rate Formulas: Suppose If the Effective Interest Rate or APY is 8.25% compounded monthly then the Nominal Annual Interest Rate or "Stated Rate" will be about 7.95%. An
effective interest rate of 8.25% is the result of monthly compounded rate x such that i = x * 12.
Real Interest Rate Formula (Table of Contents) Formula; Examples; Calculator; What is the Real Interest Rate Formula? The term “real interest rate” refers to the interest rate that has been adjusted
by removing the effect of inflation from the nominal interest rate.In other words, it is effectively the actual cost of debt for the borrower or actual yield for the lender. The real interest rate is
the nominal rate of interest minus inflation, which can be expressed approximately by the following formula: Real Interest Rate = Nominal Interest Rate – Inflation Rate = Growth of Purchasing Power.
For low rates of inflation, the above equation is fairly accurate. Inflation rate calculator solving for nominal interest rate given real interest rate and inflation Inflation Rate Equations
Calculator Finance - Real Interest Rates - Formulas. Solving for nominal interest rate. Inputs: real interest rate (r) inflation rate (i) Conversions: real interest rate (r) = 0 = 0. inflation rate
(i) = 0 = 0.
What is the nominal rate payable monthly if the effective rate is 10%? Solution. Re-arranging the formula to make i(12) Calculating simple and compound interest rates are Press Enter to expand
sub-menu, click to visit Math and Logic pageMath and Logic. Personal Calculating Nominal Interest Rate. Nominal interest rate for a period with effective interest rates in it's sub-periods can be
calculated as. i = (1 + ie)n - 1 (1). where. | {"url":"https://topbinhqwtne.netlify.app/andren86885fa/nominal-interest-rates-formulas-kahy","timestamp":"2024-11-03T18:38:26Z","content_type":"text/html","content_length":"34329","record_id":"<urn:uuid:8b39b99d-0bad-4ec1-bcc7-29f3c500a63d>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00334.warc.gz"} |
Solving More Decimal Word Problems
Analysis: We need to estimate the product of $14.50 and 15.5. To do this, we will round one factor up and one factor down.
Answer: The cost of 15.5 weeks of school lunches would be about $200.
Analysis: To solve this problem, we will multiply $11.75 by 21.
Answer: The student will earn $246.75 for gardening this month.
Analysis: To solve this problem, we will multiply 29.7 by 10.45
Answer: Rick can travel 310.365 miles with one full tank of gas.
Analysis: We need to estimate the quotient of 179.3 and 61.5.
Answer: He averaged about 3 miles per day.
Analysis: We will divide 7.11 lbs. by 9 to solve this problem.
Answer: Each jar will contain 0.79 lbs. of candy.
Analysis: To solve this problem, we will divide $19,061.00 by 36, then round the quotient to the nearest cent (hundredth).
Answer: Paul will make 36 monthly payments of $529.47 each.
Analysis: We will divide 956.4 by 15.9, then round the quotient to the nearest tenth.
Step 1:
Step 2:
Answer: Rounded to the nearest tenth, the average speed of the car is 60.2 miles per hour.
Summary: In this lesson we learned how to solve word problems involving decimals. We used the following skills to solve these problems:
1. Estimating decimal products
2. Multiplying decimals by whole numbers
3. Multiplying decimals by decimals
4. Estimating decimal quotients
5. Dividing decimals by whole numbers
6. Rounding decimal quotients
7. Dividing decimals by decimals
Directions: Read each question below. You may use paper and pencil to help you solve these problems. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a
message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.
1. Estimate the amount of money you need to pay for a tank of gas if one gallon costs $3.04 and the tank holds 11.9 gallons.
2. The sticker on Dean’s new car states that the car averages 32.6 miles per gallon. If the fuel tank holds 12.3 gallons, then how far can Dean travel on one full tank of gas?
3. Larry worked 15 days for a total of 116.25 hours. How many hours did he average per day?
4. Six cases of paper cost $159.98. How much does one case cost? Round your answer to the nearest cent.
5. There are 2.54 centimeters in one inch. How many inches are there in 51.78 centimeters? Round your answer to the nearest thousandth. | {"url":"https://mathgoodies.com/lessons/solve_more_problems/","timestamp":"2024-11-04T19:53:49Z","content_type":"text/html","content_length":"45475","record_id":"<urn:uuid:3c705d5b-f5ad-4128-abc7-bd7b315f8de7>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00427.warc.gz"} |
How do you solve 6x-3<9? | HIX Tutor
How do you solve #6x-3<9#?
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-solve-6x-3-9-8f9af936e5","timestamp":"2024-11-01T22:13:11Z","content_type":"text/html","content_length":"565366","record_id":"<urn:uuid:131110cc-61e3-45a1-89ad-7d1b6c250adb>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00473.warc.gz"} |
Misconceptions in Math
Upper elementary teachers frequently see their students’ misconceptions in math. Have you ever taught a math concept where you had to spend as much time undoing misconceptions as you do teaching new
concepts? We often inadvertently teach misconceptions when we offer our students tricks and shortcuts to math concepts. This issue isn’t isolated to any one group of teachers or grade levels. In
fact, research shows a significant drop in math test scores as students reach middle school and beyond. Of course, there many many factors in the decline of test scores, but I can’t help but think of
the possibility that students are approaching higher levels of math with an abundance of math misconceptions. I recently read Math Misconceptions, which you can find here, and it solidified my
thinking that there are things we can do as elementary educators to prevent misconceptions in math.
This blog post shares eight phrases teachers frequently say that may be responsible for causing misconceptions in math. I know that I’m guilty of using more than one, and I’m sure that I’ve missed
some phrases too.
1. The Equal Sign Means “The answer is…”
Many students view the equal sign as a symbol that means “the answer is” or “ta-da”. Instead, the equal sign in math means what is to the right and left of the symbol have the same value.
Understanding the equal sign as a relational symbol becomes more important when students begin solving equations with a missing addend, minuend, subtrahend, factor, dividend, or divisor when the
equal sign is not in a standard position To promote an understanding of the equal sign, students need exposure to a variety of equation types (3 = 8 − 5; 2 + 3 = 1 + 4; 9 − 3 = 6). Students need an
understanding of the equal sign to work on higher-level math problems such as word problems and algebraic equations. If students do not learn the correct interpretation of the equal sign performing
other higher-level math calculations will become more difficult as school progresses.
2. Subtract the Smaller Number from the Larger Number
A common misconception in math is when learning subtraction, students often learn to write the big number first. However, this is not always true. While this subtraction rule may work for students in
elementary school, this is not the case when students begin to add and subtract positive and negative integers. This rule may also create a problem when students subtract multi-digit numbers.
Students often overgeneralize previous learning and subtract the smaller number from the larger one digit-by-digit. For example in the problem 453-172. Students may correctly subtract 3-2, but when
they reach the tens place, they subtract 7-5. Students need to be taught that subtraction is not commutative and the importance of order in subtraction.
3. Multiplication Makes Numbers Bigger
Most of the time in the elementary classroom, multiplication makes numbers bigger. However, this is a common misconception in math. Once students reach fifth grade and are not exclusively multiplying
whole numbers, this is not always true. For example in the problem 0.5 x 0.2 = 0.1, the product has a smaller value. Students must develop multiple strategies to support understanding of
multiplication and to develop number sense that extends beyond whole numbers. A great way to begin a multiplication of fractions lesson would be to ask students if it’s true that multiplication
always makes numbers bigger.
4. Misconceptions in Math – Division Makes Numbers Smaller
Just as multiplication does not always make numbers larger, the same can be said for division. Division does not always make numbers smaller. For example in the problem 6 ÷ ½ = 12, the quotient is
larger than the dividend. It’s amazing that students will be solving that type of problem as early as fifth grade, and as a third and fourth grade teacher, I want my students prepared for that type
of math reasoning. Even though I don’t necessarily teach that concept, I don’t want to create rules or frame concepts that will lend themselves to confusion in the future.
5. Always Divide the Larger Number by the Smaller Number
Students are often told that they always divide the larger number by the smaller number. However in fifth grade, this is not always the case. For example in the problem 4 ÷ 6 = 2/3, the dividend is
smaller than the divisor. When this concept “sticks”, it is extremely difficult to undo later. An example that students can understand is having 2 pounds of cat food that has to be split among the 4
cats. Working on equal sharing situations like this helps students develop the number sense to ultimately handle operations with fractions. One way to avoid the misconception of putting the larger
number first is to focus instead on the parts of an equal sharing problem. Is anyone else feeling the need to offer some words of encouragement to the fifth grade teachers at your school?
6. When You Multiply by 10, Add Zero to the Number
A major part of third and fourth grade math is teaching students how to multiply by multiples of ten. Teachers often tell students that when you multiply a number by ten, just add a zero to the end
of the number. However, once again, this isn’t always true and leads to misconceptions in math. When students multiply decimals by ten and powers of ten, they do not add a zero to the end of the
number. You can see an example in the problem 7.47 x 10 ≠ 7.470. Not only does the rule lead students to misconceptions, but it also prevents students from developing a deeper understanding of our
base-ten system. Students should learn that when multiplying by 10 or 100, each digit is shifted to the left on a place value table, because they are adding another place to the number. You can learn
more about how I teach multiplying and dividing by multiples of ten in this blog post.
7. In a Fraction, the Big Number Always Goes on the Bottom
Many times, we tell student to always write the big number on bottom in fractions. Not only does this rule prevent authentic understanding of the concept of fractions, it is not an accurate rule.
When students write improper fractions in fourth grade, the numerator is greater than the denominator. Rather than teaching the rule, students should develop an understanding of a whole, numerator,
and denominator. Students tend to manipulate fractions by rote rules and memorization, rather than try to make sense of the concepts and procedures. During one of my fraction lessons last year,
students made the generalization that if the numerator is larger than the denominator, the fraction is greater than one whole. This generalization was generated by students through hands-on
exploration of fractions-not by me stating the rule.
8. Misconceptions in Math – If it’s a Rectangle, it Can’t Be a Square
Students are often taught that the categories of shapes cannot overlap. For example, a square cannot also be a rectangle. Once students enter third grade and begin to study the hierarchy of
quadrilaterals, these statements do not apply. A square is a special type of rectangle, and a square is also a special type of rhombus. Then to complicate things even further, a square, rectangle,
and rhombus are all types of quadrilaterals. Students have to be flexible in their reasoning about math terms and concepts.
Misconceptions in Math Keywords
I couldn’t write this blog post without mentioning keywords. There are some schools of thought that are totally and completely against key words. I understand the reasoning, but I don’t think that
teaching math keywords has to be a completely bad thing. I do believe it’s possible for students to overly rely on keywords or to over generalize key words. However, I also recognize that keywords
can become problematic when students begin to solve multi-step word problems, because they must decide which keywords work with which component of the problem.
Even so, I believe keywords can play a positive role in the reading comprehension aspect of word problems. Students must understand the vocabulary and meaning of keywords in the problem. So while I
continue to introduce them to my students, I do not introduce them in isolation. When students are taught the underlying structure of a word problem, they not only have greater success in problem
solving but can also gain insight into the deeper mathematical ideas in word problems. This is why I always make types of word problems a major part of my math instruction. I’ve added types of word
problems for Addition and Subtraction and Multiplication and Division to my Math Reference Notes.
I’ve also created a small Types of Word Problems bulletin board in my classroom for an easy reference. As I introduce and teach the different types of word problems, I have students complete a
different page in their Types of Word Problems booklet. Throughout the year, I also add various types of word problems to students’ work station activities.
21 thoughts on “Misconceptions in Math”
Is there a way to purchase/download your math reference notes?
Third Grade
Fourth Grade
1. Neeli Barwick
Do you have 5th grade?
I don’t at this time.
2. Shannon Caldwell
As I was reading this post, I realized how many of those misconceptions I have used. It was how I was taught and now I have an inkling as to why math has always been a struggle for me.
That said, do you have a version for second grade, or is the third grade version applicable?
Are you looking for a second grade version of the Types of Word Problems posters & activities?
1. Shannon Caldwell
3. Shelley
As an Algebra 2 teacher, I have to undo so many misconceptions from earlier grades. I wanted to add that, while -1 + 1 CANCELS to equal zero, 5/5 DOESN’T CANCEL. Rather, the 5s DIVIDE OUT to
equal 1. When the term “cancel” is misused in division, students with math struggles habitually say 5/5 = 0.
Great post, by the way. I’m going to send a copy to the math curriculum specialist in my school district, asking that she convey this information to elementary school teachers in the district.
1. Melissa
Yes! As an Algebra I Teacher, I say the property name EVERY time we solve an equation as well. “This is the additive inverse property, we say it “cancels out” but it actually goes to zero”.
“This is multiplicative inverse property, we say it “cancels out” but it actually goes to one”. I make my students repeat it when they are explaining as well.
4. Katie
I am so guilty of the multiplying by multiples of 10 misconception!! Where can I find a link to that blog post??
5. Rachel Skinner
The adding a zero one is HUGE. I didn’t realize it until I moved from 4th grade to 5th grade. I immediately went to my old team and was like, STOP SAYING ADD ZERO! I think this also shows why
time for vertical planning is an essential, but lacking, component in schools. It is completely understandable to teach a “trick” without understanding the later misconceptions when you don’t
actually know what comes later.
6. Elizabeth
Great post! Is there any chance you’d make a Reference Note (Types of Word Problems) for 5th grade?
7. Darlene Rogers
Thank you! I teach high school math (geometry, algebra 2, pre-cal, and calculus) and there are definitely times that I have to undo these misconceptions. And that you can’t take the square root
of a negative. The more elementary teachers you can reach with this will make everybody’s lives easier.
8. Melissa
As an upper level math teacher, I cringe at the word reduce; reduce the fraction. Students are under the impression that the fraction is smaller when they “reduce it”, but in fact it is
equivalent (as we know). Every time I have a student say, “ I reduced it”, I come back with, “you simplified it” (not making a huge deal). All my instructions say simplify or put in simplistic
form. This is a great thing to share with teachers!!
Many times in elementary grades younger students have such difficulty understanding the actual mathematical concepts that in the essence of time the teacher resorts to a trick that works for the
1. Denise
I admit that, as a younger teacher, I did this, not realizing at the time that there are better ways to teach… that of helping children develop understanding!
Getting a teaching job is just the beginning of our careers! It is our duty to keep learning how to better teach (especially math), all of our professional lives. I have retired after 31
years of teaching primary grades, and still kept learning, right up to the end. The beautiful thing is that my kids were always teaching ME – I just provided the problems and opportunities,
and they showed me strategies and patterns that I didn’t think of! I was always learning, along side my students!
10. Denise
Bravo! Every teacher should learn this!
11. Denise
Bravo! I totally agree. Teachers need to learn this, if they haven’t had experience with these misconceptions before.
12. Lisa
I caution my students (4th graders) when they say, “6-9; you can’t do that so you have to regroup,” because you CAN do 6-9. It’s just not a positive number.
13. Kristin
I’ve had many of these arguments with my daughters teachers. Hence the reason we started homeschooling in 3rd grade. I started her totally from scratch and when we got to division, she’d say
can’t be done when she had say 5/6. So I grabbed $5 in change and 6 stuffed animals. Told her to divide the money evenly between the stuffies. I bright in the concept of negatives with
subtraction. She’s 10 and we aren’t starting pre-algebra on time but I’m ok with that. It’s easier to have a solid foundation and start algebra a little late. Makes me sad many teachers don’t
have the luxury of understanding math a lot deeper. I know my daughters teacher finally got frustrated and basically said she wasn’t good at math and the district determined this was how she had
to teach it. I’m lucky I only teach as a tutor. I have a lot more freedom. More schools should share this information. I didn’t really learn math until I took a remedial college course at 19.
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Unscramble HABANERA
How Many Words are in HABANERA Unscramble?
By unscrambling letters habanera, our Word Unscrambler aka Scrabble Word Finder easily found 65 playable words in virtually every word scramble game!
Letter / Tile Values for HABANERA
Below are the values for each of the letters/tiles in Scrabble. The letters in habanera combine for a total of 17 points (not including bonus squares)
• H [4]
• A [1]
• B [3]
• A [1]
• N [1]
• E [1]
• R [5]
• A [1]
What do the Letters habanera Unscrambled Mean?
The unscrambled words with the most letters from HABANERA word or letters are below along with the definitions.
• habanera () - Sorry, we do not have a definition for this word | {"url":"https://www.scrabblewordfind.com/unscramble-habanera","timestamp":"2024-11-02T00:18:03Z","content_type":"text/html","content_length":"47937","record_id":"<urn:uuid:b9c99f06-5ada-449c-8d78-91e456523c6d>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00327.warc.gz"} |
Extremal Permutations in Routing Cycles
Extremal Permutations in Routing Cycles
Keywords: Routing number, Permutation, Sorting algorithm, Cayley graphs
Let $G$ be a graph whose vertices are labeled $1,\ldots,n$, and $\pi$ be a permutation on $[n]:=\{1,2,\ldots, n\}$. A pebble $p_i$ that is initially placed at the vertex $i$ has destination $\pi(i)$
for each $i\in [n]$. At each step, we choose a matching and swap the two pebbles on each of the edges. Let $rt(G, \pi)$, the routing number for $\pi$, be the minimum number of steps necessary for the
pebbles to reach their destinations.
Li, Lu and Yang proved that $rt(C_n, \pi)\le n-1$ for every permutation $\pi$ on the $n$-cycle $C_n$ and conjectured that for $n\geq 5$, if $rt(C_n, \pi) = n-1$, then $\pi = 23\cdots n1$ or its
inverse. By a computer search, they showed that the conjecture holds for $n<8$. We prove in this paper that the conjecture holds for all even $n\ge 6$. | {"url":"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i3p47","timestamp":"2024-11-03T13:10:29Z","content_type":"text/html","content_length":"15535","record_id":"<urn:uuid:7673688a-242a-48aa-b9fb-b71f745bab37>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00317.warc.gz"} |
Early Solar System Solar Wind Implantation of 7Be into Calcium-Alumimum Rich Inclusions in Primitive Meteortites
International Journal of Astronomy and Astrophysics Vol.09 No.01(2019), Article ID:90161,9 pages
Early Solar System Solar Wind Implantation of ^7Be into Calcium-Alumimum Rich Inclusions in Primitive Meteortites
Glynn E. Bricker^
Department of Physics, Purdue University Northwest, Westville, USA
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
Received: November 27, 2018; Accepted: January 21, 2019; Published: January 24, 2019
The one time presence of short-lived radionuclides (SLRs) in Calcium-Aluminum Rich inclusions (CAIs) in primitive meteorites has been detected. The solar wind implantation model (SWIM) is one
possible model that attempts to explain the catalogue of SLRs found in primitive meteorites. In the SWIM, solar energetic particle (SEP) nuclear interactions with gas in the proto-solar atmosphere of
young stellar objects (YSOs) give rise to daughter nuclei, including SLRs. These daughter nuclei then may become entrained in the solar wind via magnetic field lines. Subsequently, the nuclei,
including SLRs, may be implanted into CAI precursors that have fallen from the main accretion flow which had been destined for the proto-star. This mode of implanting SLRs in the solar system is
viable, and is exemplified by the impregnation of the lunar surface with solar wind particles, including SLRs. X-ray luminosities have been measured to be 100,000 times more energetic in YSOs,
including T-Tauri stars, than present-day solar luminosities. The SWIM scales the production rate of SLRs to nascent SEP activity in T-Tauri stars. Here, we model the implantation of ^7Be into CAIs
in the SWIM, utilizing the enhanced SEP fluxes and the rate of refractory mass inflowing at the X-region, 0.06 AU from the proto-Sun. Taking into account the radioactive decay of ^7Be and spectral
flare variations, the ^7Be/^9Be initial isotopic ratio is found to range from 1 × 10^−5 to 5 × 10^−5.
Radio-Nuclide, ^7Be, Early Solar System, Solar Wind, CAI, Solar Wind Implantation Model, X-Wind
1. Introduction
Studies report evidence for the one-time presence of SLRs, through decay product systematics, including ^10Be, ^26Al, ^36Cl, ^41Ca, and ^53Mn, in CAIs in primitive carbonaceous meteorites at the
nascence of the solar system [1] . The possible origins of these SLRs are widely varied and include stellar sources (AGB Stars, Wolf-Rayet stars, nova, and super nova) and energetic particle
interaction, from either SEPs, or galactic cosmic rays (GCRs). Bricker & Caffee [2] [3] proposed the solar wind implantation model (SWIM) for the incorporation of ^10Be and ^36Cl into CAIs early in
primitive meteorites.
In the SWIM, the SLRs come into existence via SEP nuclear reactions in the proto-solar atmosphere of the young Sun, characterized by X-ray emissions orders of magnitude greater than main sequence
stars. Studies of the Orion Nebulae indicate that pre-main sequence (PMS) stars exhibit X-ray luminosity, and hence SEP fluxes on the order of ~10^5 over contemporary SEP flux levels [4] . The
irradiation produced SLRs are then trapped by magnetic field lines, and these solar wind SLRs eventually impregnate CAI precursors. This mode of production of SLRs, entrainment of SLRs in the solar
wind, and implantation of SLR into solar system material is seen in the implantation of solar wind particles, e.g. ^10Be [5] [6] and ^14C [6] [7] , on the Moon.
^10Be is produced via SEP spallation reactions, with oxygen serving as the chief target particle in the SWIM. Similar to ^10Be, ^7Be, half-life of 53 days [8] , is also primarily produced through SEP
nuclear reactions with oxygen as the primary target particle, and ^7Be has also recently been detected in stellar photospheres [9] In addition, the one-time presence of ^7Be has been measured in CAIs
in primitive meteorites (through the study of Li, the decay product of ^7Be, systematics) [10] [11] . Owing to the 53 day half-life, local irradiation is the only possible operation pathway for ^7Be
production. As such, the large difference in half-lives between ^7Be and ^10Be is of interest in terms of chronological processes associated with early solar system and CAI formation and evolution.
In this work, we consider the possible incorporation of ^7Be into CAIs in primitive carbonaceous meteorites in the SWIM. Table 1 below characterizes berylliumisotopes found in CAIs.
2. Solar Wind Implantation Mode
2.1. Synopsis
In the SWIM, SLRs are produced in the solar nebula via SEP nuclear reactions on gaseous target material in the solar atmosphere ~4.6 Gyr, during the formation
Table 1. Beryllium isotopes found in CAIs.
Note: Radionuclide content in g^−1 calculated from initial isotopic ratio and ^9Be content in ppb. The ^9Be content in CAIs is estimated 100 ppb [14] [15] .
of the solar system. These newly produced nuclei are incorporated in the solar wind. The SLRs flow along magnetic field lines in the solar wind, and this particle flow intersects with materials which
have fallen out of the main accretion flow, which was headed to hot-spots on the Sun. At the intersection of outflowing SLRs, and inflowing fallen CAI precursor material, the SLRs may become
impregnated into the inflowing materials. The fundamental geometry for the implantation process described above and transportation of implanted CAIs to the asteroid zone can be gleaned from the
X-wind model of Shu et al. [16] [17] [18] . Figure 1 below illustrates of the basic magnetic field geometry, ^7Be production via SEP flaring activity, and subsequent implantation into CAI-precursor
material from the main funnel flow onto the proto-Sun.
2.2. Refractory Mass Inflow Rate
The effective refractory mass inflow rate, S, i.e. the refractory mass that falls from the main funnel flow which was accreting onto the star at the X-region, is calculated from equation (1):
$S={\stackrel{˙}{M}}_{D}\cdot {X}_{r}\cdot F$(1)
where ${\stackrel{˙}{M}}_{D}$ is disk mass accretion rate, X[r] is the cosmic mass fraction, and F is the fraction of material that enters the X-region from the main funnel flow [19] . For ${\
stackrel{˙}{M}}_{D}$ , we adopt 1 × 10^−7 solar masses year^−1. Disk mass accretion rates range from ~10^−7 to ~10^−10 solar masses year^−1 for T Tauri stars from 1 - 3 Myr [20] , whereas embedded
class 0 and class I PMS stars have mass accretion rates of ~10^−5 to ~10^−6 solar masses year^−1 [21] . Here we adopt for ${\stackrel{˙}{M}}_{D}$ , a rate 1 × 10^−7 solar masses year^−1,
corresponding to class II or III PMS stars. From Lee et al. [19] we utilize a cosmic mass fraction, X[r][,] and fraction of refractory material fraction F, of 4 × 10^−3 and 0.01, respectively, in our
model. X[r] represents the fraction of refractory content in the inflowing material, and F represents the fraction of inflowing mass that does not accrete onto the proto-sun. The choice 0.01
Figure 1. SWIM magnetic field geometry for SLRs production via SEP nuclear reactions. The gray area represents the main accretion flow onto “hot spots” on the PMS star. SLRs produced close to the
proto-solar surface are incorporated into CAI precursor material which has fallen from the accretion flow (figure after Shu et al. [17] ).
maximizes F, and corresponds to all the mass which comprises the planets falling from the accretion flow. F = 0.01 is the preferred value of Lee et al. [19] in their model. (See Lee et al. [19] for a
detailed discussion of X[r] and F) Employing Equation (1) and the parameters detailed above, we find the rate at which this refractory material reaches the x-region, called here the refractory mass
inflow rate, S, is 2.5 × 10^14 g s^−1. In consideration of the extreme values of, S, S could be two orders of magnitude greater if the accretion rates of ~10^−5 to ~10^−6 solar masses year^−1, or S
could also be four orders of magnitude less if the mass accretion rate was ~10^−8 to 10^−10 solar masses year^−1 and F ~0.0001.
2.3. Effective Ancient Production Rate
The effective ancient ^7Be outflow rate, P in units of s^−1, is given by:
$P=p\cdot f$(2)
where p is the ancient production rate and f is the fraction of the solar wind ^7Be that enters the CAI-forming region; f = 0.1. (See Bricker & Caffee [2] [3] for a discussion of factor f). The ^7Be
production rate is calculated assuming that SEPs are characterized by a power law relationship:
where r ranges from 2.5 to 4. For impulsive flares, i.e. r = 4, we use ^3He/H = 0.1 and ^3He/H = 0.3, and for gradual flares, i.e. r = 2.5, we use ^3He/H = 0. For all spectral indices, we assume α/H
= 0.1. Contemporary SEP flux rates at the Sun-Earth distance of 1 AU are ~100 protons cm^−2×s^−1 for E > 10 MeV [22] . We assume an increase in ancient particle fluxes over the current particle flux
of ~4 × 10^5 [2] [4] , yielding an energetic particle flux rate of 3.7 × 10^12 protons cm^−2×s^−1 for E > 10 MeV at the surface of the proto-Sun.
The production rates for cosmogenic nuclides can be calculated via:
$p=\underset{i}{\sum }{N}_{i}\int {\sigma }_{ij}\frac{dF\left(E\right)}{d{E}_{j}}dE$(4)
where i represents the target elements for the production of the considered nuclide, N[i] is the abundance of the target element (g×g^−1), j indicates the energetic particles that cause the reaction,
${\sigma }_{ij}\left(E\right)$ is the cross section for the production of the nuclide from the interaction of particle j with energy E from target i for
the considered reaction (cm^2), and $\frac{dF\left(E\right)}{d{E}_{j}}dE$ is the differential energetic particle
flux of particle j at energy E (cm^−2×s^−1) [22] . We assume gaseous oxygen target particles of solar composition [23] .
The cross-section we use to calculate ^7Be production from protons and ^4He pathways is from Sisterson et al. [24] , and the cross-section we use for production from ^3He is from Gounelle et al. [25]
. The Sisterson et al. [24] cross-section is experimental obtained, and the Gounelle et al. [25] cross-section is a combination of experimental data, fragmentation and Hauser-Feshbach codes. The
uncertainty associated with model codes are at best a factor of two. Taking into account both target abundance and nuclear cross-sections, the reaction with oxygen as the target is the primary
production pathway. Any other nuclear reaction would add little to the overall ^7Be production rate. Table 2 shows the nuclear reactions considered in the calculations.
3. Results
The content of ^7Be found in refractory material, in atoms g^−1, predicted by SWIM is given by:
${N}^{7\text{Be}}=\frac{P}{S}=\frac{p\cdot f}{{\stackrel{˙}{M}}_{D}\cdot {X}_{r}\cdot F}$(5)
where P is given atoms s^−1 and S is given in g×s^−1.
Using the refractory mass inflow rate, S, of 2.5 × 10^14 g×s^−1 from Equation (1), and calculations of P, the effective ancient ^7Be outflow rate, from Equation (2) & Equation (4), we determine the
content of ^7Be in CAIs in atoms g^−1 using Equation (5), and find the associated isotopic ratio for different flare parameters given in Table 3. Figure 2 depicts the ^7Be isotopic ratio predicted by
the SWIM from SEPs.
4. Discussion
Similar to ^10Be, the primary target for SEP production of ^7Be is oxygen. As such, the SEP origin of ^7Be and ^10Be are uniquely intertwined. The estimated ^7Be/^10Be production ratio from MeV SEPs
in the early solar system is estimated to be ~70 [14] . Using the production rate from Equation (4) and the production rate for ^10Be from Bricker & Caffee [2] from SEP interaction with oxygen
targets, we obtain a production ratio of ~50, which is similar to Leya [14] . It would then be expected that the original ratio of ^7Be/^9Be found in CAIs would be ~50 times greater than the ^10Be/^
9Be ratio, assuming the simple SWIM mechanism described above. Using 9.5 × 10^−4 [13] as the canonical ^10Be/^9Be ratio, the ^7Be/^9Be ratio would scale to 4.8 × 10^−2. We find this ratio is
reproducible within a factor of ~5, the uncertainty associated with SWIM, for spectral indices r > 3.2. The SWIM can account for the scaled up ^7Be/^9Be ratio. Figure 3 below details the ratio of ^
7Be/^9Be from SWIM to 4.8 × 10^−2.
Experimentally obtained measurements for the original ^7Be/^9Be ratio in CAIs are limited and a matter of considerable debate. Limited experimentally determined values for the ratio range from about
1.2 × 10^−3 [11] to 6.1 × 10^−3 [10] . The experimentally obtained ratios are at least a factor of 10 less than SWIM
Table 2. Nuclear reactions considered in this paper.
Table 3. Predicted ^7Be content in CAIs.
Figure 2. Predicted ^7Be content in CAIs from energetic protons as a function of solar flare parameter.
Figure 3. Ratio of ^7Be/^9Be found from SWIM.A ratio of one indicates exact match, a ratio greater than one indicates overproduction, and a ratio less than one indicates underproduction.
calculations, and also a factor of at least 10 less than the scaled up ^7Be/^9Be found from scaling the canonical ^10Be/^9Be ratio to ^7Be and ^10Be production rates. Figure 4 depicts the ratio of
SWIM obtained ratio to the canonical ^7Be/^9Be ratio.
Clearly, some other mechanism is needed to explain the overproduction of the
Figure 4. Ratio of SWIM ^9Be/^10Be ratio to canonical ^7Be/^9Be ratio. A factor greater than one indicates overproduction relative to canonical.
Figure 5. Days to canonical ratio vs. spectral index.
^7Be/^9Be ratio, both in terms of SWIM calculations and the scaling of the ^10Be/^9Be to relative ^7Be and ^10Be production rates.
An assumption of SWIM is that radionuclides are produced via SEP interaction and then immediately incorporated into CAI precursor materials. With a half-life of 53 days, it is possible that some
temporal evolution occurs before ^7Be becomes implanted. Figure 5 shows days to canonical ratio for spectral index.
Figure 5 shows that with a delay on the order of ~100 days from the time of production of ^7Be to implantation in to CAI precursor materials, the canonical ratio is replicated. Taking into account
the time from production of the radionuclide to implantation into CAI precursors, i.e., two half-lives of ^7Be, explains the deficit in ^7Be/^10Be measured ratio in comparison to the ^7Be/^10Be
production ratio. It is possible and likely for nuclei to have some finite residence time in the photosphere. Calculations of this residence time have not been performed and are beyond the scope of
this paper. Our ad hoc choice of two half-lives of residence time for ^7Be was to explain the ^7Be/^10Be measured ratio in comparison to the ^7Be/^10Be production ratio.
Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.
Cite this paper
Bricker, G.E. (2019) Early Solar System Solar Wind Implantation of ^7Be into Calcium-Alumimum Rich Inclusions in Primitive Meteortites. International Journal of Astronomy and Astrophysics, 9, 12-20. | {"url":"https://www.scirp.org/html/2-4500821_90161.htm","timestamp":"2024-11-07T18:42:47Z","content_type":"application/xhtml+xml","content_length":"58498","record_id":"<urn:uuid:f9fc1568-4e18-4bc6-b82d-e6f395c77d89>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00649.warc.gz"} |
Fact Class
Class DoubleSymFact represents the factorization of a symmetric, matrix of double-precision floating point numbers.
Namespace: CenterSpace.NMath.CoreAssembly:
NMath (in NMath.dll) Version: 7.4
The DoubleSymFact type exposes the following members.
Name Description
DoubleSymFact(DoubleSymmetricMatrix) Constructs a DoubleSymFact instance by factoring the given matrix. By default the condition number for the matrix will not be computed and will not be
available from the ConditionNumber method.
DoubleSymFact(DoubleSymmetricMatrix, Constructs a DoubleSymFact instance by factoring the given matrix.
Name Description
Cols Gets the number of columns in the matrix represented by the factorization.
IsGood Gets a boolean value which is true if the matrix factorization succeeded and the factorization may be used to solve equations, compute determinants, inverses, and so on; otherwise false.
IsSingular Gets a boolean value which is true if the matrix is Singular and the factorization may NOT be used to solve equations, compute determinants, inverses, and so on; otherwise true.
Rows Gets the number of rows in the matrix represented by the factorization.
Name Description
Clone Creates a deep copy of this factorization.
ConditionNumber Computes an estimate of the reciprocal of the condition number of a given matrix in the 1-norm.
Determinant Computes the determinant of the factored matrix.
Factor(DoubleSymmetricMatrix) Factors the matrix A so that self represents the factorization of A. By default the condition number for the matrix will not be computed and will not be available
from the ConditionNumber method.
Factor(DoubleSymmetricMatrix, Factors the matrix A so that self represents the factorization of A.
Inverse Computes the inverse of the factored matrix.
Solve(DoubleMatrix) Uses this factorization to solve the linear system AX = B.
Solve(DoubleVector) Uses the factorization of self to solve the linear system Ax = b. | {"url":"https://www.centerspace.net/doc/NMath/ref/html/T_CenterSpace_NMath_Core_DoubleSymFact.htm","timestamp":"2024-11-05T20:01:28Z","content_type":"text/html","content_length":"16000","record_id":"<urn:uuid:715585c2-9d75-4885-b675-36d8dad3e632>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00821.warc.gz"} |
ASA Community
Questions on Bayesian Model Averaging
1. Questions on Bayesian Model Averaging
Hi everyone,
I am working on a project where I have individuals whom I track in relation to "before", "during" and "after" they take a certain action (i.e., phase). The year when they take this action is also
recorded. This means that the categories of phase have the following meaning: "before" means "the year before they took the action", "during" means the "the year when they took the action" and
"after" means "the year after they took the action".
In my modelling, phase and year are predictor variables and the response variable is a count of votes cast by these individuals throughout the year in questions out of a total number of votes.
The model includes a random effect for individual.
In a first stage, I fit 4 different Bayesian models to the data using just phase as a predictor of the vote count (out of the total); each of these models uses the same model formula but a
different family: binomial, beta-binomial, zero-inflated binomial and zero-inflated beta-binomial. All models are fitted with the brm() function from the brms package of R, using default priors.
Question 1: From a Bayesian perspective, is it appropriate to compute posterior model weights for these 4 models given they use different families?
Question 2: Given that I am interested in characterizing the effect of phase (hence not in predicting from the model), what type of model weights would be most appropriate to use? (I have used
the brms function post_prob() to compute posterior model probabilities from marginal likelihoods, though I read that these probabilities are sensitive to the choice of priors; by default, this
function assumes the models are equally likely a priori.)
Question 3: If one of the models receives most weight (e.g., its weight is something like 0.9), does it still make sense to average all the model or is it ok to retain just this dominating model
for further inference?
In a second stage, I fit 4 * 3 = 12 different Bayesian models to the data, consisting of 3 sets of models. The first set of 4 models uses year on its own as a predictor and all 4 families listed
above. The second set of 4 models uses year and phase as predictors, but not their interaction, with each family in turn. The third set of 4 models uses year, phase and their interaction
year:phase as predictors, with each family in turn. All 12 models are fitted with the brm() function from the brms package of R, using default priors. The questions below mirror the questions
above, except that now there is the added complication of not just the families possibly changing across candidate models, but also predictors included in the model.
Question 4: From a Bayesian perspective, is it appropriate to compute posterior model weights for these 12 models given they use different families? (Again, here I used post_prob() from brms.)
Question 5: If one of the models receives most weight (e.g., its weight is something like 0.9), does it still make sense to average all the model or is it ok to retain just this dominating model
for further inference?
Any comments or answers would be appreciated - I would like to make sure I am not doing something totally nonsensical.
Many thanks.
Email: isabella@ghement.ca
[Isabella] [Ghement][Ghement Statistical Consulting Company Ltd.] | {"url":"https://community.amstat.org/discussion/questions-on-bayesian-model-averaging","timestamp":"2024-11-13T21:43:20Z","content_type":"text/html","content_length":"203238","record_id":"<urn:uuid:97c15bc0-11d3-41ff-ba68-60b09ffcd5d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00726.warc.gz"} |
Measure Theory and Differentiation (Part 1)Measure Theory and Differentiation (Part 1)
Measure Theory and Differentiation (Part 1)
So I had an analysis exam [S:yesterday:S] [S:last week:S] a while ago (this post took a bit of time to finish writing). It roughly covered the material in chapter 3 of Folland’s “Real Analysis:
Modern Techniques and Their Applications”. I’m decently comfortable with the material, but a lot of it has always felt kind of unmotivated. For example, why is the Lebesgue Differentiation Theorem
called that? It doesn’t look like a derivative… At least not at first glance.
A big part of my studying process is fitting together the various theorems into a coherent narrative. It doesn’t have to be linear (in fact, it typically isn’t!), but it should feel like the theorems
share some purpose, and fit together neatly. I also struggle to care about theorems before I know what they do. This is part of why I care so much about examples – it’s nice to know what problems a
given theorem solves.
After a fair amount of reading and thinking^1, I think I’ve finally fit the puzzle pieces together in a way that works for me. Since I wrote it all down for myself as part of my studying, I figured I
would post it here as well in case other people find it useful. Keep in mind this is probably obvious to anyone with an analytic mind, but it certainly wasn’t obvious to me!
Let’s get started!
To start, we need to remember how to relate functions and measures. Everything we say here will be in $\mathbb{R}$, and $m$ will be the ($1$-dimensional) Lebesgue Measure.
If $F$ is increasing and continuous from the right, then there is a (unique!) regular borel measure $\mu_F$ (called the Lebesgue-Stieltjes Measure associated to $F$) so that
\[\mu_F((a,b]) = F(b) - F(a)\]
Moreover, given any regular borel measure $\mu$ on $\mathbb{R}$, the function
\[F_\mu \triangleq \begin{cases} \mu((0,x]) & x \gt 0 \\ 0 & x = 0 \\ -\mu((x,0]) & x \lt 0 \end{cases}\]
is increasing and right continuous.
This is more or less the content of the Carathéodory Extension Theorem. It’s worth taking a second to think where we use the assumptions on $F$. The fact that $F$ is increasing means our measure is
positive. Continuity from the right is a bit more subtle, though. Since $F_\mu$ is always right continuous, we need to assume our starting function is right continuous in order to guarantee $F_{\
mu_F} = F$.
This is not a big deal, though. A monotone function is automatically continuous except at a countable set (see here for a proof) and at its countably many discontinuities, we can force
right-continuity by defining
\[\tilde{F}(x_0) \triangleq \lim_{x \to x_0^+} F(x)\]
which agrees with $F$ wherever $F$ is continuous. If we put our probabilist hat on, we say that $F_\mu$ is the Cumulative Distribution Function of $\mu$. Here $F_\mu(x)$ represents the total
(cumulative) mass we’ve seen so far.
It turns out that Lebesgue-Stieltjes measures are extremely concrete, and a lot of this post is going to talk about computing with them^2. After all, it’s entirely unclear which (if any!) techniques
from a calculus class carry over when we try to actually integrate against some $\mu_F$. Before we can talk about computation, though, we have to recall another (a priori unrelated) way to relate
functions to measures:
Given a positive, locally $L^1$ function $f$, we can define the regular measure $m_f$ by
\[m_f(E) \triangleq \int_E f dm\]
Moreover, if $m_f = m_g$, then $f=g$ almost everywhere.
The locally $L^1$ conditions says that $\int_E f dm$ is finite whenever $E$ is bounded. It’s not hard to show that this is equivalent to the regularity of $m_f$, which we’ll need shortly.
Something is missing from the above theorem, though. We know sending $F \rightsquigarrow \mu_F$ is faithful, in the sense that $F = F_{\mu_F}$ and $\mu_{F_\mu} = \mu$. We’ve now introduced the
measure $m_f$, but we didn’t say how to recover $f$ from $m_f$… Is it even possible? The answer is yes, as a corollary of a much more powerful result:
Lebesgue-Radon-Nikodym Theorem
Every measure $\mu$ decomposes (uniquely!) as
\[\mu = \lambda + m_f\]
for some measure $\lambda \perp m$ and some function $f$.
Moreover, we can recover $f$ from $\mu$ as^3
\[f(x) = \lim_{r \to 0} \frac{\mu(B_r(x))}{m(B_r(x))}\]
for almost every $x$. Here, as usual $B_r(x) = (x-r,x+r)$ is the ball of radius $r$ about $x$.
People often write $f = \frac{d \mu}{dm}$, and call it the Radon-Nikodym Derivative. Let’s see why.
In the case $\mu = m_f$, then this shows us how to recover $f$ (uniquely) from $m_f$, and life is good:
\[\frac{d m_f}{dm} = f\]
The converse needs a ~bonus condition~. In order to say $\mu = m_{\frac{d\mu}{dm}}$, we need to know that $\mu$ is absolutely continuous with respect to $m$, written $\mu \ll m$.
As an exercise, do you see why this condition is necessary? If $\mu \not \ll m$, why don’t we have a chance of writing $\mu = m_f$ for any $f$?
In the case of Lebesgue-Stieltjes measures, Lebesgue-Radon-Nikodym buys us something almost magical. For almost every $x$, we see:
\[\begin{aligned} \frac{d\mu_F}{dm}(x) &= \lim_{r \to 0} \frac{\mu_F(B_r(x))}{m(B_r(x))} \\ &= \lim_{r \to 0} \frac{F(x+r) - F(x-r)}{x+r - (x-r)} \\ &= \lim_{r \to 0} \frac{F(x+r) - F(x-r)}{2r} \\ &=
F'(x) \end{aligned}\]
Now we see why we might call this $f$ the Radon-Nikodym derivative. In the special case of Lebesgue-Stieltjes measures, it literally is the derivative. It’s immediate from the definitions that $F =
F_{m_f}$ acts like an antiderivative of $f$, since $F_{m_f}(x) = \int_0^x f\ dm$. Now we see $f = \frac{d \mu_F}{dm}$ works as a derivative of $F$ as well!
In fact, we can push this even further! Let’s take a look at the Lebesgue Differentiation Theorem
For almost every $x$, we have:
\(\lim_{r \to 0} \frac{1}{m B_r(x)} \int_{B_r(x)} f(t) dm = f(x)\)
Why is this called the differentiation theorem? Let’s look at $F_{m_f}$, which you should remember is a kind of antiderivative for $f$.
For $x > 0$ (for simplicity), we have $F_{m_f}(x) = m_f((0,x]) = \int_{(0,x]} f dm$. If we rewrite the theorem in terms of $F_{m_f}$, what do we see?
\[\begin{aligned} f(x) &= \lim_{r \to 0} \frac{1}{m B_r(x)} \int_{B_r(x)} f dm \\ &= \lim_{r \to 0} \frac{1}{(x+r) - (x-r)} \int_{x-r}^{x+r} f dm \\ &= \lim_{r \to 0} \frac{1}{2r} \left ( \int_{0}^
{x+r} f dm - \int_{0}^{x-r} f dm \right )\\ &= \lim_{r \to 0} \frac{F_{m_f}(x+r) - F_{m_f}(x-r)}{2r} \\ &= F_{m_f}'(x) \end{aligned}\]
So this is giving us part of the fundamental theorem of calculus^4! This theorem (in the case of Lebesgue-Stieltjes measures) says exactly that (for almost every $x$)
\[\left ( x \mapsto \int_0^x f dm \right )' = f(x)\]
Let’s take a moment to summarize the relationships we’ve seen. Then we’ll use these relationships to actually compute with Lebesgue-Stieltjes integrals.
\[\bigg \{ \text{increasing, right-continuous functions $F$} \bigg \} \leftrightarrow \bigg \{ \text{regular borel measures $\mu_F$} \bigg \}\] \[\bigg \{ \text{positive locally $L^1$ functions $f$}
\bigg \} \leftrightarrow \bigg \{ \text{regular borel measures $m_f \ll m$} \bigg \}\]
• By considering $F_{m_f}$ we see functions of the first kind are antiderivatives of functions of the second kind.
• By considering $\frac{d \mu_F}{dm}$, we see functions of the second kind are (almost everywhere) derivatives of functions of the first kind.
• Indeed, $\frac{d \mu_F}{dm} = F’$ almost everywhere.
• And $F_{m_f}’ = f$ almost everywhere.
Why should we care about these theorems? Well, Lebesgue-Stieltjes integrals arise fairly regularly in the wild, and these theorems let us actually compute them! It’s easy to integrate against $m_f$,
since monotone convergence gives us $\int g dm_f = \int g f dm$.
Then this buys us the (very memorable) formula:
\[\int g d \mu_F = \int g \frac{d \mu_F}{dm} dm = \int g F' dm\]
and now we’re integrating against lebesgue measure, and all our years of calculus experience is applicable!
Of course, I’ve left out an important detail: Whatever happened to that measure $\lambda$? The above formula is true exactly when $F$ is continuous everywhere. At points where it is discontinuous we
need to change it slightly by using $\lambda$. These are called singular measures, and they can be pretty pathological. A good first intuition, though, is to think of them like dirac measures, and
that’s the case that we’ll focus on in this post^5.
Let’s write \(H = \begin{cases} 0 & x \lt 0 \\ 1 & 0 \leq x \end{cases}\). This us usually called the heaviside function.
Recall our interpretation of this function: $H(x)$ is supposed to represent the mass of $(-\infty, x]$. So as we scan from left to right, we see the mass is constantly $0$ until we hit the point $0$.
Then suddenly we jump up to mass $1$. But once we get there, our mass stays constant again.
So $H$ thinks that $0$ has mass $1$ all by itself, and thinks that there’s no other mass at all!
Indeed, we see that
\[\mu_H((a,b]) = H(b) - H(a) = \begin{cases} 1 & 0 \in (a,b] \\ 0 & 0 \not \in (a,b] \end{cases}\]
So $\mu_H$ is just the dirac measure at $0$ (or $\delta_0$ to its friends)! Notice this lets us say the “derivative” of $H$ is $\delta_0$, by analogy with the Lebesgue-Stieltjes case. Or conversely,
that $H$ is the “antiderivative” of $\delta_0$. This shows us that recasting calculus in this language actually buys us something new, since there’s no way to make sense of $\delta_0$ as a
traditional function.
It’s finally computation time! Since we know $\int g d\delta_0 = g(0)$, and (discrete) singular measures look like (possibly infinite) linear combinations of dirac measures, this lets us compute all
increasing right-continuous Lebesgue-Stieltjes measures that are likely to arise in practice. Let’s see some examples! If you want to see more, you really should look into Carter and van Brunt’s “The
Lebesgue-Stieltjes Integral: A Practical Introduction”. I mentioned it in a footnote earlier, but it really deserves a spotlight. It’s full of concrete examples, and is extremely readable!
Let’s start with a continuous example. Say \(F = \begin{cases} 0 & x \leq 0 \\ x^2 & x \geq 0 \end{cases}\).
So $\mu_F$ should think that everything is massless until we hit $0$. From then on, we start gaining mass faster and faster as we move to the right. If you like, larger points are “more dense” than
smaller ones, and thus contribute more mass in the same amount of space.
Say we want to compute
\[\int_{-\pi}^\pi \sin(x) d \mu_F = \int_{-\pi}^\pi \sin(x) \cdot F' dm\]
We can compute \(F' = \begin{cases} 0 & x \leq 0 \\ 2x & x \geq 0 \end{cases}\), so we split up our integral as
\[\int_{-\pi}^0 \sin(x) \cdot 0 dm + \int_0^\pi \sin(x) \cdot 2x dm\]
But both of these are integrals against lebesgue measure $m$! So these are just “classical” integrals, and we can use all our favorite tools. So the first integral is $0$, and the second integral is
$2\pi$ (integrating by parts). This gives
\[\int_{-\pi}^\pi \sin(x) d \mu_F = 2\pi\]
That wasn’t so bad, right?
Let’s see another, slightly trickier one. Let’s look at \(F = \begin{cases} x & x \lt 0 \\ e^x & x \geq 0 \end{cases}\)
You should think through the intuition for what $\mu_F$ looks like. You can then test your intuition against a computation:
\[\mu_F = \lambda + m_f\]
In the previous example, $\lambda$ was the $0$ measure since our function was differentiable everywhere. Now, though, we aren’t as lucky. Our function $F$ is not differentiable at $0$, so we will
have to work with some nontrivial $\lambda$.
Let’s start with the places $F$ is differentiable. This gives us the density function \(f = F' = \begin{cases} 1 & x \lt 0 \\ e^x & x \gt 0 \end{cases}\).
We can also see the point $0$ has mass $1$. In this case we can more or less read this off the graph (since we have a discontinuity where we jump up by $1$), but in more complex examples we would
compute this by using $\mu_F({ 0 }) = \lim_{r \to 0^+} F(r) - F(-r)$. You can see that this does give us $1$ in this case, as expected. So we see (for $f$ as before)
\[\mu_F = \delta_0 + m_f\]
So to compute
\[\int_{-1}^1 4 - x^2 d\mu_F = \int_{-1}^1 4 - x^2 d(\delta_0 + m_f) = \int_{-1}^1 4 - x^2 d \delta_0 + \int_{-1}^1 (4 - x^2)f dm\]
we can handle the $\delta_0$ part and the $f dm$ part separately!
We know how to handle dirac measures:
\[\int_{-1}^1 4 - x^2 d \delta_0 = \left . (4 - x^2) \right |_{x = 0} = 4\]
And we also know how to handle “classical” integrals:
\[\int_{-1}^1 (4 - x^2) f dm = \int_{-1}^0 (4 - x^2) dm + \int_0^1 (4 - x^2) e^x dm = \frac{11}{3} + (3e-2)\]
So all together, we get \(\int_{-1}^1 4 - x^2 d\mu_F = 4 + \frac{11}{3} + (3e-2)\).
As an exercise, say \(F = \begin{cases} e^{3x} & x \lt 0 \\ 2 & 0 \leq x \lt 1 \\ 2x+1 & 1 \leq x \end{cases}\)
Can you intuitively see how $\mu_F$ distributes mass?
Can you compute
\(\int_{-\infty}^2 e^{-2x} d\mu_F\)
As another exercise, can you intuit how $\mu_F$ distributes mass when $F(x) = \lfloor x \rfloor$ is the floor function?
What is $\int_1^\infty \frac{1}{x^2} d\mu_F$? What about $\int_1^\infty \frac{1}{x} d\mu_F$?
Ok, I hear you saying. There’s a really tight connection between increasing (right-)continuous functions $F$ on $\mathbb{R}$ and positive integrable functions $f$. This connection is at its tightest
wherever $F$ is actually continuous, as then the measures $\mu_F$ and $m_f$ have a derivative relationship, which is reflected in the same derivative relationship of functions $F’ = f$. Not only does
this give us a way to generalize the notion of derivative to functions that might not normally have one (as in the case of the heaviside function and the dirac delta), it gives us a concrete way of
evaluating Lebesgue-Stieltjes integrals.
But doesn’t this feel restrictive? There are lots of functions $F$ which aren’t (right-)continuous or increasing that we might be interested in differentiating. There are also lots of nonpositive
functions $f$ which we might be interested in integrating. Since we got a kind of “fundamental theorem of calculus” from these measure theoretic techniques, if we can show how to apply these
techniques to a broader class of functions, we might be able to get a more general fundamental theorem of calculus.
Of course, to talk about more general functions $F$, we’ll need to allow our measures to assign negative mass to certain sets. That’s ok, though, and we can even go so far as to allow complex valued
measures! In fact, from what I can tell, this really is the raison d’être for signed and complex measures. I was always a bit confused why we might care about these objects, but it’s beginning to
make more sense.
This post is getting pretty long, though, so we’ll talk about the signed case in a (much shorter, hopefully) part 2!
1. I was mainly reading Folland (Ch. 3), since it’s the book for the course. I’ve also been spending time with Terry Tao’s lecture notes on the subject (see here, and here), as well as this PDF from
Eugenia Malinnikova’s measure theory course at Stanford. I read parts of Axler’s new book, and while I meant to read some of Royden too, I didn’t get around to it. ↩
2. As an aside, I really can’t recommend Carter and van Brunt’s “The Lebesgue-Stieltjes Integral: A Practical Introduction” enough. It spends a lot of time on concrete examples of computation, which
is exactly what many measure theory courses are regrettably missing. Chapter 6 in particular is great for this, but the whole book is excellent. ↩
3. We can actually relax this from balls $B_r(x)$ to a family ${E_r}$ that “shrinks nicely” to $x$, though it’s still a bit unclear to me what that means and what it buys us. It seems like one
important feature is that the $E_r$ don’t have to contain $x$ itself. It’s enough to take up a (uniformly) positive fraction of space near $x$. ↩
4. There’s another way of viewing this theorem which is quite nice. I think I saw it on Terry Tao’s blog, but now that I’m looking for it I can’t find it… Regardless, once we put on our nullset
goggles, we can no longer evaluate functions. After all, for any particular point of interest, I can change the value of my function there without changing its equivalence class modulo nullsets.
However, even with our nullset goggles on, the integral $\frac{1}{m B_r(x)} \int_{B_r(x)} f dm$ is well defined! So for almost every $x$, we can “evaluate” $f$ through this (rather roundabout)
approach. The benefit is that this notion of evaluation does not depend on your choice of representative! ↩
5. In no small part because I’m not sure how you would actually integrate against a singular continuous measure in the wild… ↩ | {"url":"https://grossack.site/2021/02/21/lebesgue-ftc-1.html","timestamp":"2024-11-14T07:36:05Z","content_type":"text/html","content_length":"30761","record_id":"<urn:uuid:7eaeda15-9ff1-48f9-aa8e-b615dbc959c6>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00371.warc.gz"} |
Transition to zero cosmological constant and phantom dark energy as solutions involving change of orientation of spacetime manifold
The main conclusion of long-standing discussions concerning the role of solutions with degenerate metric (g ≡ det(g[μν]) = 0 and even with g[μν] = 0) was that in the first-order formalism they are
physically acceptable and must be included in the path integral. In particular, they may describe topology changes and reduction of the 'metrical dimension' of spacetime. The latter implies
disappearance of the volume element of a 4D spacetime in a neighborhood of the point with g = 0. We pay attention to the fact that besides , the 4D spacetime differentiable manifold also possesses a
'manifold volume measure' (MVM) described by a 4-form which is sign indefinite and generically independent of the metric. The first-order formalism proceeds with an originally independent connection
and metric structures of the spacetime manifold. In this paper we bring up the question of whether the first-order formalism should be supplemented with degrees of freedom of the spacetime
differentiable manifold itself, e.g. by means of the MVM. It turns out that adding the MVM degrees of freedom to the action principle in the first-order formalism one can realize very interesting
dynamics. Such a two measures field theory (TMT) enables radically new approaches to the resolution of the cosmological constant problem. We show that fine tuning free solutions describing a
transition to the Λ = 0 state involve oscillations of g [μν] and MVM around zero. The latter can be treated as a dynamics involving changes of orientation of the spacetime manifold. As we have shown
earlier, in realistic scale invariant models (SIM), solutions formulated in the Einstein frame satisfy all existing tests of general relativity (GR). Here we reveal surprisingly that in SIM, all
ground-state solutions with Λ ≠ 0 appear to be degenerate either in g[00] or in MVM. Sign indefiniteness of MVM in a natural way yields a dynamical realization of a phantom cosmology (w < -1). It is
very important that for all solutions, the metric tensor rewritten in the Einstein frame has regularity properties exactly as in GR. We discuss new physical effects which arise from this theory and
in particular the strong gravity effect in high energy physics experiments.
ASJC Scopus subject areas
• Physics and Astronomy (miscellaneous)
Dive into the research topics of 'Transition to zero cosmological constant and phantom dark energy as solutions involving change of orientation of spacetime manifold'. Together they form a unique | {"url":"https://cris.bgu.ac.il/en/publications/transition-to-zero-cosmological-constant-and-phantom-dark-energy","timestamp":"2024-11-05T20:15:17Z","content_type":"text/html","content_length":"63182","record_id":"<urn:uuid:109345bf-c4e9-48f6-8709-7fbc339bf3d1>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00259.warc.gz"} |
The Training Sets
Next: The Algorithm Up: Modeling Previous: Modeling
As test bed for our forecasting system we used two well known time series from [BJ76]: The monthly totals of international airline passengers (thousand of passengers) from 1949 to 1960 (see figure 4
), and the daily closing prices of IBM common stock from May 1961 to November 1962 (see figure 5).
Table 1 gives some characteristics of these two time series: n the number of observations. The airline time series is an example of time series data with a clear trend and multiplicative seasonality,
whereas the IBM share price shows a break in the last third of the series and no obvious trend and/or seasonality.
The next section is concerned with the question: How can a neural network learn a time series? | {"url":"http://godefroy.sdf-eu.org/apl94/node6.html","timestamp":"2024-11-13T14:45:59Z","content_type":"text/html","content_length":"4738","record_id":"<urn:uuid:bb3478b6-0d8f-4962-9262-e6a287cbf21a>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00175.warc.gz"} |
How do you find the derivative using implicit differentiation?
How To Do Implicit Differentiation
1. Take the derivative of every variable.
2. Whenever you take the derivative of “y” you multiply by dy/dx.
3. Solve the resulting equation for dy/dx.
What do higher-order derivatives mean?
A higher-order derivative means the derivatives other than the first derivative and are used to model real-life phenomena like most transportation devices such as: Cars. Planes. Rollercoasters.
What is the higher derivative?
The process of differentiation can be applied several times in succession, leading in particular to the second derivative f″ of the function f, which is just the derivative of the derivative f′. The
second derivative often has a useful physical interpretation.
What is D 2y dx 2?
The second derivative is what you get when you differentiate the derivative. The second derivative is written d2y/dx2, pronounced “dee two y by d x squared”. …
What is implicit differentiation used for?
The technique of implicit differentiation allows you to find the derivative of y with respect to x without having to solve the given equation for y. The chain rule must be used whenever the function
y is being differentiated because of our assumption that y may be expressed as a function of x.
Why is implicit differentiation important?
Implicit differentiation is the special case of related rates where one of the variables is time. Implicit differentiation has an important application: it allows to compute the derivatives of
inverse functions. It is good that we review this, because we can use these derivatives to find anti-derivatives.
Is D DX the same as dy dx?
d/dx is an operator that says to take the derivative of something when it is multiplied. In more advanced settings sometimes it will be written D^n this means take the n’th derivative with respect to
a given variable. So in answer to your question the only time d/dx is the same as dy/dx is when you apply d/dx to y.
What do you mean by higher derivatives?
What do you mean by higher-order derivatives?
What is implicit differentiation and how does it work?
Differentiate the function with respect to x
Collect all dy/dx on one side
Finally,solve for dy/dx
What is an implicit differentiation?
Implicit differentiation is the process of finding the derivative of an implicit function. There are two types of functions: explicit function and implicit function. An explicit function is of the
form y = f (x) with the dependent variable “y” is on one of the sides of the equation.
What is implicit differentiation formula?
Differentiate both sides of f (x,y) = 0 with respect to x
Apply usual derivative formulas to differentiate the x terms
Apply usual derivative formulas to differentiate the y terms along with multiplying the derivative by dy/dx
Solve the resultant equation for dy/dx (by isolating dy/dx).
What is the difference between dy/dx and d/dx?
In the terms dx and dy, the d is for delta or “change in”. So they represent the change in y and the change in x as a function, usually in terms of each other but sometimes another parameter. So dy/
dx as you said is the slope, or change in x divided by the change in y, dy/dx is simply the inverse slope. | {"url":"https://tracks-movie.com/how-do-you-find-the-derivative-using-implicit-differentiation/","timestamp":"2024-11-04T22:01:54Z","content_type":"text/html","content_length":"52342","record_id":"<urn:uuid:257af13c-be1a-4539-9668-316f4f245ed8>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00041.warc.gz"} |
PyBites Bite 375. Find All Letter Combinations of a Phone Number
The typical phone keypad (pictured) features numbers (0 - 9), letters mapped to some of those numbers (2 - 9, inclusive), and two non-numeric characters (* and #).
Now imagine it's the 1950s, you live in a small village with 900 other villagers, any of whom you can reach simply by dialing the last four digits of their telephone number (in other words, you
only need to dial "5309" to reach Jenny, not "867-5309").
Given a string of up to four digits, return a list of strings where each string represents a valid combination of letters that can be formed from the input.
Raise a ValueError if the input digits string contains non-digit characters or more than four digits.
Example 1:
>>> from combinations import generate_letter_combinations
>>> digits = "24"
>>> generate_letter_combinations(digits)
['ag', 'ah', 'ai', 'bg', 'bh', 'bi', 'cg', 'ch', 'ci']
Example 2:
>>> from combinations import generate_letter_combinations
>>> digits = "79"
>>> generate_letter_combinations(digits)
['pw', 'px', 'py', 'pz', 'qw', 'qx', 'qy', 'qz', 'rw', 'rx', 'ry', 'rz', 'sw', 'sx', 'sy', 'sz']
Example 3:
>>> from combinations import generate_letter_combinations
>>> digits = "232"
>>> generate_letter_combinations(digits)
'ada', 'adb', 'adc', 'aea', 'aeb', 'aec', 'afa', 'afb', 'afc',
'bda', 'bdb', 'bdc', 'bea', 'beb', 'bec', 'bfa', 'bfb', 'bfc',
'cda', 'cdb', 'cdc', 'cea', 'ceb', 'cec', 'cfa', 'cfb', 'cfc'
- The strings contained in the list returned by your function can be in any order.
- Since the digits 1 and 0 are not associated with any letters, the phone number should include only digits 2 - 9, inclusive.
- There are different ways to solve this problem.
- One way is to use Python's itertools module.
Keep calm and code in Python! | {"url":"https://codechalleng.es/bites/375/","timestamp":"2024-11-07T13:19:58Z","content_type":"text/html","content_length":"85753","record_id":"<urn:uuid:e7962c67-2244-4d43-bd1c-806f416174a5>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00420.warc.gz"} |
error Archives - Christoph Bartneck, Ph.D.
In 1993, the German city of Wemding celebrated its 1200 anniversary.
They decided to start a long-term art project. Every decade, they would place a concrete block on a base. After adding 120 blocks, the pyramid would be complete.
The only problem is that the pyramid will be completed in the year 3183. Which is only 1190 in the future. Not 1200. The project made a basic mathematical error, called the picket fence error. For a
picket fence of n elements, you need to have n+1 posts.
Instead of waiting for the first decade to complete before placing the first block, they started immediately. This is like putting the first candle on the birthday cake on your child’s actual day of
Matt Parker pointed out this error during his visit to the placement of the fourth block in 2023. He also proposed an alternative design that would take 121 blocks to complete. Unfortunately, his
design is not a pyramid and would be 19.8 meters tall. That is certainly not safe in a storm.
There must be a better design. I took 121 of my beloved LEGO bricks and started on a seven-by-seven base. After some experimentation, I came up with a beautiful pyramid that is only one block taller
than the original design. It is still a proper pyramid with complete symmetry.
We can only speculate what Manfred Laber, the artist, had in mind. According to Barbara Schlecht, head of the Zeitpyramide trust, Mr. Laber was fully aware of the consequences of his design. It is
certainly much easier to design a sculpture with 120 bricks since it is divisible by 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 and 120. 121, on the other hand, is only divisible by 1,11 and
When public projects make glaring mistakes, they invite schadenfreude. We can speculate that Manfred Laber decided to sacrifice the opportunity to set a block at the beginning and at the end of the
1200 years period for having a direct relationship between the 120 blocks and the 1200 years.
There are alternative pyramid design for 121 blocks. He could have also decided to only place the foundation in 1993. In any case, this art project has become famous not for its original concepts,
but for the controversy around its maths. Which is unlikely to have been the intention of the artist.
Comparison of color measurement accuracy of ColorMunki Design and FRU WR-10QC Colorimeter
Review of the measurement accuracy of the ColorMunki Design and the FRU WR-10.
I am working on a colour project and had purchased the WR10 colorimeter to complement my long serving work horse, the X-Rite Color Munki Design. My ColorMunki is already several years old and I was
concerned that its accuracy might have declined. When I measured several hundreds of samples, I noticed that both colorimeters gave me considerably different LAB values.
To determine which device was closer to the truth I measured the 48 defined colours of Datacolor’s SpyderCHECKR 48. I calculated the absolute error both devices made. The results of a paired-sample
t-test showed that the ColorMunki is producing significantly less measurement errors on L (t(47)=-9.229, p<0.001), L (t(47)=-4.590, p<0.001) and L (t(47)=-4.871, p<0.001). However, both devices
measure colours that are significantly different from the target colour of the SpyderCheckr card on all three measurements. Figure 1 shows the means and standard deviation for all measurement errors.
Figure 1: Mean and Standard Deviation of all measurements for both devices.
There does seem to be some structure in the errors that WR-10 is producing. Have a look at the heat map (Figure 2). The data for my little experiment is available at the Open Science Framework (DOI:
Figure 2: Heat Map of the absolute errors
Although both devices show some significant deviation from the original, it is not far off from what can be expected of devices in this price range. The ColorMunki Design produces significantly
better results than the FRU’s WR-10QC. | {"url":"https://www.bartneck.de/tag/error/","timestamp":"2024-11-09T22:37:44Z","content_type":"text/html","content_length":"55622","record_id":"<urn:uuid:a13371c9-fb8c-4df6-a99a-af9d633a92ac>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00255.warc.gz"} |
How To Solve Linear Equations With Fractions - Odd CultureHow To Solve Linear Equations With Fractions
How To Solve Linear Equations With Fractions
posted by Chris Valentine
solving complex equations. Students often get lost in the several steps that have to be solved before reaching the correct answer and, therefore, need the constant guidance of teachers and parents to
be able to achieve the desired results. This is why our web pages have been designed to provide students with assistance on important topics in Mathematics in an efficient and effective manner. Read
on to find out everything you need to know about how to solve linear equations with fractions.
However, before we move on to discuss the steps that are required to solve linear equations with fractions, we must first understand the terms individually and separately. Let us understand what each
of these terms means. We also need to learn the math of numbers and how it can help us solve linear equations. It is time to befriend numbers instead of shying away from them.
Linear Equations
Linear equations are those equations of the first order that represent lines of a coordinate system. Therefore, in other words, linear equations are equations of a straight line and are formulated by
y=mx+b, where m represents the slope of the line while b stands for the y-intercept.
However, it might also be wise here to look at the history of linear equations and how they evolved to become such an integral component of Mathematics. The evolution of linear equations is very
closely linked to the studies and developments in linear algebra. The earliest studies of linear equations can be traced back to have been done by the European mathematician René Descartes in 1637
after coordinates were introduced in geometry. These changes in geometry led to the rise of a new kind of mathematical geometry that was termed Cartesian geometry. Since lines and planes were
important elements in this kind of geometry, there was an urgent need to devise equations to represent the same. This is how linear equations came into being and gradually developed to form a complex
branch of mathematics with different systems of such equations existing at their intersections that need to be solved.
Now that we have understood what a linear equation is, it is also important to go through the definition of a fraction to get a better grasp of the topic.
Just like in real life, a fraction is a small portion of a larger piece. In mathematics, a fraction is a value that represents parts of a whole value. These parts hold an equal value that constitutes
the whole and is termed numerator and denominator for the top and the bottom value, respectively. The former represents the value of the parts taken, while the former stands for the total number of
equal parts in the whole value.
Once again, it is only advisable to be aware of the history of fractions before we proceed further. It is fascinating to learn that work on fractions goes back to the ancient Egyptian civilization.
The first evidence of a study on fractions by several Egyptian mathematicians appears around 1600 B.C. in the Rhind Papyrus. However, the fractions that we find in these ancient works are different
from our own understanding of fractions. These mathematicians treated fractions more as ratios and unit fractions.
Fractions were also studied and worked on by mathematicians living in ancient India. This version of fractions is closer to how we present fractions today, and is believed by many mathematicians to
be the origin from which modern fractions have evolved. The first depiction of fractions in ancient India is recorded to have been done by Brahmagupta in A.D. 630, which was done by writing the
numerator and denominator in separate lines without the bar.
The bar in fractions is believed to have originated from the Arabs, who used the bar due to constraints of tying innovations at the time, while the numerator and denominator were differentiated by
Latin mathematicians. Until the sixteenth century, multiplication was applied to find the common denominator, before which adding and subtracting fractions had been employed for the same function
since the seventh century. Division of fractions was added much later and has continued to date as a common operation of fractions, and might have been the first way to look for a common denominator.
Today, the way to find the common multiple in fractions differs largely from these older operations. But to better understand fractions and its function today, we need to next be well-versed with
what a solution is in order to begin solving linear equations with fractions.
Mathematically, a solution is the process of assigning values to variables in an equation that can result in the equation holding true. This means that when a solution is applied to an equation, both
sides become equal in value as denoted by the ‘=’ symbol.
Steps to Solve Linear Equations with Fractions
Linear equations can be solved in five simple steps that, when applied to an equation, results in a solution to hold both sides equal. These steps are:
• Clear the fractions in the equation by multiplying both sides with the Least Common Denominator (L.C.D).
• Remove parentheses on each side by using the Distributive Property formula, x(y+z)=xy+yz.
• Combine like terms on each side.
• Undo addition or subtraction present in the equation.
• Undo multiplication or division present in the equation to make the coefficient of the variable equal.
• Undoing these actions simplifies both sides of the equation.
• Solve the equation by isolating the variable on one side and the constant on the other side.
Tips for Solving Linear Equations with Fractions
While we have labelled out the steps that can be applied to solve linear equations in the above section, there are also some tips and points that will be useful for students to remember while
attempting to solve linear equations with fractions.
• Any changes made on one side of the equation also have to be made on the other side since the left side is always equal to the right side in an equation.
• Single-variable equations can be solved by isolating the unknown variable on one side to find a number that is equal on the other side.
Things to Remember
Many students can make common mistakes, such as not multiplying both sides by the Least Common Denominator while solving linear equations with fractions. To avoid making such mistakes, here the
things you should keep in mind:
• In case of an impossible equality in an equation like x=0, there will be no solution.
• Solutions are always real for equations where the equality holds true in the case of every possible solution.
• To avoid or cancel denominators in an equation, multiply the entire equation with the Least Common Denominator.
• In an equation, parentheses can be removed by multiplying the coefficient present before them to the elements contained within them.
• In the case of a nested parentheses, which refers to a parenthesis inside another parenthesis, the exterior parenthesis is removed first by multiplying whatever value is contained in it with the
Frequently Asked Questions
Here are the questions that are frequently asked by students studying linear equations and fractions:
Name the three forms of linear equations.
The three forms of linear equations are standard form, slope-intercept form, and point-slope form.
What is the formula for representing the standard form of a linear equation?
The standard form of linear equations is given by:
Ax + By + C = 0,
where A, B, and C are constants, x and y are variables, and A ≠ 0 as well as B ≠ 0.
How to represent the slope form of linear equations?
Slope form of linear equations is represented by:
where m stands for the steepness of line while c is the y-intercept.
What is the difference between linear and nonlinear equations?
A linear equation represents straight lines, whereas a nonlinear equation does not form a straight line and can be a curve that has a variable slope value.
What are the seven types of fractions?
The seven different types of fractions are – Proper fractions, Improper fractions and Mixed fractions, like fractions, unlike fractions, equivalent fractions, and unit fractions.
How would you define proper fractions?
A fraction where the numerator is smaller than the denominator is called a proper fraction.
How would you define improper fractions?
A fraction is called an improper fraction when the numerator is greater than the denominator.
What is a mixed fraction?
A fraction that contains a combination of a whole number and a fraction is called a mixed fraction.
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Using ‘Averages’ While Thinking About Social Behavior? That’s Probably Wrong!
Ed H. Chi - Google Research Scientist
I’ve been meaning to write this blog post for a while. It’s one of my pet peeves about understanding distributions describing social behaviors online. I get annoyed whenever I read a headline like “
The Average Facebook Post Lives 22 Hours And 51 Minutes.” The key here is the word “average.” This is quite likely the wrong way to think about social behavior. This is actually quite common in
social behavior reporting that I see online. Take another example from Pew Research: “The average American has just over two discussion confidants (2.16) – that is, people with whom they discuss
important matters. This is a modest, but significantly larger number than the average of 1.93 core ties reported when we asked this same question in 2008.” [1] I am also guessing that the use of
average here is probably wrong. Why?
Averages are often not a good descriptor for many of the things we want to measure in social systems, since many of the distributions we deal with are not normal (Gaussian) distributions. I’m simply
making an observation that the above two distributions are probably not normal distributions. In the above examples, how could it have been improved? Well, use a different metric that relates to the
idea of “half-life,” or the median.
In describing human behavior, we often find a log-normal distribution instead, which means that the geometric mean is a better metric for “normal” (which for a log-normal distribution happens to
equal to its median). Thus, assuming you believe the distribution underlying your behavior is log-normal, it is better to ask for the median (that is, ask for the geometric mean and geometric
standard deviation), instead of asking for the arithmetic mean (aka the average) [2].
The below figure from Wikipedia gives a good illustration of the problem [2]. Many of our social system distributions look more like the one on the left. In this example, the median is near 1.0,
while the mean is on the right of the median at 1.7-ish, demonstrating how the arithmetic mean is a poor descriptor of “normal” behavior.
So why is the median better? Because intuitively it tells you that half of the people is above this point, and the other half below this point. It’s interesting to note that the “mode” (defined as
the highest point on the curve) is sometimes also used as a good descriptor for a log-normal distribution. By definition that the distribution is skewed, then we have mean > median > mode.
In particular, we know that frequency of posting activity, frequency of visits in many social systems are definitely not normally distributed, and the other metrics in social systems are likely to be
similarly long-tailed. And it’s worth pointing out that thinking about distributions should be part of many software design solutions, such as sorting reviews [3].
Real stat researchers will likely point out that many social system distributions are actually Pareto (or Power-law, or Zipf-distributed) [4] [5]. I won’t bore you with the details of the differences
[which are hard to explain here anyway], so see the below links for more information.
PS: BTW, sorry for the radio silence on social computing. I’m now at Google, and the transition took me away from blogging for awhile.
[1] http://www.pewinternet.org/Reports/2011/Technology-and-social-networks/Summary.aspx
[2] http://en.wikipedia.org/wiki/Log-normal_distribution
[3] http://www.evanmiller.org/how-not-to-sort-by-average-rating.html
[4] http://en.wikipedia.org/wiki/Pareto_distribution
[5] http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html
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GRAPES workshop 2010 - Abstracts
GRAPES workshop 2010 - Abstracts
Empirical Model Discovery
David Model evaluation is re-interpreted as discovering what is wrong with a specification; robust statistics as discovering which sub-sample is reliable; non-parametrics as discovering what
Hendry, functional form best characterizes the evidence; and model selection as discovering which model best matches the criteria. Yet each is addressed in isolation of the others. Empirical
University of Model Discovery seeks to tackle all of these jointly. Automatic methods enable formulation, selection, estimation and evaluation on a scale well beyond the powers of human intellect,
Oxford including when there are more candidate variables than observations. The lectures explain how major recent developments facilitate the discovery of empirical models within which the
best theoretical formulation is embedded, when the high dimensionality, non-linearity, inertia, endogeneity, evolution, and abrupt change characteristic of economic data interact to
make modelling so difficult in practice. Live computer illustrations using Autometrics show the remarkable power and feasibility of the approach.
Moudud Alam, Likelihood prediction with generalized linear and mixed models under covariate uncertainty
Örebro This paper demonstrates the techniques of likelihood prediction with the generalized linear mixed models. It also presents a way to deal with the covariate uncertainty while producing
University the measure of the prediction uncertainty. Several rather non-trivial prediction problems from the existing literature are reviewed and their likelihood solutions are presented.
Linear and non-linear causality tests in a LSTAR model: wavelet decomposition in a non-linear environment
In this paper, we use simulated data to investigate the power of different causality tests in a two-dimensional vector autoregressive (VAR) model. The data are presented in a non-linear
Yushu Li, environment that is modelled using a logistic smooth transition autoregressive (LSTAR) function. We use both linear and non-linear causality tests to investigate the unidirection
Linné causality relationship and compare the power of these tests. The linear test is the commonly used Granger causality test. The non-linear test is a non-parametric test based on Baek
University and Brock (1992) and Hiemstra and Jones (1994). When implementing the non-linear test, we use separately the original data, the linear VAR filtered residuals, and the wavelet decomposed
series based on wavelet multiresolution analysis (MRA). The VAR filtered residuals and the wavelet decomposition series are used to extract the non-linear structure of the original
data. The simulation results show that the non-parametric test based on the wavelet decomposition series (which is a model free approach) has the highest power to explore the causality
relationship in the non-linear models.
Peter The Incompleteness Problem of the APT Model
Karlsson, The Arbitrage Pricing Theory provides a theory to quantify risk and the reward for taking it. While the theory itself is sound from most perspectives, its empirical version is connected
Jönköping with several shortcomings. One extremely delicate problem arises because the set of observable asset returns rarely has a history of complete observations. Traditionally, this problem
International has been solved by simply excluding assets without a complete set of observations from the analysis. Unfortunately, such a methodology may be shown to (i) lead for any fixed time period
Business to selection bias in that only the largest companies will remain and (ii) lead to an asymptotically empty set containing no observations at all. This paper discusses some possible
School solutions to this problem and also provides a case study containing Swedish OMX data for demonstration.
International Stock Market Integration and Market Risk: The Nordic Experience
Yuna Liu, In this paper we study whether the creation of a uniform stock trading platform(OMX, NASDAQ-OMX) for the Nordic countries (Sweden, Finland, Denmark and Iceland), facilitating
Umeå cross-border trading, has changed the long-run structure of stock market volatilities and correlations on the Nordic stock markets. To accomplish this, the trend in time-varying
University volatilities and correlations are fi?ltered out in a ?first step using a non-parametric decomposition building on Loess. In a second step, possible changes in these trends due to the
integration of the markets are then analyzed. The analysis is complemented by use of parametric C-GARCH models and extensions of these (Component Correlation-GARCH models). The results
indicate, among other, that the long-run trend in volatility decreased on the Swedish and Finish stock markets when a uniform trading platform was introduced for these countries.
GARCH-Type models and Performance of Information Criteria
Farrukh GARCH models have been gaining popularity since the last two decades probably because of their ability to capture non-linear dynamics in the real life data which we often observe
Javed, especially in financial markets. This paper discusses the relative ability of some common information criteria (AIC, AICc, SIC and HQ) using their probability of correct selection, as a
Lund measure of performance, in the presence of GARCH effect. The investigation has been performed using Monte Carlo simulation of conditional variance GARCH processes with 6 different kinds
University of DGPs including for and , GARCH (1,1)-Leverage, and GARCH(1,1)-Spillover. All these models are further simulated with different parameter combinations to study the possible effect of
volatility structures on these information criteria. We noticed an impact of volatility structure of time series on the performance of these criteria.
Shutong Model Selection in Dynamic Factor Models
Ding, Dynamic factor models have become popular in applied macroeconomics with the increased availability of large data sets. We consider model specification, i.e. the choice of lag length
Örebro and the number of factors, in the setting of factor augmented VAR models. In addition to the standard Bayesian approach based on Bayes factors and marginal likelihood, we also study
University model choice based on the predictive likelihood which is particularly appealing in a forecasting context. As a benchmark we compare the performance of the Bayesian procedures with
frequentists approaches, such as the factor selection method of Bai and Ng.
Rashid Estimating mean-variance ratios of financial data
Mansoor, The Sharp ratio defined as the ratio of excess return to its risk provides a measure of excess relative return.The ratio link the first raw moment to the second central moment. In this
Jönköping paper we are suggesting some estimators of the mean-variance ratio of financial data.The study is motivated by considering a functional form between mean and standard deviation of the
International stocks by assuming multivariate normal distribution.Three potential estimators of the ratio are developed and the asymptotic properties of the different estimators are derived.An
Business empirical investigation is then performed on the several US stocks returns in order to compare the different estimators of the ratio for different sectors and we test if these are
School significantly different.
Added variable plots in nonlinear regression
An added variable plot is a commonly used plot in linear regression diagnostics. The plot provides information about the addition of a further regressor to the model. The plot can lead
Karin Stål, to the identification of nonlinearity in the selected regressor, and outliers and influential observations that may seriously impact the least-squares estimate of the parameter that
Stockholm corresponds to the selected regressor. In this paper added variable plots are derived for a nonlinear regression model with an additive error term. The added variable plot for this
University nonlinear regression model is different from the plot in the linear regression case. The plot is not created for a specific explanatory variable, but for a parameter. Thus, the plot can
be called an added parameter plot, since it provides information about the modification of the model by adding a parameter. The plot also gives a more formal tool to decide the
importance of the parameter, since it is closely connected to the score test of the null-hypothesis that the added parameter is zero. It is proved that the value of the score test
statistic is equal to SSR of the regression through the origin in the added parameter plot, divided by the estimated variance under the null hypothesis.
A Generalized Rank Based Polychoric Correlation
Depending of the properties of the variables different measures for correlation have desirable properties. If the variables are ordinal then the Pearson product moment correlation is
Petra not valid while it is still possible to use the Spearman rank correlation. When interest is in a hypothesized underlying continuous variable however, the Spearman rank correlation does
Ornstein, not work well. If the underlying variables follow a bivariate normal distribution, the polychoric correlation can recover the Pearson product moment correlation. The purpose of this
Uppsala paper is to generalize the polychoric correlation such that it is more robust against the distributional assumption. We propose fitting the polychoric correlation using the Spearman
University rank correlation adjusted for discrete data. Performing a Monte Carlo simulation study, we find that our measure performs almost identically well to the polychoric correlation when the
assumptions hold, but outperforms it in the case of skewness. We show that it is unbiased, and that its properties can be derived from the Spearman rank correlation. For testing under
the null hypothesis of zero correlation, our statistic is consistent and asymptotically normal.
Inference in Second Price Auctions with Gamma Distributed Common Values
Our paper explores possible limitations of the Gaussian model in Wegmann and Villani (2008, WV) due to intrinsically non-negative values. The relative performance of the Gaussian model
Bertil is compared to an extension of the Gamma model in Gordy (2008) within the symmetric second price common value model. A key feature in our approach is the derivation of an accurate
Wegmann, approximation of the bid function for the Gamma model, which can be inverted and differentiated analytically. This is extremely valuable for fast and numerically stable evaluations of
Stockholm the likelihood function. The general MCMC algorithm in WV is utilized to estimate WV’s eBay dataset from 1000 auctions of U.S. proof coin sets, as well as simulated datasets from the
University Gamma model with different degrees of skewness in the value distribution. The Gaussian model fits the data slightly better than the Gamma model for the particular eBay dataset, which
can be explained by the fairly symmetrical value distribution. The superiority of the Gamma to the Gaussian model is shown to increase for higher degrees of skewness in the simulated
Algorithms to find exact inclusion probabilities for 2Pπps designs
The statistical literature contain several proposals for methods generating fixed size without replacement πps sampling designs. Methods for strict πps designs have rarely been used due
to difficulties with implementation. On the other hand, approximate πps designs as the Conditional Poisson sampling design (Hajek) is a popular alternative.
Jens Laitila and Olofsson presented an easily implemented sampling design, the 2Pπps sampling design, using a two-phase approach. The first-order inclusion probabilities of the 2Pπps design
Olofsson, are asymptotically equal to the target inclusion probabilities of a strict πps design.
Örebro This paper extends the work on the 2Pπps design and presents algorithms for calculation of exact first- and second-order inclusion probabilities. Starting from a probability mass
University function (pmf) of the sum of N independent, but not equally distributed Bernoulli variables, the algorithms are based on derived expressions for the pmfs of sums of N-1 and N-2
variables, respectively.
Exact inclusion probabilities facilitate standard design-based inference and provide a tool for studying the properties of the 2Pπps design. The Conditional Poisson sampling design is
shown to be a special case of the 2Pπps design. However, empirical results presented show that the properties of the suggested point estimator can be improved using a more general 2Pπps
Nonlinear Cointegration in Nonlinear Vector Autoregressive Models
Dao Li, Existing cointegration studies mainly concern with integrated time series. This is often not applicable while the time series processes are global process. In this paper, we propose a
Örebro definition of smooth-transition (ST) type nonlinear cointegration for a group of individually global time series. We study smooth-transition vector autoregressive (STVAR) model to
University consider the proposed nonlinear cointegration. Our model is also suitable to study common nonlinear factors in an econmic system. We study the properties of STVAR models and test for
common nonlinear factors or nonlinear cointegration. Simulation studies have been carried out to show the asymptotic characteristic of the tests. Finally, we apply our work to
consumption and income data (United States, monthly from 1959:1 to 2010:3), and compare the forecasting results with linear VAR model. | {"url":"https://grapestat.se/grapes-workshop-2010-abstracts","timestamp":"2024-11-08T16:05:58Z","content_type":"application/xhtml+xml","content_length":"30906","record_id":"<urn:uuid:e34936b3-5714-46c6-8949-ca0c8ee06bbb>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00496.warc.gz"} |
National Polls
Yesterday, a national poll showing Clinton +8 was released, and it significantly impacted the model's estimate. Based mostly on the strength of that poll, Clinton's chance of winning moved from 73%
to 77%! In one day!
What a perfect time to talk about how national and state polls interact in my model. The short answer is that national polls can be thought of as 50 individual state polls, and modeled accordingly.
Figuring out the right way to incorporate national polls into the model has been very difficult. Not just
for me
, but I'm
sure I've read
Nate Silver write in 2012 that his model treated national polls "holistically."
My approach to national polls in 2012 was to calculate a poll average of national polls, then in each simulation of the election, simulate a national outcome then vary state outcomes relative the
national outcome. It worked well enough, and obviously the result was great, but it lacked elegance.
This election I've improved on that technique substantially, and my inspiration for how to do so came from thinking about what polls really are - a collection of individual preferences. National
polls are a collection of those preferences spread out across 50 states.
That's the key to how national polls are handled, so I'll say it again: national polls are a collection of preferences spread across 50 states. They can be modeled as such.
Just like for state polls, I collect every poll from the RCP average, then aggregate them using the same methodology I described
on Monday.
Once I have that average, I apportion it out to the states using population and adjusting for how red or blue the state is on a fundamental level.*
*I do this using Cook PVI, which you can read all about here
To demonstrate, let's return to my Missouri example from Monday. My national poll average was Clinton 45.5%, Trump 42.9%, with an effective sample size of 44,855 voters. To turn that data into
something specifically for Missouri requires 3 additional steps:
1. Adjust the national poll to reflect Missourian political leanings
2. Calculate how much of the national poll sample came from Missouri
3. Combine steps one and two to estimate how the national polling translates to actual voters expressing preferences in Missouri
The graphic below explains how, using trusty Missouri as our example, that looked before yesterday's big poll was released:
This national aggregate poll implies 367 votes for Clinton, and 387 votes for Trump.
Next is to combine national and state polling. Easy! Just add up the votes.
I add the state poll totals to the national poll totals, and that's my aggregate poll. This is the poll I use to calculate the candidate's chance of winning, to simulate elections, to categorize the
state, and so on.
The national poll showing Clinton +8 had a sample size of 12,742 voters. Needless to say it's shaken things up a bit. Here's how Missouri looked after this poll was included:
It added 245 Implied Missouri votes, increasing Clinton's total from 367 to 481, and Trump's from 387 to 494. Because the poll was favorable to Clinton, it decreased her deficit in Missouri from 6.4%
to 5.8%, and increased her chance to win the state from 10% to 12%.
When a big national poll is released it can move a lot of states. This one created exactly the same movement as if the following state polls were all published on the same day:
• 800-person poll in FL showing Clinton +6%
• 500-person poll in PA showing Clinton +9%
• 400-person poll in GA showing Clinton +2%
• 400-person poll in NC showing Clinton +5%
• and so it goes, all the way down to a 25-person poll in VT showing Clinton +24%, and a 23-person poll in WY showing Trump plus 22%
Big national polls can tell us a lot.
To sum up, national polls interact with state polls in the following way:
1. National polls are aggregated
2. The result is adjusted for each state according to its PVI
3. The PVI-adjusted national poll is apportioned state-by-state using population share
4. Those apportioned national votes are added to the aggregate state poll to create a final aggregated poll for each state | {"url":"http://www.basedonactualmath.com/2016/08/national-polls.html","timestamp":"2024-11-07T11:05:10Z","content_type":"text/html","content_length":"45542","record_id":"<urn:uuid:65b2a78b-45ea-4d25-943c-8fcc1aaa69f4>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00297.warc.gz"} |
5 Digit By 4 Digit Multiplication Worksheets
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5 Digit By 4 Digit Multiplication Worksheets
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Multiplication Worksheets No Carrying PrintableMultiplication
Multiplication Worksheets No Carrying PrintableMultiplication
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Python Notes
Python Notes¶
Introduction to Python for Econometrics, Statistics and Numerical Analysis: Fourth+ Edition
Python is a widely used general purpose programming language, which happens to be well suited to econometrics, data analysis and other more general numeric problems. These notes provide an
introduction to Python for a beginning programmer. They may also be useful for an experienced Python programmer interested in using NumPy, SciPy, matplotlib and pandas for numerical and statistical
analysis (if this is the case, much of the beginning can be skipped).
New material added to the fifth edition on September 2021.
Current Edition¶
Introduction to Python for Econometrics, Statistics and Numerical Analysis: Fourth Edition
Archived Versions¶
Data and Notebooks¶
• Data and Code from the notes. These files are needed to run some of the code in the notes.
• The Fama-French data set is used in the asset-pricing examples.
• The FTSE 100 data from 1984 until 2012 is used in the GJR-GARCH example.
These notebooks contains the four extended examples from the Examples chapter. | {"url":"https://bashtage.github.io/kevinsheppard.com/teaching/python/notes/","timestamp":"2024-11-06T15:38:59Z","content_type":"text/html","content_length":"13608","record_id":"<urn:uuid:d9264d98-d2bc-4f40-90b7-ca27616777b7>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00148.warc.gz"} |
Algorithms with PredictionsAlgorithms with Predictions
Credit: Andrij Borys Associates; Shutterstock
The theoretical study of algorithms and data structures has been bolstered by worst-case analysis, where we prove bounds on the running time, space, approximation ratio, competitive ratio, or other
measure that holds even in the worst case. Worst-case analysis has proven invaluable for understanding aspects of both the complexity and practicality of algorithms, providing useful features like
the ability to use algorithms as building blocks and subroutines with a clear picture of the worst-case performance. More and more, however, the limitations of worst-case analysis become apparent and
create new challenges. In practice, we often do not face worst-case scenarios, and the question arises of how we can tune our algorithms to work even better on the kinds of instances we are likely to
see, while ideally keeping a rigorous formal framework similar to what we have developed through worst-case analysis.
A key issue is how we can define the subset of "instances we are likely to see." Here we look at a recent trend in research that draws on machine learning to answer this question. Machine learning is
fundamentally about generalizing and predicting from small sets of examples, and so we model additional information about our algorithm's input as a "prediction" about our problem instance to guide
and hopefully improve our algorithm. Of course, while ML performance has made tremendous strides in a short amount of time, ML predictions can be error-prone, with unexpected results, so we must take
care in how much our algorithms trust their predictors. Also, while we suggest ML-based predictors, predictions really can come from anywhere, and simple predictors may not need sophisticated machine
learning techniques. For example, just as yesterday's weather may be a good predictor of today's weather, if we are given a sequence of similar problems to solve, the solution from the last instance
may be a good guide for the next.
What we want, then, is merely the best of both worlds. We seek algorithms augmented with predictions that are:
• Consistent: when the predictions are good, they are near-optimal on a per instance basis;
• Robust: when the predictions are bad, they are near-optimal on a worst-case basis;
• Smooth: the algorithm interpolates gracefully between the robust and consistent settings; and
• Learnable: we can learn whatever we are trying to predict with sufficiently few examples.
Our goal is a new approach that goes beyond worst-case analysis.^14 We identify the part of the problem space that a deployed algorithm is seeing and automatically tune its performance accordingly.
As a natural starting example, let us consider binary search with the addition of predictions. When looking for an element in a large sorted array, classical binary search compares the target with
the middle element and then re-curses on the appropriate half (see Figure 1). Consider, however, how we find a book in a bookstore or library. If we are looking for a novel by Isaac Asimov, we start
searching near the beginning of the shelf, and then look around, iteratively doubling our search radius if our initial guess was far off (see Figure 2). We can make this precise to show that there is
an algorithm with running time logarithmic in the error of our initial guess (measured by how far off we are from the correct location), as opposed to being logarithmic in the number of elements in
the array, which is the standard result for binary search. Since the error is no larger than the size of the array, we obtain an algorithm that is consistent (small errors allow us to find the
element in constant time) and robust (large errors recover the classical O(log n) result, albeit with a larger constant factor).
Figure 1. The execution of traditional binary search.
Figure 2. The execution of binary search, starting with a prediction.
Many readers may notice this is a variation on the idea of interpolation search, using only a predicted starting point. (Interpolation search uses the data to estimate the next comparison point,
instead of always picking the middle as in binary search.) With this view, algorithms with predictions have been in the air for some time, and the ML explosion has simply provided motivation to both
expand the idea and develop richer formalizations.
A recent success along these lines formalizes the idea of 'warm start.' When repeatedly solving similar optimization problems, practitioners often don't start from scratch each time, but instead
start searching near a previous solution. Dinitz et al.^4 analyze the performance gains of treating such a solution as a prediction in the context of min cost perfect matchings. In their setting, one
solves a number of problems on the same graph, but with different edge weights for each instance, where the edge weights may, for example, come from a distribution. They show that given a prediction
for the dual solution of a corresponding linear program, they can compute a feasible dual solution from it, improving the overall running time and expanding upon the "use the solution from yesterday"
Predictions have also been suggested as a means for reducing space usage for several data structures, for example notably in the seminal work of Kraska et al.^7 for learned indices. As an example of
how predictions can save space, we first explain the later work of Hsu et al.^6 on data structures for frequency estimation that use learning.
Frequency estimation algorithms are used to approximately count things, such as the number of packets a router sees sent from each IP address. Since it can be expensive in both space and time to keep
a separate counter for each address, estimation algorithms use techniques such as hashing each address into a table of shared counters (usually hashing each address into several locations for
robustness), and then deriving an estimate when queried for an IP address from its associated counters. The largest count estimate errors occur when an address with a small count hashes to the same
locations as addresses with a large count, as it then appears that the address should itself have a high count. If we somehow knew the addresses with large counts ahead of time, we could assign them
their own counters and handle them separately from the sketch, avoiding such large errors and obtaining better frequency estimation with smaller overall space. The paper by Hsu et al.^6 introduces
the idea of using machine learning to predict which objects (in this example, IP addresses) have large counts, and separate them out in this way. They prove bounds for specific cases and demonstrate
empirically both that high-count elements are predictable and that using such predictions can lead to improved practical performance.
As another example of how predictions can save space, Kraska et al.^7 propose a framework for learned Bloom filters. Bloom filters are compressed data structures for set membership; for a set X of
keys, a Bloom filter correctly returns yes for any x that is truly in X, but may give a false positive for keys not in the set. Bloom filters have a space-accuracy trade-off, where more space allows
for fewer false positives. The work of Kraska et al.^7 suggests that if a set can be learned, that is, a predictor can imperfectly predict whether an element is or is not in the set, that can be used
to derive a learned Bloom filter that combines the predictor with a standard Bloom filter in a way that improves the space-accuracy trade-off. We leave the details of the various improved learned
Bloom filters to the relevant papers.^7,9,17
Perhaps unsurprisingly, one area where using predictions is having a tremendous impact is for online algorithms, where the algorithm responds to an incoming data stream and the future is unknown. The
theoretical framework of competitive analysis considers the worst-case ratio between the performance of an online algorithm and the optimal algorithm as a measure, so a "two-competitive" algorithm is
always within a factor of two of optimal. Coping with the worst-case possible future is often difficult, and thus taking advantage of predictions in this setting is often quite powerful. For example,
some recent results consider scheduling problems. Jobs arrive over time at a single server and have to be scheduled; the cost for each job is the time between when it arrives and when it finishes,
and one wants to minimize the total cost over all jobs. If a job's required processing time is known on arrival, then scheduling by Shortest Remaining Processing Time (SRPT) is optimal. But what if
only estimates of job times are known? Recent work shows that if every job with true size s has an estimate between [bs,as] for constants a,b with 0 < b < 1 < a, there is an algorithm with
competitive ratio O((a/b)log^2(a/b)), even if the algorithm does not know a and b in advance. That is, one can achieve performance close to optimal, and the performance gracefully degrades with the
estimate quality.^2 Scheduling with predictions has similarly been studied in the context of queueing theory, where the models have probabilistic assumptions, such as Poisson arrivals and independent
and identically distributed service times. In this setting, when using estimates, SRPT can perform quite badly even when estimates are again bounded in [bs, as] for a job of size s, but a variation
of SRPT using estimates converges to the performance of SRPT with full information as a and b go to 1, and is within O(a/b) of SRPT always, again without knowing a and b in advance.^15 Other work
looking at the queueing setting has shown that even one bit of advice, predicting whether a job is short or long for some suitable notion of short or long, can greatly improve performance.^10 Several
other online problems have been studied with predictions, including caching,^8 online clustering, and the historically fun and enlightening ski rental^13 and secretary problems.^1,5
Predictions have also been suggested as a means for reducing space usage for several data structures.
It is worth noting there is also a great deal of recent work in the closely related area of data-driven algorithm design. At a high level, this area often studies the tuning of an algorithm's
hyperparameters, such as the step-size in a gradient descent, or which of the many possible initializations for k-means clustering is best. (The survey by Balcan^3 provides a deep dive into this
The research area of Algorithms with Predictions has really only just started, but it seems to be booming, as researchers reexamine classical algorithms and see where they can be improved when good
predictions are available. This marriage of classical algorithms and data structures with machine learning may lead to significant improvements in systems down the road, providing benefits when good
predictions are available (as they seem to be in the real world) but also limiting performance downsides when predictions go wrong (as, inevitably, also seems to happen in the real world).
For those interested in more technical detail, we have a short survey available,^11 and there are related recent workshops with talks online.^12,16
1. Antoniadis, A. et al. Secretary and online matching problems with machine learned advice. In Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information
Processing Systems 2020, H. Larochelle et al., Eds. NeurIPS 2020 (Dec. 6–12 (virtual) 2020).
2. Azar, Y., Leonardi, S., and Touitou, N. Distortion-oblivious algorithms for minimizing flow time. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, S. Naor and N.
Buchbinder, Eds. (Jan. 9–12, 2022), 252–274.
3. Balcan, M. Data-driven algorithm design. CoRR abs/2011.07177 (2020).
4. Dinitz, M. et al. Faster matchings via learned duals. In Advances in Neural Information Processing Systems (2021), M. Ranzato, A. et al., Eds., vol. 34, Curran Associates, Inc., 10393–10406.
5. Dütting, P. et al. Secretaries with advice. In EC '21: The 22^nd ACM Conference on Economics and Computation. P. Biró, S. Chawla, and F. Echenique, Eds., Budapest, Hungary, July 18–23, 2021t al.
(2021), ACM, 409–429.
6. Hsu, C. et al. Learning-based frequency estimation algorithms. In Proceedings of the 7^th International Conference on Learning Representations, ICLR 2019, (New Orleans, LA, USA, May 6–9, 2019);
7. Kraska, T. et al. The case for learned index structures. In Proceedings of the 2018 International Conference on Management of Data. G. Das, C.M. Jermaine, and P.A. Bernstein, Eds. SIGMOD
Conference 2018 (Houston, TX, USA, June 10–15, 2018), 489–504.
8. Lykouris, T., and Vassilvitskii, S. Competitive caching with machine learned advice. J. ACM 68, 4 (2021).
9. Mitzenmacher, M. A model for learned BLOOM filters and optimizing by sandwiching. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing
Systems 2018, NeurIPS 2018, S. Bengio et al., Eds. (Dec. 3–8, 2018, Montréal, Canada (2018), 462–471.
10. Mitzenmacher, M. Queues with small advice. In Proceedings of the 2021 SIAM Conference on Applied and Computational Discrete Algorithms, ACDA 2021, M. Bender, et al., Eds., (July 19–21, (virtual)
2021), 1–12.
11. Mitzenmacher, M., and Vassilvitskii, S. Algorithms with predictions. In Beyond the Worst-Case Analysis of Algorithms, T. Roughgarden, Ed. Cambridge University Press, 2020, 646–662.
12. ML4A 2021—Machine Learning for Algorithms (July 2021); https://bit.ly/3wThaVs
13. Purohit, M., Svitkina, Z., and Kumar, R. Improving online algorithms via ML predictions. In Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information
Processing Systems 2018. S. Bengio et al., Eds. NeurIPS 2018 (Dec. 3–8, 2018), Montréal, Canada (2018), 9684–9693.
14. Roughgarden, T., Ed. Beyond the Worst-Case Analysis of Algorithms. Cambridge University Press, 2020.
15. Scully, Z., Grosof, I., and Mitzenmacher, M. Uniform bounds for scheduling with job size estimates. In 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, Jan.–Feb. 3, 2022,
Berkeley, CA, USA (2022), M. Braverman, Ed., vol. 215 of LIPIcs, Schloss Dagstuhl-Leibniz-Zentrum für Informatik, pp. 114:1–114:30.
16. STOC 2020 - Workshop 5: Algorithms with Predictions. https://bit.ly/3wThgwi
17. Vaidya, K. et al. Partitioned learned BLOOM filters. In 9^th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3–7, 2021 (2021), OpenReview.net.
Michael Mitzenmacher was supported in part by NSF grants CCF-2101140, CNS-2107078, and DMS-2023528.
The Digital Library is published by the Association for Computing Machinery. Copyright © 2022 ACM, Inc.
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Negative Binomial Distribution. Probability density function, cumulative distribution function, mean and variance
This calculator calculates negative binomial distribution pdf, cdf, mean and variance for given parameters
In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of
success is the same every time the experiment is conducted. Wikipedia
When we want to know the probability of k successes in n such trials, we should look into binomial distribution.
When we want to know the probability of getting the first success on k-th trial, we should look into geometric distribution.
When we want to know the probability that the k-th success is observed on the n-th trial, we should look into negative binomial distribution.
Probability density function of negative binomial distribution is
• p is the probability of success of a single trial,
• x is the trial number on which the k-th success occurs.
• $C^{n}_{m}=\frac{n!}{m!(n-m)!}$ is the number of combinations of m from n
Cumulative distribution function of negative binomial distribution is
• $I_x(a,b)$ is the regularized incomplete beta function
Note that $f(k)=p^k$, that is, the chance to get the k-th success on the k-th trial is exactly k multiplications of p, which is quite obvious.
Mean or expected value for the negative binomial distribution is
Variance is
The calculator below calculates the mean and variance of the negative binomial distribution and plots the probability density function and cumulative distribution function for given parameters: the
probability of success p, number of successes k, and the number of trials to plot on chart n.
Note that there are other formulations of the negative binomial distribution. They are created using the following notation: n - number of trials, r - number of failures, k - number of successes,
with n=k+r. These are:
• k successes, given r failures
• n trials, given r failures
• r failures, given k successes
• n trials, given k successes (a case described above)
• k successes, given n trials (binomial distribution).
They have slightly different formulas.
Digits after the decimal point: 2
Negative Binomial Distribution
The file is very large. Browser slowdown may occur during loading and creation.
Cumulative distribution function
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Marketing Insights: Recall %, how is this calculated?
Marketing Insights: Percentage of Patients in Recall
How is this calculated in the Legwork Software?
This percentage is the current monthly average of patients in recall to date.
Let me provide you a visual.
In the Reports tab>In Recall, what is the total number of patients in Recall? Let's say you have 1,580.
What is your total number of Active Patients (Reports tab>Active)? Let's say you have 3,611.
Take the amount of patients In Recall and divide by the total number of Active Patients: 1580/3611=.437 or 44%.
Now, to calculate the percentage of the current monthly average of patients In Recall to date:
Let's say today is 4/24/19, so there have been 24 days in the month so far, take the 44% In Recall and divide by the amount of days in the month so far 44/24=1.83 or a rounded average of 2%. | {"url":"https://support.lwcrm.com/hc/en-us/articles/13391310735003-Marketing-Insights-Recall-how-is-this-calculated","timestamp":"2024-11-14T08:44:32Z","content_type":"text/html","content_length":"22523","record_id":"<urn:uuid:dad212c4-360c-4d5b-87ef-10d7fb88d81b>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00569.warc.gz"} |
Environment-induced decoherence
Environment-induced decoherence
Is this soup eaten as hot as it is being served? Well, let's try to investigate a little further.
Imagine a photon in the box on its way from the inner wall toward the body of a person that is present in the box. This photon doesn't see that person yet. For the photon the person is in a box in a
superposition of all its possible states. When the photon hits the person's body this acts as a measurement, opening that box. While at the same time another box has formed around the person and the
photon hitting it and has closed. In fact, when the photon hits the person, it is as if the photon has permeated through the wall of the box that person is in, adding itself to its content. Somehow
it has doubled itself as many times as there were different states in the box and then each single photon enters one specific possible state. It must be like that since when in one of the possible
states in the box the photon was missing this would violate the law of energy conservation, which is one of the most founded laws in nowadays physics. Remember any of those possible states is a world
on its own, with its own reality experience, but not seeing the other worlds. And every state obeys the same laws of physics. Up until now there are no signs of a different view about that.
The inner observations in the box cannot significantly alter the motion of the objects in the box but they do "nudge" the wavefunction of the things they encounter. These encounters disturb the
wavefunction's coherence, its neat order of wave crests and troughs, necessary for generating neat interference effects. In doing so messing it up to some average smeared-out state.
The more smeared out it is, the lesser one can infer from it the existence of different worlds in superposition. But this doesn't say those worlds are not there. You have the smeared-out state
interference pattern and in the box is the superposition of all possible worlds that possibly could have led to it. That includes the worlds we know are in there PLUS the worlds that too can lead to
precisely the same smeared-out pattern observed by us, Outside Observers.
Environment induced decoherence, as the smearing out is called, works fast. A grain of dust in empty space decoheres in about a millionth of a second, due to the photons of the 2.7 K microwave cosmic
background radiation present everywhere in outer space. I now regard CMB at page 4 of EXPANSION OF THE UNIVERSE as favorite view. In Standstill at page 5 of THE EXPANSION OF THE UNIVERSE the
background radiation is presented as a kind of riddle.
The famous two-slit experiment doesn't suffer from decoherence. Decoherence is not a fundamental feature of empty space. In principle one can think of ways making decoherence as slow as desired.
Decoherence explains why special interference effects of wavefunctions do not show off at the earth surface or such place, long enough to measure them.
Is it possible to sufficiently slow down decoherence? We shall see. Decoherence bridges between the quantum realm and our every day reality as we experience it. But, as Brian Green points out at the
end of his chapter about decoherence, “far from everyone is convinced that the bridge has yet been fully built.”
did you know the cat escaped?
Gravitation is the biggest box-opener, choice-maker, realizator of the wavefunction's superposition. In doing so gravity tends to annul quantum mechanics' superpositions. GR fights QM, as to speak.
In page 3, 4 and 5 of the storyline NEWTON EINSTEIN GRAVITATION I develop a new approach to gravity. In brief, mass absorption from the Higgs field curves space and the subsequent act of gravity
curves it back. As a result the background grid of space is not curved, it only looks like that.
How does gravity opens boxes? Consider a mass in empty space. At every coupling of its constituting particles is absorbed mass from the Higgs field. According to my theory of gravitation , a tiny
parcel of vacuum is absorbed along with the mass absorption and subsequently the resulting hole is filled in by the surrounding parcels of the vacuum. This is the act of gravity. Every coupling in
the mass (that it must have to maintain its coherence) betrays the location and orientation of the coupling particles in the mass by means of the (in principle) observable changes in motion or
observable forces (like weight) of the surrounding matter (like gas molecules). The changes and forces may be too tiny for us to actually measure, but in principle they are. The result is that the
wavefunction of masses collapses every single moment by means of the act of gravity.
More about it in paragraph Tone at page 6 of THE DIRECTION OF TIME.
Does the box gain extra mass in the process of photon-doubling? Well, according to NEG photons have no rest mass and do not gravitate. The photon has no net absorption from the Higgs field. But for
other particles one must regard this in a quite straightforward way. The interference pattern of the wavefunctions in the box is all we can know about the box's content until we open the box. From
that interference pattern is derived a probability division of objects and particles that precisely predicts how much mass of particles and objects can be expected where and when. The observed mass
of the closed box is just the end result of this probability division.
Subsequently it depends on e.g. the (in principle observable) change of movement of e.g. air molecules surrounding the box, caused by the gravitation of the mass inside the box. Each of that
movements betrays the content of the box to a certain extend and to that extend the wavefunction of the inner part of the box collapses partially.
QED uses renormalization, connecting every spacetime point of our four-dimensional universe with every other spacetime point in the same universe along routes that follow the curves of spacetime.
That is the usual assumption, as far as I know. But in NEG spacetime is not curved, it only looks like that. Then the routes used in renormalization are just straight lines through a flat vacuum and
time. Isn't it?
QED brings back all things with photons and electrons (in fact electric charges in general) to three fundamental actions.
1) an electron goes from point A to point B;
2) a photon goes from point C to point D;
3) an electron and a photon couple, the photon is gone then.
Let's take the photon from 2) to be equal to the photon from 3), so point B = point D. Higgs absorption only takes place at 3), the coupling between an electric charge and a photon. There are no
reactions at 1) or 2), so it can only be at 3).
Key point of QED is that the electron traverses from A to B along every possible route through spacetime. In QED this means that the electron that is going from A to B, splits at A in an infinite
number of electrons and each individual electron visits one point in spacetime, couples there with a virtual photon and returns to point B. ALL points of spacetime are visited in this way, also the
most distant ones in the uttermost far future or past.
(Also the photon traverses all possible routes from C to D and visits all points of spacetime. There it couples to a virtual electric charge, a virtual electron. But the photon is absorbed by the
electric charge and then is gone. Re-emission of the photon by the virtual electron gives the Feynman diagram one coupling extra, diminishing its amplitude by a factor 10.)
The infinite number of split-off electrons and photons are “immoral” as Feynman called them. Where necessary the electrons go back in time and go faster than light. The photons can bow into curved
paths where necessary. It are not the usual particles, not quite. Or the picture is incomplete yet.
The immoral electrons don't see each other, they don't react with each other. They are superposed to each other. It is for sure the virtual photons they couple with, do not exist all in the same
universe. Such a universe would be filled with photons at every point in spacetime.
In QED renormalization procedure, in each Feynman diagram the amplitude is reciprocal to r, where r is covered distance. The paths of the infinite split-off immoral electrons lay in spacetime, so
they have a definite shape in our universe. So for the renormalization procedure of QED itself it is important whether the routes traversed in each contribution, in each Feynman diagram, are dragged
along in the action of gravity as described in the NEG storyline. Or that the traversing is to be taken through flat spacetime, the paths being superposed to all gravitational activity, existing in
its own empty flat vacuum where in thought all particles that are not needed, are erased. Mind in NEG the Higgs field absorption is taken from one single field out of THE FIELD OF ALL POSSIBLE
VELOCITIES. All those velocity fields are superpositions relative to each other. The paths of the immoral particles are just extra superpositions next to the fields of all possible velocities. And in
each point of spacetime all fields are not curved, except for one, IF there is a Higgs field absorption taking place there. I conclude the immoral particles traverse along straight lines in
superposed flat spaces.
In NEG space starts out flat. Then at point B a coupling takes place between the electron and the photon. Along with absorption from the Higgs field, necessary for the electron to gain mass, a tiny
parcel of space is absorbed. Space now is locally curved according to the Einstein equations. The photon is absorbed and is gone now. The electron starts its traversing from point B to the next
photon absorption spot E along all possible paths. This takes the time spanned by point of time of B to point of time of E. In the same time the action of gravity does undo the curvature of spacetime
until space is flat again. This is assumed to take some time too. So it seems the immoral photons and electrons perform their tasks at the same time that gravitation does. They lay their routes
through a spacetime that is continuously involved in the act of gravitation of innumerable other particles, other masses everywhere in the universe. It is not before or after the
curve-and-gravitate-back act, but meanwhile. The immoral particles must follow the curves of spacetime.
My intuition says reasoning 1 is the true one. But actual decision cannot be made yet. Because the character of the immorality of the electric charges and photons in QED renormalization is unknown.
And I don't have a neat estimation how much time the act of gravity takes, the flattening of spacetime after each Higgs field absorption. I tried to issue the problem at paragraph Propagation speed
at page 5 of NEWTON EINSTEIN GRAVITATION.
At the end of State 3 at page 5 of NEG is stated that “Renormalization procedure is set up as new every time a coupling is made”. It is used there as reason to decide renormalization performs in flat
space. Is that right?
Oktober 2023
When having red the storyline NEWTON EINSTEIN GRAVITATION page 3, 4 and 5, one might have concluded that unision of GR and QM is just that space isn't curved after all and QM renormalization just
performs in flat spactime. But it might not be that simple. Not always. | {"url":"http://leandraphysics.nl/sea4.html","timestamp":"2024-11-01T20:54:51Z","content_type":"text/html","content_length":"15984","record_id":"<urn:uuid:a52a4a9e-bffa-4a25-94e5-543a13017b00>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00142.warc.gz"} |
MAT 1222 Algebra Section 5 Rasmussen College Jack is mowing a lawn that has a shed. Jack wants to know the area if the lawn he has to mow. Here are the d - Graduatesplug.com
MAT 1222 Algebra Section 5 Rasmussen College Jack is mowing a lawn that has a shed.
Jack wants to know the area if the lawn he has to mow.
Here are the dimensions of the yard and the shed (see picture below)
Find a polynomial that describes the area of the lawn he has to mow (i.e. find the area of the yard minus the area of the shed.)
The polynomial in your final answer should only have two terms. This means you have to combine like terms and simplify.) Mowing a Lawn
13x – 4
2x +4
Here are the dimensions of
the yard and the shed.
Mowing a Lawn
13x – 4
2x + 4
Find a polynomial that
describes the area of the
lawn he has to mow
(i.e. find the area of the
yard minus the area of the
Mowing a Lawn
13x – 4
2x + 4
The polynomial in your
final answer should only
have two terms. This
means you have to
combine like terms and
Mowing a Lawn
Show all of your steps and use
Microsoft Equation Editor to
render all of your algebraic
Purchase answer to see full
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https://graduatesplug.com/wp-content/uploads/2021/02/logo.png 0 0 Ben https://graduatesplug.com/wp-content/uploads/2021/02/logo.png Ben2021-08-25 12:09:362021-08-25 12:09:36MAT 1222 Algebra Section 5
Rasmussen College Jack is mowing a lawn that has a shed. Jack wants to know the area if the lawn he has to mow. Here are the d | {"url":"https://graduatesplug.com/mat-1222-algebra-section-5-rasmussen-college-jack-is-mowing-a-lawn-that-has-a-shed-jack-wants-to-know-the-area-if-the-lawn-he-has-to-mow-here-are-the-d/","timestamp":"2024-11-12T03:56:58Z","content_type":"text/html","content_length":"62180","record_id":"<urn:uuid:3c73c78c-dc91-4b7c-abcf-a505ba9a69d0>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00091.warc.gz"} |
For any triangle, the radius of its inscribed circle, the radius of its circumcircle and the distance of their centers are related through Euler's theorem in geometry (but earlier already published
by Chapple). In one dimension higher, the Grace-Danielsson inequality gives a condition for the three values, so that a (non-regular) tetrahedron between the spheres exists, hence is completely
contained inside the larger sphere and completely encloses the smaller sphere. In higher dimensions, Greg Egan conjectured a generalized Grace-Danielsson inequality and proved it to be sufficient for
a simplex to exist between the spheres under a blog post of John Baez. A few weeks ago, the inequality was also proven to be necessary by Sergei Drozdov. | {"url":"https://events.ccc.de/congress/2023/hub/en/event/egan-conjecture-holds/","timestamp":"2024-11-05T03:36:24Z","content_type":"text/html","content_length":"22683","record_id":"<urn:uuid:7df09435-a434-4e16-811a-d80c0af83daf>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00114.warc.gz"} |
scipy.optimize.least_squares() runs 5 times and gives back initial guess everytime
In the realm of numerical optimization, scipy.optimize.least_squares() is a powerful function that aims to find the parameters that minimize the sum of the squares of residuals between observed and
computed values. However, some users have reported that the function runs multiple times but returns the initial guess each time without yielding an improved solution.
The Problem Scenario
Consider the following original code snippet, where the function least_squares() is invoked multiple times but fails to provide a meaningful output:
from scipy.optimize import least_squares
def residuals(params, x, y):
return y - (params[0] * x + params[1])
x = [1, 2, 3]
y = [2, 4, 6]
initial_guess = [0, 0]
result = least_squares(residuals, initial_guess, args=(x, y))
for _ in range(5):
In this code, the function is called only once, but if you expect it to run multiple times or with different initial guesses, you may be doing something wrong, leading to getting the same result
multiple times.
Analyzing the Problem
The issue you're experiencing, where least_squares() returns the initial guess every time, could be attributed to several factors, including:
1. Convergence Issues: The optimization algorithm might be unable to converge to a solution from the initial guess you provided. This could happen if the problem is poorly scaled, or if the function
is not well-defined.
2. Function Implementation: If the residuals function is incorrect, or not defined properly, the optimization process may not work as expected.
3. Input Values: Sometimes the data provided may not contain sufficient variation for the optimizer to find a suitable fit.
Debugging Steps
1. Check Residual Function: Ensure that your residual function is implemented correctly. It should represent the difference between the observed data and the model's prediction.
2. Try Different Initial Guesses: Test various initial guesses to see if the optimization behavior changes. This can be crucial in non-convex problems where local minima are prevalent.
3. Inspect Input Data: Make sure that your input data has the necessary characteristics for optimization, such as variability and relevance to the fitting model.
Practical Example
To provide a more concrete example, let's enhance the scenario by invoking least_squares() in a loop with various initial guesses to check if the output improves:
import numpy as np
from scipy.optimize import least_squares
def residuals(params, x, y):
return y - (params[0] * x + params[1])
x = np.array([1, 2, 3])
y = np.array([2, 4, 6])
# Trying different initial guesses
initial_guesses = [[0, 0], [1, 1], [2, 2], [0.5, 0.5], [3, 3]]
for guess in initial_guesses:
result = least_squares(residuals, guess, args=(x, y))
print(f'Initial Guess: {guess} | Optimized Parameters: {result.x}')
By running the above code, you can observe how different initial guesses affect the results of the optimization process.
In conclusion, scipy.optimize.least_squares() is a versatile tool for solving least-squares problems, but it requires careful consideration of the initial parameters, the design of the residual
function, and the input data to work effectively.
If you find yourself continuously returning to the initial guess, remember to debug your model and experiment with different approaches.
Additional Resources
By following these best practices and suggestions, you will enhance your optimization skills and get better results from your computations. Happy optimizing! | {"url":"https://laganvalleydup.co.uk/post/scipy-optimize-least-squares-runs-5-times-and-gives-back","timestamp":"2024-11-13T01:58:21Z","content_type":"text/html","content_length":"83922","record_id":"<urn:uuid:e3f6b6fa-fa7c-49f7-ac4b-42f29bc3c792>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00615.warc.gz"} |
"Calculating the Sum of the Squares of the First 50 Counting Numbers"
Given int variables k and total that have already been declared, use a while loop to compute the sum of the squares of the first 50 counting numbers, and store this value in total. Thus your code
should put 1*1 + 2*2 + 3*3 +… + 49*49 + 50*50 into total. Use no variables other than k and total.
LANGUAGE: C++
Given int variables k and total that have already been declared, use a while loop to compute the sum of the squares of the first 50 counting numbers, and store this value in total. Thus your code
should put 1*1 + 2*2 + 3*3 +… + 49*49 + 50*50 into total. Use no variables other than k and total.
while (k<=50){ | {"url":"https://matthew.maennche.com/2014/01/given-int-variables-k-and-total-that-have-already-been-declared-use-a-while-loop-to-compute-the-sum-of-the-squares-of-the-first-50-counting-numbers-and-store-this-value-in-total-thus-your-code-shou/","timestamp":"2024-11-09T12:57:17Z","content_type":"text/html","content_length":"93257","record_id":"<urn:uuid:eeca63b0-626f-479c-83c6-ef18e1817e41>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00091.warc.gz"} |
Kilometre per hour to Metre per minute Converter (km/h to m/min)
Kilometre per hour to Metre per minute Converter
“Quick conversion: Transform kilometers per hour to meters per minute effortlessly!”
Kilometre per hour to Metre per minute Conversion Formula
To convert kilometres per hour to metres per minute, you can use the following formula:
1 kilometre per hour = 16.6667 metres per minute
Example of Kilometre per hour to Metre per minute Conversion
Example 1: 50 Kilometre per hour to Metre per minute Conversion
For example, let’s convert 50 kilometres per hour to metres per minute:
50 kilometres per hour * 16.6667 = 833.335 metres per minute
Example 2: 54 Kilometre per hour to Metre per minute Conversion
For example, let’s convert 54 kilometres per hour to metres per minute:
54 kilometres per hour * 16.6667 = 899.998 metres per minute
Example 3: 66 Kilometre per hour to Metre per minute Conversion
For example, let’s convert 66 kilometres per hour to metres per minute:
66 kilometres per hour * 16.6667 = 1099.9982 metres per minute
Kilometre per hour to Metre per minute Conversion Table
Here’s a conversion table for kilometres per hour to metres per minute for the first 20 entries:
Kilometres per Hour Metres per Minute
1 16.6667
2 33.3334
3 50.0001
4 66.6668
5 83.3335
6 100.0002
7 116.6669
8 133.3336
9 150.0003
10 166.667
11 183.3337
12 200.0004
13 216.6671
14 233.3338
15 250.0005
16 266.6672
17 283.3339
18 300.0006
19 316.6673
20 333.334
Kilometre per hour to Metre per minute Converter FAQs
What is a Kilometre per hour to Metre per minute Converter?
It's a tool that converts speed from kilometres per hour (km/h) to metres per minute (m/min).
Why use a Kilometre per hour to Metre per minute Converter?
It helps in converting speed measurements between km/h and m/min, useful in different contexts like physics, engineering, or everyday calculations.
How does it work?
It multiplies the speed in km/h by 1000/60to convert it to m/min, since 1 km = 1000 m and 1 hour = 60 minutes.
What types of speeds can it convert?
It converts any speed given in kilometres per hour to metres per minute, maintaining accuracy in the conversion process.
Are there limitations to using a Kilometre per hour to Metre per minute Converter?
It's straightforward for converting linear speeds, but it's important to ensure units are correctly interpreted and applied in relevant contexts (e.g., linear motion calculations).
Related Posts
Related Tags
km/hr to m/min in fraction, how to convert km/h to m/h, convert 66 km hr to m/min, 54 km/h to m/min, km hr to m s 5/18, how to convert km/h to m/s in physics, how to convert km/h to m/s class 9, how
to convert km/h into minutes
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EViews Help: Overview
The conventional linear regression model assumes that the
The GLM framework of Nelder and McCullagh (1972) generalizes linear regression by allowing the mean component
• A linear predictor or index offset term.
• A smooth, invertible link function,
A wide range of familiar models may be cast in the form of a GLM by choosing an appropriate distribution and link function. For example:
Linear Regression Normal Identity:
Exponential Regression Normal Log:
Logistic Regression Binomial Logit:
Probit Regression Binomial Probit:
Poisson Count Poisson Log:
For a detailed description of these and other familiar specifications, see McCullagh and Nelder (1981) and Hardin and Hilbe (2007). It is worth noting that the GLM framework is able to nest models
for continuous (normal), proportion (logistic and probit), and discrete count (Poisson) data.
Taken together, the GLM assumptions imply that the first two moments of
where dispersion constant prior weight that corrects for unequal scaling between observations.
Crucially, the properties of the GLM maximum likelihood estimator depend only on these two moments. Thus, a GLM specification is principally a vehicle for specifying a mean and variance, where the
mean is determined by the link assumption, and the mean-variance relationship is governed by the distributional assumption. In this respect, the distributional assumption of the standard GLM is
overly restrictive.
Accordingly, Wedderburn (1974) shows that one need only specify a mean and variance specification as in
Equation (32.2)
to define a quasi-likelihood that may be used for coefficient and covariance estimation. Not surprisingly, for variance functions derived from exponential family distributions, the likelihood and
quasi-likelihood functions coincide. McCullagh (1983) offers a full set of distributional results for the quasi-maximum likelihood (QML) estimator that mirror those for ordinary maximum likelihood.
QML estimators are an important tool for the analysis of GLM and related models. In particular, these estimators permit us to estimate GLM-like models involving mean-variance specifications that
extend beyond those for known exponential family distributions, and to estimate models where the mean-variance specification is of exponential family form, but the observed data do not satisfy the
distributional requirements (Agresti 1990, 13.2.3 offers a nice non-technical overview of QML).
Alternately, Gourioux, Monfort, and Trognon (1984) show that consistency of the GLM maximum likelihood estimator requires only correct specification of the conditional mean. Misspecification of the
variance relationship does, however, lead to invalid inference, though this may be corrected using robust coefficient covariance estimation. In contrast to the QML results, the robust covariance
correction does not require correction specification of a GLM conditional variance. | {"url":"https://help.eviews.com/content/glm-Overview.html","timestamp":"2024-11-05T07:04:11Z","content_type":"application/xhtml+xml","content_length":"27025","record_id":"<urn:uuid:780ee434-ee2c-421b-b610-2238ccb5cac6>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00515.warc.gz"} |
Uniform Motion Problems
About me and why I created this physics website.
Uniform Motion Problems
On this page I put together a collection of uniform motion problems to help you understand uniform motion better. Uniform motion is motion in a straight line at constant velocity.
Problem # 1
A train is traveling on a straight section of track at constant speed. In 60 seconds it covers a distance of 1800 meters. What is the speed of the train?
30 m/s
Problem # 2
A car is traveling down a highway at a speed of 100 km/h. How far does the car travel in 24 minutes?
40 km
Problem # 3
A car accelerates from 0 to 100 km/h in 10 seconds, and then travels at constant speed. What is the speed of the car after 1 minute?
100 km/h
Problem # 4
A car travels a distance of
in 60 seconds and a distance of 2
in 120 seconds. At 30 seconds the car has traveled a distance of 650 meters. What is the value of
1300 meters
Problem # 5
The figure below shows the road that a car travels and the corresponding speed versus time graph for the car. Between which two points is the car experiencing uniform motion?
See answer Problem # 6
The figure below shows the position versus time graph for a car traveling on a straight road. Between which two points is the car experiencing uniform motion?
See answer Problem # 7
In the previous problem, at which point does the car change its direction of motion?
See answer Answers For Uniform Motion Problems Answer for Problem # 5
For the car to experience uniform motion, it has to move in a straight line AND travel at constant speed. This is only true between the points A and B.
Answer for Problem # 6
The car experiences uniform motion in locations where the slope of the line is constant, which is between points B and C, and between points C and D.
Answer for Problem # 7
At point C, the slope of the line changes, which means that the velocity of the car transitions from positive to negative. This means that the car reverses its travel direction. Note that the cusp at
point C is not very realistic. In reality, there would be a more gradual transition in velocity as the slope changes from positive to negative.
Return to Physics Questions page Return to Real World Physics Problems home page | {"url":"https://www.real-world-physics-problems.com/uniform-motion-problems.html","timestamp":"2024-11-04T05:36:18Z","content_type":"text/html","content_length":"49449","record_id":"<urn:uuid:b1b63fd6-89b2-414e-80be-ea45d9fdf4de>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00654.warc.gz"} |
There are 4 steps to follow in order to get 3in1:
1) Determine the windless power to be used.
2) Power Adjustment.
3) Checking distance.
4) Angle Adjustment.
Step 1) Determine the windless power to be used.
Windless 3in1 power and angle chart:
4 Spin (need tail wind or up wind):
Attacking right -> 4.5-4.7
Attacking left <- 4.1-4.3
Step 2) Power Adjustment.
Calculating the correct power to use.
Height difference:
Low power: +0.05 for higher height of 1 bots height, -0.05 for lower height of 1 bots height.
High power: +0.05 for higher height of 2 bots height, -0.05 for lower height of 2 bots height.
Calculation: Orginal Power +/- wind x Factor
Why is there 2 Diagrams?
Because the longer the bullet stays on air, the affection of wind will be larger. So, needed more adjustment to overcome the force.
Step 3) Checking distance (checking what is the basic angle to use with the calculated power).
Power Chart:
for high tail wind, use the larger angle.
Noticing there are 2 angles in some boxes?
For against wind, use the smaller angle. For tail wind, use the larger angle
Step 4) Angle Adjustment (to overcome the wind force).
This angle adjustment chart is by - k n a t - ShenPu.
Calculation: Original +/- Wind x Factor | {"url":"http://creedo.gbgl-hq.com/random_trico_info.php","timestamp":"2024-11-11T14:02:52Z","content_type":"text/html","content_length":"2443","record_id":"<urn:uuid:1b7bfe70-7e46-4c07-9003-b9990bf1f156>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00791.warc.gz"} |
Clinical measures of adiposity and percentage fat loss: which measure most accurately reflects fat loss and what should we aim for?
Objective: To determine which clinical measure of childhood obesity should be monitored to best reflect change in adiposity in a weight management programme and estimate the degree of change needed
to be relatively certain of fat reduction.
Subjects: 92 obese children with a mean (range) age of 12.8 (6.9–18.9) years and a mean body mass index standard deviation score (BMI SDS) of +3.38 (+2.27 to +4.47) attending a hospital-based clinic
on a regular, 3 monthly basis.
Measurements: Pairs of weight and height measured up to 2.41 years apart used to derive BMI as kg/m^2, and adjusted for age and gender to give weight and BMI SDS (BMI-z score) using British 1990
Growth Reference Data. Contemporaneous adiposity estimated by fatness measured by a bioimpedance segmental body composition analyser.
Results: Changes in BMI-z scores, compared to BMI, weight and weight SDS, most accurately reflected loss of fat. Reductions of 0.25, 0.5, 0.75, and 1 BMI SDS equate to expected mean falls in total
body fat percentage of 2.9%, 5.8%, 8.7% and 11.6%. Approximate 95% prediction intervals indicated that a fall in BMI SDS of at least 0.6 over 6–12 months (or 0.5 over 0–6 months) is consistent with
actual fat loss.
Conclusion: Change in BMI-z score best reflects percentage fat loss compared to BMI, weight and weight SDS. The wide variation in likely percentage fat loss for a given BMI SDS reduction means a loss
of 0.5–0.6 is required to be relatively certain of definite percentage fat reduction.
• AIC, Akaike information criterion
• BMI, body mass index
• IOTF, International Obesity Task Force
• MCMC, Markov chain Monte Carlo
• PI, prediction interval
• SDS, standard deviation score
• BMI-z scores
• adiposity
• bio-impedance
• obesity
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Childhood obesity is now a significant global problem. The increasing prevalence of childhood obesity in the United Kingdom, together with the stated government aim to reduce the year on year
increase of obesity in under 11 year olds^1 by 2010, will inevitably focus attention in this country on the development of community, primary care and school-based weight management programmes for
treating childhood obesity. The paucity of randomised trials in this area and the absence of sufficiently powered interventions so far reported^2 will inevitably necessitate a thorough evaluation of
future interventions to identify which have the greatest efficacy at least economic cost. Many of these interventions are likely to be conducted in areas with very basic clinical evaluation tools,
allowing simply an estimate of height and weight rather than more sophisticated measures such as skinfold thickness. We have used our data from a hospital-based clinic, which experiences a wide
spectrum of success in weight management, to explore the best measure of body composition to represent actual loss of adiposity as percentage fat mass reduction, as determined by bioimpedance. We
sought a simple, empirical relationship that could be used to indicate the likely reduction in percentage fat. When evaluating success in weight management for obesity, the evidence points to the
need to reduce adiposity (fat mass as a percentage of whole body weight) rather than any other measures such as improved fitness.^3,^4 Adiposity or fat mass is intimately linked to both blood
pressure and insulin sensitivity in childhood populations, whereas fitness is probably associated with any improvements through its modulating effect on adiposity. For this reason, it is likely that
a reduction in the level of adiposity is required to improve morbidity and long-term health, while still acknowledging the importance and positive benefits of regular exercise through improved
skeletal muscle function.^5,^6
The Care of Childhood Obesity Clinic is a hospital-based intervention aimed solely at children with International Obesity Task Force (IOTF) defined obesity. Through a diet and exercise lifestyle
modulation scheme, 83% of our patients attending the clinic for a year or more reduce body mass index (BMI)-z scores, with 28% achieving a loss of greater than 0.5 BMI standard deviation score (SDS).
Weight was measured on a digital scales (Seca, Hamburg, Germany) in light clothing with shoes removed and height was measured using a Harpenden stadiometer (Holtain, Crymych, UK).^8 BMI was
calculated as kg/m^2 and adjusted for age and gender to give a BMI standard deviation score (SDS) using British 1990 Growth Reference Data from the Child Growth Foundation.^9 A relatively recent
addition to our evaluation of obese children over 7 years of age has been an estimate of adiposity using the Tanita bioimpedance segmental body composition analyser (model BC-418MA; Tanita, Yiewsley,
UK). This model has been validated against more complex and expensive investigations to assess adiposity.^10,^11 More recently, it has been further validated across the pubertal age range (11±3.6
(males) and 11±3.0 (females) years) against DXA and air-displacement plethysmography (BOD POD).^12 At each 3 monthly standard clinic visit, bioimpedance is therefore estimated as a measure of
adiposity and we used this as our “gold standard” measure.
Repeated Tanita data over time were available on 92 obese children and young people (41 male). At first assessment the mean (range) age was 12.8 (6.9–18.9) years. All patients were obese by IOTF
guidelines (BMI SDS >2.25 for females and >2.37 for males)^13 with a mean BMI SDS of +3.38 (+2.27 to +4.47) and a mean percentage total body fat of 44.7% (27.7% to 62.4%). Eighty six (93%) patients
were of white ethnic origin (two Black, two South-Asian, two mixed race) and 19% were pre-pubertal. The first and most recent Tanita measurements were used, with a median interval between them of
0.83 years (range 0.01–2.41 years).
Which measure of adiposity best predicts reduction in percentage fat mass over this period?
Exploratory analyses were carried out to investigate the relationships between percentage fat and BMI, BMI SDS, weight and weight SDS (see fig 1 for the raw data), firstly using one result per child,
then using both results but allowing for correlated results for each child (data not shown). The relationships between percentage fat and both BMI SDS and weight SDS were approximately linear, with a
slight improvement in the model for BMI SDS if adjustment was made for age. The relationships between percentage fat and BMI and weight were more complex; the former was approximately quadratic in
BMI, requiring adjustment for age, while the latter required square root transformation of weight and adjustment for both age and sex. In general, as better fits were obtained (in terms of the
residual variation) for BMI and BMI SDS than for weight or weight SDS, only the former were explored in the following analysis.
Estimation of the fall in percentage fat from the fall in BMI SDS
The linear relationship between percentage fat and BMI SDS suggested to us that the change in percentage fat (defining ”change” as ”initial minus final”) should be linearly related to the
corresponding change in BMI SDS, with a possible adjustment for the increase in age, that is, the time interval (in decimal years) between the two measurements, as follows: change in %fat=b[1]
×change in BMI SDS+b[2]×time interval+e, where b[1] and b[2] were coefficients to be estimated and e was the residual for the individual, assumed to be approximately normally distributed with mean 0
and variance σ^2. The variance σ^2 would be expected to vary according to the time interval between the two measurements; for a given individual, pairs of percentage fat results closer in time would
be expected to be more highly correlated than those further apart in time, and therefore the variance of the difference would be smaller. The children were divided arbitrarily into three subgroups
according to the time interval between each child’s pair of measurements: up to 0.5 year (median 0.23 years; n=27), from 0.5 to 1 year (median 0.71 year; n=28) and over 1 year (median 1.22 year;
n=37). The model above was fitted using REML in the SAS PROC MIXED procedure (SAS release 8.2; SAS, Cary, NC, USA), with the same regression coefficients (b[1] and b[2]) for the three subgroups but
different variances. The time interval coefficient (b[2]) was not statistically significant (p=0.400) and this term was dropped from the model. The coefficient for the model using change in BMI SDS
alone (b[1]) was estimated to be 11.60 (SE 1.155) and the variances for the three subgroups were 7.68, 11.69 and 19.80, respectively. The mean predicted change in percentage fat, for a given fall in
BMD SDS, thus could be estimated from 11.60×change in BMI SDS and the standard error (SE) of this estimate was √{(1.155×change in BMI SDS)^2}. Figure 2 shows the data for the three subgroups, the
predicted changes and approximate 95% prediction intervals (PIs).
Values for clinical usefulness
In the above model, reductions of 0.25, 0.5, 0.75, and 1 BMI SDS equate to expected mean (SE) falls in total body fat percentage of 2.9% (0.3%), 5.8% (0.6%), 8.7% (0.9%) and 11.6% (1.2%). If these
reductions are observed over an interval of time from 6 months to 1 year, then the approximate 95% PIs are −3.9% to 9.7%, −1.1% to 12.7%, 1.7% to 15.7% and 4.4% to 18.8%, respectively. One can
calculate that the minimum BMI SDS change to be consistent with a fat reduction (ie, with the lower limit of the 95% PI >0) over this period is 0.60–7% (95% PI 0.0% to 13.9%). Over a shorter interval
(less than 6 months), the equivalent value BMI SDS change would be 0.49–mean percentage fat reduction 5.7% (95% PI 0.1% to 11.3%)
Estimation of the fall in percentage fat from the fall in BMI
If only raw BMI data, and not BMI SDS data, are available, preliminary work (above) suggested that fall in percentage fat might also be predicted from changes in BMI, the square of the BMI and
increase in age, that is, the time interval, as follows:
Change in %fat=b[1]×change in BMI+b[2]×change in BMI^2+b[3] time interval+e
The model fitted using REML, with the same three variance subgroups as before (ie, time intervals <0.5, 0.5–1 and >1 year). Although the coefficient for the time interval was not significant in this
case (p=0.070), the term was retained because of its impact on the Akaike information criterion (AIC; ie, 494.8 compared with 498.2), which assessed the goodness of fit of the model while taking
into account the number of parameters. This model fitted slightly better than the model using BMI SDS above (AIC=498.3). The estimates of the coefficients b[1], b[2] and b[3] were 3.90 (SE 0.995),
−0.0339 (SE 0.0145) and 0.818 (SE 0.446), respectively, and the variances of the three subgroups were 6.61, 10.44 and 17.43.
Table 1 gives examples of the fall in percentage fat estimated from the fall in BMI and BMI^2, for time intervals of 0.75 years and 1.25 years, respectively. Falls in BMI of between 1 and 5 units are
used for illustration and are shown separately for initial BMI values of 30, 35, 40 and 45 units, since the change in BMI^2 depends on the initial BMI.
We believe this study makes some important observations regarding measuring changes in adiposity in the clinical setting and how much we should aim for to be relatively certain of beneficial effects.
In clinical terms the change in BMI-z score appears to give the simplest surrogate measure of percentage loss in fat mass or adiposity. In a setting where BMI SDS scores are not available, a model
derived from changes in BMI and BMI^2 could be used. This model fitted our data slightly better but was more cumbersome to use than the change in BMI-z score.
In 2005, Cole et al presented data suggesting that BMI might be a slightly better measure of adiposity change over time than BMI-z scores.^14 However, there are a number of important differences
between our study and that of Cole et al. Their study was observational and examined reproducibility of measures over 9 months, thereby not assuming a downward trend over time, while our children
were mainly “improving” in a weight management programme. Furthermore, we have been able to study changes in BMI/BMI SDS in relation to a validated measure of actual percentage fat loss. Although we
acknowledge that bioimpedance is slightly less accurate than more sophisticated research tools such as DXA or MRI scans, and is incapable of differentiating visceral (central) from subcutaneous fat
which is of great relevance to obesity co-morbidities,^15 its ease of use and cost made serial measurements affordable and simple for regular clinical use.
We have developed equations that could be used prospectively. While the variability in percentage fat loss for a given BMI or BMI SDS reduction is wide, our data do suggest that any intervention in
childhood obesity needs to be able to demonstrate a fall of at least 0.6 BMI SDS to be more or less certain of reducing adiposity. This is interesting because other researchers have identified a
minimum reduction of 0.5 as being required to produce significant improvements in indices of blood pressure, lipid profile and measures of insulin resistance.^16,^17 Furthermore, others have
identified that a rise in BMI SDS around the 0.5 level increases the risk of developing metabolic syndrome.^18 We believe it likely that the almost certain improvement in adiposity associated with
this level of BMI SDS reduction leads to the documented improvement in cardiovascular and endocrine outcome measures. When planning intervention studies for treating childhood obesity, we suggest
that this should be the level at which success is defined as the basis for working out statistical power.
In deriving our equations we used arbitrary subgrouping of the time intervals. We further explored a Bayesian approach that allowed us to relate the variance with the actual time interval between the
two measurements. Although the variance model was not easily validated, this approach led to similar findings. Details of this approach are given in the Appendix and figs 3 and 4.
What is already known on this topic
• Childhood obesity prevalence continues to increase in the UK.
• Effective management strategies are urgently needed.
What this study adds
• For a given reduction in BMI SDS or BMI, the range of percentage fat loss is wide.
• For BMI SDS, a reduction of between 0.5–0.6 is required to be relatively certain of actually reducing adiposity.
In essence, we believe that we have shown that it is possible to use either measurement of BMI or BMI-z scores to predict actual changes in percentage fat. In purely clinical terms, however, it is
likely that staff may have to rely on BMI estimates to evaluate success. We believe this provides a useful, additional tool to evaluate clinical success, although our equations will need validation
for different ages, racial groups and longer time intervals than those used in this study.
In the methods described in this paper, uncertainty in the estimation of the three variances was not taken into account in the prediction. This was achievable using a Bayesian approach, and as an
alternative approach we explored the use of Markov chain Monte Carlo (MCMC) methods, using WinBUGS V.1.4.1 (Imperial College and MRC, London, UK). The variance σ^2 was assumed to be related to the
time interval between a child’s pairs of measurements (t) in the following way: σ^2=1/α[1]×(1−α[2]^t), but limited exploratory analysis did seem to support this. Posterior distributions were
obtained for α[1] and α[2], as well as for the regression coefficients (the b’s.). Wide normal priors were used for the regression coefficients, a gamma prior for α[1], a uniform prior for α[2,] and
a long burn-in of 10 000 iterations (despite quite rapid convergence).
For the first model, to estimate the fall in percentage fat from the fall in BMI SDS, median posteriors after a further 40 000 iterations were calculated to be 11.03 (95% credible interval
8.46–13.58), 0.053 (0.035–0.074) and 0.0082 (0.0002–0.0912) for b[1], α[1] and α[2], respectively. Posterior predicted values were obtained for the changes in percentage fat associated with
incremental changes of 0.1–0.9 in BMI SDS, over intervals of 0.25 and 0.75 years, and the median posteriors are plotted in fig 3. The 95% credible intervals are expected to be wider than the PIs in
fig 2, because in the Bayesian analysis the predicted values incorporated uncertainty in all three coefficients. Figure 3 suggested a fall in BMI SDS of 0.7 or more over 9+ months would be consistent
with percentage fat reduction, and a smaller change (0.6) over 3 months.
For the second model, which used change in BMI, BMI^2 and the time interval, median posteriors (and 95% credible region) for the coefficients b[1], b[2] and b[3] were 3.83 (1.56 to 6.05), −0.0339
(−0.0663 to −0.0010) and 0.796 (−0.134 to 1.726), respectively, and for the coefficients α[1] and α[2] were 0.060 (0.041 to 0.084) and 0.0048 (0.0001 to 0.0640). Median posterior predicted values
(and 95% credible regions) are shown in fig 4 for integral changes in BMI from 1 to 5, and over time intervals of 0.25, 0.75 and 1.25 years.
We would like to thank Nicky Welton for helpful discussions about MCMC posterior predicted values. Anna Ford is supported by a grant from The BUPA Foundation.
• ↵* Both authors contributed equally to this work.
• Published Online First 29 January 2007
• Competing interests: None.
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The Basics of Cos and Sin: Understanding Their Significance - SoftHandTech
The Basics of Cos and Sin: Understanding Their Significance
Understanding the concepts of cosine (cos) and sine (sin) is essential for anyone involved in mathematics, physics, engineering, or any field that involves the manipulation of waves and oscillations.
These two trigonometric functions underpin a wide range of mathematical principles and have significant applications in various scientific and technical disciplines. Whether you’re a student trying
to grasp the fundamental principles of trigonometry or a professional seeking to apply these concepts to real-world problems, gaining a solid understanding of cos and sin is crucial.
In this article, we will delve into the basics of cos and sin, exploring their definitions, properties, and practical implications. By demystifying these essential trigonometric functions, we aim to
provide a clear and comprehensive overview that will empower readers to utilize cos and sin with confidence and expertise in their academic and professional pursuits.
Key Takeaways
Cosine (cos) and sine (sin) are trigonometric functions used in mathematics to relate the angles and sides of a right-angled triangle. The cosine of an angle in a right-angled triangle is the ratio
of the length of the adjacent side to the length of the hypotenuse, while the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. These functions are
fundamental in trigonometry and have applications in physics, engineering, and various other fields.
Trigonometric Functions And Their Definitions
Trigonometric functions are essential tools in mathematics and science for understanding the relationships between the angles and sides of triangles. The primary trigonometric functions are sine
(sin), cosine (cos), and tangent (tan), which are used to calculate the ratios of the sides of a right-angled triangle. The sine of an angle in a right-angled triangle is the ratio of the length of
the side opposite the angle to the hypotenuse, while the cosine is the ratio of the adjacent side to the hypotenuse. The tangent is the ratio of the opposite side to the adjacent side.
These functions are fundamental in trigonometry and are used to solve a wide range of problems related to angles and distances. By understanding their definitions and properties, one can effectively
analyze and solve various real-world problems involving periodic phenomena, oscillations, and waveforms. Additionally, trigonometric functions are indispensable in fields such as physics,
engineering, and astronomy, making them essential for anyone studying or working in these disciplines. Overall, having a clear understanding of the definitions and applications of trigonometric
functions is crucial for mastering the principles of mathematics and its applications across different fields.
Understanding The Unit Circle
Understanding the unit circle is crucial in grasping the concepts of cosine and sine. The unit circle is a circle with a radius of 1 unit centered at the origin of a coordinate system. It allows us
to visualize the relationship between the coordinates on the circle and the values of sine and cosine for those angles.
When we plot a point (cos θ, sin θ) on the unit circle for a given angle θ, we can see how the cosine and sine values correspond to the x and y coordinates of that point. This aids in understanding
the periodic nature of these trigonometric functions and their relevance in various fields like physics, engineering, and mathematics.
Moreover, the unit circle helps simplify the understanding of trigonometric functions and provides a geometric interpretation of their values. It is a fundamental tool for solving trigonometric
equations, analyzing periodic phenomena, and making connections between different trigonometric identities. This visualization tool enhances the comprehension of cosine and sine, making it an
essential concept in trigonometry and its applications.
Properties And Graphs Of Cosine And Sine Functions
Properties and Graphs of Cosine and Sine Functions:
The cosine and sine functions exhibit several key properties and characteristics that are fundamental to understanding their significance in mathematics and other fields. Both functions are periodic,
meaning they repeat their values at regular intervals, which gives rise to their distinct wave-like graphs. The period of the cosine and sine functions is 360 degrees, or 2π radians, and their
amplitude determines the range of values they can attain.
The graphs of the cosine and sine functions are symmetric about the y-axis, reflecting their even and odd properties, respectively. The cosine function starts at its maximum value of 1 and decreases
until it reaches its minimum value of -1, while the sine function begins at 0, increases to its maximum of 1, and then decreases back to 0. These functions also feature phase shifts, which cause
their graphs to shift horizontally. Understanding these properties is crucial in analyzing and solving problems in trigonometry, physics, engineering, and other areas where periodic phenomena are
Applications Of Cosine And Sine In Mathematics
The applications of cosine and sine in mathematics are vast and varied. In trigonometry, these functions are essential for solving problems involving angles and triangles. Cosine and sine are used to
calculate the lengths of sides and the measures of angles in right-angled triangles, as well as in non-right-angled triangles using the law of sines and the law of cosines. Additionally, these
functions are fundamental in analytic geometry and calculus, where they are used to represent periodic phenomena and oscillatory motion, as well as to model and analyze various physical quantities
and phenomena.
Moreover, cosine and sine play a crucial role in Fourier analysis, a mathematical technique used to decompose complex periodic signals into simpler components. This decomposition process has
widespread applications in signal processing, data analysis, and various fields of engineering and science. In differential equations, the solutions often involve trigonometric functions, making
cosine and sine indispensable in the study of oscillatory motion, electrical circuits, and various natural phenomena. Overall, the applications of cosine and sine in mathematics are fundamental and
far-reaching, underpinning many important concepts and techniques across different areas of study.
Trigonometric Identities Involving Cos And Sin
Trigonometric identities involving cos and sin are fundamental to understanding the interrelationships between these functions. These identities, such as the Pythagorean identity, are crucial in
simplifying expressions involving cos and sin. The Pythagorean identity states that sin^2θ + cos^2θ = 1, providing a fundamental connection between the two functions and enabling the conversion of
one trigonometric function into another.
Furthermore, the double-angle identities, such as sin(2θ) = 2sinθcosθ and cos(2θ) = cos^2θ – sin^2θ, are essential in double-angle formulas and are frequently used in various applications of
trigonometry. These identities allow for the transformation of trigonometric functions into simpler forms, aiding in the simplification of complex trigonometric expressions and equations.
Understanding and mastering these trigonometric identities involving cos and sin are crucial for solving trigonometric equations, simplifying expressions, and tackling advanced topics such as
calculus and physics. As such, a clear comprehension of these identities is integral to the study of trigonometry and its applications in various fields.
Solving Problems Using Cos And Sin
In practical terms, solving problems using cosine and sine involves applying these trigonometric functions to real-world scenarios. One common application is in solving problems related to angles and
distances, such as in navigation, physics, engineering, and astronomy. By using the values of cos and sin, one can calculate the height of buildings, the distance between two points, the angle of
elevation, and even the orbital paths of celestial bodies.
In the field of engineering, cos and sin are frequently used to analyze forces and motions in mechanical systems, allowing engineers to design structures and machines with precision and accuracy.
Furthermore, in physics, these trigonometric functions are utilized to calculate the trajectory of projectiles, the motion of waves, and the behavior of oscillating systems. Understanding how to
solve problems using cos and sin is essential for anyone pursuing a career in these fields and is an invaluable skill in practical problem-solving across various disciplines.
Understanding Periodicity And Amplitude
Periodicity and amplitude are fundamental concepts in understanding the behavior of cosine and sine functions. Periodicity refers to the repetition of the values of a function within a certain
interval. For both cosine and sine functions, the period is 2π, meaning that their values repeat every 2π units along the x-axis.
Amplitude, on the other hand, measures the maximum displacement of a wave from its equilibrium position. For the cosine and sine functions, the amplitude represents the maximum distance the graphs of
these functions reach from the x-axis, which is equal to 1. Understanding these concepts is crucial in analyzing and interpreting the behavior of trigonometric functions and is fundamental in various
mathematical and scientific applications, such as physics, engineering, and signal processing.
In summary, periodicity and amplitude are essential aspects of trigonometric functions that provide insights into their repetitive nature and maximum displacement. As such, mastering these concepts
is essential for anyone seeking to understand and apply the principles of trigonometry in different fields of study and professional practice.
Real-Life Applications Of Cos And Sin
The real-life applications of cosine and sine extend to various fields such as engineering, physics, astronomy, and even everyday activities. In engineering, these trigonometric functions are used to
analyze the forces and stresses acting on structures, aiding in the design and construction of buildings, bridges, and tunnels. In physics, the motion of waves and oscillations, as well as the
behavior of alternating current in electrical circuits, can be described and predicted using cosine and sine functions.
Moreover, in astronomy, these functions are crucial in predicting the positions and motions of celestial bodies and understanding phenomena such as the tides. Even in everyday activities, cosine and
sine come into play when using GPS navigation systems, where the location and distance calculations are based on trigonometric principles. Additionally, these functions are relevant in fields such as
music, computer graphics, and even medical imaging, where they are employed in analyzing and creating sound, visual effects, and diagnostic images, respectively. The real-world applications of cosine
and sine highlight the profound impact of these trigonometric functions across diverse disciplines.
The Bottom Line
In understanding the basics of cosine (cos) and sine (sin), it becomes clear that these fundamental mathematical concepts hold significant practical importance in various fields such as physics,
engineering, and computer science. By grasping the underlying principles and real-world applications of these trigonometric functions, professionals can more effectively solve complex problems and
optimize their work. Furthermore, acquiring a deeper comprehension of cos and sin enables individuals to gain a new perspective and appreciation for the mathematical foundations that shape our
understanding of the world around us. In essence, the study and application of cos and sin not only enhance technical proficiency but also contribute to a broader intellectual appreciation of the
interplay between mathematics and the physical universe.
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Confused on how many rows
I am working on a Tom Baker Scarf using this pattern.
It says 3 rows of purple but then at the top it says "measurements in garter ribs (2 rows each). So am I doing 3 rows or 6 rows? I am so confused, I did 3 but then I am looking at the images of the
scarf online and it looks so much bigger, is that after blocking. Any help would be great. Thanks. | {"url":"https://forum.knittinghelp.com/t/confused-on-how-many-rows/76343","timestamp":"2024-11-14T21:38:09Z","content_type":"text/html","content_length":"13862","record_id":"<urn:uuid:2d217c56-bcff-4850-a9c6-bbd1e511dcb0>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00891.warc.gz"} |
How To Calculate Growth Rate: An Essential Guide for Success - CalculatorBox
How To Calculate Growth Rate: An Essential Guide for Success
Growth rate is a metric used to measure the change in a value over a period of time. It can be applied to various aspects of a business, such as revenue, market share, or customer base, and is
essential for understanding the progress and success of a business or investment.
Fun Fact: The “Rule of 70” is a quick, mental math trick used to estimate the number of years required to double the value of an investment or a quantity at a constant growth rate. You simply divide
70 by the growth rate (expressed as a percentage), and the result is approximately the number of years it will take for the value to double! For instance, if you have a growth rate of 7%, it would
take about 10 years for your value to double (70 ÷ 7 = 10).
To calculate the growth rate, you’ll need data from two points in time: the beginning value and the ending value. There are a few methods to calculate growth rate depending on the complexity and
desired accuracy of the result.
The simplest method to calculate growth rate is the straight-line percent change. This method helps you understand the basic growth rate without comparing it to other results. This method works best
for calculating simple growth rates, such as year-over-year (YoY) revenue growth.
Another method is the compound annual growth rate (CAGR), which provides a smoothed growth rate that accounts for changes in value over time. CAGR is a more accurate method for calculating growth
rates that rise or fall in value over multiple periods.
Keep in mind that these methods are only a starting point for understanding growth rates. Depending on the specific context and the data you are analyzing, other methods or adjustments may be
required to more accurately measure growth over time.
Principles of Calculation
Key Factors for Calculations
To calculate growth rate, you need to understand the essential elements involved in the process. Firstly, you have to obtain data that shows a change in a particular quantity over time. You’ll need
two numbers: the starting value and the ending value of the quantity.
There are various methods to calculate growth rates, but the most common formula is the Compound Annual Growth Rate (CAGR). It is a widely used metric that describes an investment’s growth rate. The
formula for calculating CAGR is:
CAGR = (\frac{Ending \space Value}{Beginning \space Value})^{\frac{1}{n}} - 1
• Ending Value (EV) represents the final value of the quantity
• Beginning Value (BV) represents the initial value of the quantity
• n is the number of periods (usually in years) between the two values
Essential Variables
To get accurate results while calculating growth rates, make sure to consider all necessary variables related to the given data. The key factors you need to pay attention to include:
Key Factor Description
Timeframe Ensure that the period you choose for calculating growth rate is consistent and relevant to the data you’re analyzing. For example, if you’re looking at annual growth, use a
one-year time span.
Make sure the data you use for calculating growth rate is reliable and accurate. Inaccurate data can lead to incorrect results, which may mislead your analysis.
Data reliability
Appropriate Different situations may require different formulas for calculating growth rates. For instance, to calculate the annual growth rate, you can use the CAGR formula mentioned above.
Calculating the Basic Growth Rate
In order to calculate a basic growth rate, you only need two numbers: the starting value and the ending value of the quantity you are analyzing. By using a simple formula, you can easily determine
the growth rate of your investment, revenue, or any other metric that has changed over time. Here’s how you can calculate the basic growth rate:
Key Factor Description
Determine the starting value (Initial This is the initial amount of the quantity you are analyzing, such as the initial investment in a financial instrument or the revenue of a company in the
Value) beginning period.
Determine the ending value (Final Value) This is the final amount of the quantity you are analyzing, such as the current worth of an investment or the revenue of the company in the ending period.
Now that you have the starting and ending values, you can use the following formula to calculate the basic growth rate:
Growth \space Rate = \frac{(Final\space Value-Initial \space Value)}{Initial \space Value}
Here are the steps to apply the formula:
Step Description
Determine the starting value (Initial Value) Subtract the initial value from the final value to determine the change in the quantity over time.
Determine the ending value (Final Value) After finding the difference, divide it by the initial value to obtain the growth rate.
Convert the growth rate to a percentage Multiply the growth rate by 100 to express it as a percentage.
For example: If your starting investment was $1,000 and, after a year, it has grown to $1,200, the basic growth rate calculation would be as follows:
Growth \space Rate = \frac{(1,200-1,000)}{1,000} = \frac{200}{1,000} = 0.2 × 100 = 20\%
In this example, the basic growth rate of your investment is 20%. This method should be applied when calculating simple growth rates without considering other factors or comparing them with other
Calculating Annual Growth Rate
When you need to calculate the annual growth rate, it’s essential to understand the basic formula and approach. The annual growth rate indicates the percentage change in a value over one year.
To calculate the annual growth rate, first, you’ll need the initial and final values of the metric you’re considering for a specified period, like population, revenue, or investments. Here’s the
formula for calculating the annual growth rate:
Annual \space Growth \space Rate = [(\frac{Final\space Value}{Initial \space Value}) - 1] × 100
Follow these steps to calculate the annual growth rate:
Step Description
Determine the starting value (Initial Value) The starting point of your data set, which is usually the value of your metric at the beginning of the year or period under consideration.
Determine the ending value (Final Value) The end point of your data set, i.e., the value of your metric at the end of the year or period.
Divide the final value by the initial value To get the growth factor, divide the final value by the initial value.
Subtract 1 from the growth factor This gives you the growth rate in decimal form.
Multiply by 100 to get the percentage Convert the growth rate to a percentage by multiplying it by 100.
For example, suppose you want to calculate the annual growth rate of your company’s revenue. If the revenue at the beginning of the year was $50,000 and it reached $60,000 by the end of the year, the
calculation would be:
Annual \space Growth \space Rate = [(\frac{60,000}{50,000}) - 1] = 0.2 × 100 = 20\%
Your company experienced a 20% annual growth rate in revenue.
Keep in mind that this formula calculates the basic annual growth rate. For more complex scenarios, like when considering compounded growth over multiple years, you may need to use a different
approach, such as the Compound Annual Growth Rate (CAGR) formula. But for a single year or period, this method remains a straightforward and reliable way to calculate annual growth rates.
Interpreting Growth Rate Results
Positive and Negative Growth Rates
When you calculate growth rates, it’s essential to understand the implications of positive and negative growth rates. A positive growth rate indicates that an entity, such as revenue, population, or
investments, is increasing over time. On the other hand, a negative growth rate shows that the entity is decreasing over time.
For example, if you calculate the annual growth rate of a company’s revenue and find it to be 5%, this means that the revenue is increasing by 5% each year. Conversely, if the annual growth rate is
-3%, the revenue is decreasing by 3% each year.
Implications of Changing Growth Rates
Changing growth rates can have various implications, depending on the context. For instance, a business experiencing an increase in its revenue growth rate might be benefiting from successful
marketing strategies, new product releases, or favorable market conditions. In contrast, a declining growth rate could signal internal issues, increased competition, or adverse market trends.
In the case of population growth, a high growth rate might result in strains on resources, infrastructure, and social services in a region. A negative population growth rate could signify an aging
population or a higher out-migration rate.
Application of Growth Rate in Different Sectors
In this section, we will discuss the application of growth rate calculation in different sectors, such as economics, business, and population studies. Understanding the growth rate in these areas
will help you better analyze their development and progress over time.
In Economics
In economics, growth rate is a key indicator to measure the performance of an economy. It is often calculated as the percentage change in Gross Domestic Product (GDP) over a specified period of time.
The growth rate of GDP is widely used to compare the economic health of different countries and to identify trends in the global economy.
To calculate the GDP growth rate, you can use the following formula:
GDP \space Growth \space Rate = (\frac{Current\space GDP - Previous\space GDP}{Previous\space GDP}) × 100
In Business
In business, growth rate is crucial to evaluate the performance of a company over time. It helps investors, management, and other stakeholders to make informed decisions based on the past success of
the business. The most commonly used growth rate in business is the revenue growth rate. You can calculate the revenue growth rate using the formula:
Revenue\space Growth \space Rate = (\frac{Current\space Period \space Revenue- Previous\space Period \space Revenue}{Previous\space Period \space Revenue}) × 100
Other important growth rates in business include profit growth rate, customer base growth rate, and market share growth rate. To calculate these growth rates, you can follow a similar formula by just
replacing the revenue with the relevant metric.
In Population Studies
In population studies, the growth rate is used to measure the change in the size of a population over time. It takes into account the factors of births, deaths, and migration. The population growth
rate is an essential factor when planning for resources and infrastructure to accommodate a growing population.
To calculate the population growth rate, you can use the following formula:
Population\space Growth \space Rate = (\frac{Ending\space Population- Starting\space Population}{ Starting\space Population}) × 100
By understanding how to calculate growth rate and its application in different sectors, you can effectively analyze the development and progress over time in economics, business, or population
Frequently Asked Questions
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Cambridge International Mathematics 0607 | AP Learning Star
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Online Tutorial Services for IB, Cambridge & Pearson Edexcel
International Mathematics 0607
The Cambridge IGCSE International Mathematics (0607) course is designed to enhance students' mathematical skills, problem-solving abilities, and critical thinking within a global context. This
curriculum offers a comprehensive approach to mathematics, integrating core principles with practical applications, preparing students for diverse academic and career pathways.
Content Overview
Cambridge IGCSE International Mathematics (0607) covers a wide range of mathematical topics suitable for students with varying abilities and backgrounds. It includes both core and extended curriculum
options to cater to different learning needs.
Core Curriculum: The core curriculum includes fundamental topics such as number, algebra, geometry, probability, and statistics. This curriculum provides a solid foundation in mathematics, essential
for understanding and applying mathematical concepts in everyday situations.
Extended Curriculum: The extended curriculum extends beyond the core topics, covering advanced algebra, functions, calculus, and other higher-level mathematical concepts. It is designed for students
who are aiming for deeper mathematical understanding and preparing for more advanced studies in mathematics or related fields.
Syllabus Outline
Assessment Structure
The assessment for Cambridge IGCSE International Mathematics (0607) is structured as follows:
Core Curriculum:
Paper 1: Non-calculator (60 marks, 1 hour 15 minutes)
Paper 3: Calculator (60 marks, 1 hour 15 minutes)
Paper 5: Investigation (40 marks, 1 hour 15 minutes)
Extended Curriculum:
Paper 2: Non-calculator (60 marks, 1 hour 30 minutes)
Paper 4: Calculator (60 marks, 1 hour 30 minutes)
Paper 6: Investigation and Modelling (50 marks, 1 hour 30 minutes)
Key Differences from Cambridge IGCSE Mathematics (0580)
• Scope of Topics: Cambridge IGCSE International Mathematics (0607) covers a broader spectrum of topics, including advanced algebra, calculus, and modelling, which are not extensively covered in
Cambridge IGCSE Mathematics (0580).
• Assessment Variety: Unlike Cambridge IGCSE Mathematics (0580), which primarily consists of two papers, International Mathematics (0607) includes additional papers focusing on investigation and
modelling, enhancing students' analytical and practical skills.
Career Opportunities
Studying Cambridge IGCSE International Mathematics (0607) prepares students for a wide range of career paths, including:
• STEM Fields: Applying advanced mathematical principles in science, technology, engineering, and mathematics.
• Business and Economics: Using quantitative analysis for financial planning and market research.
• Research and Academia: Pursuing higher education and careers in mathematics, physics, or other related fields.
By choosing Cambridge IGCSE International Mathematics (0607), students develop essential mathematical skills and critical thinking abilities necessary for academic success and future career
advancement in a globalized world.
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Roll the chance cube: Armada, dice, and probability
Everything's perfect: your attack is set up just right, you've got the enemy ship dead to rights. The dice roll is just a formality. Here come the dice, and... crap.
At least you won a young Jake Lloyd as a consolation prize?
Dice can and will betray you in any game that utilizes them. On the other hand, they add an element of excitement and vary the play experience. What to expect from them and how far to push them can
take some getting used to, and this article aims to further explore that element of Armada.
The basics
Let's start with a fairly basic breakdown of each different color of dice.
Red dice
Red dice have the following sides:
• 1 double-hit
• 1 accuracy
• 2 hit
• 2 crit
• 2 blank
With 5 different kinds of results (compared to the 3 possible outcomes on blue and black dice), red dice have a well-deserved reputation for being the most swingy and unreliable color of dice -
sometimes they produce disastrously bad results, but every now and then those double-hit and/or accuracy results show up when you weren't counting on them, surprising you.
Against ships, red dice do an average of 0.75 damage (6 total damage points divided by 8 sides), although it should be noted that 3/8 of their sides produce no damage at all, being either blank or
having an accuracy icon, so their damage is inconsistent.
Against squadrons, red dice do an average of 0.5 damage (4 total damage points divided by 8 sides), although it's important to note here that 5/8 of its sides don't do any damage against squadrons,
making them extremely unreliable against them.
Because only 1/8 of the sides on red dice have an accuracy icon, it can be frustrating (only showing up 12.5% of the time on a die-by-die basis). In small doses, the inclusion of an accuracy icon
into the pool is often a waste - you're not doing enough damage to make locking down defense tokens as good as simply doing more damage would've been or don't get enough icons to lock down a pair of
identical defense tokens, making the icon effectively a blank. Conversely, when you're pouring on red dice in large doses (for example, with
) you're really hoping to get an accuracy result to make your damage "stick" better, but if your pool is reds-only, you can't rely on that accuracy icon showing up under normal circumstances.
Blue dice
Blue dice have the following sides:
Blue dice are the only dice color with no blank sides, making them very reliable for producing results, even if they're not necessarily the results you wanted.
Against ships, blue dice do an average of 0.75 damage, the same as red dice. They gain from only 2/8 of their sides being damage-free, but they're never going to luck into a 2-hit side for a
serendipitous damage spike.
Against squadrons, blue dice do an average of 0.5 damage, again the same as red dice. They're again more reliable, doing damage 50% of the time (4/8 sides), but without a lucky double-hit possibility
you'll never see the occasional "how much damage, again?" silliness you get from
Zertik Strom
when they luck out; then again, you're also much less likely to see your attacks completely fail, either.
With 2/8 of their sides featuring an accuracy icon, blue dice have the best odds (25% per die) of rolling an accuracy result naturally. In enough numbers and/or with the help of red dice, you can
generally count on decent odds of getting an accuracy result.
Black dice Black dice
have the following sides:
Black dice are the kings of damage-dealing, although they lack accuracy icons and are typically only available at close range.
Against ships, black dice deal an average of 1 damage, making them 33% better at doing so than red or blue dice.
Against squadrons, black dice do an average of 0.75 damage, making them
50% better
at doing so than red or blue dice.
Because they lack accuracy icons and have 2 blank sides, black dice often appreciate a bit of help from other dice colors or upgrades to assist them with locking down brace and scatter defense tokens
as well as some upgrade support to provide them with a reroll to unlock their full destructive potential (more on that soonish).
Easy for you to say, Obi-Wan; you never had a game's outcome depend on a red dice roll!
Average damage
Determining how much damage an attack should do is simple on its face. Take the average of each type of dice, multiply it appropriately and then voila!
For example, an ISD-I front arc at close range attacking a ship should do (without rerolls):
3 red = (3*0.75) 2.25
2 blue = ( 2*0.75) 1.50
3 black = (3*1) 3
For a total of 6.75 damage.
There are a few important provisions to note about doing this kind of thing, however:
• This is only the average amount of damage. Your results may vary, sometimes wildly. You can expect most rolls to stick near the number here (varying up or down in this example by 1-2 points).
• This doesn't factor in any rerolls or dice modification.
• This cares only about average damage and won't give you odds on generating accuracy results.
• Obviously the effectiveness of the attack will be modified by your opponent's defense tokens, game state, shields, hull, etc. All this number is giving you is average raw damage. It's a
somewhat-helpful tool for making quick decisions but it doesn't in any way replace reading the table well.
Because of all those provisions, getting to a more nuanced average can be done but it takes more time (much more in some cases). In-game, being able to fire off a basic average mentally can be done
without too much trouble, but getting more detailed results usually needs to be done outside of the game.
"For luck" and also for awkward future Skywalker family reunions
You can modify averages if you factor in a reroll, although it's important to be
really specific
about under what circumstances you're using the reroll and how many dice are being rerolled. We'll start with the easiest:
Rerolling all dice [of one type] in a pool
Red dice are the best example: say that you have an effect (like
Darth Vader
) that allows for easy rerolls of as many dice as you like (these are the easiest rerolls to plan for). When it comes to red dice, you'll obviously want to reroll the blanks. Do you want to reroll
the accuracy results too? In some cases, you probably do: with small pools of red dice the accuracy results don't usually do much (but in some cases, such as
when the defender has one high-value defense token
, they do) and generating damage is probably better than locking down a low-impact defense token; alternatively, maybe you generated a lot of accuracy results already and you only want to keep one or
two of them. Conversely, you might be starved for accuracy results and want to keep them. It's really difficult to say ahead of time exactly what you "should" do, which makes averages difficult to
predict. Let's get that done mathematically to show you the difference.
A red die has a 2/8 (25%) chance of producing a blank, so by rerolling the 25% chances of a blank, you get another chance at damage, which averages out to 0.75 (as we covered earlier). So the math
works out to:
0.75 (the original average damage) + [0.25(the chance of a blank, which will be rerolled)*0.75(a red die's average damage)] = 0.9375 average damage, a 25% improvement.
However, if you reroll the accuracy results as well as the blanks, the math changes (using the same formula as above, just with a 37.5% chance of being rerolled) to an average of 1.03125 damage, a
37.5% improvement (obviously).
So basically if you're trying to guesstimate the average damage of a pool of rerollable red dice, it will really depend on the rules you're applying to the rerolls (for example, rerolls on blue dice
can be pointless if you get the number of accuracy icons you want the first time around - they don't have any blank sides).
Another important consideration is squadrons. Specifically, when ships attack them or non-Bomber squadrons attack back, you have much fewer useful sides on the dice. Let's use red dice again as an
example, and let's assume it's a single red flak die that won't benefit from a single accuracy result (like the
0.5 average damage + [0.625 (chance of not producing damage) * 0.5] = 0.8125 average damage, a 62.5% improvement.
If you want the short version, check out the table below, but it has the following assumptions:
• Accuracy icons aren't rerolled against ships (which is a really smart thing to do under the right circumstances)
• Accuracy icons are rerolled against squadrons, which in most cases is correct (as one or two dice flak attacks don't usually benefit much from accuracy, barring a hit+accuracy from two-dice
attacks against scatter aces).
• Multiple dice are involved. It gets a little different when considering just one, as you'll have no desire to keep accuracy icons on a single-die attack, which would improve the reroll vs. ship
□ Blue would be 0.94 against ships and red would be 1.03 against ships with a reroll.
□ The "vs. ship" values also apply to Bombers attacking ships, as they get damage from critical icons just like ships do against other ships.
• Values were rounded to the nearest hundredth because who cares about it past that, right?
│Dice type│Vs. ship│Reroll vs. ship│Vs. squadrons│Reroll vs. squads │
│Red │ 0.75 │ 0.94 │ 0.5 │ 0.81 │
│Blue │ 0.75 │ 0.75 │ 0.5 │ 0.75 │
│Black │ 1 │ 1.25 │ 0.75 │ 0.94 │
If you're looking for averages outside of those specific assumptions, I've hopefully given you the tools so far to get there. If not, we still have other things to cover before we're done!
Stacked rerolls
In some circumstances (for example, Darth Vader +
Ordnance Experts
for black dice) you can get multiple entire-pool rerolls (usually for one color of dice). Determining averages for those just requires checking on the total number of dice that still qualify for a
second reroll after the first. Changing up your qualifications for what should be rerolled the first time versus the second can produce better results than applying the same criteria to both rerolls.
For example, let's look at black dice as they're easiest (accuracies aren't a worry about rerolling versus not) here:
If you reroll all non hit+crit results the first time, you get:
(25% of kept hit+crit dice are 2 damage = 0.5 average damage) + (75% of dice are rerolled with an average of 1 damage for the end result = 0.75 damage) = 1.25 average damage, same as the earlier
However, you'll find that with this reroll method, you still have an (75% rerolled chance*25% chance reroll turns up blank =) 18.75% chance per die of it being blank at the end of your first reroll.
Even if your damage average remains the same as the more conservative reroll method (1.25), your highs are higher (better odds of getting hit+crits) and your lows are lower (better odds of doing
nothing). We'll talk about this a bit more later when it comes to fishing for specific icons, but anyways...
With the second reroll you can now "correct" for all those extra blanks by rerolling only those. We maintain our first-reroll average damage of 1.25 but we add (18.75% chance of being blank*1 average
damage =)0.1875, rounded to 0.19 additional damage
for an end total of 1.44 average damage if you use a reckless black dice reroll followed by a "regular" black dice reroll.
Using a "regular" black dice reroll both times will produce an average that is lower:
You'll only have (25% chance of initial blank * 25% chance of turning up blank again =) 6.25% of "regular reroll" black dice turning up blank again on the first reroll, so you only add 1.25 + (6.25%*
average 1 damage =) 0.0625 damage, rounded to 0.06;
therefore the average damage of a black die using the "regular reroll" method twice in a row is only 1.31, about 10% lower.
There's a lot of trickiness you can probability out with stacked rerolls using different reroll methods for the subsequent rerolls to get end results a bit more fine-tuned (especially combined with
stuff we haven't gotten to yet). Thankfully (for all of us), the ability to stack multiple-dice rerolls is fairly rare.
Rerolling only one die in a pool of multiple dice
Effects like
or a
concentrate fire token
can allow you to reroll a single die. If you're trying to use this to calculate average damage, it works a little differently than effects that allow you to reroll a whole pool. You can math it out
long-form by considering each die that qualifies individually, which I'll demonstrate below. Say you want to see what kind of a benefit your
TIE Fighter
gets from using Swarm and you're considering accuracy icons as reroll candidates. Therefore each individual die has a 50% chance of generating the crit or accuracy icon you would want to use Swarm
on. Because you only need one candidate die, it works out like:
1. The first die has a 50% chance of being a candidate.
2. Should the first die fail (50% chance) the second die also has a 50% chance (50%*50% = +25% chance)
3. Should the first two dice fail (25% chance), the third die has a 50% chance (25%*50%= +12.5% chance)
4. We come to the conclusion that (50+25+12.5=)87.5% of the time, Swarm will be beneficial for a TIE Fighter rolling 3 blue dice. Because it allows a reroll of a 0.5 average damage blue die, we
would multiply 0.875 (the chance of using Swarm) by 0.5 (the average damage of the rerolled die) = 0.4375, or 0.44 when rounded to the nearest hundredth.
5. Therefore, a Swarm-using TIE Fighter improves from 1.5 to 1.94 average damage, a 29% improvement.
Obviously this is a bit of a hassle, so let me introduce you to my friend, the
binomial distribution calculator
, which does all the steps of determining total chance of 1 or more (whatevers) for you. Here's a bit of help on getting the hang of it. Just enter numbers into the top 3 boxes and hit "calculate":
• Probability of success on a single trial: how many dice sides have what you want on them? It's a 12.5% chance per dice side, which would be entered as 0.125.
• Number or trials: the number of dice of that type being rolled.
• Number of successes (x): How many of a given result do you want from your roll? The resulting calculation will show you what the percentage chance is of getting exactly that many results, less
than that number, equal to or more, etc.
You can use a binomial distribution calculator for a lot of probability-based things, but for our purposes it's easier to get into it with situations like this first before doing anything more
complex. We'll talk more about the binomial distribution calculator later.
Other assorted reroll weirdness
By which I mean "reroll abilities that cost you something." I'm thinking primarily of
Leading Shots
here, as it costs you a guaranteed icon on a blue die for some rerolls. It's difficult to factor Leading Shots into pre-made calculations, because which blue die to give up for the reroll is going to
be very context-dependent. Sometimes it's not worth it to give up a guaranteed (whatever) for small gains. For example, giving up a blue hit die to reroll a single blank black die is pretty much an
even trade - you're giving up one damage for a die that should on average produce one damage. It's generally not worth it unless you're fishing for a black hit+crit result (we're getting to fishing
for results later!). Similarly, giving up a blue hit to reroll a single blank red die is usually not worth it because you're giving up 1 for an average of 0.75. On the other hand, rerolling two blank
red die by giving up a hit is usually worth it (1 versus 1.5). Giving up an unnecessary blue accuracy icon is a very easy choice, though, but it will definitely depend on the circumstance.
If you really
figure out your average damage considering Leading Shots, I'd subtract 0.75 (the average damage of a blue die, reflecting that in some circumstances you'd give up a hit, in others an accuracy) from
the initial roll and then "upgrade" the remaining dice to their rerolled average values. It's still a bit weird, though.
You've made it this far and you're going farther?
Dice modification
Okay, so how do we factor in dice modification? When it comes to damage modifications, it's not too tough. When it comes to something like accuracy generation, it's very context-dependent and so I
generally steer clear (sometimes you'll want/need that extra accuracy from
H9 Turbolasers
, sometimes you won't, factoring it into probabilities gets really weird unless you have a specific target in mind under specific circumstances and you're willing to go crazy with the binomial
distribution calculator).
Anyways, let's use an example that combines some of the reroll concepts we've included already. Say, for example, we're trying to decide if using Turbolaser Reroute Circuits on an
is worth it with Darth Vader in the fleet compared to using Enhanced Armament (or Slaved Turrets, sure) instead. We make a few assumptions about our experiment and then it's number crunching time!
• Darth Vader will reroll all blank and accuracy icons (so we're assuming we're not shooting flotillas)
• We're considering one side arc attack only. The Arquitens can sometimes get front/rear attacks or shoot out of both side arcs at different targets (if it's feeling brave/suicidal), but for our
purposes we care only about one side arc shot.
• The TRCs would be used on a blank dice if possible or to upgrade a single hit to a double hit if that is not possible.
• We will also assume that the Arquitens is using a concentrate fire dial in both cases, as it likes to do with Vader especially whenever it can.
Just in case it wasn't clear, the assumptions made in these kinds of trials are very rigid and don't always reflect what's right to do in a given game state, so they are limited by their nature.
Still, it gives some kind of basis for comparison.
Enhanced Armament experiment
5 total red dice where all accuracy and blank icons are rerolled do an average of 1.03 damage apiece (as we covered earlier), so the average comes out to 5.15 damage.
That was easy, and now it's time to get a little more complicated with the TRC comparison.
Turbolaser Reroutes experiment
4 total red dice where all accuracy and blank icons are rerolled do an average of 1.03 damage apiece, so the average pre-TRCs is 4.12 damage.
Because there is a 3/8 chance (2 blanks, 1 accuracy) of rerolling any given dice and then a 3/8 chance those rerolled dice are once again not what we wanted, the odds of an individual "bad" dice
showing up that can be easy fodder for TRCs is (3/8*3/8 =) 14.06% per die. If we toss that into our buddy the binomial distribution calculator, we find that we then have a 45.4% chance of that ideal
situation occurring on at least one of our 4 dice. We also need to weed out the chance that we reach the promised land of a perfect roll of all double-hits on 4 dice(which means the TRCs would do
nothing useful), which happens less than 0.02% of the time, so it can be disregarded.
So what that means is 45.4% of the time, TRCs will add 2 damage to this attack. The remaining 54.6% of the time, they will only add 1 by upgrading a hit or crit to a double-hit. We factor that in by
doing the following two equations:
• 0.454*2 = 0.91 (rounded up)
• 0.546*1 = 0.55 (rounded up)
And we find that 4.12 (our pre-TRCs damage) + 0.91 + 0.55 = 5.58.
Now is it worth it to use TRCs and tap out the Arquitens' evade token for +0.46 average damage more? That's an exercise I leave to you, but at least we now have a grounds for comparison in our
imaginary fleet.
By using similar methods you can determine what effect dice modification effects will have on the average damage of your ships compared to other upgrades, or perhaps as part of a suite of upgrades or
other reroll/modify effects present in your fleet.
Chasing specific results
The final portion of this article will cover the odds of producing a specific result while rolling, sometimes when factoring in rerolls. This is often useful for chasing after/fishing for a specific
icon: that can be accuracy icons for locking down specific defense tokens (often scatters on scatter-equipped units) or critical icons on specific-colored dice for triggering critical-dependent
Simply determining your chances on an initial roll are fairly simple by just plugging the information into the binomial distribution calculator. For example, a Gladiator-I's side arc (with no extra
help) has a 68.4% chance of generating 1+ hit+crit results to trigger a black crit ordnance upgrade. Once you factor in rerolls, though, you need to determine under what conditions you're rerolling.
If you assume that the Gladiator in question can reroll every dice that's not a hit+crit (with commander Vader let's say), then we're looking at each dice's percentage chance of getting us that sweet
sweet crit we're chasing is:
25% (initial chance) + [75% (chance of rerolling) * 25% (trying again for that 2/8 chance)] = 43.75% chance of getting a hit+crit per black die, which improves our Gladiator-I's chances to 89.9%.
Simply rerolling blanks in the above experiment gives us different chances of:
25% + [25% (rerolling only blanks) * 25%] = 31.25% chance of getting hit+crit per black die, which improves our Gladiator-I's chances to 77.7%.
The takeaway from our Gladiator number-crunching above is if you're using a black crit ordnance upgrade, you're generally better off going for the high-risk high-reward option of rerolling every
black dice you can if you don't roll a hit+crit initially. If that makes you uncomfortable, maybe you're better off either accepting that your black crit upgrade won't fire quite as often or using a
ordnance upgrade
You can also put together some of the things we've already covered to deduce a more complex average. Let's say that you have an
with Leading Shots that has a flotilla sitting in its front arc at close range, trying to just get in the way and jam up its flight path. We're not concerned with how much damage it's going to do
because the answer is going to be "enough." We're concerned with whether it can get at least 1 accuracy result to lock down that annoying scatter. Here's how we would go through everything.
The 3 red dice in the front each have a 1/8 chance of getting an accuracy icon and their chances of producing at least 1 as a group are 33%.
The 2 blue dice in the front each have a 2/8 chance of getting an accuracy icon and their chances of producing at least 1 as a group are 43.75%
Because we don't care about our blue dice unless the red dice fail, that means our initial roll has a 33% + [43.75% * 67% (the chance of the red dice failing)] = 62.31% (rounded) chance of getting at
least one accuracy against the offending flotilla.
Now let's factor in Leading Shots, which we assume will be used to reroll all red dice and the one remaining blue die if our initial volley fails to produce any accuracy results. The 3 red dice again
have a 33% chance of doing what we want. The 1 blue die has a 25% chance, but we only care about that if the red dice fail, so our second roll looks like 33%+ [25%*67%] = 49.75%
Now before we get too excited, we need to also remember that this "frantically reroll, looking for accuracy icons once more" contingency only happens if our original attempt fails. So it's also
dependent on the percentage chance of even being necessary. That makes our
final equation
work out to:
62.31% (initial chance) + [49.75% (chance on the reroll, if it's necessary) * 37.69% (chance the initial roll failed and the reroll is required)] = 81.06% chance.
So our example ISD has just a bit better than a 4 in 5 chance of blowing that flotilla to kingdom come. If that's not high enough for your liking, then build some better accuracy-generation tech into
your fleet/that ISD.
Final thoughts
Hopefully this article didn't bore too many of you to tears, but I find the number-crunching occasionally rather insightful when it comes to challenging ideas I have about what to include in my fleet
and what upgrades/support to use with it.
It's impossible to go through every possible use for probability in a game like Armada, but hopefully the examples provided are sufficient to help interested players start crunching some numbers for
8 comments:
1. Doing the Lord's work here ...
2. Wait one the table of reroll math it says Blue dice vs ships are average of .75 damage with or without reroll is that right?
1. I explained the assumptions immediately above the chart. It assumes that you're not rerolling accuracy results, which sometimes you are and sometimes you aren't depending on circumstances. If
you are rerolling accuracy results, it changes things, and that different results are given above the chart.
2. Oh sorry my bad
3. Just found this, and I love it. Thanks for this.
1. You're welcome!
4. Could you explain how you calculated this here exactly?
The 3 red dice in the front each have a 1/8 chance of getting an accuracy icon and their chances of producing at least 1 as a group are 33%.
The 2 blue dice in the front each have a 2/8 chance of getting an accuracy icon and their chances of producing at least 1 as a group are 43.75%
1. Binomial distribution calculator, is the short answer. You can do the math long-hand (and it's not that bad with fewer dice), but it's faster with the calculator. I can give you an example
with the blue dice, though:
You've got a 1/4 chance of the first blue die getting an accuracy.
There's a 3/4 chance that the first one didn't get an accuracy (if it rolled one already, then our 1+ accuracies have already occurred), at which point the second die has the same 1/4 chance.
We're looking for the odds of 1 or more accuracy results so the math comes out to:
1/4 + (3/4x1/4) = 4/16 + 3/16 = 7/16 = 43.75% | {"url":"https://cannotgetyourshipout.blogspot.com/2017/12/roll-chance-cube-armada-dice-and.html","timestamp":"2024-11-13T19:21:33Z","content_type":"text/html","content_length":"131890","record_id":"<urn:uuid:d6bff324-0440-426b-b297-d282cb1a42d3>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00539.warc.gz"} |
Optimize a ML Model for Fast Inference on Ethos-U microNPU
Deep Neural Networks (DNNs) are trained using 32-bit IEEE single-precision to represent the floating-point model weights and activation tensors. This compute budget is typically acceptable during
training by taking advantage of CPU and especially GPUs with large compute capabilities. However, these models are often required to run on systems with limited memory and compute capability such as
embedded devices. Therefore, running a DNN inference on resource-constrained embedded systems using 32-bit representation is not always practical due to the massive number of multiply accumulate
operations (MACs) required.
TensorFlow Lite is an open-source deep learning framework enabling on-device machine learning. It allows you to convert a pre-trained TensorFlow model into a TensorFlow Lite flat buffer file
(.tflite) which is optimized for speed and storage. During conversion, optimization techniques can be applied to accelerate an inference and reduce model size.
TensorFlow Model Optimization Toolkit provides optimization techniques such as quantization, pruning, and clustering that are compatible with TensorFlow Lite. Based on the optimization technique, the
complexity and the size of the model can be reduced which results in less memory usage, smaller storage size and download size.
Also, optimization is necessary for some hardware accelerators such as Arm Ethos-U microNPU as it performs calculation in 8-bit integer precision. So, to deploy any model on Arm Ethos-U microNPU, it
is required to first be optimized.
Ethos-U55 is a first generation microNPU designed to accelerate neural networks inference in a low-area with low-power consumption. Paired with a Cortex-M processor, Ethos-U microNPU deliver up to a
480x ML performance uplift compared to previous Cortex-M generations. Its configurability allows developers to target a wide range of AI applications with:
• Four different configurations: 32/64/128/256 MACs/cycle
• Maximum performance up to a 0.5 TOP/s in 16nm process
• Fours possible host processors: Cortex-M55, Cortex-M7, Cortex-M33 and Cortex-M4
Quantization reduces the precision of model's parameter values (that being, weights). By default, are 32-bit floating point numbers. This results in a smaller model size and faster computation, often
with minimal or no loss in accuracy. However, depending on a model architecture and a quantization method, the impact on the accuracy may be significant. Therefore, the trade-off between model
accuracy and size should be considered during the application development process.
Quantization can take place during model training or after model training. Based on that, quantization is classified into two principal techniques:
• Post-training: quantize the weights after training the model
• Quantization-aware training (QAT): quantize the weights during training
Post-training integer quantization
You can quantize a trained 32-bit float TensorFlow model during conversion into a TensorFlow Lite model using post-training integer quantization techniques. Post-training integer quantization not
only increases inferencing speed on microcontrollers but also is compatible with fixed-point hardware accelerators such as Arm Ethos-U microNPUs. It converts models’ parameters from 32-bit floating
points to nearest 8-bit fixed-point numbers while getting reasonable quantized model accuracy with 3-4x reduction in model size.
There are two modes of post-training integer quantization:
• Post-training integer quantization with int8 activation and weights
• Post-training integer quantization with int16 activation and int8 weights (16x8 quantization mode)
Quantizing using integer-only converts weights, variables, input, and output tensors to integers. However, int16 activations could result in better accuracy at expense of slower compute times, while
maintaining nearly the same model size as int8. Some examples of models that benefit from 16x8 quantization mode of the post-training quantization include:
• Super-resolution,
• Audio signal processing such as noise cancelling
• Image de-noising
• HDR reconstruction from a single image
• Speech recognition and NLP models
How does full integer quantization work?
For full integer quantization, first the weights of the model are quantized to 8bit integer values. Then the variable tensors such as layer activations are quantized. To calculate the potential range
of values that all these tensors can take, a small subset of training or validation data is required. This representative dataset can be made using the following representative_data_gen() generator
def representative_data_gen():
for input_value in tf.data.Dataset.from_tensor_slices(train_images).batch(1).take(100):
yield [input_value]
Model inference is then performed using this representative dataset to calculating minimum and maximum values for variable tensors.
Integer with float fallback: To convert float32 activations and model weights into int8 and use float operators for those that have not an integer implementation, use the following snipped code:
converter = tf.lite.TFLiteConverter.from_saved_model(saved_model_dir)
converter.optimizations =[tf.lite.Optimize.DEFAULT]
converter.representative_dataset = representative_dataset
tflite_quant_model = converter.convert()
Alternatively, to quantize the model to 16x8 quantization mode:
Setting the optimizations flag to use default optimization and then specify 16x8 quantization mode in the target specification as follow.
converter.optimizations = [tf.lite.Optimize.DEFAULT]
converter.target_spec.supported_ops = [tf.lite.OpsSet.EXPERIMENTAL_TFLITE_BUILTINS_ACTIVATIONS_INT16_WEIGHTS_INT8]
converter.representative_dataset = representative_dataset
tflite_quant_model = converter.convert()
As a result, optimized operators and unoptimized operators are replaced with supported quantized operators and unsupported operators respectively.
Integer only: Enforcing full integer quantization for all operations including input and output and return error for those operations that cannot quantize:
converter = tf.lite.TFLiteConverter.from_keras_model(model)
converter.optimizations = [tf.lite.Optimize.DEFAULT]
converter.representative_dataset = representative_data_gen
# Ensure that if any ops can't be quantized, the converter throws an error
converter.target_spec.supported_ops = [tf.lite.OpsSet.TFLITE_BUILTINS_INT8]
# Set the input and output tensors to int8 (APIs added in r2.3)
converter.inference_input_type = tf.int8
converter.inference_output_type = tf.int8
tflite_model_quant = converter.convert()
With these post-training optimization methods, it is important to ensure that our model design is as efficient and optimized as possible while maintaining zero or minimum accuracy loss.
You will have a fully quantized TensorFlow Lite model after post-training integer quantization which is compatible with integer-based devices such as Ethos-U. However, to deploy your model on a
system using an Ethos-U NPU, the quantized TensorFlow Lite file should be compiled with Vela for further optimization. #Compile NN #Arm Ethos NPU #optimize
You can learn how to run your model through the Vela optimizer on the Vela page.
Quantization-aware training (QAT)
The accuracy of the model can drop as we move to lower precision (for example, 8-bit) from 32-bit float using post-training quantization. However, for minimum or even zero accuracy loss in critical
applications such as security systems, quantization-aware training technique may be required.
Quantization-aware training simulates inference-time quantization errors during training, so the model learns robust parameters around that loss. Quantized error is the error associated with
quantizing of the weights and activations to lower precision and then converting back to 32-bit float. Note that quantization is only simulated in the forward pass to induce the quantization error
while the backward pass remains the same and only floating-point weights are updated.
Define a model and applying quantization aware training to the trained model:
quantize_model = tfmot.quantization.keras.quantize_model
# q_aware stands for for quantization aware.
q_aware_model = quantize_model(model)
# `quantize_model` requires a recompile.
You can train the model for an epoch with quantization aware training only on a subset of training data and evaluate the model against baseline:
train_input_subset = train_input[0:1000]
train_labels_subset = train_ labels[0:1000]
q_aware_model.fit(train_input_subset, train_labels_subset,
batch_size=500, epochs=1, validation_split=0.1)
_, baseline_model_accuracy = model.evaluate(
test_images, test_labels, verbose=0)
_, q_aware_model_accuracy = q_aware_model.evaluate(
test_images, test_labels, verbose=0)
print('Baseline test accuracy:', baseline_model_accuracy)
print('Quant test accuracy:', q_aware_model_accuracy
Finally, create fully quantized model with int8 weights and int8 activations for TFLite:
converter = tf.lite.TFLiteConverter.from_keras_model(q_aware_model)
converter.optimizations = [tf.lite.Optimize.DEFAULT]
quantized_tflite_model = converter.convert()
You will have a fully quantized TensorFlow Lite model after quantization aware training compatible with integer only devices such as Ethos-U. However, to deploy your model on a system using an
Ethos-U NPU, the quantized TensorFlow Lite file should be compiled with Vela for further optimization.
Which quantization method to choose for your ML model?
Choosing a quantized method for your application is a trade-off between model performance and size. For example, with 16x8 quantization technique, you sacrifice speed and size while getting better
performance compared to full integer quantization. But if you only want to benefit from speed, full int8 quantization is the best.
Also depending on your ML application and availability of training data, you can choose a quantized method for your ML model. For example, a critical application like security systems that zero or
minimum accuracy loss is required, quantization-aware training is beneficial.
Weight pruning
After training deep learning models, it is common to see that the model is over-parameterized with parameters that have values of zero or close to zero. Setting a certain percentage of those values
to zero during training and using a subset of trained parameters generates sparsity in the model. The sparse model preserves the high-dimensional features of the original network after pruning those
Training the model parameters with network pruning technique helps achieve high compression rates with minimal accuracy loss and enables execution of the model on embedded devices with only a few
kilobytes of memory. Also, model sparsity can further accelerate inference within Arm Ethos-U NPU.
Note that similar to quantization there is a trade-off between the size of the model and the accuracy of the optimized model.
Define a model to fine-tune pre-trained model with pruning starting with 50% sparsity and ending with 80% sparsity.
import tensorflow_model_optimization as tfmot
prune_low_magnitude = tfmot.sparsity.keras.prune_low_magnitude
# Compute end step to finish pruning after 2 epochs.
batch_size = 128
epochs = 2
validation_split = 0.1 # 10% of training set will be used for validation set.
num_images = train_images.shape[0] * (1 - validation_split)
end_step = np.ceil(num_images / batch_size).astype(np.int32) * epochs
# Define model for pruning.
pruning_params = {
'pruning_schedule': tfmot.sparsity.keras.PolynomialDecay(initial_sparsity=0.50,final_sparsity=0.80,begin_step=0, end_step=end_step)
model_for_pruning = prune_low_magnitude(model, **pruning_params)
# `prune_low_magnitude` requires a recompile.
from_logits=True), metrics=['accuracy'])
Training and evaluating the model against baseline
logdir = tempfile.mkdtemp()
callbacks = [
model_for_pruning.fit(train_images, train_labels,
batch_size=batch_size, epochs=epochs, validation_split=validation_split,
_, model_for_pruning_accuracy = model_for_pruning.evaluate(
test_images, test_labels, verbose=0)
print('Baseline test accuracy:', baseline_model_accuracy)
print('Pruned test accuracy:', model_for_pruning_accuracy)
Weight pruning can further combine with quantization to improve memory footprint from both techniques and speed inference. Quantization then allows pruned model to be used with Ethos-U Machine
Learning processors.
converter = tf.lite.TFLiteConverter.from_keras_model(model_for_export)
converter.optimizations = [tf.lite.Optimize.DEFAULT]
quantized_and_pruned_tflite_model = converter.convert()
_, quantized_and_pruned_tflite_file = tempfile.mkstemp('.tflite')
with open(quantized_and_pruned_tflite_file, 'wb') as f:
To deploy your model on a system using an Ethos-U NPU, the quantized TensorFlow Lite file should be compiled with Vela for further optimization.
Weight clustering
Another technique of model optimization is a weight clustering which proposed and contributed by Arm ML team to TensorFlow Model Optimization Toolkit. Clustering reduces the storage and the size of
the model leading to benefits for deployments on resource-constrain embedded systems. With this technique, first a fixed number of cluster centers for each layer is defined. Next, the weights of each
layer are grouped into N clusters which later be replaced by their closest center.
Therefore, the size of the model will be reduced by replacing similar weights in each layer with the same value. These values are found by running a clustering algorithm over the weights of a trained
model. Depending on the model and number of chosen clusters, the accuracy of the model could drop after clustering. To reduce the impact on accuracy, you must pass a pre-trained model with acceptable
accuracy before clustering.
Before passing the model to the clustering API, it needs to be fully trained with acceptable accuracy.
import tensorflow_model_optimization as tfmot
cluster_weights = tfmot.clustering.keras.cluster_weights
CentroidInitialization = tfmot.clustering.keras.CentroidInitialization
clustering_params = {
'number_of_clusters': 3,
'cluster_centroids_init': CentroidInitialization.LINEAR
# Cluster a whole model
clustered_model = cluster_weights(model, **clustering_params)
# Use smaller learning rate for fine-tuning clustered model
opt = tf.keras.optimizers.Adam(learning_rate=1e-5)
Cluster sequential and functional models
Tips to get better model accuracy after clustering:
• Cluster layers with more redundant parameters such as tf.keras.layers.Dense , tf.keras.layers.Conv2D in contrast with early layers.
# Create a base model
base_model = setup_model()
clustered_model = tf.keras.Sequential([
bias_initializer=pretrained_bias), **clustering_params), Dense(...)])
• Avoid clustering certain layers such as attention mechanism which influences model accuracy and skip clustering those layers
Weight clustering also can further combine with quantization to improve memory footprint from both techniques and speed inference. Quantization then allows clustered model to be used with Ethos-U
machine learning processors.
converter = tf.lite.TFLiteConverter.from_keras_model(final_model)
converter.optimizations = [tf.lite.Optimize.DEFAULT]
converter.representative_dataset = representative_dataset
tflite_quant_model = converter.convert()
quantized_and_clustered_tflite_file = 'quantized_clustered.tflite'
with open(quantized_and_clustered_tflite_file, 'wb') as f:
However, to deploy your model on a system using an Ethos-U NPU, the quantized TensorFlow Lite file should be compiled with Vela for further optimization.
You can download the complete code sample of weight clustering combining with quantization technique from here: https://github.com/ARM-software/ML-examples/tree/master/ethos-u-microspeech
Collaborative optimization
Collaborative optimization is a process of stacking different optimization techniques to improve inference speed on special hardware accelerators such as Ethos-U microNPUs.
This technique keeps the balance between compression and accuracy for deployment by taking advantages of accumulated optimization effect. Various combinations of the quantization techniques for
deployment are possible such as:
• Weight pruning
• Weight clustering
• Quantization (post-training quantization and quantization aware training)
Therefore, you can apply one or both of pruning and clustering following by post-training or QAT. However, combining these techniques does not preserve the results of the preceding technique. This
leads to losing the overall benefits of simultaneously applying them.
For example, the sparsity of the pruned model will not preserve after deploying clustering. To address this problem the following collaborative optimization techniques can be used.
• Sparsity preserving clustering
□ clustering API that ensures a zero-cluster, preserving the sparsity of the model
• Sparsity preserving quantization aware training (PQAT)
□ QAT training that preserves the sparsity of the model
• Cluster preserving quantization aware training (CQAT)
□ QAT training API that does re-clustering and preserves the same number of centroids.
• Sparsity and cluster preserve quantization aware training (PCQAT)
□ QAT training API that preserves the sparsity and number of clusters of a model trained with sparsity-preserving clustering.
Sparsity preserving clustering example
To apply sparsity preserving clustering, first you need to prune the model using pruning API. Next, chain the model with clustering using the sparsity-preserving API. Finally, quantize the model with
post-training quantization for deployment on Ethos-U microNPU.
Prune and fine-tune the model to 50% sparsity:
prune_low_magnitude = tfmot.sparsity.keras.prune_low_magnitude
pruning_params = {
'pruning_schedule': tfmot.sparsity.keras.ConstantSparsity(0.5, begin_step=0, frequency=100)
callbacks = [tfmot.sparsity.keras.UpdatePruningStep()]
pruned_model = prune_low_magnitude(model, **pruning_params)
# Use smaller learning rate for fine-tuning
opt = tf.keras.optimizers.Adam(learning_rate=1e-5)
optimizer=opt, metrics=['accuracy'])
# Fine-tune model
To check the model kernels was correctly pruned, we need to strip the pruning wrapper first.
stripped_pruned_model = tfmot.sparsity.keras.strip_pruning(pruned_model)
Apply sparsity preserving clustering:
# Sparsity preserving clustering
from tensorflow_model_optimization.python.core.clustering.keras.experimental import (
cluster_weights = cluster.cluster_weights
clustering_params = {
'number_of_clusters': 8,
'cluster_centroids_init': CentroidInitialization.KMEANS_PLUS_PLUS,
'preserve_sparsity': True
sparsity_clustered_model = cluster_weights(stripped_pruned_model, **clustering_params)
# Train sparsity preserving clustering model
sparsity_clustered_model.fit(train_data, train_labels,epochs=3, validation_split=0.1)
Create a TFLite model from combining sparsity preserving weight clustering and post-training quantization:
stripped_sparsity_clustered_model = tfmot.clustering.keras.strip_clustering(sparsity_clustered_model)
converter = tf.lite.TFLiteConverter.from_keras_model(stripped_sparsity_clustered_model)
converter.optimizations = [tf.lite.Optimize.DEFAULT]
sparsity_clustered_quant_model = converter.convert()
_, pruned_and_clustered_tflite_file = tempfile.mkstemp('.tflite')
with open(pruned_and_clustered_tflite_file, 'wb') as f:
The previous example demonstrates the process of training to get an optimized sparsity preserving model. For the other techniques, please refer to the CQAT, PQAT, and PCQAT. #tensorflow, #
tensorflowlite, #collaborative optimization.
Compression and accuracy results
The following table shows the results of running experiments on DS-CNN-L and Mobilenet-V2, demonstrating the compression benefits vs. accuracy loss incurred. It also summarizes the number of microNPU
cycles that were used for computations using Ethos-U55 NPU accelerator configured with 128 MAC units.
• DS-CNN-L baseline accuracy (non-optimized model, fp32): 95.03%
• Mobilenet-V2 baseline accuracy (non-optimized model, fp32): 72.36%
Model Technique Top-1 Accuracy % NPU Active Cycles Size of Compressed .tflite (Mb)
8-bit post-training quantization 94.73 639,856 0.43
8-bit quantization with re-training 94.73 649,773 0.41
DS-CNN-L Clustering + 8-bit quantization with re-training 94.31 645,373 0.35
Pruning + clustering + 8-bit quantization with re-training 94.67 582,973 0.32
8-bit post-training quantization 70.58 7,377,159 3.41
MobileNet-V2 8-bit quantization with re-training 72.27 7,284,184 3.40
Clustering + 8-bit quantization with re-training 71.01 7,070,184 2.53
Pruning + clustering + 8-bit quantization with re-training 71.59 6,591,097 2.55 | {"url":"https://community.arm.com/arm-community-blogs/b/ai-and-ml-blog/posts/optimize-a-ml-model-for-inference-on-ethos-u-micronpu","timestamp":"2024-11-08T12:29:41Z","content_type":"text/html","content_length":"145422","record_id":"<urn:uuid:cadb7516-96a9-4771-add4-3b6617e58939>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00754.warc.gz"} |
Chemistry and chemical thermodynamics
Clayton First semester 2008 (Day)
Sunway First semester 2008 (Day)
This unit covers thermodynamics from a chemical engineering viewpoint. Content will cover basic concepts and the use of: thermodynamic functions such as free energy, enthalpy, and entropy; estimation
of properties of pure compounds and mixtures; description of solution thermodynamics and its applications, equilibrium phase diagrams and chemical reaction equilibria.
On successful completion of this course students should:
1. be able to apply mass, energy and entropy balances to flow processes
2. be able to calculate the properties of ideal and real mixtures based on thermodynamic principles
3. be able to determine changes in the properties of gases, fluids and solids undergoing changes in temperature and volume
4. be able to explain the underlying principles of phase equilibrium in binary and multi-component systems
5. understand the concepts involved in describing the extent to which chemical reactions proceed, and the determination of composition attained at equilibrium.
Individual tests: 15%
Laboratory: 15%
Examination (3 hours): 70%
Students are required to achieve at least 45% in the total continuous assessment component (assignments, tests, mid-semester exams, laboratory reports) and at least 45% in the final examination
component. Students faiing to achieve this requirement will be given a maximum of 44% in the unit.
Contact hours
3 hours lectures, 1 hour tutorial classes, 2 hours laboratory classes and 6 hours of private study per week | {"url":"https://www3.monash.edu/pubs/2008handbooks/units/CHE3161-pr.html","timestamp":"2024-11-03T18:50:00Z","content_type":"text/html","content_length":"5999","record_id":"<urn:uuid:f5773591-4120-466b-b584-1083fb3cf711>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00255.warc.gz"} |
Air pressure in bubbles
One could ask: Why does the bubble not grow over time, if the pressure inside is larger than the pressure outside? We have to take another pressure into account: The pressure from the soap film
itself, towards the centre of the bubble. As a result of surface tension the soap film will minimize its surface area by making the bubble as small as possible. Hence the bubble does not grow because
there is a balance between the pressure inside the bubble and the pressure from the soap film plus the air pressure from the outside.
The bigger the bubble, the lower the pressure! The pressure in an infinitely small bubble will, in principle, be infinitely large. Therefore there are limit to how small a bubble that can be
produced. The pressure difference between two bubbles becomes apparent, if they are produced at a flat surface touching each other, so that they have a shared soap film. The pressure difference in
two equally sized bubbles must be zero, and the shared soap film will be flat. If the bubbles have different sizes, and therefore different air pressures, this will be seen as the shared soap film
will bulge out from the one with the highest pressure. If you make the experiment, you will actually see that the small bubble is bulging out into the big one. This illustrates that the pressure in
small bubbles is bigger than in larger bubbles.
Two bubbles of the same size has the same air pressure and therefore there is a straight soap film between the two bubbles
A small bubbles has higher air pressure and therefore the soap film bulges into the big bubble
En lille boble har et større tryk og derfor buler sæbehinden imellem dem ind i den store boble
The air pressure of two equally sized bubbles is the same. Thiscan be seen from the flat shared soap film between the two equally sized bubbles If the bubbles have different sizes the shared soap
film will bulge from the small bubble into the larger. This shows us that the air pressure in small bubbles is higher than the air pressure in larger bubbles.
The air pressure, p inside a soap bubble can be described mathematically as,
[latex] p = \frac{ 4 \sigma }{ r } [/latex]
where [latex] \sigma [/latex] is the surface tension of the liquid and r is the bubble radius. The equation is based on the famous Laplace-Young equation which was discovered in 1806 by Marquis de
[latex] \Delta p = \sigma ( \frac{ 1 }{ R1 } + \frac{ 1 }{ R2 } ) [/latex]
The Laplace-Young equation describes the pressure difference across a surface which acts as a border for liquids or gasses. Every point on such a surface has two characteristic curvatures: a maximal
and a minimal curvature. These will always be normal (perpendicular) to each other. In the equation the curvature is expressed as a radius (R1 or R2) of the circle that can be fitted to the surface.
The excess pressure inside a soap bubble is related to the size of the bubble, as can be seen from the graph.
The pressure is higher in smaller bubbles than in larger ones.
The maximal and minimal curvature in a point at the surface of an egg. The curvatures will always be normal (perpendicular) to each other
A soap film with no air trapped inside it is a so-called minimal surface. The air pressure on each side of the film will be the same, and from the Laplace-Young equation we can see that the total
curvature in a given point is zero:
[latex] 0 = \Delta p = \sigma ( \frac{ 1 }{ R1 } + \frac{ 1 }{ R2 } ) = \frac{ 1 }{ R1 } + \frac{ 1 }{ R2 }[/latex]
This means that for every point on a minimal surface, the maximal and minimal curvature will be equal in size and in opposite directions. This is of course true when the curvature in both directions
is zero, as it is for a flat surface. It is more surprising that it is true for more complex minimal surfaces such as a catenoid. | {"url":"https://www.soapbubble.dk/en/articles/lufttryk","timestamp":"2024-11-13T05:00:56Z","content_type":"text/html","content_length":"10451","record_id":"<urn:uuid:7a849963-7ffb-482d-9f6a-5850429857f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00275.warc.gz"} |
Quick experiment, boosting vs annoy
While following through the Statistical Learning course, I came across this part on doing regression with boosting. Then reading through the material, and going through it makes me wonder, the same
method may be adapted to Erik Bernhardsson‘s annoy algorithm.
Then I started to prototype again, but unlike my previous attempts (here and here), I am not interested in finding the nearest neighbours, so I did not reuse the code there. Instead I built a very
rough proof-of-concept code with purely numpy code.
from functools import reduce
def distance(points, plane_normal, d):
return np.divide(np.add(d, np.sum(np.multiply(points, plane_normal), axis=1).reshape(-1,1)),
np.sqrt(np.sum(np.power(plane_normal, 2))))
def split_points(x):
idx = np.random.randint(x.shape[0], size=2)
points = x[idx,]
plane_normal = np.subtract(*points)
return split_points(x) if np.all(plane_normal == 0) else (points, plane_normal)
def tree_build(penalty, x, y):
points, plane_normal = split_points(x)
plane_point = np.divide(np.add(*points), 2)
d = 0 - np.sum(np.multiply(plane_normal, plane_point))
dist = distance(x, plane_normal, d)
return {'plane_normal': plane_normal,
'd': d,
'points': (y[dist &amp;gt;= 0].size, y[dist &amp;lt; 0].size),
'mean': (penalty * np.mean(y[dist &amp;gt;= 0]),
penalty * np.mean(y[dist &amp;lt; 0]))}
def boost_annoy(B, penalty, x, y):
return [tree_build(penalty, x, y) for b in range(B)]
def boost_annoy_transform(forest, x):
result = np.zeros((x.shape[0], 1))
for tree in forest:
dist = distance(x, tree['plane_normal'], tree['d'])
result = np.add(result, np.where(dist &amp;gt;= 0, tree['mean'][0], tree['mean'][1]))
return result
So the tree is built to somehow mirror the behavior of a boosting tree building algorithm, and hence it is somewhat limited by a penalty tuning parameter. Another parameter that affects the prediction is also taken from the boosting model, which is the number of trees, B. Also because this was meant to be a quick prototype, so I only branch out once.
Supplying these values to the model for training is shown as follows
x = np.random.normal(size=(1000, 10))
y = np.random.normal(size=(1000, 1))
forest = boost_annoy(B, 0.0001, x, y)
y_hat = boost_annoy_transform(forest, x)
While I don’t have real world data to test, I tested by generating a bunch of random observations and responses, then use scikit-learn’s KFold utility function to run through a bunch of test with
different values of B for observations in 10-dimensional space.
And the testing notebook is here.
As seen in the graph, while the training error of the algorithm does not go down as significantly as boosting’s training error, the test error on the other hand is generally lower than boosting’s
test error. This is a rather interesting result for the first proof-of-concept type of prototype.
Of course there are a couple of things I need to fix, for instance proper branching (i.e. more than 1 level), some performance issues etc. However I am quite happy with the current result, hopefully
I get to put in more time in making this into something proper.
Ah well, I screwed up a little bit on the sample generation lol, so in real life data it is a lot of work to be done, will see if I have time to do a proper follow up to this post.
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Search for contact interactions in μ+μ- events in pp collisions at √s=7 TeV
Search for contact interactions in
events in pp collisions at ffiffiffi p s
¼ 7 TeV
S. Chatrchyan et al.*
(CMS Collaboration)
(Received 18 December 2012; published 1 February 2013)
Results are reported from a search for the effects of contact interactions using events with a high-mass, oppositely charged muon pair. The events are collected in proton-proton collisions at ffiffiffi
¼ 7 TeV using the Compact Muon Solenoid detector at the Large Hadron Collider. The data sample corresponds to an integrated luminosity of5:3 fb^1. The observed dimuon mass spectrum is consistent with
that expected from the standard model. The data are interpreted in the context of a quark- and muon-compositeness model with a left-handed isoscalar current and an energy scale parameter. The 95%
confidence level lower limit on is 9.5 TeV under the assumption of destructive interference between the standard model and contact-interaction amplitudes. For constructive interference, the limit is
13.1 TeV. These limits are comparable to the most stringent ones reported to date.
DOI:10.1103/PhysRevD.87.032001 PACS numbers: 12.60.Rc, 13.85.Qk
The existence of three families of quarks and leptons might be explained if these particles are composed of more fundamental constituents. In order to confine the constitu- ents (often referred to as
‘‘preons’’ [1,2]) and to account for the properties of quarks and leptons, a new strong gauge interaction, metacolor, is introduced. Below a given inter- action energy scale, the effect of the
metacolor interac- tion is to bind the preons into metacolor-singlet states. For parton-parton center-of-mass energy less than, the meta- color force will manifest itself in the form of a flavor-
diagonal contact interaction (CI) [3,4]. In the case where both quarks and leptons share common constituents, the Lagrangian density for a CI leading to dimuon final states can be written as
Lql¼ ðg^2[0]=^2Þf[LL]ðqL^q[L]Þð L[][L]Þ þ [LR]ðqL^q[L]Þð R[][R]Þ þ [RL]ðuR^u[R]Þð L[][L]Þ þ [RL]ð d[R]^d[R]Þð L[][L]Þ þ [RR]ðuR^u[R]Þð R[][R]Þ
þ [RR]ð d[R]^d[R]Þð R[][R]Þg; (1) where q[L]¼ ðu; dÞ[L]is a left-handed quark doublet, u[R]and d[R]are right-handed quark singlets, and [L]and [R]are left- and right-handed muons. By convention, g^2
[0]=4 ¼ 1. The parameter characterizes the compositeness energy scale.
The parameters [ij]allow for differences in magnitude and phase among the individual terms. Lower limits on are
set separately for each term with [ij]taken, by convention, to have a magnitude of 1.
The dimuons from the subprocesses for standard model (SM) Drell-Yan (DY) [5] production and from CI produc- tion can have the same helicity state. In this case, the scattering amplitudes are summed,
resulting in an interfer- ence term in the cross section for pp ! X þ ^þ^, as illustrated schematically in Fig. 1.
The differential cross section corresponding to the com- bination of a single term in Eq. (1) with DY production can be written as
dM[] ¼ d^DY
dM[] [ij] I
^2þ ^2[ij] C
^4; (2) where M[] is the invariant dimuon mass, I is due to interference, andC is purely due to the CI. Note that ij¼ þ1 corresponds to destructive interference and ij ¼ 1 to constructive
interference. The processes contributing to the cross section in Eq. (2) are denoted collectively by
‘‘CI/DY.’’ The difference d^CI=DY=dM[] d^DY=dM[]
is the signal we are searching for in this paper.
The contact-interaction model used for this analysis is the left-left isoscalar model (LLIM) [4], which corresponds to a left-handed current interaction described by the first
FIG. 1. Schematic representation of the addition of DY (left diagram) and CI (right diagram) amplitudes, for common helicity states, contributing to the total cross section for pp ! X þ ^þ^.
*Full author list given at the end of the article.
Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- bution of this work must maintain attribution to the author(s) and the
published article’s title, journal citation, and DOI.
term of Lql in Eq. (1). The LLIM is the conventional benchmark for CI in the dilepton channel. For this analysis, all initial-state quarks are assumed to be composite.
Previous searches for CI in the dijet and dilepton chan- nels have all resulted in limits on the compositeness scale
. Searches have been reported from experiments at LEP [6–10], HERA [11,12], the Tevatron [13–18], and recently from the ATLAS [19–22] and CMS [23–25] experiments at the LHC. The best limits in the
LLIM dimuon channel are > 9:6 TeV for destructive interference and >
12:9 TeV for constructive interference, at the 95% confi- dence level (CL) [22].
In this paper, we report a search for CI in the dilepton channel produced in pp collisions at ffiffiffi
¼ 7 TeV using the Compact Muon Solenoid (CMS) detector at the Large Hadron Collider (LHC). The data sample corresponds to an integrated luminosity of5:3 fb^1.
The basic features of the LLIM dimuon mass spectra are demonstrated with a generator-level simulation using PYTHIA [26], with appropriate kinematic selection criteria that approximate the acceptance
of the detector.
Figures2(a)and2(b)show the LLIM dimuon mass spectra for different values of for destructive and constructive interference, respectively. The curves illustrate that with increasing mass the CI leads
to a less steeply falling yield relative to DY production, with the effect steadily increas- ing with decreasing. For a given value of , the event yield is seen to be larger for constructive
interference compared to the destructive case, with the relative differ- ence increasing with.
For the results presented in this paper, the analysis is limited to a dimuon mass range from 200 to2000 GeV=c^2. The lower mass is sufficiently above the Z peak so that a deviation from DY production
would be observable. The highest dimuon mass observed is between 1300 and 1400 GeV=c^2 and, for the values of where the limits are set, less than one event is expected for dimuon masses above 2000
GeV=c^2. In order to limit the mass range in which the detector acceptance has to be evaluated, we therefore choose an upper mass cutoff of 2000 GeV=c^2. To optimize the limit on, a minimum mass M^
minis varied between the lower and upper mass values, as described in Sec.VI.
The central feature of the CMS apparatus is a super- conducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the field volume are a silicon pixel and strip tracker,
a lead tungstate crystal electromagnetic calorimeter, and a brass-scintillator had- ron calorimeter. Muons are measured in gas-ionization
detectors embedded in the steel flux-return yoke.
Extensive forward calorimetry complements the coverage provided by the barrel and endcap detectors. A detailed description of the CMS detector can be found in Ref. [27].
The tracker and muon detector are important subsystems for this measurement. The tracker measures charged par- ticle trajectories within the range jj <2:5, where pseu- dorapidity ¼ ln½tanð=2Þ, and
polar angle is measured from the beam axis. The tracker provides a transverse momentum (p[T]) resolution of about 1% at a few tens ofGeV=c to 10% at several hundred GeV=c [28], where p[T] is the
component of momentum in the plane
2) (GeV/c
500 1000 1500 2000 2500 3000 )2Events/(30 GeV/c
= 3 TeV Λ
= 5 TeV Λ
= 7 TeV Λ
= 9 TeV Λ
= 13 TeV Λ DY destructive interference left-left isoscalar model
PYTHIA simulation
2) (GeV/c
500 1000 1500 2000 2500 3000 )2Events/(30 GeV/c
105 [Λ][= 3 TeV]
= 5 TeV Λ
= 7 TeV Λ
= 9 TeV Λ
= 13 TeV Λ DY constructive interference
left-left isoscalar model
PYTHIA simulation
FIG. 2 (color online). Simulated dimuon mass spectra using the left-left isoscalar model for different values of for (a) destructive interference and (b) constructive interference.
The events are generated using thePYTHIAMonte Carlo program with kinematic selection requirements that approximate the acceptance of the detector. As increases, the effects of the CI are reduced, and
the event yield approaches that for DY production. The model predictions are shown over the full mass range, although the model is not valid as M[]c^2approaches.
The integrated luminosity corresponds to63 fb^1.
S. CHATRCHYAN et al. PHYSICAL REVIEW D 87, 032001 (2013)
perpendicular to the beam axis. Tracker elements include about 1400 silicon pixel modules, located close to the beamline, and about 15 000 silicon microstrip modules, which surround the pixel system.
Tracker detectors are arranged in both barrel and endcap geometries. The muon detector comprises a combination of drift tubes and resistive plate chambers in the barrel region and a combination of
cathode strip chambers and resistive plate chambers in the endcap regions. Muons can be recon- structed in the range jj <2:4.
For the trigger path used in this analysis, the first level (L1) selects events with a muon candidate based on a subset of information from the muon detector. The trigger muon is required to have jj
<2:1 and pTabove a thresh- old that was raised to40 GeV=c by the end of the data- taking period. This cut has little effect on the acceptance for muon pairs with masses above200 GeV=c^2. The small
effect is included in the simulation. The high level trigger (HLT) refines the L1 selection using the full information from both the tracker and muon systems.
This analysis uses the same event selection as the search for new heavy resonances in the dimuon channel, discussed in Ref. [29]. Each muon track is required to have a signal (‘‘hit’’) in at least
one pixel layer, hits in at least nine strip layers, and hits in at least two muon detector stations.
Both muons are required to have p[T]>45 GeV=c. To reduce the cosmic ray background, the transverse impact parameter of the muon with respect to the beamspot is required to be less than 0.2 cm. In
order to suppress muons coming from hadronic decays, a tracker-based isolation requirement is imposed such that the sum of p[T] of all tracks, excluding the muon and within a cone surrounding the
muon, is less than 10% of the p[T]of the muon. The cone is defined by the conditionR ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðÞ^2þ ðÞ^2
p ¼ 0:3,
where is the azimuthal angle of a track, and the differ- ences and are determined with respect to the muon’s direction.
The two muons are required to have opposite charge and to be consistent with originating from a common vertex. To suppress cosmic ray muons that are in time with the collision event, the angle
between the two muons must be smaller than 0:02 radians. At least one of the reconstructed muons must be matched (within R < 0:2 andp[T]=p[T]<1) to the HLT muon candidate.
If an event has more than two muons passing the above requirements, then the two highest-p[T]muons are selected, and the event is retained only if these muons are oppositely charged. Only three such
events are observed with selected dimuon mass above200 GeV=c^2, and in all three cases, the dimuon mass is less than300 GeV=c^2. Thus, events with multiple dimuon candidates play essentially no role
in the analysis.
This section describes the method used to simulate the mass distribution from the CI/DY process of Eq. (2), including the leading-order (LO) contributions from DY and CI amplitudes, their
interference, the effects of next-to- leading-order (NLO) QCD and QED corrections, and the response of the detector. The predicted number of CI/DY events is the product of the generated number of CI/
DY events, a QCD K-factor, a QED K-factor, and a factor denoted as ‘‘acceptance times migration’’ (A M). The factor A M is determined from the detector simulation of DY events, as explained below in
Sec.V B. The simulation of background due to non-DY SM processes is also described.
A. Event samples with detector simulation A summary of the event samples used for simulation of the detector response to various physics processes is pre- sented in Table I. The event generators used
are PYTHIA, with the CTEQ6.6M implementation [30] of parton distri- bution functions (PDF),POWHEG[31–33], andMADGRAPH5
[34]. The detector simulation is based onGEANT4[35].
B. Detector acceptance times mass migration To simplify the analysis, we use the detector simulation for DY events to determine the detector response for CI/DY events, which have a behavior similar
to that for DY events for the large values of of interest in this analysis. For a given value of M[]^min, the product of acceptance times migration (A M) is given by the ratio of the number of DY
events reconstructed with mass above M^min[] to the number of DY events generated with mass above M^min[]. Some of the reconstructed events have been generated with mass below M^min[]because of the
smearing due to the mass reconstruction, which has a resolution of 6.5% at masses around1000 GeV=c^2, rising to 12% at2000 GeV=c^2. The dependence of A M on M[]^min is plotted in Fig. 3 and values
are given in Table II. The increase of A M at lower mass is due to the increase in acceptance, while at higher mass, it is dominated by the growth in mass reso- lution. Since the cross section falls
steeply with mass, events tend to migrate from lower to higher mass over a range determined by the mass resolution.
To validate that the A M factor based on DY produc- tion is applicable to CI/DY production, we compare event yields predicted using the A M factor with those pre- dicted using a simulation of CI/DY
production. The study is performed for the cases of constructive interference with
¼ 5 and 10 TeV, which represent a wide range of possible CI/DY cross sections. The results differ by at most 3%, consistent with the statistical precision of the study. The systematic uncertainty in
A M is conserva- tively assigned this value.
. . .
1. Event pileup
During the course of the 2011 data-taking period, the luminosity increased with time, resulting in an increasing
‘‘event pileup,’’ the occurrence of multiple pp interactions recorded by the detector as a single event. The dependence of reconstruction efficiency on event pileup is studied by weighting simulated
events so that the distribution of the number of reconstructed primary vertices per event matches that in data. The reconstruction efficiency is found to be insensitive to the variations in event
pileup encoun- tered during the data-taking period.
C. Higher-order strong and electromagnetic corrections
Since we use the leading-order generator PYTHIA to simulate the CI/DY production, we must determine a QCD K-factor which takes into account higher-order initial-state diagrams. Under the assumption
that the QCD K-factor is the same for DY and CI/DY events, we determine the QCD K-factor as the ratio of DY events generated using the next-to-leading-order generator
MC@NLO [36] to those generated using PYTHIA. The
MC@NLO generator is used with the same PDF set as used withPYTHIA. The resulting QCD K-factor as a func- tion of M^min[] is given in Table II. The large sizes of the simulated event samples result in
statistical uncertainties of less than 0.5%. The systematic uncertainty is assigned the value 3%, the size of the correction [37] between next-to- next-to-leading-order (NNLO) and NLO DY cross sec-
tions. For SM processes other than DY production, the QCD K-factor is found, independent of dimuon mass, from the ratio of the cross section determined using
MC@NLOto the cross section determined fromPYTHIA. The effect of higher-order electromagnetic processes on CI/DY production is quantified by a mass-dependent QED K-factor determined using the HORACE
generator [38].
The values of the QED K-factor, as a function of M^min[], are given in TableII. The systematic uncertainty is assigned as the size of the correction, jðQED K-factorÞ 1j, since the effect of
higher-order QED corrections on the new physics of CI is unknown.
D. Non-DY SM backgrounds
Using the samples of simulated events listed in TableI, event yields are predicted for various non-DY SM back- ground processes, as shown in Table III. The yields are given as a function of M^min[],
and they are scaled to the integrated luminosity of the data, 5:28 0:12 fb^1 [39].
TABLE I. Description of event samples with detector simulation. The cross section and integrated luminosity L are given for each sample generated.
Process Generator Number of events ðpbÞ Lðpb^1Þ Order
Z=^! , M[] 120 GeV=c^2 ^PYTHIA 5:45 10^4 7:90 10^0 6:91 10^3 LO
Z=^! , M 200 GeV=c^2 ^PYTHIA 5:50 10^4 9:70 10^1 5:67 10^4 LO
Z=^! , M 500 GeV=c^2 ^PYTHIA 5:50 10^4 2:70 10^2 2:04 10^6 LO
Z=^! , M[] 800 GeV=c^2 ^PYTHIA 5:50 10^4 3:10 10^3 1:77 10^7 LO
Z=^! , M[] 1000 GeV=c^2 ^PYTHIA 5:50 10^4 9:70 10^4 5:67 10^7 LO
Z=^! PYTHIA 2:03 10^6 1:30 10^3 1:56 10^3 LO
tt MADGRAPH 2:40 10^6 1:57 10^2 1:54 10^5 NLO
tW ^POWHEG 7:95 10^5 7:90 10^0 1:01 10^5 NLO
tW POWHEG 8:02 10^5 7:90 10^0 1:02 10^5 NLO
WW PYTHIA 4:23 10^6 4:30 10^1 9:83 10^4 LO
WZ PYTHIA 4:27 10^6 1:80 10^1 2:37 10^5 LO
ZZ PYTHIA 4:19 10^6 5:90 10^0 7:10 10^5 LO
W þ jets MADGRAPH 2:43 10^7 3:10 10^4 7:82 10^2 NLO
multijet, (p[T]>15 GeV=c) ^PYTHIA 1:08 10^6 8:47 10^4 1:28 10^2 LO
2) (GeV/c
Acceptance x Migration
0.8 0.85 0.9 0.95
CMS simulation
FIG. 3 (color online). Acceptance times migration, A M, versus M^min[]. Corresponding values and uncertainties are given in TableII. The error bars indicate statistical uncertainties based on
simulation of the DY process. The systematic uncertainty is 3%, as explained in the text. The increase of A M at lower mass is due to the increase in acceptance, while at higher mass, it is dominated
by the growth in mass resolution. Since the cross section falls steeply with mass, events tend to migrate from lower to higher mass over a range determined by the mass resolution.
S. CHATRCHYAN et al. PHYSICAL REVIEW D 87, 032001 (2013)
For comparison, the expected yields are also shown for DY events. The relevant backgrounds, in decreasing order of importance, are tt, diboson ðWW=WZ=ZZÞ, W (including W þ jets and tW), and Z !
production. The back- ground from multijet events is studied using both the simulation sample listed in Table I and control samples from data, as reported in Ref. [29]. The results of either method
indicate that no multijet background events are expected for M^min[]>200 GeV=c^2. For M^min[]>
1000 GeV=c^2 the fractional statistical uncertainty in the non-DY background is large, but the absolute yield is much smaller than that for DY background.
E. Predicted event yields
Using the methods described above, the sum of the event yields for the CI/DY process and the non-DY SM back- grounds, for the integrated luminosity of the data sample, are predicted as a function of
M^min[] and. The predicted event yields for destructive and constructive interference are given in TablesIVandV.
For destructive interference, there is a region of the M[]^min parameter space where the predicted number of events is less than for SM production. This ‘‘reduced- yield’’ region is indicated in
Table IV. The region of parameter space, M[]^min>600 GeV=c^2 and 12 TeV, where our expected limit is most stringent [see Fig.5(a)], lies outside the reduced-yield region. For constructive
interference, the predicted number of events is always larger than for SM production.
VI. EXPECTED AND OBSERVED LOWER LIMITS ON A. Dimuon mass distribution from data
The observed numbers of events versus M^min[] are given in TableIV. The observed distribution of M[]is plotted in Fig.4along with the expected distributions from the SM and for CI/DY plus non-DY SM
processes, for three illus- trative values of . The data are consistent with the pre- dictions from the SM, dominated by DY production.
B. Limit-setting procedure
Since the data are consistent with the SM, we set lower limits on in the context of the LLIM. The expected and observed 95% CL lower limits on are determined using the CL[S] modified-frequentist
procedure described in [40,41], taking the profile likelihood ratio as a test statistic [42]. The expected mean number of events for a signal TABLE II. Multiplicative factors used in the prediction
of the
expected number of events from the CI/DY process. The un- certainties shown are statistical. The systematic uncertainty is 3% for A M and 3% for the QCD K-factor, as explained in the text. The
uncertainty in the QED K-factor is dominated by the systematic uncertainty that is assigned as the size of the correc- tion, jðQED K factorÞ 1j, to allow for systematic uncertainty in the generator.
M[]^min(GeV=c^2) A M QCD K-factor QED K-factor
200 0:80 0:01 1:303 0:005 1.01
300 0:82 0:01 1:308 0:005 0.99
400 0:83 0:01 1:299 0:005 0.97
500 0:86 0:02 1:305 0:005 0.95
600 0:86 0:01 1:299 0:005 0.94
700 0:87 0:01 1:298 0:005 0.92
800 0:88 0:01 1:288 0:005 0.91
900 0:89 0:01 1:280 0:004 0.90
1000 0:89 0:01 1:278 0:004 0.89
1100 0:89 0:01 1:275 0:004 0.88
1200 0:91 0:01 1:268 0:004 0.88
1300 0:92 0:01 1:262 0:004 0.87
1400 0:94 0:01 1:260 0:004 0.87
1500 0:97 0:01 1:261 0:004 0.86
TABLE III. Expected event yields for DY and non-DY SM backgrounds. The uncertainties shown are statistical. A systematic uncertainty of 2.2% arises from the determination of integrated luminosity
M[]^min(GeV=c^2) DY tt Diboson W þ Jets & tW Z ! Sum non-DY
200 3630 18 454 3 123:0 2 47:90 1:35 6:96 4:14 632:3 5:9
300 870:6 8:8 104 2 38:6 1:2 12:82 0:70 0 155:9 2:1
400 301:6 5:1 26:0 0:8 12:7 0:7 3:32 0:35 0 42:0 1:1
500 123:8 3:3 8:19 0:46 5:07 0:41 1:02 0:20 0 14:3 0:6
600 55:31 0:19 2:92 0:27 2:42 0:28 0:29 0:11 0 5:63 0:41
700 27:35 0:13 1:12 0:17 0:86 0:16 0:07 0:05 0 2:06 0:24
800 14:23 0:10 0:34 0:09 0:51 0:12 0:07 0:05 0 0:92 0:16
900 7:72 0:07 0:05 0:03 0:25 0:08 0:07 0:05 0 0:36 0:10
1000 4:32 0:05 0:05 0:03 0:10 0:05 0:07 0:05 0 0:21 0:08
1100 2:46 0:04 0:05 0:03 0:09 0:05 0:07 0:05 0 0:20 0:08
1200 1:48 0:03 0 0:01 0:01 0:07 0:05 0 0:08 0:05
1300 0:91 0:02 0 0:01 0:01 0:07 0:05 0 0:08 0:05
1400 0:56 0:02 0 0:01 0:01 0:07 0:05 0 0:08 0:05
1500 0:33 0:02 0 0 0:07 0:05 0 0:07 0:05
. . .
TABLE V. Observed and expected number of events as in TableIV. Here CI/DY predictions are for constructive interference. Shown with bold-italic font is the expected event yield corresponding to the
value of closest to the observed 95% CL lower limit on of 13.1 TeV (12.9 TeV expected) for M^min[] selected to be800 GeV=c^2.
M[]^min(GeV=c^2) 400 500 600 700 800 900 1000 1100 1200 1300 1400
Source Number of events
Data 338 141 57 28 14 13 8 3 2 1 0
SM MC (TeV) MC 343.6 138.1 60.9 29.4 15.2 8.1 4.5 2.7 1.6 1.0 0.6
18 359.2 147.7 67.8 34.1 18.7 10.6 6.4 4.0 2.5 1.6 1.1
17 358.9 149.3 69.1 35.1 19.3 11.1 6.7 4.3 2.7 1.8 1.2
16 365.2 153.7 70.3 36.1 20.2 11.7 7.2 4.6 3.0 2.0 1.3
15 365.6 156.3 71.9 37.2 20.9 12.3 7.6 4.9 3.1 2.1 1.4
14 368.5 154.9 74.6 39.1 22.4 13.3 8.4 5.5 3.6 2.4 1.7
13 377.8 164.4 77.9 41.7 24.2 14.7 9.4 6.3 4.2 2.9 2.0
12 379.2 170.5 82.5 45.2 26.9 16.7 11.0 7.4 5.0 3.5 2.4
11 388.9 174.6 88.6 49.9 30.4 19.3 12.9 8.8 6.1 4.2 3.0
10 406.0 184.5 97.9 57.1 36.0 23.7 16.2 11.3 7.9 5.6 4.0
9 440.3 214.8 113.2 68.8 44.8 30.3 21.2 15.0 10.7 7.7 5.5
8 470.0 237.1 138.2 87.7 59.6 41.6 29.9 21.8 15.7 11.4 8.1
7 563.9 307.3 181.0 120.4 86.7 62.1 44.8 31.5 23.3 16.9 12.3
6 696.8 415.0 269.2 187.3 136.9 101.7 75.3 57.4 41.8 30.7 23.2
5 1007 675.0 467.8 345.8 268.0 202.3 153.3 116.9 87.3 64.6 47.0
4 1839 1346 997.4 765.1 586.6 451.1 349.6 266.8 200.3 147.8 109.5
3 4800 3762 2861 2251 1754 1358 1041 791.0 597.1 453.6 338.5
TABLE IV. Observed and expected number of events for illustrative values of M^min. The expected yields are shown for SM production and for the sum of CI/DY production (for destructive interference
and for a given) and non-DY SM backgrounds. For each column of M^min, the expected yield forCI=DY þ non-DY SM production that is just above that expected for SM production is in bold font. Entries
above the bold ones correspond to values of for which the expected yield is less than that for SM production, because of the destructive interference term in the cross section. As discussed in Sec.VI
C, the best expected limit is obtained for M[]^min¼ 1100 GeV=c^2. For this choice, the expected event yield, in bold-italic font, corresponds to the value of closest to the observed 95% CL lower
limit on of 9.5 TeV (9.7 TeV expected).
M[]^min(GeV=c^2) 500 600 700 800 900 1000 1100 1200 1300 1400 1500
Source Number of events
Data 141 57 28 14 13 8 3 2 1 0 0
SM MC (TeV) MC 138.1 60.9 29.4 15.2 8.1 4.5 2.7 1.6 1.0 0.6 0.4
18 134.2 58.0 27.9 14.3 7.7 4.3 2.6 1.5 1.0 0.6 0.4
17 134.5 57.9 27.7 14.4 7.8 4.4 2.6 1.6 1.0 0.6 0.4
16 134.9 58.0 27.8 14.5 7.8 4.5 2.7 1.6 1.0 0.7 0.5
15 135.6 58.3 28.1 14.7 8.0 4.7 2.9 1.7 1.1 0.8 0.5
14 133.7 58.3 28.3 15.0 8.4 5.0 3.1 1.9 1.3 0.9 0.6
13 134.1 59.3 29.1 15.7 8.9 5.4 3.5 2.2 1.5 1.0 0.7
12 138.6 60.1 30.2 16.7 9.8 6.1 4.1 2.7 1.9 1.3 0.9
11 135.7 62.5 32.1 18.4 11.2 7.3 5.0 3.5 2.4 1.7 1.2
10 141.1 66.7 35.7 21.2 13.6 9.2 6.6 4.6 3.3 2.4 1.7
9 148.5 73.8 42.4 27.1 18.3 13.1 9.5 6.9 5.0 3.7 2.6
8 164.7 88.1 54.4 36.8 26.2 19.3 14.3 10.6 7.8 5.8 4.1
7 198.1 117.5 79.4 57.6 43.3 31.6 24.0 17.5 13.1 9.0 6.2
6 278.1 182.3 131.7 100.1 76.7 57.9 45.1 33.0 22.6 16.9 11.3
5 469.2 338.7 261.7 204.4 158.6 123.2 96.7 74.6 56.8 41.5 29.2
4 1025 784.1 620.1 494.2 384.3 302.6 232.8 174.6 127.2 94.5 68.5
3 3199 2517 2012 1599 1242 975.7 744.7 575.4 437.7 320.1 231.4
S. CHATRCHYAN et al. PHYSICAL REVIEW D 87, 032001 (2013)
from CI is the difference of the number of CI/DY events expected for a given, and the number of DY events. The expected mean number of background events is the sum of events from the DY process and
the non-DY SM back- grounds. The observed and expected numbers of events are given in TablesIVandV.
Systematic uncertainties in the predicted signal and background event yields are estimated from a variety of sources and included as nuisance parameters in the limit-setting procedure. Significant
sources of systematic uncertainty are given in Table VI. The uncertainty in the integrated luminosity is described in Ref. [39]. The uncer- tainty in the CI/DY acceptance is explained in Sec.V B.
The uncertainties in the prediction of backgrounds depend on the value of M^min[]. These uncertainties are given in Table VI for the values of M^min[] chosen for limits on with destructive and
constructive interference. The PDF uncertainty in the expected yield of DY events is evaluated using the PDF4LHC procedure [43]. The uncertainties in the QED and QCD K-factors are explained in Sec. V
The uncertainty from non-DY backgrounds is due to the statistical uncertainty associated with the simulated event samples. The systematic uncertainties which decrease the limit on by the largest
amounts are the uncertainties on the PDF and QED K-factor. When both these uncertainties are set to zero, the limit for destructive interference is increased by 0.4% and the limit for constructive
interfer- ence is increased by 3.0%. Thus, the systematic uncertain- ties degrade the limits by only small amounts.
We considered possible systematic uncertainties in mod- eling the detector response by comparing kinematic dis- tributions between data and simulation of DY and non-DY SM processes. There are no
differences in these distribu- tions that could lead to significant systematic uncertainties through their effect on selection efficiency and mass resolution.
C. Results for limits on
The observed and expected lower limits on at 95% CL as a function of M^min[] for destructive and constructive interference are shown in Figs. 5(a) and5(b). The value of M^min[], chosen to maximize
the expected sensitivity, is 1100 GeV=c^2 for destructive interference, and 800 GeV=c^2 for constructive interference. The observed (expected) limit is 9.5 TeV (9.7 TeV) for destructive
2) (GeV/c
200 400 600 800 1000 1200 1400 1600 1800 2000 )2Events/(20 GeV/c
= 7 TeV, 5.3 fb-1
s CMS,
data = 8 TeV (const.) Λ = 8 TeV (destr.) Λ = 10 TeV (const.) Λ = 10 TeV (destr.) Λ
= 12 TeV (const.) Λ = 12 TeV (destr.) Λ
DY t ttW diboson
τ τ
→ Z W+Jets
FIG. 4 (color online). Observed spectrum of M[]and predic- tions for SM and CI/DY plus non-DY SM production.
Predictions are shown for three illustrative values of , for constructive and destructive interference. The error bars for data are 68% Poisson confidence intervals.
2) (GeV/c
expected limit σ
±1 expected limit
±2 expected limit observed limit best expected limit
= 7 TeV, 5.3 fb-1
s CMS,
destructive interference (a)
2) (GeV/c
expected limit σ
±1 expected limit
±2 expected limit observed limit best expected limit
= 7 TeV, 5.3 fb-1
s CMS,
constructive interference (b)
FIG. 5 (color online). Observed and expected limits as a function of M^min[] for (a) destructive interference and (b) constructive interference. The value of M^min[], chosen to maximize the expected
sensitivity, is1100 GeV=c^2for destruc- tive interference and800 GeV=c^2for constructive interference.
The observed (expected) limit is 9.5 TeV (9.7 TeV) for destruc- tive interference and 13.1 TeV (12.9 TeV) for constructive interference. The observed limit at the value chosen for M^min[]
is indicated with a red plus sign. The variations in the observed limits lie almost entirely within the1- bands, consistent with statistical fluctuations.
. . .
interference and 13.1 TeV (12.9 TeV) for constructive interference. The variations in the observed limits lie almost entirely within the1- (standard deviation) uncer- tainty bands in the expected
limits, consistent with statis- tical fluctuations. The number of expected events corresponding to the observed limits on are shown in TablesIVandV.
The CMS detector is used to measure the invariant mass distribution of ^þ^pairs produced in pp collisions at a center-of-mass energy of 7 TeV, based on an integrated luminosity of5:3 fb^1. The
invariant mass distribution in the range 200 to2000 GeV=c^2 is found to be consistent with standard model sources of dimuons, which are domi- nated by Drell-Yan production. The data are interpreted
in the context of a quark- and muon-compositeness model with a left-handed isoscalar current and an energy scale parameter. The 95% confidence level lower limit on is 9.5 TeV under the assumption of
destructive interference between the standard model and contact-interaction ampli- tudes. For constructive interference, the limit is 13.1 TeV.
These limits are comparable to the most stringent ones reported to date.
We congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC and
thank the technical and administrative staffs at CERN and at other CMS institutes for their contributions to the suc- cess of the CMS effort. In addition, we gratefully acknowl- edge the computing
centers and personnel of the Worldwide LHC Computing Grid for delivering so effec- tively the computing infrastructure essential to our analy- ses. Finally, we acknowledge the enduring support for
the construction and operation of the LHC and the CMS detector provided by the following funding agencies:
BMWF and FWF (Austria); FNRS and FWO (Belgium);
CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MEYS (Bulgaria); CERN; CAS, MoST, and NSFC (China);
COLCIENCIAS (Colombia); MSES (Croatia); RPF (Cyprus); MoER, SF0690030s09 and ERDF (Estonia);
Academy of Finland, MEC, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NKTH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN
(Italy); NRF and WCU (Korea); LAS (Lithuania); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); MSI (New Zealand); PAEC (Pakistan); MSHE and NSC (Poland); FCT (Portugal);
JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan);
MON, RosAtom, RAS and RFBR (Russia); MSTD (Serbia); SEIDI and CPAN (Spain); Swiss Funding Agencies (Switzerland); NSC (Taipei); ThEP, IPST and NECTEC (Thailand); TUBITAK and TAEK (Turkey);
NASU (Ukraine); STFC (United Kingdom); DOE and NSF (USA). Individuals have received support from the Marie-Curie programme and the European Research Council (European Union); the Leventis Foundation;
the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office;
the Fonds pour la Formation a` la Recherche dans l’Industrie et dans l’Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); the Ministry of
Education, Youth and Sports (MEYS) of Czech Republic; the Council of Science and Industrial Research, India; the Compagnia di San Paolo (Torino); and the HOMING PLUS programme of Foundation for
Polish Science, cofinanced by European Union, Regional Development Fund.
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TABLE VI. Systematic uncertainties affecting the limit on, evaluated for the values of M^min[] that provide the best expected limits for constructive and destructive interference.
Uncertainty (%)
Source Constructive Destructive
Integrated luminosity 2.2 2.2
Acceptance times migration (A M) 3.0 3.0
PDF 13.0 16.0
QED K-factor 9.0 11.8
QCD K-factor 3.0 3.0
DY MC statistics 1.2 1.6
Non-DY backgrounds 1.1 2.9
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S. Chatrchyan,^1V. Khachatryan,^1A. M. Sirunyan,^1A. Tumasyan,^1W. Adam,^2E. Aguilo,^2T. Bergauer,^2 M. Dragicevic,^2J. Ero¨,^2C. Fabjan,^2,bM. Friedl,^2R. Fru¨hwirth,^2,bV. M. Ghete,^2J. Hammer,^2N.
Ho¨rmann,^2 J. Hrubec,^2M. Jeitler,^2,bW. Kiesenhofer,^2V. Knu¨nz,^2M. Krammer,^2,bI. Kra¨tschmer,^2D. Liko,^2I. Mikulec,^2
M. Pernicka,^2,aB. Rahbaran,^2C. Rohringer,^2H. Rohringer,^2R. Scho¨fbeck,^2J. Strauss,^2A. Taurok,^2 W. Waltenberger,^2G. Walzel,^2E. Widl,^2C.-E. Wulz,^2,bV. Mossolov,^3N. Shumeiko,^3J. Suarez
Gonzalez,^3 M. Bansal,^4S. Bansal,^4T. Cornelis,^4E. A. De Wolf,^4X. Janssen,^4S. Luyckx,^4L. Mucibello,^4S. Ochesanu,^4 B. Roland,^4R. Rougny,^4M. Selvaggi,^4Z. Staykova,^4H. Van Haevermaet,^4P. Van
Mechelen,^4N. Van Remortel,^4 A. Van Spilbeeck,^4F. Blekman,^5S. Blyweert,^5J. D’Hondt,^5R. Gonzalez Suarez,^5A. Kalogeropoulos,^5M. Maes,^5 A. Olbrechts,^5W. Van Doninck,^5P. Van Mulders,^5G. P. Van
Onsem,^5I. Villella,^5B. Clerbaux,^6G. De Lentdecker,^6
V. Dero,^6A. P. R. Gay,^6T. Hreus,^6A. Le´onard,^6P. E. Marage,^6A. Mohammadi,^6T. Reis,^6L. Thomas,^6 G. Vander Marcken,^6C. Vander Velde,^6P. Vanlaer,^6J. Wang,^6V. Adler,^7K. Beernaert,^7A.
Cimmino,^7S. Costantini,^7 G. Garcia,^7M. Grunewald,^7B. Klein,^7J. Lellouch,^7A. Marinov,^7J. Mccartin,^7A. A. Ocampo Rios,^7D. Ryckbosch,^7
N. Strobbe,^7F. Thyssen,^7M. Tytgat,^7P. Verwilligen,^7S. Walsh,^7E. Yazgan,^7N. Zaganidis,^7S. Basegmez,^8 G. Bruno,^8R. Castello,^8L. Ceard,^8C. Delaere,^8T. du Pree,^8D. Favart,^8L. Forthomme,^8A.
Giammanco,^8,cJ. Hollar,^8
V. Lemaitre,^8J. Liao,^8O. Militaru,^8C. Nuttens,^8D. Pagano,^8A. Pin,^8K. Piotrzkowski,^8N. Schul,^8 J. M. Vizan Garcia,^8N. Beliy,^9T. Caebergs,^9E. Daubie,^9G. H. Hammad,^9G. A. Alves,^10M. Correa
Martins Junior,^10
D. De Jesus Damiao,^10T. Martins,^10M. E. Pol,^10M. H. G. Souza,^10W. L. Alda´ Ju´nior,^11W. Carvalho,^11 A. Custo´dio,^11E. M. Da Costa,^11C. De Oliveira Martins,^11S. Fonseca De Souza,^11D. Matos
Figueiredo,^11 L. Mundim,^11H. Nogima,^11V. Oguri,^11W. L. Prado Da Silva,^11A. Santoro,^11L. Soares Jorge,^11A. Sznajder,^11 T. S. Anjos,^12bC. A. Bernardes,^12bF. A. Dias,^12a,dT. R. Fernandez
Perez Tomei,^12aE. M. Gregores,^12bC. Lagana,^12a F. Marinho,^12aP. G. Mercadante,^12bS. F. Novaes,^12aSandra S. Padula,^12aV. Genchev,^13,eP. Iaydjiev,^13,eS. Piperov,^13
M. Rodozov,^13S. Stoykova,^13G. Sultanov,^13V. Tcholakov,^13R. Trayanov,^13M. Vutova,^13A. Dimitrov,^14 . . .
R. Hadjiiska,^14V. Kozhuharov,^14L. Litov,^14B. Pavlov,^14P. Petkov,^14J. G. Bian,^15G. M. Chen,^15H. S. Chen,^15 C. H. Jiang,^15D. Liang,^15S. Liang,^15X. Meng,^15J. Tao,^15J. Wang,^15X. Wang,^15Z.
Wang,^15H. Xiao,^15M. Xu,^15
J. Zang,^15Z. Zhang,^15C. Asawatangtrakuldee,^16Y. Ban,^16S. Guo,^16Y. Guo,^16W. Li,^16S. Liu,^16Y. Mao,^16 S. J. Qian,^16H. Teng,^16D. Wang,^16L. Zhang,^16B. Zhu,^16W. Zou,^16C. Avila,^17J. P.
Gomez,^17B. Gomez Moreno,^17
A. F. Osorio Oliveros,^17J. C. Sanabria,^17N. Godinovic,^18D. Lelas,^18R. Plestina,^18,fD. Polic,^18I. Puljak,^18,e Z. Antunovic,^19M. Kovac,^19V. Brigljevic,^20S. Duric,^20K. Kadija,^20J. Luetic,^
20S. Morovic,^20A. Attikis,^21 M. Galanti,^21G. Mavromanolakis,^21J. Mousa,^21C. Nicolaou,^21F. Ptochos,^21P. A. Razis,^21M. Finger,^22
M. Finger, Jr.,^22Y. Assran,^23,gS. Elgammal,^23,hA. Ellithi Kamel,^23,iM. A. Mahmoud,^23,jA. Radi,^23,k,l M. Kadastik,^24M. Mu¨ntel,^24M. Raidal,^24L. Rebane,^24A. Tiko,^24P. Eerola,^25G. Fedi,^25M.
Voutilainen,^25 J. Ha¨rko¨nen,^26A. Heikkinen,^26V. Karima¨ki,^26R. Kinnunen,^26M. J. Kortelainen,^26T. Lampe´n,^26K. Lassila-Perini,^26
S. Lehti,^26T. Linde´n,^26P. Luukka,^26T. Ma¨enpa¨a¨,^26T. Peltola,^26E. Tuominen,^26J. Tuominiemi,^26E. Tuovinen,^26 D. Ungaro,^26L. Wendland,^26K. Banzuzi,^27A. Karjalainen,^27A. Korpela,^27T.
Tuuva,^27M. Besancon,^28 S. Choudhury,^28M. Dejardin,^28D. Denegri,^28B. Fabbro,^28J. L. Faure,^28F. Ferri,^28S. Ganjour,^28A. Givernaud,^28 P. Gras,^28G. Hamel de Monchenault,^28P. Jarry,^28E.
Locci,^28J. Malcles,^28L. Millischer,^28A. Nayak,^28J. Rander,^28 A. Rosowsky,^28I. Shreyber,^28M. Titov,^28S. Baffioni,^29F. Beaudette,^29L. Benhabib,^29L. Bianchini,^29M. Bluj,^29,m
C. Broutin,^29P. Busson,^29C. Charlot,^29N. Daci,^29T. Dahms,^29L. Dobrzynski,^29R. Granier de Cassagnac,^29 M. Haguenauer,^29P. Mine´,^29C. Mironov,^29I. N. Naranjo,^29M. Nguyen,^29C. Ochando,^29P.
Paganini,^29D. Sabes,^29
R. Salerno,^29Y. Sirois,^29C. Veelken,^29A. Zabi,^29J.-L. Agram,^30,nJ. Andrea,^30D. Bloch,^30D. Bodin,^30 J.-M. Brom,^30M. Cardaci,^30E. C. Chabert,^30C. Collard,^30E. Conte,^30,nF. Drouhin,^30,nC.
Ferro,^30J.-C. Fontaine,^30,n
D. Gele´,^30U. Goerlach,^30P. Juillot,^30A.-C. Le Bihan,^30P. Van Hove,^30F. Fassi,^31D. Mercier,^31S. Beauceron,^32 N. Beaupere,^32O. Bondu,^32G. Boudoul,^32J. Chasserat,^32R. Chierici,^32,eD.
Contardo,^32P. Depasse,^32 H. El Mamouni,^32J. Fay,^32S. Gascon,^32M. Gouzevitch,^32B. Ille,^32T. Kurca,^32M. Lethuillier,^32L. Mirabito,^32 S. Perries,^32V. Sordini,^32Y. Tschudi,^32P. Verdier,^32S.
Viret,^32Z. Tsamalaidze,^33,oG. Anagnostou,^34S. Beranek,^34
M. Edelhoff,^34L. Feld,^34N. Heracleous,^34O. Hindrichs,^34R. Jussen,^34K. Klein,^34J. Merz,^34A. Ostapchuk,^34 A. Perieanu,^34F. Raupach,^34J. Sammet,^34S. Schael,^34D. Sprenger,^34H. Weber,^34B.
Wittmer,^34V. Zhukov,^34,p M. Ata,^35J. Caudron,^35E. Dietz-Laursonn,^35D. Duchardt,^35M. Erdmann,^35R. Fischer,^35A. Gu¨th,^35T. Hebbeker,^35
C. Heidemann,^35K. Hoepfner,^35D. Klingebiel,^35P. Kreuzer,^35C. Magass,^35M. Merschmeyer,^35A. Meyer,^35 M. Olschewski,^35P. Papacz,^35H. Pieta,^35H. Reithler,^35S. A. Schmitz,^35L. Sonnenschein,^
35J. Steggemann,^35 D. Teyssier,^35M. Weber,^35M. Bontenackels,^36V. Cherepanov,^36Y. Erdogan,^36G. Flu¨gge,^36H. Geenen,^36 M. Geisler,^36W. Haj Ahmad,^36F. Hoehle,^36B. Kargoll,^36T. Kress,^36Y.
Kuessel,^36A. Nowack,^36L. Perchalla,^36
O. Pooth,^36P. Sauerland,^36A. Stahl,^36M. Aldaya Martin,^37J. Behr,^37W. Behrenhoff,^37U. Behrens,^37 M. Bergholz,^37,qA. Bethani,^37K. Borras,^37A. Burgmeier,^37A. Cakir,^37L. Calligaris,^37A.
Campbell,^37E. Castro,^37
F. Costanza,^37D. Dammann,^37C. Diez Pardos,^37G. Eckerlin,^37D. Eckstein,^37G. Flucke,^37A. Geiser,^37 I. Glushkov,^37P. Gunnellini,^37S. Habib,^37J. Hauk,^37G. Hellwig,^37H. Jung,^37M. Kasemann,^
37P. Katsas,^37
C. Kleinwort,^37H. Kluge,^37A. Knutsson,^37M. Kra¨mer,^37D. Kru¨cker,^37E. Kuznetsova,^37W. Lange,^37 W. Lohmann,^37,qB. Lutz,^37R. Mankel,^37I. Marfin,^37M. Marienfeld,^37I.-A. Melzer-Pellmann,^37A.
B. Meyer,^37
J. Mnich,^37A. Mussgiller,^37S. Naumann-Emme,^37J. Olzem,^37H. Perrey,^37A. Petrukhin,^37D. Pitzl,^37 A. Raspereza,^37P. M. Ribeiro Cipriano,^37C. Riedl,^37E. Ron,^37M. Rosin,^37J. Salfeld-Nebgen,^
37R. Schmidt,^37,q
T. Schoerner-Sadenius,^37N. Sen,^37A. Spiridonov,^37M. Stein,^37R. Walsh,^37C. Wissing,^37C. Autermann,^38 V. Blobel,^38J. Draeger,^38H. Enderle,^38J. Erfle,^38U. Gebbert,^38M. Go¨rner,^38T.
Hermanns,^38R. S. Ho¨ing,^38 K. Kaschube,^38G. Kaussen,^38H. Kirschenmann,^38R. Klanner,^38J. Lange,^38B. Mura,^38F. Nowak,^38T. Peiffer,^38 N. Pietsch,^38D. Rathjens,^38C. Sander,^38H. Schettler,^
38P. Schleper,^38E. Schlieckau,^38A. Schmidt,^38M. Schro¨der,^38
T. Schum,^38M. Seidel,^38V. Sola,^38H. Stadie,^38G. Steinbru¨ck,^38J. Thomsen,^38L. Vanelderen,^38C. Barth,^39 J. Berger,^39C. Bo¨ser,^39T. Chwalek,^39W. De Boer,^39A. Descroix,^39A. Dierlamm,^39M.
Feindt,^39M. Guthoff,^39,e C. Hackstein,^39F. Hartmann,^39T. Hauth,^39,eM. Heinrich,^39H. Held,^39K. H. Hoffmann,^39S. Honc,^39I. Katkov,^39,p
J. R. Komaragiri,^39P. Lobelle Pardo,^39D. Martschei,^39S. Mueller,^39Th. Mu¨ller,^39M. Niegel,^39A. Nu¨rnberg,^39 O. Oberst,^39A. Oehler,^39J. Ott,^39G. Quast,^39K. Rabbertz,^39F. Ratnikov,^39N.
Ratnikova,^39S. Ro¨cker,^39
A. Scheurer,^39F.-P. Schilling,^39G. Schott,^39H. J. Simonis,^39F. M. Stober,^39D. Troendle,^39R. Ulrich,^39 J. Wagner-Kuhr,^39S. Wayand,^39T. Weiler,^39M. Zeise,^39G. Daskalakis,^40T. Geralis,^40S.
Kesisoglou,^40 A. Kyriakis,^40D. Loukas,^40I. Manolakos,^40A. Markou,^40C. Markou,^40C. Mavrommatis,^40E. Ntomari,^40 L. Gouskos,^41T. J. Mertzimekis,^41A. Panagiotou,^41N. Saoulidou,^41I. Evangelou,
^42C. Foudas,^42P. Kokkas,^42
S. CHATRCHYAN et al. PHYSICAL REVIEW D 87, 032001 (2013) | {"url":"https://1library.co/document/qo59grdj-search-contact-interactions-%CE%BC-%CE%BC-events-collisions-tev.html","timestamp":"2024-11-09T10:47:52Z","content_type":"text/html","content_length":"217273","record_id":"<urn:uuid:9f3c7449-0981-4c39-b497-83b83c544513>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00071.warc.gz"} |
An enumeration of labeling placement options. This is used to specify the preferred position of the text label, with respect to its feature geometry.
Default position for the label, dependent on the type of feature being labeled.
Lower-left corner of label is at final geometry coord; label extrapolates the last geometry segment.
Lower midpoint of label prefers the midpoint of the geometry; label follows the geometry segments.
Lower right corner of label is at first geometry coord; label extrapolates the first geometry segment.
Lower right corner of label is at final geometry coord; label follows the last geometry segments.
Lower left corner of label is at first geometry coord, label follows the first geometry segments.
Upper left corner of label is at final geometry coord, label extrapolates the last geometry segment.
Upper midpoint of label prefers the midpoint of the geometry, label follows the geometry segments.
Upper right corner of label is at first geometry coord, label extrapolates the first geometry segment.
Upper right corner of label is at final geometry coord, label follows the last geometry segments.
Upper left corner of label is at first geometry coord, label follows the first geometry segments.
Left midpoint of label is at final geometry coord, label extrapolates the last geometry segment.
Center of label prefers the midpoint of the geometry, label follows the geometry segments.
Right midpoint of label is at first geometry coord, label extrapolates the first geometry segment.
Right midpoint of label is at final geometry coord, label follows the last geometry segments.
Left midpoint of label is at first geometry coord, label follows the first geometry segments.
Lower-right corner of the label is offset northwest of point symbol.
Lower left corner of the label is offset North-east of point symbol.
Upper right corner of the label is offset South-west of point symbol.
Upper left corner of the label is offset South-east of point symbol.
Center of label is as far inside polygon as possible. Note that if a polygon contains holes (defined as counter-clockwise rings), labels will not be placed within those holes. | {"url":"https://developers.arcgis.com/kotlin/api-reference/arcgis-maps-kotlin/com.arcgismaps.arcgisservices/-labeling-placement/index.html","timestamp":"2024-11-12T14:12:14Z","content_type":"text/html","content_length":"57736","record_id":"<urn:uuid:48126f3f-18a3-4d70-bf1e-420e5de198c8>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00119.warc.gz"} |
Taming the AGM
You're reading: Irregulars
Taming the AGM
This post is in response to Peter’s post introducing the Approximate Geometric Mean.
The approximate geometric mean $\mathrm{(AGM)}$ is a nice approximation of the geometric mean $\mathrm{(GM)}$, but it has some quirks as we will see. After a discussion at the MathsJam gathering, I
was intrigued to find out how good an approximation it is.
To get a better understanding, we first have to look again at its definition. For $A=a\cdot 10^x$ and $B=b \cdot 10^y$, we set
\[ \mathrm{AGM}(A,B):=\mathrm{AM}(a,b)\cdot 10^{\mathrm{AM}(x,y)} \]
where $\mathrm{AM}$ stands for the arithmetic mean. This makes also sense when $a$ and $b$ are not just integers between 1 and 10, but any real numbers. Note that we won’t consider negative $A$ and
$B$ (i.e. negative $a$ and $b$), as the geometric mean runs into issues if we do so. The values of $x$ and $y$ may be negative, though. The $\mathrm{AGM}$ looks like a mix between the $\mathrm{AM}$
and the $\mathrm{GM}$, so what can possibly go wrong?
Same mean, different numbers
In contrast to the $\mathrm{AM}$ and the $\mathrm{GM}$, the $\mathrm{AGM}$ depends on the number base (10 in this case) and the presentation of $A$ and $B$.
If we write $A=(10a) \cdot 10^{(x-1)}$, we get a different value for $\mathrm{AGM}(A,B)$. This looks rather unfortunate, but it will turn out to be helpful. To ease notation we will assume in the
following that $a\geq b$ unless otherwise stated. This can be done without loss of generality as $\mathrm{AGM}(A,B)=\mathrm{AGM}(B,A)$.
Peter Rowlett proved in his post that $\mathrm{GM}\leq \mathrm{AGM}$. The question is, how far can the $\mathrm{AGM}$ exceed the $\mathrm{GM}$? In other words, what’s the supremum of the ratio $R=\
Using the notation of $A$ and $B$ as above we get
\begin{align*} R=\frac{\mathrm{AGM}(A,B)}{\mathrm{GM}(A,B)}= \frac{\mathrm{AM}(a,b)}{\mathrm{GM}(a,b)} = \frac{1}{2}\cdot (\sqrt{a/b}+\sqrt{b/a}).\end{align*}
So, the ratio $R$ doesn’t depend on $x$ and $y$ but only on $a$ and $b$. That’s convenient. Taking $a$ and $b$ in the interval $[1,10)$, as is usual, we can look at the plot of $R(a,b)$.
As long as we are in the blue part of the graph, $\mathrm{AGM}$ looks to be a sensible approximation of the $\mathrm{GM}$. So let’s look at the bad combinations of $a$ and $b$.
The worst case happens when $a$ and $b$ are maximally far apart: The supremum of $R(a,b)$ is its limit for $a \rightarrow 10$ and $b=1$. So in general, $1\leq R \lt 5.5/\sqrt{10} \approx 1.74$.
This supremum doesn’t look too bad at first, but unfortunately, the result can be unusable in extreme cases. For example, if $a=999=9.99\cdot 10^2$ and $b=1000=1 \cdot 10^3$, we have $\mathrm{GM}
(A,B)\approx \mathrm{AM}(A,B)=999.5$ and $\mathrm{AGM}(A,B)\approx 1738$ – not only is the $\mathrm{AM}$ a better approximation of the $\mathrm{GM}$ than the $\mathrm{AGM}$ in this instance, the $\
mathrm{AGM}$ is bigger than both the numbers $A$ and $B$ of which it is supposed to give some kind of mean!
Let’s analyse this a bit deeper. The ratio $R$ only depends on the ratio $r=a/b$. In closed form we can write $R(r)=1/2\cdot (\sqrt{r}+\sqrt{r}^{-1})$ and we are left to study this function in the
range $[1,10]$. Its maximum is $R(10)$, but smaller $r$ give better results. And we will see, that we don’t have to put up with $r=10$.
Here, the flexibility of the definition of the $\mathrm{AGM}$ comes into play. Due to the choice of a suitable presentation of the numbers we can guarantee that $r$ isn’t too big. If we have $r\leq \
sqrt{10}$ which is equivalent to $\sqrt{10}b\geq a \geq b$ we calculate $\mathrm{AGM}(A,B)$ as above. If $a>\sqrt{10}b$, we change the presentation of the number:
\begin{align*}B=b \cdot 10^y = (10b)\cdot 10^{y-1}=:b’ \cdot 10^{y-1} \end{align*}
and continue from there.
So, let’s redefine the $\mathrm{AGM}$ for $10>a\geq b\geq 1$ like this:
\mathrm{AM}(a,b)\cdot 10^{\mathrm{AM}(x,y)}, & \sqrt{10}b\geq a,\\
\mathrm{AM}(10b,a)\cdot 10^{\mathrm{AM}(x,y-1)}, & \text{otherwise}.
Note, that in the second case we have $\sqrt{10}a>10b>a$, so that the roles of the pair $(a,b)$ are taken over by the pair $(10b,a)$. Setting $r=10b/a$ in the second case, we have in both cases $1\
leq r\leq \sqrt{10}$, so we only have to study $R(r)$ in the interval $[1,\sqrt{10}]$, which will turn out to be rather benign.
Note also, that this new $\mathrm{AGM}$ can still be calculated without a calculator when using the approximation $\sqrt{10}\approx 3$, as Colin Beveridge suggested in Peter’s post.
In the example above with $A=999$ and $B=1000$ we write $B=10\cdot 10^2$ and find with this new definition of the $\mathrm{AGM}$:
\begin{align*}\mathrm{AGM}(999;1000)=\mathrm{AM}(9.99;10) \cdot 10^{\mathrm{AM}(2;2)}=999.5, \end{align*}
This coincides with the arithmetic mean of the two numbers and is really close to the geometric mean. This is looking promising.
If we define the $\mathrm{AGM}$ of two numbers $A$ and $B$ in the way explained above, we get the following two inequalities:
\begin{align*} (I) \quad & \mathrm{GM}(A,B)\leq \mathrm{AGM}(A,B) \leq \mathrm{GM}(A,B) \cdot 1.2 \\
(II) \quad & \mathrm{GM}(A,B) \leq \mathrm{AGM}(A,B) \leq \mathrm{AM}(A,B) \end{align*}
\]Both inequalities together mean that not a lot can go wrong when using the $\mathrm{AGM}$ with the appropriate presentation of the numbers: The $\mathrm{AGM}$ is bigger than the $\mathrm{GM}$, but
exceeds it by maximally 20%, and it is always smaller than the $\mathrm{AM}$.
As a consequence, the $\mathrm{AGM}$ will always be between $A$ and $B$. So it is indeed a “mean” of some kind.
A proof of these two inequalities
(I) We only have to find the maximum of $R=\mathrm{AGM}/\mathrm{GM}$. Due to the discussion above we can assume that $\sqrt{10}b\geq a \geq b$, but $a$ can now be bigger than 10. The latter is not a
problem though.
The maximum of $R=\mathrm{AGM}(A,B)/\mathrm{GM}(A,B)=\mathrm{AM}(a,b)/\mathrm{GM}(a,b)$ is attained when $a$ and $b$ are maximally apart, i.e. $r=\sqrt{10}$, so
\max\left(\frac{\mathrm{AGM}(A,B)}{\mathrm{GM}(A,B)}\right)=\frac12 \cdot (10^{1/4}+{10^{-1/4}}) \approx 1.2.
(II) We will show that $\mathrm{AGM}(A,B)/\mathrm{AM}(A,B) \leq 1$. Let’s drop the assumption that $a\geq b$. Instead we assume, again without loss of generality, that $x\geq y$, so that we can set
$z=:x-y \geq 0$. For the ratio $r=a/b$ we have $\sqrt{10} \geq r \geq 1/\sqrt{10}$. If $r$ fell outside this interval, we would have had to change the presentation of one of the numbers before
calculating the $\mathrm{AGM}$. Dividing the numerator and denominator in the above inequality by $B$ we get:
\frac{\mathrm{AGM}(A,B)}{\mathrm{AM}(A,B)}=\frac{(1+r) \cdot 10^{z/2}} {1+r \cdot 10^z}.
So we look for an upper bound of the function $f_z(r):=\frac{(1+r) \cdot 10^{z/2}} {1+r \cdot 10^z}$ when varying $z$ and $r$ and want to show that this upper bound is smaller or equal to 1. Note,
that we only have to check for integer $z\geq 0$ (The result is actually false if we allow any real $z$).
For $z=0$, we have $f_0(r)=1$ for any $r$ and hence $\mathrm{AGM}=\mathrm{AM}$. For a fixed $z \geq 1$ we can derive the function $f_z(r)$ with respect to $r$ and find that the slope is always
negative. Hence for a fixed $z$, the function $f_z(r)$ attains a maximum when $r$ is smallest, i.e. $r=1/\sqrt{10}$, so we are left to show that
f_z(1/\sqrt{10})=\frac{(1+10^{-1/2})10^{z/2}}{1+10^{z-1/2}}\leq 1.
For $z=1$ we have equality again and $\mathrm{AGM} = \mathrm{AM}$. For $z\geq 2$ we can write $z=2+z’$ with $z’$ being an integer $\geq 0$. We get the following chain of inequalities
f_z\left(\frac{1}{\sqrt{10}}\right)=\frac{(1+10^{-1/2})10^{(2+z’)/2}}{1+10^{3/2 + z’}}<\frac{2\cdot 10^{1+z’/2}}{10^{3/2+z’}}\leq \frac{2}{\sqrt{10}}<1.
This proves the second inequality. ☐
In summary, modifying the definition of the $\mathrm{AGM}$ to assure that the ratio of the “leading characters” is as close to 1 as possible, makes sure that the $\mathrm{AGM}$ works well, even in
the bad cases.
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Symmetry of solutions in discrete and continuous structural topology optimization
In this paper symmetry and asymmetry of optimal solutions in symmetric structural topology optimization problems are investigated, based on the choice of variables. A group theory approach is used to
formally define the symmetry of the structural problems. This approach allows the set of symmetric structures to be described and related to the entire search space. It is shown that, given a
symmetric problem with continuous variables, an optimal symmetric solution (if any) necessarily exists. However, it is shown that this does not hold for the discrete case. Finally a number of
examples are investigated to demonstrate the findings of the research.
Original language English (US)
Title of host publication Proceedings of the 11th International Conference on Computational Structures Technology, CST 2012
Publisher Civil-Comp Press
Volume 99
ISBN (Print) 9781905088546
State Published - 2012
Event 11th International Conference on Computational Structures Technology, CST 2012 - Dubrovnik, Croatia
Duration: Sep 4 2012 → Sep 7 2012
Other 11th International Conference on Computational Structures Technology, CST 2012
Country/Territory Croatia
City Dubrovnik
Period 9/4/12 → 9/7/12
All Science Journal Classification (ASJC) codes
• Environmental Engineering
• Civil and Structural Engineering
• Computational Theory and Mathematics
• Artificial Intelligence
• Asymmetric topology
• Group theory
• Structural topology optimization
• Symmetric topology
• Symmetry operation
• Truss topology optimization
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Aerodynamic Forces Calculator
❤️ Aerodynamic Forces Calculator
🎯 Table of contents:
📕 Contents
What are Aerodynamic Forces?
Aerodynamic Forces are the forces that act on an object or a body when it moves through the air. These forces depend on factors such as the shape, size, and motion of the object. Aerodynamic Forces
are crucial in various fields, including engineering, aviation, and transportation, and they impact objects like airplanes, automobiles, ships, and artificial satellites.
• Lift Force: This force is responsible for lifting an object upward. It is what allows airplanes to stay in the air and is generated by differences in air pressure between the upper and lower
surfaces of the object.
• Drag Force: Drag force acts in the opposite direction of an object's motion and opposes its movement. It slows down the object and consumes energy. It's a critical factor for vehicles and
aircraft, as it makes it more challenging for them to move forward.
• Lateral Forces and Moments: These forces cause lateral movement or rotation of an object. They are essential for maneuvering and controlling the direction of vehicles and aircraft.
Definition: Aerodynamic forces are the forces exerted on an object as it moves through a fluid, typically air. These forces are a result of the interaction between the object's surface and the
surrounding air molecules.
What are Aerodynamic Forces Calculator
Aerodynamic Forces Calculator is a tool used to compute the aerodynamic forces acting on an object or aircraft moving through the air. It is a valuable tool in aviation and engineering for
understanding the impact of air density, velocity, and reference area on lift and drag forces.
• Air Density: Air density (measured in kilograms per cubic meter, kg/m³) is a crucial parameter in aerodynamics. It represents the mass of air particles in a given volume of air. Higher air
density results in greater lift and drag forces.
Aerodynamic Forces Calculator Formula is used to determine the aerodynamic forces acting on an object or aircraft moving through the air. It involves the calculation of two primary forces: lift
force and drag force, which are essential for understanding the object's behavior in aerodynamic conditions.
Lift Force Formula: The formula for calculating the lift force is given by:
Lift Force (L) = 0.5 * Air Density (ρ) * Velocity (V)² * Reference Area (A) * Lift Coefficient (Cl)
The lift force is influenced by factors such as air density, velocity, reference area, and the lift coefficient. It represents the force that allows an object to rise or stay aloft in the air.
Drag Force Formula: The formula for calculating the drag force is given by:
Drag Force (D) = 0.5 * Air Density (ρ) * Velocity (V)² * Reference Area (A) * Drag Coefficient (Cd)
The drag force opposes the object's motion through the air and is influenced by air density, velocity, reference area, and the drag coefficient.
These formulas are fundamental in aerodynamics and are used in various engineering and aviation applications to predict and analyze the behavior of objects in flight.
• Velocity: Velocity (measured in meters per second, m/s) is the speed at which the object is moving through the air. It significantly influences the aerodynamic forces. Higher velocity increases
both lift and drag forces.
• Reference Area: Reference area (measured in square meters, m²) is the specific area of the object that is exposed to the airflow. It plays a critical role in calculating aerodynamic forces, as
different shapes and sizes of objects have varying reference areas.
Aerodynamic Forces Calculator allows engineers and researchers to perform complex calculations to design and analyze the behavior of aircraft, vehicles, and other objects in various aerodynamic
Aerodynamic Forces on a Car
Aerodynamic forces on a car are the forces that act on the vehicle as it moves through the air. These forces can have a significant impact on a car's performance, fuel efficiency, and handling. The
primary aerodynamic forces acting on a car are:
Drag (Aerodynamic Resistance):
• Drag is the resistance encountered by a car as it moves through the air. It opposes the vehicle's forward motion and is a major factor in determining a car's top speed and fuel efficiency.
• Drag force depends on several factors, including the car's shape, size, speed, and the air density. Streamlined, aerodynamic designs help reduce drag.
• Car manufacturers often use wind tunnels and computer simulations to optimize a vehicle's aerodynamics and reduce drag.
• While lift is primarily associated with aircraft, it can also affect cars, especially at high speeds or on vehicles with specific designs.
• Lift on a car can lead to reduced traction on the road, affecting stability and handling. Engineers aim to minimize lift through design features like spoilers and diffusers.
• Downforce is the opposite of lift; it is a force that pushes the car downward, increasing traction and stability.
• Downforce is essential for high-performance and racing cars, as it helps them maintain grip on the road or track, especially during high-speed cornering.
• Features like wings, splitters, and underbody designs are used to generate downforce.
Side Forces:
• Side forces are lateral forces acting on the car due to crosswinds or uneven airflow around the vehicle.
• These forces can affect the car's stability, especially on highways with strong crosswinds.
Yaw Forces:
• Yaw forces occur when the car is subjected to changes in its direction or orientation, such as during cornering or skidding.
• Yaw forces can influence a car's stability and handling characteristics, and they are controlled through factors like tire grip and suspension design.
Car manufacturers and engineers strive to strike a balance between these aerodynamic forces to optimize a car's performance, efficiency, and safety. They use wind tunnel testing, computational fluid
dynamics (CFD) simulations, and other tools to design cars with aerodynamic properties that minimize drag, enhance stability, and provide the desired handling characteristics. Reducing aerodynamic
drag is especially crucial for improving fuel efficiency in everyday passenger cars.
Frequently Asked Questions
1. What Are Aerodynamic Forces?
Aerodynamic Forces are the forces that act on an object or aircraft when it moves through the air. They include lift force, drag force, and lateral forces, which are essential in understanding how
objects behave in aerodynamic conditions.
2. How Is Lift Force Calculated?
The lift force is calculated using the formula: Lift Force (L) = 0.5 * Air Density (ρ) * Velocity (V)² * Reference Area (A) * Lift Coefficient (Cl). It represents the upward force that allows objects
like airplanes to stay in the air.
3. What Is the Role of Drag Force in Aerodynamics?
Drag force opposes the object's motion through the air and is calculated using the formula: Drag Force (D) = 0.5 * Air Density (ρ) * Velocity (V)² * Reference Area (A) * Drag Coefficient (Cd). It
influences an object's speed and energy consumption.
4. How Do Aerodynamic Forces Impact Vehicle Design?
Aerodynamic forces play a crucial role in vehicle design, affecting factors such as fuel efficiency and stability. Engineers consider these forces when designing cars, trucks, and other vehicles to
optimize their performance.
5. What Are the Practical Applications of Understanding Aerodynamic Forces?
Understanding aerodynamic forces is essential in fields like aviation, automotive engineering, and sports equipment design. It helps improve the efficiency, safety, and performance of various objects
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Lateral Surface Area of a Triangular Pyramid Calculator Online
Home » Simplify your calculations with ease. » Mathematical Calculators »
Lateral Surface Area of a Triangular Pyramid Calculator Online
A triangular pyramid, a type of polyhedron, consists of a triangular base and three triangular faces converging at a single point above the base. Calculating the lateral surface area (LSA) of such a
pyramid is crucial for various applications, from architectural design to crafting and educational projects. Our LSA calculator provides a quick, reliable way to determine this area, ensuring
accuracy for any project or educational need.
Formula for Lateral Surface Area
To compute the lateral surface area of a triangular pyramid, use the formula:
Here's how to apply it:
• Measure the sides of the base triangle: Let the sides be 𝑎a, 𝑏b, and 𝑐c.
• Calculate the perimeter of the base: 𝑃=𝑎+𝑏+𝑐P=a+b+c
• Determine the slant height: This is the perpendicular distance from the apex to the midpoint of any base side.
• Apply the formula: Substitute the values into the formula to compute the LSA.
This approach ensures that anyone, regardless of their mathematical background, can calculate the LSA of a triangular pyramid easily.
Table of General Terms
Term Definition
Base Triangle The triangle at the base of the pyramid.
Slant Height The perpendicular distance from the apex to the midpoint of a base side.
Perimeter Total length around the base triangle.
Lateral Surface Area The total area of all the triangular faces excluding the base.
Consider a triangular pyramid with base sides of 3 cm, 4 cm, and 5 cm, and a slant height of 6 cm. The perimeter 𝑃P of the base is 3+4+5=123+4+5=12 cm.
Using the formula:
LSA = (1/2) * 12 * 6 = 36 cm²
This example demonstrates how to use the calculator to find the LSA, simplifying the process for educational or practical applications.
Most Common FAQs
What is slant height?
Slant height is the distance from the apex of the pyramid to the midpoint of any side of the base triangle, crucial for calculating the lateral surface area.
How do you find the perimeter of the base?
To find the perimeter, simply add the lengths of all the sides of the base triangle.
Can the LSA calculator be used for any triangular pyramid?
Yes, the calculator is versatile and can be used for any triangular pyramid, regardless of the dimensions of the base or the slant height.
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Liquid vibrations in a container with a membrane at the free surface
In the paper, the vibrations of liquid in a rigid and circular cylindrical container have been investigated. The effects of an elastic membrane placed on the free surface of the fluid have been
analyzed on sloshing frequencies. The cylindrical container is assumed to be partially filled with an incompressible fluid. The potential theory is used to formulate the mathematical problem and set
up the boundary value problem. The solutions are obtained for velocity potential and the deflection of elastic membrane in form of Fourier-Bessel series. The results are compared for the proposed
analytical approach and BEM.
• An analytical approach is used to solve the boundary value problem.
• Sloshing in presence of membrane on liquid free surface is discussed.
• Vibrations modes are reported.
1. Introduction
“Sloshing is known as the free surface motion in a tank which is filled partially” [8]. Slosh motion is a potentially dangerous situation to engineering structures and environment and can lead to the
failure of structural units and their stability. The dynamic behaviour of structures carrying fuel tanks or storage reservoirs is significantly affected by fluid structure-interaction and is very
dangerous to the safety and stability of the structure. It is necessary to control the vibrations of fluid-structure interaction to maintain the stability of the structures used in various
engineering applications such as transporting liquid, petroleum reservoirs and space vehicles. Control of fluid sloshing inside a container has always been a challenge while designing any tank due to
uninvited vibrations which are dangerous for the stability of the system. The Gas-free liquid is a key function of propellant tanks/containers in spacecraft and offshore oil industry where floating
roofed oil tanks are used Propellant sloshing in tanks also influences the rigid body motion of the spacecraft or launch vehicles, which then needs to be controlled by the reaction control system.
Sloshing is a worrisome situation in oil storage tanks as well as in natural gas reservoirs. In recent years, floating liquified natural gas (FLNG) has gained popularity as innovative technology when
it comes to natural gas exploitation. These floating platforms as well as huge tanks filled with LNG/ Oil are vulnerable to the vibrations due to ground motion during an earthquake. Suppression
devices (rigid/elastic) are considered most effective to dampen the impact loads due to sloshing in a tank as shown in the studies reported in [1-6]. The effects of rigid baffles on sloshing
frequencies in circular cylindrical containers are investigated analytically in [1, 2]. The non-linear sloshing problem is reported in [3] and stress analysis is investigated. The effects of
vertically placed baffles have been reported in [4] and [6]. Some studied on floating roof (considered as membrane) effects on sloshing frequencies since Tokachi-Oki (2003) earthquake are reported in
[7-9], and [12]. The effects of perforated baffles on sloshing are reported in [10] and [11]. The vibrations of liquid in cylindrical tanks with and without baffles on the free surface under
horizontal and vertical excitation are reported in [13]. The problem of sloshing in horizontal elliptical tanks with T-shaped baffles is investigated in [14]. Many researchers have been investigating
the membrane’s dynamic response due to sloshing. In this paper, the fundamental sloshing modes of liquid with a membrane placed on the free surface are reported.
2. Problem statement
Fig. 1 shows a schematic diagram of sloshing in a container with elastic membrane placed at the free surface. A vertical circular cylindrical tank with radius $R$ containing a fluid free surface at
rest is considered at height $h$ from the rigid bottom of the container. To model the sloshing problem, the following assumptions are made: (i) the container is rigid and flow field is considered
irrotational, (ii) the fluid is considered incompressible, (iii) small displacement of free surface of liquid are considered. The velocity potential theory is used to formulate the problem. The
boundary value problem in terms of the velocity potential is given as follows.
Fig. 1Schematic diagram of the problem
For an irrotational flow motion, the velocity $V$ can be written as a gradient of potential $\varphi \left(r,\theta ,z,t\right)$ such that $V=abla \varphi$. Based on above assumptions of the
potential theory, velocity potential satisfies the Laplace equation:
${abla }^{2}\varphi =0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}0<r<R,\mathrm{}\mathrm{}0\le \theta \le 2\pi ,\mathrm{}\mathrm{}0\le z\le h.$
The boundary conditions at the rigid boundary of the cylinder are at the cylinder wall $r=R$ and at the rigid bottom:
$z=0,\mathrm{}\mathrm{}\mathrm{}\frac{\partial \varphi }{\partial r}=0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\frac{\partial \varphi }{\partial z}=0.$
The boundary condition for motion of membrane at the free surface is given by:
$\begin{array}{c}\frac{{\partial }^{2}\stackrel{-}{w}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial \stackrel{-}{w}}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}\stackrel{-}{w}}{\partial {\theta
}^{2}}-\frac{\mu }{T}\frac{{\partial }^{2}\stackrel{-}{w}}{\partial {t}^{2}}=-\frac{p}{T},\end{array}$
where $\stackrel{-}{w}$ denotes the deflection of the membrane, $T$ the tension per unit length, $\mu$ mass/unit area of the membrane and $p$ denotes the liquid pressure on the membrane. The membrane
deflection satisfies the following boundary conditions at the rigid wall of the container $r=R$:
$\frac{\partial \stackrel{-}{w}}{\partial r}=\stackrel{-}{w}=0.$
The continuity of normal velocity components at the free surface $z=h$ should be satisfied:
$\frac{\partial \varphi }{\partial z}=\frac{\partial \stackrel{-}{w}}{\partial t},$
and the dynamic boundary condition at $z=h$ gives:
$p=-\rho \frac{\partial \varphi }{\partial z}-\rho g\stackrel{-}{w}.$
So, the coupled boundary value problem given by Eqs. (1)-(6), is required to solve to find the unknown functions $\varphi$ and $\stackrel{-}{w}$. First, we describe the analytical approach. The
separation of variables method is used to find the velocity potential in the following form:
$\varphi \left(r,\theta ,z,t\right)=\sum _{m=0}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{A}_{mn}{J}_{m}\left(\frac{{k}_{mn}r}{R}\right)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\frac{{k}_
{mn}z}{R}\right)\mathrm{c}\mathrm{o}\mathrm{s}m\theta {e}^{i\omega t},$
where ${k}_{mn}$ are roots of ${J\mathrm{"}}_{m}\left(kR\right)=0$. Using velocity potential function for Eq. (6), have the expression for pressure can be written as:
$p={e}^{i\omega t}\left[-pi\omega \sum _{m=0}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{A}_{mn}{\mathrm{J}}_{m}\left(\frac{{k}_{mn}r}{R}\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(m\theta \
right)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\frac{{k}_{mn}h}{R}\right)\right]-\rho g\stackrel{-}{w}.$
The thickness of the membrane is considered negligible. Eq. (4) gives rise to:
$\frac{{\partial }^{2}{w}_{1}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial {w}_{1}}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}{w}_{1}}{\partial {\theta }^{2}}+\left(\frac{\mu {\stackrel{-}{\
omega }}^{2}-g\rho }{T}\right){w}_{1}=\frac{i\rho \stackrel{-}{\omega }}{T}\sum _{m=0}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{A}_{mn}{\mathrm{J}}_{m}\left(\frac{{k}_{mn}r}{R}\right)\mathrm
{c}\mathrm{o}\mathrm{s}\left(m\theta \right)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\frac{{k}_{mn}h}{R}\right),$
which is a nonhomogeneous differential equation in $\stackrel{-}{w}$. Solution for the deflection of membrane is following:
$\stackrel{-}{w}\left(r,\theta ,t\right)={e}^{i\omega t}\left[\sum _{m=0}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{A}_{mn}\left(\frac{i\rho \omega {R}^{2}}{T}\right)\mathrm{c}\mathrm{o}\
mathrm{s}\mathrm{h}\left(\frac{{k}_{mn}h}{R}\right)\frac{{J}_{m}\left(\frac{{k}_{mn}r}{R}\right)\mathrm{c}\mathrm{o}\mathrm{s}m\theta }{\left[{c}^{2}-{k}_{mn}^{2}\right]}\right$$\mathrm{}\mathrm{}\
mathrm{}\mathrm{}\mathrm{}\mathrm{}+\sum _{m=0}^{\mathrm{\infty }}{B}_{m}{\mathrm{J}}_{m}\left(\frac{cr}{R}\right)\mathrm{c}\mathrm{o}\mathrm{s}m\theta ],\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}
\mathrm{}\mathrm{}{c}^{2}=\frac{\mu {\omega }^{2}-\rho g}{T}.$
Using Eq. (5) at $z=h$, we obtain:
$\sum _{m=0}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{A}_{mn}\frac{{k}_{mn}}{R}{J}_{m}\left(\frac{{k}_{mn}r}{R}\right)\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}\left(\frac{{k}_{mn}h}{R}\right)\
mathrm{c}\mathrm{o}\mathrm{s}m\theta$$={\left(i\omega \right)}^{2}×\left[\sum _{m=0}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}\frac{i\omega {R}^{2}}{T}{A}_{mn}{J}_{m}\left(\frac{{k}_{mn}r}{R}\
right)\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\frac{{k}_{mn}h}{R}\right)\frac{\mathrm{c}\mathrm{o}\mathrm{s}m\theta }{\left[{c}^{2}-{{k}^{2}}_{mn}\right]}\right]×\sum _{m=0}^{\mathrm{\infty }}
{B}_{m}{\mathrm{J}}_{m}\left(\frac{{k}_{mn}r}{R}\right)\mathrm{c}\mathrm{o}\mathrm{s}m\theta .$
Boundary condition $w=$0, at $r=R$ gives:
$\sum _{m=0}^{\mathrm{\infty }}\sum _{n=1}^{\mathrm{\infty }}{A}_{mn}\left(\frac{i\rho \omega {R}^{2}}{T}\right)\frac{{J}_{m}\left({k}_{mn}\right)\mathrm{c}\mathrm{o}\mathrm{s}m\theta }{\left[{c}^{2}
-{k}_{mn}^{2}\right]}\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{h}\left(\frac{{k}_{mn}h}{R}\right)+\sum _{m=0}^{\mathrm{\infty }}{B}_{m}{J}_{m}\left(c\right)\mathrm{c}\mathrm{o}\mathrm{s}m\theta =0.$
To determine the unknowns ${A}_{mn}$, ${B}_{m}$ and unknown frequency $\stackrel{-}{\omega }$, given by the Eqs. (11) and (12) is tedious task. This problem was solved using method developed in [1,
2]. For simulations using BEM, first we consider the problem of free vibrations of membrane without interaction with the liquid. We have:
$\frac{{\partial }^{2}\stackrel{-}{w}}{\partial {r}^{2}}+\frac{1}{r}\frac{\partial \stackrel{-}{w}}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}\stackrel{-}{w}}{\partial {\theta }^{2}}-\frac{\
mu }{T}\frac{{\partial }^{2}\stackrel{-}{w}}{\partial {t}^{2}}=0,$
with $\frac{\partial \stackrel{-}{w}}{\partial r}=\stackrel{-}{w}=0$.
Supposing that $\stackrel{-}{w}={w}_{1}\left(r,\theta \right){e}^{i\stackrel{-}{\omega }t}$, one can obtain:
$L\left[{w}_{1}\right]+{\stackrel{-}{\omega }}^{2}\frac{\mu }{T}{w}_{1}=0,\mathrm{}\mathrm{}\mathrm{}\mathrm{}L\left[{w}_{1}\right]=\frac{{\partial }^{2}{w}_{1}}{\partial {r}^{2}}+\frac{1}{r}\frac{\
partial {w}_{1}}{\partial r}+\frac{1}{{r}^{2}}\frac{{\partial }^{2}{w}_{1}}{\partial {\theta }^{2}}$
The solutions of this problem are own modes and own frequencies, ${\stackrel{-}{\omega }}_{mk}\text{,}$${w}_{1mk}={w}_{mk}\left(r\right)\mathrm{c}\mathrm{o}\mathrm{s}m\theta$.
These modes and frequencies are reported in [14]. To obtain the solution of coupled problem the next series is used for each $m$:
$\stackrel{-}{w}\left(r,\theta ,z,t\right)=\mathrm{c}\mathrm{o}\mathrm{s}m\theta {\sum }_{k=1}^{{N}_{w}}{c}_{k}\left(t\right){w}_{mk}\left(r\right).$
Then the next representation for function $\varphi$ is obtained:
$\varphi \left(r,\theta ,z,t\right)=\mathrm{c}\mathrm{o}\mathrm{s}m\theta {\sum }_{k=1}^{{N}_{w}}{\stackrel{˙}{c}}_{k}\left(t\right){\stackrel{-}{\mathrm{\Phi }}}_{mk}\left(r\right).$
We obtain the system of ODEs of second order for each $m$ in the following form:
${A}_{kl}^{m}=\frac{\mu }{T}\left({w}_{mk},{w}_{ml}\right)+\rho \left({\stackrel{-}{\Phi }}_{mk},{w}_{ml}\right),\mathrm{}\mathrm{}\mathrm{}\mathrm{}{B}_{kl}^{m}=\left({\stackrel{-}{\omega }}_{mk}^
{2}+\rho g\right)\left({w}_{mk},{w}_{ml}\right).$
For receiving functions ${\stackrel{-}{\mathrm{\Phi }}}_{mk}$ the boundary element method is in used [13]. If we suppose that ${c}_{k}\left(t\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left(i\omega t\
right)$, we get the eigenvalue problem. We will seek the harmonic functions ${\stackrel{-}{\mathrm{\Phi }}}_{mk}$ in the form of the sum of simple and double layer potentials [16].
We shall indicate a moistened surface of a shell through ${S}_{1}$. The free surface of the liquid ${S}_{0}$ coincides with the plane $xy\left(orr,\theta \right)$ in unperturbed state. So, we have
the singular integral equations for receiving ${\stackrel{-}{\mathrm{\Phi }}}_{mk}$ in the following form:
$2\pi \stackrel{-}{\mathrm{\Phi }}\left({P}_{0}\right)+\underset{{S}_{1}}{\iint }\stackrel{-}{\mathrm{\Phi }}\frac{\partial }{\partial n}\frac{1}{\left|P-{P}_{0}\right|}d{S}_{1}=\underset{{S}_{0}}{\
iint }w\frac{1}{\left|P-{P}_{0}\right|}d{S}_{0}.$
This equation is reduced to one-dimensional one as in [15]. For numerical simulation, the boundary elements with constant density are applied. Here, ${N}_{0}$ is the number of boundary elements along
the free surface radius; ${N}_{w}$ along the shell wall, and ${N}_{bot}$ along the shell bottom. Consider the rigid circular cylindrical shell with the radius and height as $R=$ 1 m, and $h=$1 m,
respectively. Table 1 below provides the numerical values of the natural frequencies of liquid sloshing for $m=$1.
Table 1Slosh frequency parameters ωmk2/g of the fluid-filled rigid cylindrical shell
$m$ BEM $k=$1 $k=$2 $k=$3 $k=$4 $k=$5
${N}_{0}$ ${N}_{w}$ ${N}_{bot}$ $k=$1 $k=$2 $k=$3 $k=$4 $k=$5
25 20 20 1.6590 5.3301 8.5385 11.7071 14.8684
1 50 40 40 1.6579 5.3297 8.5372 11.7082 14.8655
100 80 80 1.6573 5.3293 8.5366 11.7066 14.8635
Analytical solution 1.6573 5.3293 8.5363 11.7060 14.8635
The results of Table 1 testify convergence of proposed BEM. In should be noted that the accuracy $\epsilon =$ 10^-4 has been achieved here for ${N}_{0}={N}_{w}={N}_{bot}=$ 80.
3. Results
Several numerical experiments have been performed to validate the accuracy of the proposed analytical method and determine the frequency. Eqs. (11) and (12) are solved to determine frequencies $\
Table 2Validation of numerical and analytical results
Frequency (Hz)
BEM, 40 Elements 0.95582 1.62832 2.05986 2.41284
Analytical Solution 0.95597 1.62777 2.05970 2.41198
The validation of numerical results using BEM with the deployed analytical approach is shown in Table 2. The results show a good convergence. For computations, the parameters of clamped Silicon
membrane are considered as the following: $R=$ 0.5 m, $\rho =$ 998 kg/m^3, thickness ${h}_{m}=$ 0.001 m, material density ${\rho }_{m}=$ 2800kg/m^3, Young modulus $E=$ 50 MPa, Poison’s ratio $v=$
0.49, and $h=$ 1 m. Fundamental modes of sloshing of liquid vibrations are shown in Fig. 2.
Fig. 2Fundamental slosh modes of liquid in a circular cylindrical container
Fig. 3Sloshing modes with membrane vibrations as Eva plastic roof on free surface
To see the modes of membrane, Eva plastic membrane is used at the free surface. The radius of the membrane is taken with 0.5 m, thickness ${h}_{m}=$ 0.001 m, material density as 950 kg/m^3. The
sloshing modes of the membrane are shown in Fig. 3.
4. Conclusions
In this paper, the sloshing in a right vertical circular cylindrical container in the presence of a membrane placed at the free surface is investigated. The fundamental sloshing modes of liquid with
and without membrane at the free surface are reported. An analytical approach used to determine sloshing frequencies is validated using Boundary Element Method (BEM). A good agreement is shown
between two, even for a small number of boundary elements. The main aim of this research was to validate both analytical and numerical methods using comparison of the results.
• Choudhary N., Bora S. N. Liquid sloshing in a circular cylindrical container containing a two-layer fluid. International Journal of Advances in Engineering Sciences and Applied Mathematics, Vol.
8, Issue 4, 2016, p. 240-248.
• Choudhary N., Bora S. N. Linear sloshing frequencies in the annular region of a circular cylindrical container in presence of a rigid baffle. Sadhana-Academy Proceedings in Engineering Sciences,
Springer, Vol. 42, Issue 5, 2017, p. 805-815.
• Falahaty H., Khayyer A., Gotoh H. Enhanced particle method with stress point integration for simulation of incompressible fluid-nonlinear elastic structure interaction. Journal of Fluids and
Structures, Vol. 81, 2018, p. 325-360.
• George A., Cho I. H. Anti sloshing effects of a vertical porous baffle in a rolling rectangular tank. Ocean Engineering, Vol. 214, 2020, p. 107871.
• Gotoh H., Khayyer A. On the state of the art of particle methods for coastal and ocean engineering. Coastal Engineering Journal, Vol. 60, Issue 1, 2018, p. 79-103.
• Goudarzi M. A., Danesh P. N. Numerical investigation of a vertically baffled rectangular tank under seismic excitation. Journal of Fluids and Structures, Vol. 61, 2018, p. 450-460.
• Hosseini M., Goudarzi M. A., Soroor Reduction of seismic sloshing in floating roof liquid storage tanks by using a suspended annular baffle (SAB). Journal of Fluids and Structures, Vol. 71, 2017,
p. 40-55.
• Ibrahim R. A. Liquid sloshing dynamics: theory and applications. Cambridge University Press, p. 970, 2005.
• Koh C. G., Luo M., Gao M., Bai W. Modelling of liquid sloshing with constrained floating baffle. Computers and Structures, Vol. 122, 2013, p. 270-279.
• Molin B., Remy F. Experimental and Numerical Study of the sloshing motion in a rectangular tank with a perforated screen. Journal of Fluids and Structures, Vol. 43, 2013, p. 463-480.
• Molin B., Remy F. Inertia effects in TLD sloshing with perforated screens. Journal of Fluids and Structures, Vol. 59, 2015, p. 165-177.
• Sakai F., Inouse R. Some considerations on seismic design and controls of sloshing floating roofed oil tanks. The 14th World conference on Earthquake Engineering, 2008.
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Engineering Sciences, Vol. 122, 2019, p. 13-25.
• Brebbia C. A., Telles J. C. F., And Wrobel L. C. Boundary Element Techniques. Springer-Verlag, Berlin and New York, 1984.
About this article
Modal analysis and applications
cylindrical container
The authors acknowledge the support of the Indo-Ukraine Joint Research Project Grant funded by DST, India, and Education and Science Ministry of Ukraine.
Copyright © 2021 Neelam Choudhary, et al.
This is an open access article distributed under the
Creative Commons Attribution License
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Avicorp has a
$ 13.2
million debt issue outstanding, with a
6.2 %
coupon rate. The...
Avicorp has a $ 13.2 million debt issue outstanding, with a 6.2 % coupon rate. The...
Avicorp has a
$ 13.2
million debt issue outstanding, with a
6.2 %
coupon rate. The debt has semi-annual coupons, the next coupon is due in six months, and the debt matures in five years. It is currently priced at
94 %
of par value.
a. What is Avicorp's pre-tax cost of debt? Note: Compute the effective annual return.
b. If Avicorp faces a
40 %
tax rate, what is its after-tax cost of debt?
Note: Assume that the firm will always be able to utilize its full interest tax shield.
Calculation is shown below
Working is shown below | {"url":"https://justaaa.com/finance/271358-avicorp-has-a-132-million-debt-issue-outstanding","timestamp":"2024-11-01T18:53:54Z","content_type":"text/html","content_length":"41312","record_id":"<urn:uuid:c09d9ec1-cf54-411f-bb0b-bf4d2ca0e511>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00559.warc.gz"} |
Science:Math Exam Resources/Courses/MATH110/April 2018/Question 06 (a)
MATH110 April 2018
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) •
Q7 (a) • Q7 (b) • Q7 (c) • Q7 (d) • Q7 (e) • Q8 (a) • Q8 (b) • Q8 (c) • Q8 (d) • Q9 • Q10 •
Question 06 (a)
Read the problem below and answer the questions in part (a) and (b). Makes sure your solution includes a sketch labeled consistently with variables in your calculations. Your asnwers should be a
numerical value, but you do not need to simplify it.
A small spider is crawling along the graph of a parabola ${\displaystyle y=x^{2}}$ in the first quadrant (where ${\displaystyle x}$ and ${\displaystyle y}$ are measured in cm) in such a way that its
${\displaystyle x}$-coordinate increases at a constant rate of ${\displaystyle 1{\text{cm}}/{\text{s}}}$. The spider is pulling a thin thread of silk with it that is fixed at the origin.
(a) When the spider is at the point ${\displaystyle (3,9)}$, how fast is the silk thread lengthening?
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is
correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still
stuck, go for the next hint.
Hint 1
Find the relation between the length of the silk thread and the ${\displaystyle x}$-coordinate of the location of the spider.
Hint 2
Denoting the ${\displaystyle x}$-coordinate of the location of the spider by ${\displaystyle x}$, we have ${\displaystyle {\frac {dx}{dt}}=1({\text{cm}}/{\text{s}})}$.
Hint 3
Use either the chain rule (Solution 1) or the implicit differentiation (Solution 2).
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
• If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you
are stuck or if you want to check your work.
• If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem
and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.
Solution 1
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
Let ${\displaystyle x}$ be the ${\displaystyle x}$-coordinate of the location of the spider in time ${\displaystyle t}$. Since the spider moves along the graph ${\displaystyle y=x^{2}}$, the location
of the spider in time ${\displaystyle t}$ is ${\displaystyle (x,x^{2})}$. Note that the length ${\displaystyle l}$ of the silk thread is the distance between two points ${\displaystyle (x,x^{2})}$
and ${\displaystyle (0,0)}$. Using the distance formula, we get
Since ${\displaystyle l}$ is a function of ${\displaystyle x}$ and ${\displaystyle x}$ is the function of ${\displaystyle t}$, by the chain rule, we have
By the Hint 2, we have ${\displaystyle {\frac {dx}{dt}}=1}$.
Since ${\displaystyle l}$ can be written as the composite of two functions ${\displaystyle l(x)=f(g(x))}$, where ${\displaystyle f(u)={\sqrt {u}}}$ and ${\displaystyle g(x)=x^{2}+x^{4}}$, we apply
the chain rule with ${\displaystyle f'(u)={\frac {1}{2{\sqrt {u}}}}}$ and ${\displaystyle g'(x)=2x+4x^{3}}$ to get
Plugging ${\displaystyle {\frac {dl}{dx}}}$ and ${\displaystyle {\frac {dx}{dt}}}$ into the formula for ${\displaystyle {\frac {dl}{dt}}}$, we obtain
At the location of the spider ${\displaystyle (3,9)}$, i.e., ${\displaystyle x=3}$, we therefore have
Answer: ${\displaystyle \color {blue}{\frac {19}{\sqrt {10}}}{\text{cm}}/{\text{s}}}$
Solution 2
Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies.
Let ${\displaystyle x}$ be the ${\displaystyle x}$-coordinate of the location of the spider in time ${\displaystyle t}$. Since the spider moves along the graph ${\displaystyle y=x^{2}}$, the location
of the spider in time ${\displaystyle t}$ is ${\displaystyle (x,x^{2})}$. Note that the length ${\displaystyle l}$ of the silk thread is the distance between two points ${\displaystyle (x,x^{2})}$
and ${\displaystyle (0,0)}$. Using the distance formula, we get
Now, we find ${\displaystyle {\frac {dl}{dt}}}$ using the implicit differentiation. Since
we take the derivative with respect to ${\displaystyle {\frac {d}{dt}}}$ to get On the other hand, we compute the derivative of the function ${\displaystyle u(x)=x^{2}+x^{4}}$ with respect to ${\
displaystyle t}$ using chain rule (note that x depends on time t): Therefore
Noting that ${\displaystyle {\frac {dx}{dt}}=1}$ and ${\displaystyle l(3)={\sqrt {3^{2}+9^{2}}}={\sqrt {90}}=3{\sqrt {10}}}$, we have
Finally we isolate the derivative of ${\displaystyle l}$ with respect to ${\displaystyle t}$ to get
Answer: ${\displaystyle \color {blue}{\frac {19}{\sqrt {10}}}{\text{cm}}/{\text{s}}}$
Click here for similar questions
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Aggregation: Mean
Aggregation: Mean#
Any constructors that have not completed the proof-writing and vetting process may still be accessed if you opt-in to “contrib”. Please contact us if you are interested in proof-writing. Thank you!
import opendp.prelude as dp
Known Dataset Size#
The much easier case to consider is when the dataset size is known:
input_space = dp.vector_domain(dp.atom_domain(bounds=(0., 10.)), size=10), dp.symmetric_distance()
sb_mean_trans = dp.t.make_mean(*input_space)
sb_mean_trans([5.] * 10)
The sensitivity of this transformation is the same as in make_sum (when dataset size is known), but is divided by size.
That is, \(map(d_{in}) = (d_{in} // 2) \cdot max(|L|, U) / size\), where \(//\) denotes integer division with truncation.
# since we are in the bounded-DP model, d_in should be a multiple of 2,
# because it takes one removal and one addition to change one record
Note that this operation does not divide by the length of the input data, it divides by the size parameter passed to the constructor. As in any other context, it is expected that the data passed into
the function is a member of the input domain, so no promises of privacy or correctness are guaranteed when the data is not in the input domain. In particular, the function may give a result with no
error message.
sb_mean_trans = dp.t.make_mean(*input_space)
You can check that a dataset is a member of a domain by calling .member:
In this case, [5.] is not a member because the input domain consists of vectors of length ten.
Unknown Dataset Size#
There are several approaches for releasing the mean when the dataset size is unknown.
The first approach is to use the resize transformation. You can separately release an estimate for the dataset size, and then preprocess the dataset with a resize transformation.
data = [5.] * 10
bounds = (0., 10.)
input_space = dp.vector_domain(dp.atom_domain(T=float)), dp.symmetric_distance()
# (where TIA stands for Atomic Input Type)
count_meas = input_space >> dp.t.then_count() >> dp.m.then_laplace(1.)
dp_count = count_meas(data)
mean_meas = (
input_space >>
dp.t.then_clamp(bounds) >>
dp.t.then_resize(dp_count, constant=5.) >>
dp.t.then_mean() >>
The total privacy expenditure is the composition of the count_meas and mean_meas releases.
from opendp.combinators import make_basic_composition
make_basic_composition([count_meas, mean_meas]).map(1)
Another approach is to compute the DP sum and DP count, and then postprocess the output.
dp_sum = input_space >> dp.t.then_clamp(bounds) >> dp.t.then_sum() >> dp.m.then_laplace(10.)
dp_count = input_space >> dp.t.then_count() >> dp.m.then_laplace(1.)
dp_fraction_meas = dp.c.make_basic_composition([dp_sum, dp_count])
dp_sum, dp_count = dp_fraction_meas(data)
print("dp mean:", dp_sum / dp_count)
print("epsilon:", dp_fraction_meas.map(1))
dp mean: 7.778118283305409
epsilon: 2.000000009313226
The same approaches are valid for the variance estimator. The Unknown Dataset Size notebook goes into greater detail on the tradeoffs of these approaches. | {"url":"https://docs.opendp.org/en/v0.8.0/user/transformations/aggregation-mean.html","timestamp":"2024-11-02T20:11:57Z","content_type":"text/html","content_length":"44474","record_id":"<urn:uuid:806d4b2a-cf90-447b-8469-51524184304a>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00167.warc.gz"} |
I Need Help Boys-girls
1.) B
2.) water
The role of the presenter in a group discussion to introduce information for the group to discuss.
What do you mean by group discussion?
A discussion involving a number of people who are connected by some shared activity, interest, or quality.
Group discussion is a tool to test your teamwork skills, listening skills, discussion ability, subject knowledge, and communication.
Intrinsic skills like reasoning, speaking and time management come in very handy.
Skills that you can work upon include presentation, summarizing and people speaking.
The purpose of a group discussion is not to win an argument or to amuse your classmates.
The purpose of a discussion is to help each group member explore and discover personal meanings of a text through interaction with other people.
Learn more about group discussion here: | {"url":"https://diemso.unix.edu.vn/question/i-need-help-boys-girls-br-narj","timestamp":"2024-11-15T00:31:11Z","content_type":"text/html","content_length":"68497","record_id":"<urn:uuid:48b773b9-33fa-4072-bf07-55050881b9b9>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00160.warc.gz"} |
template<typename Traits, typename V = Arr_vertex_base<typename Traits::Point_2>, typename H = Arr_halfedge_base<typename Traits::X_monotone_curve_2>, typename F = Arr_face_base>
class CGAL::Arr_dcel< Traits, V, H, F >
The DCEL class used by the Arrangement_2, Arr_bounded_planar_topology_traits_2, Arr_unb_planar_topology_traits_2 class templates and other templates.
It is parameterized by a geometry traits type and optionally by a vertex, halfedge, or face types. By default, the Arr_dcel class template uses the Point_2 and X_monotone_curve_2 types nested in the
traits type to instantiate the vertex and base halfedge types, respectively. Thus, by default the DCEL only stores the topological incidence relations and the geometric data attached to vertices and
edges. Any one of the vertex, halfedge, or face types can be overridden. Notice that if the vertex and halfedge types are overridden, the traits type is ignored.
Is model of
Template Parameters
Traits a geometry traits type, which is a model of the ArrangementBasicTraits_2 concept.
V the vertex type, which is a model of the ArrangementDcelVertex concept.
H the halfedge type, which is a model of the ArrangementDcelHalfedge concept.
F the face type, which is a model of the ArrangementDcelFace concept.
See also
Arr_dcel_base<V, H, F> | {"url":"https://doc.cgal.org:443/latest/Arrangement_on_surface_2/classCGAL_1_1Arr__dcel.html","timestamp":"2024-11-06T07:52:23Z","content_type":"application/xhtml+xml","content_length":"11940","record_id":"<urn:uuid:5d549805-cfda-41e4-b0d4-0e14178c1077>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00844.warc.gz"} |
In Java, what does NaN mean?
The NaN property means a "Not-a-Number" value. This property tells us that a value is not a legal number.
When there is an operation to produce some undefined result,Nan is produced. For example, any number divided by 0.0 is arithmetically undefined.Also, the square root of a negative number is also
undefined in maths, so it is Nan.
According to IEEE 754, there are two types of NaN: quiet and signaling.
"NaN" stands for "not a number".
"Nan" is created if a computer operation has input parameters that generate the procedure to create some indefinable result.
Get a detailed understanding of Java from scratch! Refer to this video provided by Intellipaat:
For example, 0.0 divided by zero.0 is arithmetically indefinable.
Taking the root of a negative number is also indefinable. | {"url":"https://intellipaat.com/community/287/in-java-what-does-nan-mean","timestamp":"2024-11-12T03:27:26Z","content_type":"text/html","content_length":"100290","record_id":"<urn:uuid:37f68e3c-b809-4219-b230-8b2b890fcbaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00506.warc.gz"} |
4.5 Comparing Distance Preserving Methods | An Introduction to Spatial Data Science with GeoDa
4.5 Comparing Distance Preserving Methods
Traditionally, the fit of a distance preserving dimension reduction method is assessed by means of the objective function used in the algorithm, such as the stress function for MDS or cost for t-SNE.
However, such measures are not comparable across methods. Another commonly used metric, the rank correlation between the original inter-observation distances and the corresponding distances in the
embedded space is comparable. However, this indicator may be less appropriate for t-SNE, since that method optimizes the match between close observations. In contrast, rank correlation is a global
correlation coefficient.
An alternative approach can be based on the concept of common coverage percentage, introduced in Section 3.5.2.2 as a way to measure the overlap between geographical k-nearest neighbors and k-nearest
neighbors in the embedded space. It was also used to assess the overlap between different MDS solutions. The common coverage percentage is a simple ratio of the density obtained by the intersection
of two k-nearest neighbor graphs to the maximum overlap possible. In the case of k-nearest neighbors, the maximum is \(k/n\), the property listed as % non-zero in the Weights Manager.
In addition to comparing different embedded solutions, this idea can also be applied to obtain an alternative overall measure of the fit of the embedded solution, relative to the k-nearest neighbors
in the multivariate attribute space. The maximum coverage is the same, since it is based on the same k for k-nearest neighbors. The only difference with the spatial case is that the reference weights
are based on the neighbor structure in multi-attribute variable space. The ratio is then computed of the density of the intersection with the weights for the embedded solution to this maximum.
4.5.1 Comparing t-SNE Options
As was the case for different MDS solutions, it is not straightforward to visually compare the scatter plots that result from manipulating the parameters of the t-SNE algorithm. As before, a solution
is to measure the degree of overlap between k-nearest neighbor graphs computed from the scatter plots (i.e., the scatter plot coordinates form the locational information).
For example, comparing the default t-SNE result in Figure 4.4 to the outcome from tuning the momentum and perplexity parameters in Figure 4.9 yields 1.43% non-zero elements in the corresponding
k-nearest neighbor graph intersection for k=6. This results in a common coverage percentage of 62.2%. This value is larger than the 46.5% found between classic metric MDS and SMACOF in Section
4.5.2 Local Fit with Common Coverage Percentage
As mentioned, the common coverage percentage logic can be used to compute a local measure of goodness of fit between the embedded space nearest neighbors and the multi-attribute nearest neighbors.
Figure 4.11 shows the results for classic metric MDS (Figure 3.3), SMACOF (using Euclidean distances, Figure 3.8) and default t-SNE (Figure 4.4). The common coverage percentage is computed for values
of k equal to 1, 2, 4, 6 and 8, corresponding with increasingly less local neighborhoods. In addition, for the sake of completeness, the rank correlation is listed as well.
As we had already seen previously, SMACOF does best on the rank correlation measure, with t-SNE yielding the worst results. In contrast, t-SNE does much better on the measures of local fit. The focus
on the local is illustrated by a slightly decreasing coverage percentage, from 54.81% for the nearest neighbor (k=1) to 50.90% for k=8. These percentages indicate more than 50% match between the
nearest neighbors. The classic MDS methods do much worse, with SMACOF consistently outperforming metric MDS, but achieving clearly inferior measures of local fit relative to t-SNE. As is to be
expected, the common coverage percentage improves with the number of neighbors, almost tripling for SMACOF, from 10.44% for k=1 to 28.38% for k=8.
These findings clearly illustrate the strong focus on local matches inherent in the t-SNE algorithm. | {"url":"https://lanselin.github.io/introbook_vol2/comparing-distance-preserving-methods.html","timestamp":"2024-11-08T19:15:21Z","content_type":"text/html","content_length":"55557","record_id":"<urn:uuid:94ee3d7b-f10b-4823-9f80-1ddcd1a810df>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00310.warc.gz"} |
Let’s take a quick look at what you have learnt in this session.
Some important points can be summarised as follows:
1. Firstly, you understood the limitations of preliminary machine learning and how deep learning can be used to build complex models.
2. Next, you saw how the architecture of ANNs draws inspiration from the human brain.
3. You also learnt about the basic functioning of a perceptron.
4. Further, you learnt about the basic building block of ANNs: Neurons. The structure of an artificial neuron is shown below.
Here, ‘a’ represents the inputs, ‘w’ represents the weights associated with the inputs, and ‘b’ represents the bias of the neuron.
5. You then learnt about the architecture of ANNs, including the topology, the parameters (weights and biases) on which the neural network is trained and the hyperparameters.
6. ANNs only take numerical inputs. Hence, you need to convert all types of data into a numeric format so that neural networks can process it.
7. Next, you were introduced to the most common activation functions such as sigmoid, ReLU, Leaky ReLU and tanh, which are shown below.
8. Some simplifying assumptions in the architecture of ANNs are as follows.
1. The neurons in an ANN are arranged in layers, and these layers are arranged sequentially.
2. The neurons within the same layer do not interact with each other.
3. The inputs are fed into the network through the input layer, and the outputs are sent out from the output layer.
4. Neurons in consecutive layers are densely connected, i.e., all neurons in layer l are connected to all neurons in layer l+1.
5. Every neuron in the neural network has a bias value associated with it, and each interconnection has a weight associated with it.
6. All neurons in a particular hidden layer use the same activation function.
9. Finally, you fixed the following notations:
1. W represents the weight matrix.
2. b stands for bias.
3. x represents input.
4. y represents the ground truth label.
5. p represents the probability vector of the predicted output for the classification problem.
6. h represents the output of the hidden layers, and hL represents the output prediction for the regression problem.
7. z represents the cumulative input fed into each neuron of a layer.
8. The superscript represents the layer number.
9. The subscript represents the index of each individual neuron in a layer.
In the next segment, you will attempt some graded questions to test your understanding of the topics covered in this session. All the best!
Report an error | {"url":"https://www.internetknowledgehub.com/summary-39/","timestamp":"2024-11-08T05:55:06Z","content_type":"text/html","content_length":"80580","record_id":"<urn:uuid:ebe33a25-bc82-4dae-a235-55bb0ff07e5e>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00425.warc.gz"} |
Convert a vector from factor to numeric
unfactor {dmm} R Documentation
Convert a vector from factor to numeric
Convert a vector (such as a dataframe column) from factor to numeric. Non-numeric values will coerce to NA.
x A vector of type factor. Typically this would be one column of a dataframe.
This function may be useful when preparing a dataframe for dmm(). It is a common problem for dataframe columns to be automatically made type factor when constructing the dataframe with functions such
as read.table, due to the presence of a small number of non-numeric values. Dataframe columns used as traits or as covariates should not be of type factor.
A vector of numeric values is returned.
Neville Jackson
See Also
Functions dmm(), read.table()
tmp <- as.factor(c(1,2,3))
utmp <- unfactor(tmp)
version 2.1-10 | {"url":"https://search.r-project.org/CRAN/refmans/dmm/html/unfactor.html","timestamp":"2024-11-12T23:14:20Z","content_type":"text/html","content_length":"2702","record_id":"<urn:uuid:fe3620d6-7130-40a4-b2d3-45ba6bb66ca4>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00265.warc.gz"} |
American Mathematical Society
Nakajima’s problem for general convex bodies
HTML articles powered by AMS MathViewer
by Daniel Hug
Proc. Amer. Math. Soc. 137 (2009), 255-263
DOI: https://doi.org/10.1090/S0002-9939-08-09432-X
Published electronically: July 8, 2008
PDF | Request permission
For a convex body $K\subset \mathbb {R}^n$, the $k$th projection function of $K$ assigns to any $k$-dimensional linear subspace of $\mathbb {R}^n$ the $k$-volume of the orthogonal projection of $K$
to that subspace. Let $K$ and $K_0$ be convex bodies in $\mathbb {R}^n$, and let $K_0$ be centrally symmetric and satisfy a weak regularity assumption. Let $i,j\in \mathbb {N}$ be such that $1\le i<j
\le n-2$ with $(i,j)\neq (1,n-2)$. Assume that $K$ and $K_0$ have proportional $i$th projection functions and proportional $j$th projection functions. Then we show that $K$ and $K_0$ are homothetic.
In the particular case where $K_0$ is a Euclidean ball, we thus obtain characterizations of Euclidean balls as convex bodies having constant $i$-brightness and constant $j$-brightness. This special
case solves Nakajima’s problem in arbitrary dimensions and for general convex bodies for most indices $(i,j)$. References
• A. D. Alexandroff, Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it, Leningrad State Univ. Annals [Uchenye
Zapiski] Math. Ser. 6 (1939), 3–35 (Russian). MR 0003051
• Christina Bauer, Intermediate surface area measures and projection functions of convex bodies, Arch. Math. (Basel) 64 (1995), no. 1, 69–74. MR 1305662, DOI 10.1007/BF01193552
• Stefano Campi, Reconstructing a convex surface from certain measurements of its projections, Boll. Un. Mat. Ital. B (6) 5 (1986), no. 3, 945–959 (English, with Italian summary). MR 871707
• G. D. Chakerian, Sets of constant relative width and constant relative brightness, Trans. Amer. Math. Soc. 129 (1967), 26–37. MR 212678, DOI 10.1090/S0002-9947-1967-0212678-1
• G. D. Chakerian and H. Groemer, Convex bodies of constant width, Convexity and its applications, Birkhäuser, Basel, 1983, pp. 49–96. MR 731106
• G. D. Chakerian and E. Lutwak, Bodies with similar projections, Trans. Amer. Math. Soc. 349 (1997), no. 5, 1811–1820. MR 1390034, DOI 10.1090/S0002-9947-97-01760-1
• Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved problems in geometry, Problem Books in Mathematics, Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive
Mathematics, II. MR 1107516, DOI 10.1007/978-1-4612-0963-8
• William J. Firey, Convex bodies of constant outer $p$-measure, Mathematika 17 (1970), 21–27. MR 267465, DOI 10.1112/S0025579300002667
• Richard J. Gardner, Geometric tomography, Encyclopedia of Mathematics and its Applications, vol. 58, Cambridge University Press, Cambridge, 1995. MR 1356221
• R. J. Gardner and A. Volčič, Tomography of convex and star bodies, Adv. Math. 108 (1994), no. 2, 367–399. MR 1296519, DOI 10.1006/aima.1994.1075
• P. Goodey, R. Schneider, and W. Weil, Projection functions of convex bodies, Intuitive geometry (Budapest, 1995) Bolyai Soc. Math. Stud., vol. 6, János Bolyai Math. Soc., Budapest, 1997,
pp. 23–53. MR 1470754
• Paul Goodey, Rolf Schneider, and Wolfgang Weil, On the determination of convex bodies by projection functions, Bull. London Math. Soc. 29 (1997), no. 1, 82–88. MR 1416411, DOI 10.1112/
• P. Goodey, R. Howard, Examples and structure of smooth convex bodies of constant $k$-brightness (in preparation).
• Paul Goodey and Gaoyong Zhang, Inequalities between projection functions of convex bodies, Amer. J. Math. 120 (1998), no. 2, 345–367. MR 1613642
• Eric Grinberg and Gaoyong Zhang, Convolutions, transforms, and convex bodies, Proc. London Math. Soc. (3) 78 (1999), no. 1, 77–115. MR 1658156, DOI 10.1112/S0024611599001653
• E. Heil and H. Martini, Special convex bodies, Handbook of convex geometry, Vol. A, B, North-Holland, Amsterdam, 1993, pp. 347–385. MR 1242985
• Ralph Howard, Convex bodies of constant width and constant brightness, Adv. Math. 204 (2006), no. 1, 241–261. MR 2233133, DOI 10.1016/j.aim.2005.05.015
• Ralph Howard and Daniel Hug, Smooth convex bodies with proportional projection functions, Israel J. Math. 159 (2007), 317–341. MR 2342484, DOI 10.1007/s11856-007-0049-z
• R. Howard, D. Hug, Nakajima’s problem: convex bodies of constant width and constant brightness, Mathematika (to appear).
• Daniel Hug, Contributions to affine surface area, Manuscripta Math. 91 (1996), no. 3, 283–301. MR 1416712, DOI 10.1007/BF02567955
• Daniel Hug, Absolute continuity for curvature measures of convex sets. II, Math. Z. 232 (1999), no. 3, 437–485. MR 1719698, DOI 10.1007/PL00004765
• P. McMullen, On the inner parallel body of a convex body, Israel J. Math. 19 (1974), 217–219. MR 367810, DOI 10.1007/BF02757715
• S. Nakajima, Eine charakteristische Eigenschaft der Kugel, Jber. Deutsche Math.-Verein 35 (1926), 298–300.
• Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and its Applications, vol. 44, Cambridge University Press, Cambridge, 1993. MR 1216521, DOI 10.1017/
• Rolf Schneider, Polytopes and Brunn-Minkowski theory, Polytopes: abstract, convex and computational (Scarborough, ON, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 440, Kluwer Acad.
Publ., Dordrecht, 1994, pp. 273–299. MR 1322067
Similar Articles
• Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 52A20, 52A39, 53A05
• Retrieve articles in all journals with MSC (2000): 52A20, 52A39, 53A05
Bibliographic Information
• Daniel Hug
• Affiliation: Fakultät für Mathematik, Institut für Algebra und Geometrie, Universität Karlsruhe (TH), KIT, D-76128 Karlsruhe, Germany
• MR Author ID: 363423
• Email: daniel.hug@kit.edu
• Received by editor(s): July 12, 2007
• Received by editor(s) in revised form: November 20, 2007
• Published electronically: July 8, 2008
• Additional Notes: The author was supported in part by the European Network PHD, FP6 Marie Curie Actions, RTN, Contract MCRN-511953.
• Communicated by: N. Tomczak-Jaegermann
• © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
• Journal: Proc. Amer. Math. Soc. 137 (2009), 255-263
• MSC (2000): Primary 52A20; Secondary 52A39, 53A05
• DOI: https://doi.org/10.1090/S0002-9939-08-09432-X
• MathSciNet review: 2439448 | {"url":"https://www.ams.org/journals/proc/2009-137-01/S0002-9939-08-09432-X/?active=current","timestamp":"2024-11-02T06:07:43Z","content_type":"text/html","content_length":"74713","record_id":"<urn:uuid:c81c8e73-5a7c-484f-ba9a-430e1bf38029>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00454.warc.gz"} |
College Publications - Studies in Logic
New Directions in Term Logic
George Englebretsen, editor
The systematic account of deductive reasoning and the development of a formal logic to reveal the principles of such reasoning began with Aristotle's syllogistic. It was a term logic, a logic that
dominated the field until the rise of modern predicate logic at the end of the Nineteenth century. That system quickly supplanted the old logic of terms. However, in the middle of the Twentieth
century Fred Sommers took up the challenge to build a revised and strengthened term logic, one fit to challenge the hegemony of predicate logic. The aim was to devise a formal logic that could better
serve as the logic of natural language.
In recent years a new group of logicians have taken this version of term logic into many new directions. This book presents here for the first time some of the best examples of their work with new
essays by philosophers, logicians, mathematicians, computational theorists, and historians of logic. The topics addressed include relative terms, logical copulation, Aristotelian diagrams, modality,
truth, semantics, epistemic term logic, non-classical quantifiers, and more.
July 2024
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Extract draws of variables in a Bayesian model fit into a tidy data format — gather_draws
Imagine a JAGS or Stan fit named model. The model may contain a variable named b[i,v] (in the JAGS or Stan language) with dimension i in 1:100 and dimension v in 1:3. However, the default format for
draws returned from JAGS or Stan in R will not reflect this indexing structure, instead they will have multiple columns with names like "b[1,1]", "b[2,1]", etc.
spread_draws and gather_draws provide a straightforward syntax to translate these columns back into properly-indexed variables in two different tidy data frame formats, optionally recovering
dimension types (e.g. factor levels) as it does so.
spread_draws and gather_draws return data frames already grouped by all dimensions used on the variables you specify.
The difference between spread_draws is that names of variables in the model will be spread across the data frame as column names, whereas gather_draws will gather variables into a single column named
".variable" and place values of variables into a column named ".value". To use naming schemes from other packages (such as broom), consider passing results through functions like to_broom_names() or
For example, spread_draws(model, a[i], b[i,v]) might return a grouped data frame (grouped by i and v), with:
• column ".chain": the chain number. NA if not applicable to the model type; this is typically only applicable to MCMC algorithms.
• column ".iteration": the iteration number. Guaranteed to be unique within-chain only. NA if not applicable to the model type; this is typically only applicable to MCMC algorithms.
• column ".draw": a unique number for each draw from the posterior. Order is not guaranteed to be meaningful.
• column "i": value in 1:5
• column "v": value in 1:10
• column "a": value of "a[i]" for draw ".draw"
• column "b": value of "b[i,v]" for draw ".draw"
gather_draws(model, a[i], b[i,v]) on the same model would return a grouped data frame (grouped by i and v), with:
• column ".chain": the chain number
• column ".iteration": the iteration number
• column ".draw": the draw number
• column "i": value in 1:5
• column "v": value in 1:10, or NA if ".variable" is "a".
• column ".variable": value in c("a", "b").
• column ".value": value of "a[i]" (when ".variable" is "a") or "b[i,v]" (when ".variable" is "b") for draw ".draw"
spread_draws and gather_draws can use type information applied to the model object by recover_types() to convert columns back into their original types. This is particularly helpful if some of the
dimensions in your model were originally factors. For example, if the v dimension in the original data frame data was a factor with levels c("a","b","c"), then we could use recover_types before
Which would return the same data frame as above, except the "v" column would be a value in c("a","b","c") instead of 1:3.
For variables that do not share the same subscripts (or share some but not all subscripts), we can supply their specifications separately. For example, if we have a variable d[i] with the same i
subscript as b[i,v], and a variable x with no subscripts, we could do this:
Which is roughly equivalent to this:
Similarly, this:
Is roughly equivalent to this:
The c and cbind functions can be used to combine multiple variable names that have the same dimensions. For example, if we have several variables with the same subscripts i and v, we could do either
of these:
Each of which is roughly equivalent to this:
Besides being more compact, the c()-style syntax is currently also faster (though that may change).
Dimensions can be omitted from the resulting data frame by leaving their names blank; e.g. spread_draws(model, b[,v]) will omit the first dimension of b from the output. This is useful if a dimension
is known to contain all the same value in a given model.
The shorthand .. can be used to specify one column that should be put into a wide format and whose names will be the base variable name, plus a dot ("."), plus the value of the dimension at ... For
spread_draws(model, b[i,..]) would return a grouped data frame (grouped by i), with:
• column ".chain": the chain number
• column ".iteration": the iteration number
• column ".draw": the draw number
• column "i": value in 1:20
• column "b.1": value of "b[i,1]" for draw ".draw"
• column "b.2": value of "b[i,2]" for draw ".draw"
• column "b.3": value of "b[i,3]" for draw ".draw"
An optional clause in the form | wide_dimension can also be used to put the data frame into a wide format based on wide_dimension. For example, this:
is roughly equivalent to this:
The main difference between using the | syntax instead of the .. syntax is that the | syntax respects prototypes applied to dimensions with recover_types(), and thus can be used to get columns with
nicer names. For example:
would return a grouped data frame (grouped by i), with:
• column ".chain": the chain number
• column ".iteration": the iteration number
• column ".draw": the draw number
• column "i": value in 1:20
• column "a": value of "b[i,1]" for draw ".draw"
• column "b": value of "b[i,2]" for draw ".draw"
• column "c": value of "b[i,3]" for draw ".draw"
The shorthand . can be used to specify columns that should be nested into vectors, matrices, or n-dimensional arrays (depending on how many dimensions are specified with .).
For example, spread_draws(model, a[.], b[.,.]) might return a data frame, with:
• column ".chain": the chain number.
• column ".iteration": the iteration number.
• column ".draw": a unique number for each draw from the posterior.
• column "a": a list column of vectors.
• column "b": a list column of matrices.
Ragged arrays are turned into non-ragged arrays with missing entries given the value NA.
Finally, variable names can be regular expressions by setting regex = TRUE; e.g.:
Would return a tidy data frame with variables starting with b_ and having one dimension. | {"url":"http://mjskay.github.io/tidybayes/reference/spread_draws.html","timestamp":"2024-11-08T01:41:10Z","content_type":"text/html","content_length":"43783","record_id":"<urn:uuid:8e32d3a0-aa0d-4547-a583-f386e6bb1c7c>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00263.warc.gz"} |
Symmetry rulesSymmetry rules – Science in School
Everyone knows what symmetry is. In this article, though, Mario Livio from the Space Telescope Science Institute, Baltimore, USA, explains how not only shapes, but also laws of nature, can be
Everybody will recognize the inkblot below left as being symmetrical, but few know that the figure below right is also considered symmetrical in the precise mathematical sense. So, what is symmetry,
really? And why has this concept become so pivotal that many scientists believe it to be the basis of the laws of nature?
…but so is this!
This inkblot is obviously
When things that could have changed, don’t
Symmetry represents immunity to possible alterations — those stubborn cores of shapes, phrases, laws, or mathematical expressions that remain unchanged under certain transformations. Consider, for
instance, the phrase “Madam, I’m Adam”, which is symmetrical when read back to front, letter by letter. That is, the sentence remains the same when read backwards.
A butterfly’s bilateral
The title of the documentary, A Man, a Plan, a Canal, Panama, has the same property. Phrases with this type of symmetry are known as palindromes, and palindromes play an important role in the
structure of the male-defining Y chromosome. Until 2003, genome biologists believed that, due to the fact that the Y chromosome lacks a partner (with which it could swap genes), its genetic cargo was
about to dwindle away through damaging mutations. To their surprise, however, the researchers who sequenced the Y chromosome discovered that it fights destruction with palindromes. About 6 million
(out of 50 million) of the chromosome’s DNA letters form palindromic sequences. These ‘mirror’ copies provide backups in the case of damaging mutations, and allow the chromosome, in some sense, to
have sex with itself — strands can swap position.
For two-dimensional figures and shapes, like those drawn on a piece of paper, there are precisely four types of ‘rigid’ symmetry (when stretching and distortions are not allowed), known as:
reflection, rotation, translation, and glide reflection.
A snowflake is symmetrical
under rotation
We encounter symmetry under reflection all around us — this is the familiar bilateral symmetry that characterises animals. Draw a line down the middle of a picture of a butterfly (right). Now flip it
over, while keeping the central line in place. The resulting perfect overlap indicates that the butterfly remains unchanged under reflection about its central line.
Many letters of the alphabet also have this property. If you hold a sheet of paper up to a mirror with the phrase ‘MAX IT WITH MATH’ written vertically, it looks the same.
Symmetry under rotation is also very prevalent in nature. A snowflake (right) rotated through 60, 120, 180, 240, 300, or 360 degrees about an axis through its centre (perpendicular to its plane)
leads to an indistinguishable configuration. A circle rotated through any angle about a central, perpendicular axis will remain unaltered.
Symmetry under translation is the type of immunity to change that is encountered in recurring, repeating motifs, such as the one in the second figure. Translation means a displacement or shift, by a
certain distance, along a particular line. Many classical friezes, wallpaper designs, rows of windows in high-rise apartment buildings, and even centipedes, exhibit this type of symmetry.
Finally, the footprints generated by a left-right-left-right walk are symmetrical under glide reflection (see below). The transformation in this case consists of a translation (or glide), followed by
a reflection in a line parallel to the direction of the displacement (the dotted line).
Footprints are preserved by glide reflection
All of the symmetries discussed so far are symmetries of shape and form — ones that we can actually see with our own eyes. The symmetries underlying the fundamental laws of nature are in some sense
closely related to these, but instead of focusing on form or figure they address a different question: what transformations can be performed on the world around us that would leave unchanged the laws
describing all observed phenomena?
Symmetry rules
The ‘laws of nature’ collectively describe a body of rules that are supposed to explain literally everything we observe in the universe. That such a grand set of rules even exists was inconceivable
before the 17th century. Only through the works of scientific giants such as Galileo Galilei (1564-1642), René Descartes (1596-1650), and in particular, Isaac Newton (1642-1727), did it become clear
that a mere handful of laws could explain a wide range of phenomena. Suddenly, things as diverse as the falling of apples, tides on the beach, and the motion of the planets all fell under the
umbrella of Newton’s law of gravitation.
Similarly, building on the impressive experimental results of Michael Faraday (1791-1867), the Scottish physicist James Clerk Maxwell (1831-1879) was able to explain all the classical electric,
magnetic, and light phenomena with just four equations! Think about this for a moment – the entire world of electromagnetism in four equations.
The laws of nature were found to obey some of the same symmetries we have already encountered, as well as a few other, more esoteric, ones. To begin with, the laws are symmetrical under translation.
The manifestation of this property is simple: whether you perform an experiment in New York or Los Angeles, at the other edge of the Milky Way or in a galaxy a billion light-years from here, you will
be able to describe the results using the same laws. How do we know this to be true? Because observations of galaxies all across the universe show not only that the law of gravity is the same there
as here, but also that hydrogen atoms at the edge of the observable universe obey precisely the same laws of electromagnetism and quantum mechanics as they obey here on Earth.
Newton’s law of gravity may
be symmetrical under
rotation, but this doesn’t
mean the orbits are
The laws of nature are also symmetrical with respect to rotation – the laws look precisely the same whether we measure directions with respect to north or the nearest coffee shop – physics has no
preferred direction in space.
If it were not for this remarkable symmetry of the laws under translation and rotation, there would be no hope of ever understanding different parts of the cosmos. Furthermore, even here on Earth, if
the laws were not symmetrical, experiments would have to be repeated in every laboratory across the globe.
A word of caution is needed to distinguish between symmetries of shapes and symmetries of laws. The ancient Greeks thought that the orbits of the planets around the sun were symmetrical with respect
to rotation: circular. In fact, it is not the shape of the orbit, but Newton’s law of gravity that is symmetrical under rotation. This means that the orbits can be (and indeed are!) elliptical, but
that the orbits can have any orientation in space (left).
In my opening paragraph, I made a statement stronger than merely saying that the laws obey certain symmetries; I said that symmetry may be the source of laws. What does this mean?
The source of natural laws
Imagine that you have never heard of snowflakes before, and someone asks you to guess the shape of one. Clearly, this is an impossible task. For all you know, the snowflake may look like a teapot,
like the letter S, or like Bugs Bunny.
Trying to reconstruct a
Even if you are given the shape of one ray of the snowflake (right, a) and are told that this is part of its total shape, this is not much help. The snowflake could still look, for example, like the
configuration right, b. If you are told, on the other hand, that the snowflake is symmetrical under rotations through 60 degrees about its centre, this information can be used very effectively. The
symmetry immediately limits the possible configurations to six-cornered, twelve-cornered, eighteen-cornered, and so on, snowflakes. Assuming, based on experience, that nature would opt for the
simplest, most economical solution, a six-cornered snowflake (right, c) would be a very reasonable guess. In other words, the requirement of the symmetry of the shape has guided us in the right
In the same way, the requirement that the laws of nature would be symmetrical under certain transformations not only dictates the form of these laws, but also, in some cases, necessitates the
existence of forces or of yet undiscovered elementary particles. Let me explain, using two interesting examples.
One of Einstein’s main goals in his explanation of general relativity was to formulate a theory in which the laws of nature would look precisely the same to all observers. That is, the laws had to be
symmetrical under any change in our point of view in space and time (in physics, this is known as ‘general covariance’). An observer sitting on the back of a giant turtle should deduce the same laws
as an observer on a merry-go-round or in an accelerating rocket. Indeed, if the laws are to be universal, why should they depend on whether the observer is accelerating?
Although Einstein’s symmetry requirement was certainly reasonable, it was by no means trivial. After all, a million whiplash injuries per year in the United States alone demonstrate that we feel
acceleration. Every time an aeroplane hits an air pocket, we feel our stomachs leap into our throats — there appears to be an unmistakable distinction between uniform and accelerating motion. So how
can the laws of nature be the same for accelerating observers, when these observers appear to experience additional forces?
Consider the following situation. If you stand on bathroom scales inside an elevator that is accelerating upward, your feet exert a greater pressure on the scales — the scales will register a higher
weight (below, a). The same would happen, however, if gravity somehow became stronger in a static elevator. An elevator accelerating downward would feel just like weaker gravity (below, b). If the
elevator’s cable snapped, you and the scales would free-fall in unison, and the scales would register zero weight (below, c). Free-fall is therefore equivalent to someone miraculously switching
gravity off. This led Einstein in 1907 to a ground-breaking conclusion: the force of gravity and the force resulting from acceleration are in fact one and the same. This powerful unification has been
dubbed the ‘equivalence principle’, implying that acceleration and gravity are really two facets of the same force — they are equivalent.
Weight control, elevator style – gaining weight when the elevator accelerates upward (a), losing weight when it accelerates downward (b) and achieving weightlessness when it free-falls (c)
In a lecture delivered in Kyoto in 1922, Einstein described that moment of epiphany he had in 1907: “I was sitting in the patent office in Bern when all of a sudden a thought occurred to me: if a
person falls freely, he won’t feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.”
The equivalence principle is really a statement of a pervasive symmetry; the laws of nature — as expressed by Einstein’s equations of general relativity — are the same in all systems, including
accelerating ones. So why are there apparent differences between what is observed on a merry-go-round and in a laboratory at rest? General relativity provides a surprising answer. They are
differences only in the environment, not in the laws themselves. Similarly, the directions of up and down only appear to be different on Earth because of the Earth’s gravity. The laws of nature
themselves have no preferred direction (they are symmetrical under rotation); they do not distinguish between up and down. Observers on a merry-go-round, according to general relativity, feel the
centrifugal force that is equivalent to gravity. The conclusion is truly electrifying: the symmetry of the laws under any change in the space-time co-ordinates necessitates the existence of gravity!
This explains why symmetry is the source of forces. The requirement of symmetry leaves nature no choice: gravity must exist.
Dr. Mario Livio, a Senior Astrophysicist at the Space Telescope Science Institute, gives a very interesting account of the symmetry of the laws of nature. For figures and shapes drawn on a piece of
paper, there are four types of symmetries: reflection, rotation, translation, and glide reflection. How can these be applied to the laws of nature? Are the laws of nature symmetrical? And which
transformations can be performed on them so that the laws remain unchanged?
Although not directly connected with curriculum material for school science, this article will surely interest all science teachers who would like to improve their understanding of the laws that
govern the universe. Mathematics teachers would find this article of particular interest. | {"url":"https://www.scienceinschool.org/article/2006/symmetry/","timestamp":"2024-11-09T19:27:30Z","content_type":"text/html","content_length":"94110","record_id":"<urn:uuid:ec921206-1701-432b-83f3-91c7775ede81>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00704.warc.gz"} |
The PISA2009lite package is released | R-bloggersThe PISA2009lite package is released
The PISA2009lite package is released
[This article was first published on
SmarterPoland » PISA in english
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
This post introduces a new R package named PISA2009lite. I will show how to install this package, what is inside and how to use it.
PISA (Programme for International Student Assessment) is a worldwide study focused on measuring performance of 15-year-old school pupils. More precisely, scholastic performance on mathematics,
science and reading is measured in more than 500 000 pupils from 65 countries.
First PISA study was performed in 2000, second in 2003, and then in 2006, 2009 and the last one in 2012. Data from the last study will be made public on December 2013. Data from previous studies are
accessible through the PISA website http://www.oecd.org/pisa/.
Note that this data set is quite large. This is why the PISA2009lite package will use more than 220MB of your disk [data sets are compressed on disk] and much more RAM [data sets will be decompressed
after [lazy]loading to R].
Let's see some numbers. In PISA 2009 study, number of examined pupils: 515 958 (437 variables for each pupil), number of examined parents: 106 287 (no, their questions are not related to scholastic
performance), number of schools from which pupils were sampled: 18 641. Pretty large, complex and interesting dataset!
On the official PISA webpage there are instructions how to read data from 2000-2009 studies into SAS and SPSS statistical packages. I am transforming these dataset to R packags, to make them easier
to use for R users.
Right now, PISA2009lite is mature enough to share it. There are still many things to correct/improve/add. Fell free to point them [or fix them].
This is not the first attempt to get the PISA data available for R users. On the github you can find 'pisa' package maintained by Jason Bryer (https://github.com/jbryer/pisa) with data from PISA 2009
But since I need data from all PISA editions, namely 2000, 2003, 2006, 2009 and 2012 I've decided to create few new packages, that will support consistent way to access data from different PISA
Open the R session
The package is on github, so in order to install it you need just
# dont download 220MB of compressed data if the package is already
# installed
if (length(find.package("PISA2009lite", quiet = TRUE)) == 0) install_github("PISA2009lite",
now PISA2009lite is ready to be loaded
You will find five data sets in this package [actually ten, I will explain this later]. These are: data from student questionnaire, school questionnaire, parent questionnaire, cognitive items and
scored cognitive items.
## [1] 515958 437
## [1] 106287 90
## [1] 18641 247
## [1] 515958 273
## [1] 515958 227
For most of variables in each data set there is a dictionary which decode answers for particular question. Dictionaries for all questions for a given data set are stored as a list of named vectors,
these lists are named after corresponding data sets [just add suffix 'dict'].
For example fist six entries in a dictionary for variable CNT in the data set student2009.
## ALB ARG AUS AUT AZE
## "Albania" "Argentina" "Australia" "Austria" "Azerbaijan"
## BEL
## "Belgium"
You can do a lot of things with these data sets. And I am going to show some examples in next posts.
Country ranking in just few lines of code
But as a warmer let's use it to calculate average performance in mathematics for each country.
Note that student2009$W_FSTUWT stands for sampling weights, student2009$PV1MATH stands for first plausible value from MATH scale while student2009$CNT stands for country
means <- unclass(by(student2009[, c("PV1MATH", "W_FSTUWT")], student2009[, "CNT"],
function(x) weighted.mean(x[, 1], x[, 2])))
# sort them
means <- sort(means)
Let's add proper country names [here dictionaries are useful] and plot it.
names(means) <- student2009dict$CNT[names(means)]
dotchart(means, pch = 19)
abline(v = seq(350, 600, 50), lty = 3, col = "grey") | {"url":"https://www.r-bloggers.com/2013/06/the-pisa2009lite-package-is-released/","timestamp":"2024-11-11T14:59:42Z","content_type":"text/html","content_length":"91685","record_id":"<urn:uuid:cafb92d4-fd56-47dd-80b8-28114820fa20>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00604.warc.gz"} |
CHISQ.DIST.RT - Excel docs, syntax and examples
The CHISQ.DIST.RT function calculates the right-tailed probability of the chi-squared distribution, which is commonly used in statistical analysis. This function is useful for determining the
likelihood of observing a certain chi-squared value or higher in a chi-squared distribution.
=CHISQ.DIST.RT(x, degrees_freedom)
x The value at which to evaluate the chi-squared distribution.
degrees_freedom The number of degrees of freedom for the chi-squared distribution. It must be a positive integer.
About CHISQ.DIST.RT 🔗
When dealing with statistical analysis and the need to assess the probability associated with a chi-squared value, turn to the CHISQ.DIST.RT function in Excel. This function offers a reliable
approach to determine the likelihood of a chi-squared value equal to or greater than a specific value in a right-tailed chi-squared distribution. It aids in gauging the significance of observed
differences or relationships in statistical studies, providing valuable insights for decision-making in research and data analysis endeavors. The CHISQ.DIST.RT function relies on the chi-squared
distribution and the degrees of freedom to yield the probability associated with the given chi-squared value in the right tail of the distribution, offering a practical tool for statistical
evaluations and hypothesis testing.
Examples 🔗
Suppose a statistical study produces a chi-squared value of 8.2 with 5 degrees of freedom. To calculate the right-tailed probability of observing a chi-squared value equal to or greater than 8.2 in a
chi-squared distribution with 5 degrees of freedom, use the CHISQ.DIST.RT formula: =CHISQ.DIST.RT(8.2, 5). This will return the probability of observing a chi-squared value of 8.2 or higher in the
right tail of the chi-squared distribution with 5 degrees of freedom.
The CHISQ.DIST.RT function assumes that the provided degrees of freedom is a positive integer. Ensure that the degrees of freedom value accurately reflects the constraints on the variables in your
statistical analysis to obtain meaningful results. Additionally, the x value used in the CHISQ.DIST.RT function should correspond to a valid chi-squared value in the context of your statistical
Questions 🔗
How does the CHISQ.DIST.RT function calculate the right-tailed probability?
The CHISQ.DIST.RT function calculates the right-tailed probability based on the chi-squared distribution and the specified degrees of freedom. It evaluates the likelihood of observing a chi-squared
value equal to or greater than the given x value in the right tail of the distribution.
What is the significance of the degrees of freedom in the CHISQ.DIST.RT function?
The degrees of freedom in the CHISQ.DIST.RT function represent the constraints or independent variables in the statistical analysis. It influences the shape and variability of the chi-squared
distribution, thereby impacting the resulting probability of observing a specific chi-squared value or higher.
Can the CHISQ.DIST.RT function be used for left-tailed probabilities?
No, the CHISQ.DIST.RT function specifically calculates the right-tailed probability of the chi-squared distribution. To determine left-tailed probabilities, consider using the CHISQ.DIST function in
Excel, which provides the cumulative probability of the chi-squared distribution up to a given value.
Related functions 🔗
Leave a Comment | {"url":"https://spreadsheetcenter.com/excel-functions/chisq-dist-rt/","timestamp":"2024-11-05T01:37:33Z","content_type":"text/html","content_length":"29550","record_id":"<urn:uuid:44baa490-f04c-49b2-b435-e6f877ae4ee4>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00194.warc.gz"} |
A computational modeling of primary-microRNA expressionA computational modeling of primary-microRNA expression
MicroRNAs (miRNAs) play crucial roles in gene regulation. Most studies so far focus on mature miRNAs, which leaves many gaps in our knowledge in primary miRNAs (pri-miRNA). To fill these gaps, we
attempted to model the expression of pri-miRNAs in 1829 primary cell types and tissues in this study. We demonstrated that the expression of their associated mRNAs could model the expression of the
pri-miRNAs well. These associated mRNAs are different from their corresponding target mRNAs and are enriched with specific functions. The majority of the associated mRNAs of a miRNA are shared across
conditions, although a fraction of the associated mRNAs are condition-specific. Our study shed new light on the understanding of miRNA biogenesis and general gene transcriptional regulation. | {"url":"https://www.researchsquare.com/article/rs-1499215/v1","timestamp":"2024-11-11T14:49:57Z","content_type":"text/html","content_length":"147656","record_id":"<urn:uuid:aeb1d95b-8e6a-432d-b374-6d1b2b1693bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00865.warc.gz"} |
Multiline plots in R using ggplot2
Published on
Multiline plots in R using ggplot2
Danial Khosravi
With the help of ggplot2, creating beautiful charts is an easy task in R. However it can get a little bit tricky when you're trying to plot a set of data on a single chart, over a shared x axis.
A neat trick is using the library reshape2 which is a very useful data manipulation library for R. With the help of melt function of this library, we can combine our data into a single data frame in
the format that ggplot2 wants from us in order to draw different lines over the same axis.
In this example, in data.csv I have function values of y=x, y=x^2 and y=x^3 for x values from 1 to 10 and i'm trying to draw these 3 charts on the same axis.
Note: if you haven't installed ggplot2 and reshape2 make sure to run
``` and
``` js script.R https://github.com/DanialK/multiple-line-graph-r/blob/master/script.R
data <- read.csv('./data.csv')
chart_data <- melt(data, id='x')
names(chart_data) <- c('x', 'func', 'value')
ggplot() +
geom_line(data = chart_data, aes(x = x, y = value, color = func), size = 1)+
xlab("x axis") +
ylab("y axis")
You can find the source code here on github | {"url":"https://danialk.github.io/blog/2015/12/13/multiline-plots-in-r-using-ggplot2/","timestamp":"2024-11-04T11:51:32Z","content_type":"text/html","content_length":"53351","record_id":"<urn:uuid:1891d6c8-e42b-4292-ad2b-6e911d657d20>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00564.warc.gz"} |
Chemistry/Mathematics — Secondary, BAE
Combined major offered by the Department of Chemistry, College of Science and Engineering and the Department of Mathematics, College of Science and Engineering
103-121 credits plus supporting courses in physics
The program of study for the chemistry/mathematics majors provides many benefits to students, such as close student-faculty contact and relatively small classes. The students have direct access to
modern laboratory equipment and instrumentation, and there are opportunities for research work under the direction of a faculty advisor.
As a practical matter, Mathematics is a science of pattern and order. Its domain is not molecules or cells, but numbers, chance, form, algorithms, and change. As a science of abstract objects,
Mathematics relies on logic rather than on observation as its standard of truth, yet employs observation, simulation, and even experimentation as means of discovering truth.” From: Everybody Counts:
A Report to the Nation on the Future of Mathematics Education (c) 1989 National Academy of Sciences.
This major must be accompanied by the professional preparation program in secondary education offered through Woodring College of Education. Courses required for a state teaching endorsement must be
completed with a grade of C (2.0) or better.
Why Consider a Mathematics-Secondary in Chemistry Major?
Teaching mathematics and chemistry is a challenge, a responsibility, and an opportunity. Learning to teach mathematics and chemistry occurs through a variety of means: the study of a wide variety of
mathematics and chemistry, pedagogical preparation within a mathematical and science context, formal clinical preparation in education, an extended internship, and continual experiences as a student,
learner, and problem solver in mathematics and chemistry.
Everyone aspiring to be a mathematics and chemistry teacher is aware of the demand for qualified teachers at the secondary level, but there is an even greater need for quality mathematics and
chemistry teachers—teachers who care about students, mathematics and chemistry teachers who have a broad and deep understanding of mathematics and chemistry and teachers who are thoroughly
professional. The responsibilities are great, but the rewards are even greater.
As a prospective teacher you need to focus on expanding your personal understanding of mathematics and chemistry and capitalizing on opportunities to work with pre-college students as a tutor, as a
classroom assistant, as a practicum student, and as a novice teacher in your internship.
Chemistry Department Chair Chemistry Program Coordinator Math Department
James Vyvyan Stacey Maxwell Bond Hall 202
Chemistry Building 270A Chemistry Building 270 360-650-3785
360-650-2883 360-650-3070 mathdept@wwu.edu
James.Vyvyan@wwu.edu chemdept@chem.wwu.edu http://www.wwu.edu/depts/math
Secondary Education Secondary Education
Math Department Advisor Professional Program Program Coordinator
Jessica Cohen Information Program Manager Christina Carlson
Bond Hall 180 Janna Cecka Miller Hall 401A
360-650-3830 Miller Hall 401C 360-650-3327
360-650-3347 Christina.Carlson@wwu.edu
Secondary Education Teacher
How to Declare (Admission and Declaration Process):
The Chemistry Department has a two-step process for admission into our degree programs. Phase I students are students who have declared their intent to major in chemistry, and are in the process of
completing the general chemistry (CHEM 121, 122, 123) series. Admission to Phase II is based on academic performance in the introductory courses. Students must achieve an average grade of 2.5 or
higher in their introductory biology and general chemistry series and organic chemistry I & II courses before they can advance to Phase II and begin taking upper-division coursework.
This major must be accompanied by the professional education program in secondary education. This major meets the requirements for Washington state teaching endorsements in both chemistry and
mathematics. See the Secondary Education section of this catalog for program admission, completion, and teacher certification requirements.
As certification to teach high school now requires more than four years of study, advisement prior to or at the beginning of the third year is absolutely necessary to avoid lengthening the program.
Grade Requirements
Recommendation for teaching endorsement normally requires completion of the following major with a grade point of 2.50 or better in the required major courses.
Students must earn a grade of C (2.0) or better in the secondary education professional program and in all courses required for the endorsement. | {"url":"https://catalog.wwu.edu/preview_program.php?catoid=11&poid=4927","timestamp":"2024-11-03T01:31:03Z","content_type":"text/html","content_length":"57688","record_id":"<urn:uuid:19dd697e-a4fb-4de1-88ee-f3ce86245e68>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00057.warc.gz"} |
Solve stiff differential equations and DAEs — variable order method
[t,y] = ode15s(odefun,tspan,y0), where tspan = [t0 tf], integrates the system of differential equations $y"=f\left(t,y\right)$ from t0 to tf with initial conditions y0. Each row in the solution array
y corresponds to a value returned in column vector t.
All MATLAB^® ODE solvers can solve systems of equations of the form $y"=f\left(t,y\right)$, or problems that involve a mass matrix, $M\left(t,y\right)y"=f\left(t,y\right)$. The solvers all use
similar syntaxes. The ode23s solver only can solve problems with a mass matrix if the mass matrix is constant. ode15s and ode23t can solve problems with a mass matrix that is singular, known as
differential-algebraic equations (DAEs). Specify the mass matrix using the Mass option of odeset.
[t,y] = ode15s(odefun,tspan,y0,options) also uses the integration settings defined by options, which is an argument created using the odeset function. For example, use the AbsTol and RelTol options
to specify absolute and relative error tolerances, or the Mass option to provide a mass matrix.
[t,y,te,ye,ie] = ode15s(odefun,tspan,y0,options) additionally finds where functions of (t,y), called event functions, are zero. In the output, te is the time of the event, ye is the solution at the
time of the event, and ie is the index of the triggered event.
For each event function, specify whether the integration is to terminate at a zero and whether the direction of the zero crossing matters. Do this by setting the 'Events' property to a function, such
as myEventFcn or @myEventFcn, and creating a corresponding function: [value,isterminal,direction] = myEventFcn(t,y). For more information, see ODE Event Location.
sol = ode15s(___) returns a structure that you can use with deval to evaluate the solution at any point on the interval [t0 tf]. You can use any of the input argument combinations in previous
ODE with Single Solution Component
Simple ODEs that have a single solution component can be specified as an anonymous function in the call to the solver. The anonymous function must accept two inputs (t,y), even if one of the inputs
is not used in the function.
Solve the ODE
Specify a time interval of [0 2] and the initial condition y0 = 1.
tspan = [0 2];
y0 = 1;
[t,y] = ode15s(@(t,y) -10*t, tspan, y0);
Plot the solution.
Solve Stiff ODE
An example of a stiff system of equations is the van der Pol equations in relaxation oscillation. The limit cycle has regions where the solution components change slowly and the problem is quite
stiff, alternating with regions of very sharp change where it is not stiff.
The system of equations is:
The initial conditions are and . The function vdp1000 ships with MATLAB® and encodes the equations.
function dydt = vdp1000(t,y)
%VDP1000 Evaluate the van der Pol ODEs for mu = 1000.
% See also ODE15S, ODE23S, ODE23T, ODE23TB.
% Jacek Kierzenka and Lawrence F. Shampine
% Copyright 1984-2014 The MathWorks, Inc.
dydt = [y(2); 1000*(1-y(1)^2)*y(2)-y(1)];
Solving this system using ode45 with the default relative and absolute error tolerances (1e-3 and 1e-6, respectively) is extremely slow, requiring several minutes to solve and plot the solution.
ode45 requires millions of time steps to complete the integration, due to the areas of stiffness where it struggles to meet the tolerances.
This is a plot of the solution obtained by ode45, which takes a long time to compute. Notice the enormous number of time steps required to pass through areas of stiffness.
Solve the stiff system using the ode15s solver, and then plot the first column of the solution y against the time points t. The ode15s solver passes through stiff areas with far fewer steps than
[t,y] = ode15s(@vdp1000,[0 3000],[2 0]);
Pass Extra Parameters to ODE Function
ode15s only works with functions that use two input arguments, t and y. However, you can pass in extra parameters by defining them outside the function and passing them in when you specify the
function handle.
Solve the ODE
Rewriting the equation as a first-order system yields
odefcn.m represents this system of equations as a function that accepts four input arguments: t, y, A, and B.
function dydt = odefcn(t,y,A,B)
dydt = zeros(2,1);
dydt(1) = y(2);
dydt(2) = (A/B)*t.*y(1);
Solve the ODE using ode15s. Specify the function handle such that it passes in the predefined values for A and B to odefcn.
A = 1;
B = 2;
tspan = [0 5];
y0 = [0 0.01];
[t,y] = ode15s(@(t,y) odefcn(t,y,A,B), tspan, y0);
Plot the results.
Compare Stiff ODE Solvers
The ode15s solver is a good first choice for most stiff problems. However, the other stiff solvers might be more efficient for certain types of problems. This example solves a stiff test equation
using all four stiff ODE solvers.
Consider the test equation
${y}^{\prime }=-\lambda y.$
The equation becomes increasingly stiff as the magnitude of $\lambda$ increases. Use $\lambda =1×1{0}^{9}$ and the initial condition $y\left(0\right)=1$ over the time interval [0 0.5]. With these
values, the problem is stiff enough that ode45 and ode23 struggle to integrate the equation. Also, use odeset to pass in the constant Jacobian $J=\frac{\partial f}{\partial y}=-\lambda$ and turn on
the display of solver statistics.
lambda = 1e9;
y0 = 1;
tspan = [0 0.5];
opts = odeset('Jacobian',-lambda,'Stats','on');
Solve the equation with ode15s, ode23s, ode23t, and ode23tb. Make subplots for comparison.
tic, ode15s(@(t,y) -lambda*y, tspan, y0, opts), toc
104 successful steps
1 failed attempts
212 function evaluations
0 partial derivatives
21 LU decompositions
210 solutions of linear systems
Elapsed time is 1.016343 seconds.
tic, ode23s(@(t,y) -lambda*y, tspan, y0, opts), toc
63 successful steps
0 failed attempts
191 function evaluations
0 partial derivatives
63 LU decompositions
189 solutions of linear systems
Elapsed time is 0.183604 seconds.
tic, ode23t(@(t,y) -lambda*y, tspan, y0, opts), toc
95 successful steps
0 failed attempts
125 function evaluations
0 partial derivatives
28 LU decompositions
123 solutions of linear systems
Elapsed time is 0.192325 seconds.
tic, ode23tb(@(t,y) -lambda*y, tspan, y0, opts), toc
71 successful steps
0 failed attempts
167 function evaluations
0 partial derivatives
23 LU decompositions
236 solutions of linear systems
Elapsed time is 0.268061 seconds.
The stiff solvers all perform well, but ode23s completes the integration with the fewest steps and runs the fastest for this particular problem. Since the constant Jacobian is specified, none of the
solvers need to calculate partial derivatives to compute the solution. Specifying the Jacobian benefits ode23s the most since it normally evaluates the Jacobian in every step.
For general stiff problems, the performance of the stiff solvers varies depending on the format of the problem and specified options. Providing the Jacobian matrix or sparsity pattern always improves
solver efficiency for stiff problems. But since the stiff solvers use the Jacobian differently, the improvement can vary significantly. Practically speaking, if a system of equations is very large or
needs to be solved many times, then it is worthwhile to investigate the performance of the different solvers to minimize execution time.
Evaluate and Extend Solution Structure
The van der Pol equation is a second order ODE
${y}_{1}^{\prime \prime }-\mu \left(1-{y}_{1}^{2}\right){y}_{1}^{\prime }+{y}_{1}=0.$
Solve the van der Pol equation with $\mu =1000$ using ode15s. The function vdp1000.m ships with MATLAB® and encodes the equations. Specify a single output to return a structure containing information
about the solution, such as the solver and evaluation points.
tspan = [0 3000];
y0 = [2 0];
sol = ode15s(@vdp1000,tspan,y0)
sol = struct with fields:
solver: 'ode15s'
extdata: [1x1 struct]
x: [0 1.4606e-05 2.9212e-05 4.3818e-05 1.1010e-04 1.7639e-04 2.4267e-04 3.0896e-04 4.5006e-04 5.9116e-04 7.3226e-04 8.7336e-04 0.0010 0.0012 0.0013 0.0015 0.0017 0.0018 0.0021 0.0024 0.0027 0.0030 0.0033 0.0044 0.0055 0.0066 ... ] (1x592 double)
y: [2x592 double]
stats: [1x1 struct]
idata: [1x1 struct]
Use linspace to generate 2500 points in the interval [0 3000]. Evaluate the first component of the solution at these points using deval.
x = linspace(0,3000,2500);
y = deval(sol,x,1);
Plot the solution.
Extend the solution to ${t}_{f}=4000$ using odextend and add the result to the original plot.
tf = 4000;
sol_new = odextend(sol,@vdp1000,tf);
x = linspace(3000,tf,350);
y = deval(sol_new,x,1);
hold on
Solve Robertson Problem as Semi-Explicit Differential Algebraic Equations (DAEs)
This example reformulates a system of ODEs as a system of differential algebraic equations (DAEs). The Robertson problem found in hb1ode.m is a classic test problem for programs that solve stiff
ODEs. The system of equations is
hb1ode solves this system of ODEs to steady state with the initial conditions , , and . But the equations also satisfy a linear conservation law,
In terms of the solution and initial conditions, the conservation law is
The system of equations can be rewritten as a system of DAEs by using the conservation law to determine the state of . This reformulates the problem as the DAE system
The differential index of this system is 1, since only a single derivative of is required to make this a system of ODEs. Therefore, no further transformations are required before solving the system.
The function robertsdae encodes this DAE system. Save robertsdae.m in your current folder to run the example.
function out = robertsdae(t,y)
out = [-0.04*y(1) + 1e4*y(2).*y(3)
0.04*y(1) - 1e4*y(2).*y(3) - 3e7*y(2).^2
y(1) + y(2) + y(3) - 1 ];
The full example code for this formulation of the Robertson problem is available in hb1dae.m.
Solve the DAE system using ode15s. Consistent initial conditions for y0 are obvious based on the conservation law. Use odeset to set the options:
• Use a constant mass matrix to represent the left hand side of the system of equations.
• Set the relative error tolerance to 1e-4.
• Use an absolute tolerance of 1e-10 for the second solution component, since the scale varies dramatically from the other components.
• Leave the 'MassSingular' option at its default value 'maybe' to test the automatic detection of a DAE.
y0 = [1; 0; 0];
tspan = [0 4*logspace(-6,6)];
M = [1 0 0; 0 1 0; 0 0 0];
options = odeset('Mass',M,'RelTol',1e-4,'AbsTol',[1e-6 1e-10 1e-6]);
[t,y] = ode15s(@robertsdae,tspan,y0,options);
Plot the solution.
y(:,2) = 1e4*y(:,2);
ylabel('1e4 * y(:,2)');
title('Robertson DAE problem with a Conservation Law, solved by ODE15S');
Input Arguments
Output Arguments
ode15s is a variable-step, variable-order (VSVO) solver based on the numerical differentiation formulas (NDFs) of orders 1 to 5. Optionally, it can use the backward differentiation formulas (BDFs,
also known as Gear's method) that are usually less efficient. Like ode113, ode15s is a multistep solver. Use ode15s if ode45 fails or is very inefficient and you suspect that the problem is stiff, or
when solving a differential-algebraic equation (DAE) [1], [2].
[1] Shampine, L. F. and M. W. Reichelt, “The MATLAB ODE Suite,” SIAM Journal on Scientific Computing, Vol. 18, 1997, pp. 1–22.
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
• odeset inputs JPattern and MvPattern must be passed as function handles that return either a sparse or full matrix.
• odeset inputs Mass and Jacobian must be passed as full matrices or as functions that return full or sparse matrices. To pass a sparse matrix, you must pass a function.
• All odeset option arguments must be constant.
• Variable-sizing support and dynamic memory allocation must be enabled.
• Input types must be homogenous – all double or all single.
• You must provide at least two output arguments, t and y.
Version History
Introduced before R2006a | {"url":"https://ch.mathworks.com/help/matlab/ref/ode15s.html","timestamp":"2024-11-14T17:49:44Z","content_type":"text/html","content_length":"147626","record_id":"<urn:uuid:c8dadf4a-7a2b-42a6-8f79-9fefce8d9c8e>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00455.warc.gz"} |
Browse by Authors
Number of items: 4.
Parsa, A. M. and Kozhan, I. and Wulkow, M. and Hutchinson, R. A. (2014) Modeling of Functional Group Distribution in Copolymerization: A Comparison of Deterministic and Stochastic Approaches.
Macromolecular Theory and Simulations, 23 (3). pp. 207-217. ISSN 10221344
Nikitin, A. and Wulkow, M. and Schütte, Ch. (2013) Modeling of Free Radical Styrene/Divinylbenzene Copolymerization with the Numerical Fractionation Technique. Macromolecular Theory and Simulation,
22 (9). pp. 475-489.
Schütte, Ch. and Wulkow, M. (2010) A Hybrid Galerkin–Monte-Carlo Approach to Higher-Dimensional Population Balances in Polymerization Kinetics. Macromol. React. Eng., 4 (9-10). pp. 562-577.
Schütte, Ch. and Wulkow, M. (1992) Quantum Theory with Discrete Spectra and Countable Systems of Differential Equations - A Numerical Treatment of Raman Spectroscopy. preprint . | {"url":"http://publications.imp.fu-berlin.de/view/people/Wulkow=3AM=2E=3A=3A.html","timestamp":"2024-11-13T02:44:28Z","content_type":"application/xhtml+xml","content_length":"11358","record_id":"<urn:uuid:39c46433-1077-4100-ad93-4121d09f073c>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00790.warc.gz"} |
Leveraged ETFs
Effective Altruism Investing Strategies
Leveraged ETFs
Leveraged ETFs are entirely inappropriate for retirement investing, but may be useful for Effective Altruism investing. The volatility of leveraged ETFs is huge. Over the period 2007-2009, a 3X
leveraged ETF would have likely fallen to 4% of its peak value. Losses of close to this magnitude are not extraordinary.
At present I would recommend investing in small cap value rather than leveraged ETFs. If investing in leveraged ETFs I would currently recommend the ProShares 2X leveraged ETF SSO. The ProShares 3X
leveraged ETF UPRO has slightly better expected performance for effective altruism purposes, but this comes with higher volatility and an increased likely of negative results. This might lead to
investor regret, and the possibility of bailing on the chosen strategy.
If investing in leveraged ETFs it is important to pay attention to expected returns on the underlying index, real borrowing costs, and the lag between leveraged ETFs and model leverage. The
performance lag is made up of the expense ratio, overheads, and inefficiencies. When the key values change, so will the applicability of leveraged ETFs.
This article expands on Effective Altruism Investing Strategies by looking at the suitability of leveraged ETFs for effective altruism investments.
Effective altruism is a movement that seeks to use reason to achieve the most good in the world. The bulk of this article is relevant to anyone seeking a better understanding of the performance of
leveraged ETFs. Only the final section deals with concerns specific to effective altruism.
Leverage is an advanced investing topic, and leverage should only be considered if you have mastered other aspects of investing, especially the potential permanently negative consequences of taking
on risk.
Leveraged ETFs
Leveraged ETFs multiply the daily returns of an index; typically by a factor of 2 or 3. Leveraged ETFs return higher returns when times are good at the cost of worse returns when times are bad.
Because the bet is recommitted to each day, over longer periods, the returns can be significantly higher than the factor of 2 or 3 that might be naively expected. This is also true for negative
returns, and makes leveraged ETFs unsuitable for most investors.
The primary investment strategy of leveraged ETFs is the execution of swap agreements. These swap agreements are contracts that specify the ETF will pay a counterparty a fixed interest rate in
exchange for receiving or making payments equal to the change in value of the index through some date. The counterparty is typically a large bank or brokerage that is able to at least notionally use
the interest payments to borrow funds and then invest the funds in the stock market. The leveraged ETF will also hold cash pledged as collateral against the swap agreement, and may hold shares making
up the underlying index (either directly or through shares in an ETF that tracks the underlying index), as well as possibly index futures contracts. The composition of these later components is
likely to vary on a day-to-day basis to maintain the target leverage factor, while new swap agreements are probably only executed monthly, or less frequently.
The expected results for a leveraged ETF then are very similar to what you might expect if you borrowed on margin to invest in the market, and were rebalancing on a daily basis to maintain the target
leverage factor. If you were to borrow funds you would likely pay a higher margin rate than a leveraged ETF, but on the flip side you don't have to deal with associated management expenses and other
fund expenses, which for 3X funds are typically a high 0.95% per annum. This suggests a natural benchmark for a leveraged ETF: the daily return on the underlying total return index multiplied by the
leverage factor less the cost of borrowing funds at the risk free rate. For the risk free rate I will normally be adopting the Fedfunds rate, which is the rate on overnight loans between banks. For
long run historical analysis the Fedfunds rate isn't available, so I use the 1-month Treasury bill rate plus 0.4%, which is the difference between these two interest rates over the period 1955-2016.
For current analysis I use arithmetic mean annualized real stock market returns of 4.5% with an annualized real volatility of 16.8%. This corresponds to a geometric mean real return of 3.2%. These
numbers are justified in Effective Altruism Investing Strategies. As of April 2017 the Fedfunds rate was 0.9% nominal, and the Survey of Professional Forecasters inflation projection for 2017 was
2.3%, resulting in a real cost of borrowing of -1.4%.
For historical analysis I use the return statistics of Dimson, Marsh and Staunton's Credit Suisse yearbook weighted developed world index for the period 1900-2016, with an arithmetic mean real
annualized returns of 6.5%, and volatility 17.4%. The reported geometric mean is 5.1%. They also report the geometric mean real interest rate for short term Treasury bills as 0.8%, making the risk
free rate 0.8% + 0.4% = 1.2%.
FINRA, the financial industry regulatory authority has put out an alert warning of the risks of leveraged ETFs to buy-and-hold investors. These products are entirely inappropriate for typical
retirement portfolios, but the risk profile may be appropriate for effective altruists. In Effective Altruism Asset Allocation it was shown that a reasonable asset allocation for effective altruism
portfolios is probably around 300% stocks.
Performance of real world leveraged ETFs
Table 1 presents the available forward leveraged ETFs that are based on broad U.S. market indexes. Of the 2X ETFs, Direxion's SPUU has low assets, low volume, and tracks the index poorly, making
ProShares' SSO preferable. Of the 3X leveraged ETFs ProShares' UPRO appears to lag an idealized benchmark by less than Direxion's SPXL. This makes UPRO preferable, with the caveat that Yahoo Finance
was only able to provide two full years of financial data for UPRO with a computed lag of 1.3%, so I instead used seven full years of UPRO net asset values to compute the performance lag. SPXL
provided eight years of data.
ETF leverage factor performance lag underlying index
ProShares SSO 2 1.4% S&P 500
Direxion SPUU 2 3.5% S&P 500
ProShares UPRO 3 1.8% S&P 500
Direxion SPXL 3 2.2% S&P 500
Table 1. Characteristics of forward leveraged ETFs that are based on broad U.S. market indexes.
On a daily basis the return of leveraged ETFs do a good job of tracking the underlying index as shown in Figure 1. On an annual basis things are less attractive as the management fees, effective
costs of borrowing, and other expenses add up.
Figure 1 - Daily change in nominal UPRO closing price as a function of daily change in nominal benchmark index over the period 2014-11-11 though 2017-05-11. Some daily return values were missing from
the data series, causing multiple days returns to be represented as a single data point, such as the data point at the extreme right.
I model the level of a leveraged ETF using the applicable total return index, the leverage factor, the then Fedfunds rate, and a single fudge factor: the empirically derived annual lag of the ETF
behind this benchmark. Figure 2 shows the performance of SSO against this model. I intentionally chose SSO, because it is the only leveraged ETF with financial data prior to the 2007-2009 sub-prime
financial crisis. This is a period where the Fedfunds rate was much higher than it is today. The model tracks the actual data very closely. So closely that the two lines are largely coincident and
you really have to peer at the graph before you can see any of the red line. Model versus actual plots for UPRO and SPXL are similar. SPUU does a worse job of tracking its model, although it still
tracks it reasonably well.
Figure 2 - Real daily index levels for synthetic SSO ETF and actual SSO ETF.
Daily data for the S&P 500 total return index is available from 1988 on. This suggests a natural backtest for UPRO, in which we calculate how the leveraged ETF would have been expected to perform.
This is shown in Figure 3. Some observations are in order. First in a rising or declining market the gains are losses are far in excess of 3X. This is due to the daily resetting. Second the synthetic
UPRO significantly lags its benchmark after 30 years due to the annual underperformance lag adding up. Third the synthetic UPRO hasn't regained the level it had during the 2000 dotcom bubble, despite
the index having done so. Forth, whether the synthetic UPRO outperforms the simple 1X index depends on the ending date chosen, but overall on average it outperforms. Finally, from 2000 to 2002 the
synthetic UPRO dropped to 7.3% of its peak value value, and from 2007 to 2009 it dropped to 4.2% of that peak value. For a synthetic 2X SSO the corresponding declines were to 19.9% and 15.5% of their
peak values respectively. It takes a very strong mind to be able tolerate such losses.
Figure 3 - Real daily index levels for S&P 500 total return index, 3X benchmark, and synthetic UPRO ETF (based on historical net asset values).
A mathematical model of leveraged ETF performance
The previous synthetic backtest is for a single history, and will not re-occur in the future. To predict performance in the future I construct a simple mathematical model of what is happening.
Derivation of the model
I assume the total return index can be described using geometric Brownian motion with constant drift, constant volatility, and no-autocorrelation. Later we will see how inaccuracies in this
assumption lead to discrepancies between the mathematical model and bootstrapped projected performance.
In the presence of leverage, the geometric Brownian motion drift parameters μ[leverage_GBM] and underlying volatility σ[leveraged_GBM] would be expected to given by:
μ[leveraged_GBM] = f . μ[index_GBM] + (1 - f) . Rf[GBM] - Lag[GBM]
σ[leveraged_GBM] = f . σ[index_GBM]
where μ[index_GBM] is the drift on the underlying total return index, σ[index_GBM] is the underlying total return index volatility, f is the leverage factor (2 or 3), Rf[GBM] is the instantaneous
risk free rate, and Lag[GBM] is the instantaneous performance lag. These later two values are related to the annualized risk free rate, Rf, and the expense ratio and other factors related lag, Lag,
Rf[GBM] = ln(1 + Rf)
Lag[GBM] = - ln(1 - Lag)
Geometric Brownian motion results in a log-normally distributed returns. Returns after n years are given by the lognormal probability density function with parameters μ = n . μ[LND], and σ = sqrt(n)
. σ[LND]. Or equivalently, the annualized return is given by the lognormal probability density function with parameters μ = μ[LND], and σ = σ[LND] / sqrt(n). By comparing the definitions of geometric
Brownian motion and the lognormal distribution it can be seen:
μ[GBM] = μ[LND] + σ[LND]^2 / 2
σ[GBM] = σ[LND]
The parameters μ[LND] and σ[LND] of the log-normal distribution are given by the equations:
μ[LND] = log((1 + R) / sqrt(1 + σ^2 / (1 + R)^2))
σ[LND] = sqrt(log(1 + σ^2 / (1 + R)^2))
where R and σ are the arithmetic mean annual return and volatility respectively.
I thus proceed as follows. Compute μ[index_LND] and σ[index_LND] using equations (5) and (6). Convert them to μ[index_GBM] and σ[index_GBM] using equations (3) and (4). Compute the leveraged values μ
[leveraged_GBM] and σ[leveraged_GBM] using equations (1) and (2). Convert these values back to μ[leveraged_LND] and σ[leveraged_LND] using equations (3) and (4) in reverse. Compute and plot the
corresponding probability density function.
Inflation rate independence
It is worth noting that the model is independent of the inflation rate. This can be seen by assuming the values used in the leverage equation are all nominal, and the instantaneous inflation rate is
w, so that:
μ_nominal[leveraged_GBM] = f . μ_nominal[index_GBM] + (1 - f) . Rf_nominal[GBM] - Lag[GBM]
μ[leveraged_GBM] + w = f . (μ[index_GBM] + w) + (1 - f) . (Rf[GBM] + w) - Lag[GBM]
μ[leveraged_GBM] = f . μ[index_GBM] + (1 - f) . Rf[GBM] - Lag[GBM]
which is the leverage equation in real form, independent of the inflation rate.
Application of the model
First we will apply the model to the present. Figure 4 shows projected return probabilities for a 3X ETF for different holding periods assuming a constant cost of borrowing of -1.4%. Even for a 50
year period there is a significant chance of -5% annual returns, which would result in the original investment being cut down to 8% of its original value. Worse annualized returns are possible over
shorter periods.
Figure 4 - Real return probabilities during current times for a 3X S&P 500 total return index with a -1.4% real Fedfunds borrowing rate and a 1.8% per annum performance lag. Projected S&P 500 real
total return 4.5% arithmetic, annual volatility 16.8%.
Now we turn to how the model performed using historical return statistics and borrowing costs. Figure 5 shows the return probabilities again for a 3X ETF. The return probabilities are very similar.
Underlying index return expectations are higher, but this has largely been offset by higher borrowing costs.
Figure 5 - Real return probabilities during historical times for a 3X S&P 500 total return index with a 1.2% real Fedfunds borrowing rate and a 1.8% per annum performance lag. Projected S&P 500 real
total return 6.5% arithmetic, annual volatility 17.4%.
Figure 6 shows the return probabilities for a 2X ETF assuming the current projected market performance, cost of borrowing, and a 1.4% per annum performance lag, similar to that for SSO. The results
can still be quite averse; less than in the 3X case, but perhaps still so bad that it doesn't make a lot of psychological difference. Decimation, even over 50 years, is still a possible outcome.
Figure 6 - Real return probabilities during current times for a 2X S&P 500 total return index with a -1.4% real Fedfunds borrowing rate and a 1.4% per annum performance lag. Projected S&P 500 real
total return 4.5% arithmetic, annual volatility 16.8%.
An empirical model of leveraged ETF performance
Bootstrapping is the technique of generating large amounts of return sequence data by concatenating smaller sub-sequences from the historical record. Unlike a mathematical model, bootstrapping allows
the resulting return sequence to have variable drift, volatility, and auto-correlation phenomena such as momentum. Here I use bootstrapping to projected the performance of leveraged ETFs based on the
daily S&P 500 total return index from 1988 to 2016, using only the leverage factor, the performance lag, and the risk free rate.
Since inflation is not uniform over this period, I subtract it out, and then add in an assumed constant inflation rate. Although, this later step isn't strictly necessary.
Since this is a relatively small time the mean annual return and volatility on an annual basis might differ substantially from the expected values. The next step then is to gently and uniformly
massage the historical daily data so that it reflects the assumed annual return and volatility. To maximize the information contained in the data I compute annual returns using a 365 day wide sliding
window (or strictly speaking 252 returns per year) and take the average value. I also wrap the data, joining the start of 1988 to the end of 2016 so that all data has equal weight, which helps with
latter analysis.
Next I convert the daily return values into leveraged daily return values by using the leverage factor, the risk free rate, and the expected performance lag.
Finally I perform simulations. I construct 500000 return sequences. Each sequence is constructed by concatenating month long samples (21 returns per month) from within the wrapping leveraged daily
return sequence. For each sequence I compute the final return value and plot a histogram of the results.
Figure 7 compares 20 year annualized returns of the mathematical model and the bootstrapped results, but with the underlying S&P 500 total return index replaced by an artificially generated true
geometric Brownian motion sequence matching the assumed mean annual return and annualized volatility.
Figure 7 - Mathematical model versus bootstrapped geometric Brownian motion returns. Real return probabilities for mathematical model and 3X geometric Brownian motion return over 20 years with a 1.2%
real Fedfunds borrowing rate and a 1.8% per annum performance lag. Geometric Brownian motion real total return 6.5% arithmetic, annual volatility 17.4%.
The good fit between the mathematical model and the bootstrapped results suggests we are on the right track. That the two do not fit perfectly can be ascribed to using month long samples in the
bootstrapping process. I discovered that if I set the bootstrap sample size to one day the fit becomes perfect. I want to use a bootstrap sample size that is as large as possible so that when applied
to real world returns it captures parameter variabilities and auto-correlations, but setting it too large means that the available samples do not reflect the full range of possibilities of true
geometric Brownian motion. I found that when setting this sample size to two months or larger it started to reflect negatively on the tight relationship between the mathematical and geometric
Brownian motion bootstrapped models. Hence the use of a one month sample size.
Application of the model
Figure 8 shows the application of the bootstrap model to the S&P 500 total return index. Here there is some divergence between the simple mathematical model that assumes geometric Brownian motion and
the bootstrap model which does not, but the difference isn't that great. Using the bootstrap model is preferable, but we can probably get away with using the simple mathematical model if we need to.
Figure 8 - Real return probabilities for mathematical model and bootstrapped 3X S&P 500 total return index over 20 years with a -1.4% real Fedfunds borrowing rate and a 1.8% per annum performance
lag. Projected S&P 500 real total return 4.5% arithmetic, annual volatility 16.8%.
Volatility of leveraged ETFs
Leveraged ETFs are highly volatile. This volatility could mistakenly lead to the conclusion that they are a poor ex-ante investment opportunity for effective altruism investing. It is therefore vital
that before investing in them you know what to expect.
For the simple mathematical model the annual real volatility of a leveraged ETF can be calculated as:
σ[leveraged] = sqrt((e^σ[leveraged_LND]^2 - 1) . e^2μ[leveraged_LND] + σ[leveraged_LND]^2)
This produces the volatilities shown in Table 2.
asset allocation σ[leveraged]
100% stocks 22.2%
200% stocks 35.8%
300% stocks 58.6%
Table 2. Annual volatilities of leveraged ETFs during current times computed using the simple mathematical model. Performance lags: 1X 0.1%, 2X 1.4%, 3X 1.8%. -1.4% real Fedfunds borrowing rate.
Projected S&P 500 real total return 4.5% arithmetic, volatility 16.8%.
I turn now from the expected losses over a one year period to losses over multiple years, and switch from the simple mathematical model to the more accurate bootstrapped model. First, a basic
understanding of the role of time on the value of investments is necessary. Figure 9 shows the projected distribution of relative portfolio sizes for an investment in an ETF that tracks the massaged
S&P 500 index with a 0.1% expense ratio computed using the bootstrap model. As can be seen the greater the length of time, the greater the chance of major gains, while the chance of a major loss
doesn't vary by much. A one year 20% loss isn't extraordinary.
Figure 9 - Portfolio sizes / initial portfolio size during current times for bootstrapped S&P 500 total return index with a -1.4% Fedfunds borrowing rate and 0.1% per annum performance lag. Projected
S&P 500 real total return 4.5% arithmetic, annual volatility 16.8%.
The same is not true when leverage is used. When leverage is used the chance of a major loss increases as time increases. A 40% one year loss isn't extraordinary. Figure 10 presents the current
projected distribution of relative portfolio sizes for a 2X leveraged ETF investments after 1, 5, and 20 years using the bootstrap model and including an appropriate performance lag. As can be seen,
the greater the duration, the higher the probability of a major loss, but also the higher the probability of major gains.
Figure 10 - Portfolio sizes / initial portfolio size during current times for bootstrapped 2X S&P 500 total return index with a -1.4% Fedfunds borrowing rate and a 1.4% per annum performance lag.
Projected S&P 500 real total return 4.5% arithmetic, annual volatility 16.8%.
The corresponding graph for a 3X leveraged investment is shown in Figure 11. Here the chance of a major loss is very high especially as time increases. And even a one year 60% loss isn't
Figure 11 - Portfolio sizes / initial portfolio size during current times for bootstrapped 3X S&P 500 total return index with a -1.4% Fedfunds borrowing rate and a 1.8% per annum performance lag.
Projected S&P 500 real total return 4.5% arithmetic, annual volatility 16.8%.
To reiterate the point, Figure 12 reproduces the 20 year results from the previous figures with a change in scale. Tables 3 and 4 present the median and mean final value factors for the above
distributions. Leverage boosts the mean expected final portfolio size at the possible expense of median portfolio size. When you use leverage you are more likely to end up on the left of the graph,
but you are hoping this will be offset by the small probability of ending at the extreme right. Be careful of fixating on expected mean final portfolio sizes. Utility, not wealth is what matters, and
it displays diminishing marginal returns.
Figure 12 - Portfolio sizes / initial portfolio size during current times after 20 years for bootstrapped models. -1.4% real Fedfunds borrowing rate. Performance lags: 1X 0.1%, 2X 1.4%, 3X 1.8% per
annum. Projected S&P 500 real total return 4.5% arithmetic, annual volatility 16.8%.
asset allocation 1 year 5 year 20 year
100% stocks 1.04 1.17 1.85
200% stocks 1.05 1.19 1.92
300% stocks 1.04 1.11 1.38
Table 3. Median values for the previous figures.
asset allocation 1 year 5 year 20 year
100% stocks 1.04 1.23 2.30
200% stocks 1.08 1.48 4.74
300% stocks 1.13 1.83 11.09
Table 4. Mean values for the previous figures.
A model of effective altruism utility for leveraged ETFs
If the amount of good done was linear in donation amount, we would be done, but it is not. Because of the law of diminishing returns, the good done by a donation ten times as large is less than ten
times the good done by a donation of a particular size. This is captured by the concept of utility.
In Effective Altruism Asset Allocation I suggest a reasonable model of utility for an individual is a constant relative risk aversion (CRRA) utility function with a coefficient of relative risk
aversion of 2. However things become more complicated for an effective altruist because only a fraction of consumption is likely to be correlated with the returns of the optimal level of market
leverage, while the rest can be treated as uncorrelated. I suggest that for every $100k per year of uncorrelated consumption, there might be somewhere around $200k of assets dedicated to the cause
whose returns are correlated with the optimal level of market leverage. As it turns out the exact coefficient of relative risk aversion, and the exact amount of correlated assets doesn't matter
greatly. What matters most is the is some uncorrelated consumption that acts as a brake on the worst aspects of a CRRA utility function at small levels of consumption.
My model of effective altruism utility then is to use the empirical bootstrapped model of leveraged ETF performance, each month to donate a fraction of the portfolio to the cause, and compute the
utility associated with having done so. When computing utility I also take into account uncorrelated donations/consumption. At the end of each simulation sequence I add up all the utilities, and
compute the total utility associated with that simulation sequence. Having done this it is possible to plot the range of utilities, or compute their median or mean. Since utility values are not
meaningful to people, each utility is inversed to compute a certainty equivalence: the constant annual would be correlated donation amount that has the same utility as the given utility value.
In deciding how much to donate each year I keep things simple and used 1/N of the portfolio size where N is the number of years remaining. This is clearly sub-optimal. The optimal fraction is a
function of both portfolio size and years remaining that varies in ways that can only be computed using stochastic dynamic programming. My attempt to use a slightly more sophisticated scheme,
variable percentage withdrawal, which attempts to divide donations more uniformly by taking into account the growing nature of the portfolio, produced worse results; possibly because the extreme
volatility of leveraged ETFs makes the growth rate to use hard to determine.
I developed an asset allocator called Opal that uses stochastic dynamic programming to determine the optimal asset allocation as a function of age and portfolio size. Opal can also be run in
non-stochastic dynamic programming mode to compute the certainty equivalent utility associated with a given initial portfolio size and fixed asset allocation. Table 5 presents a comparison of the
results produced using Opal and the bootstrap model. Simulation is a random process, and the variability associated with donation amounts reported by each model is in the range $100-200. Thus the two
models are in good agreement.
asset allocation donation strategy lag annual certainty equivalent donation amount
Opal bootstrap model
100% stocks 1/N 0% $12,008 $12,109
200% stocks 1/N 0% $27,983 $28,235
300% stocks 1/N 0% $33,067 $33,326
400% stocks 1/N 0% $27,919 $28,046
optimal dynamic 1/N 0% $36,719 -
optimal dynamic optimal dynamic 0% $40,409 -
optimal dynamic optimal dynamic 1.8% $31,374 -
Table 5. Certainty equivalent donation amounts of mean utilities for various asset allocation strategies. Coefficient of relative risk aversion 2. Initial correlated assets $200,000. 50 year
non-stochastic life expectancy. $100,000 correlated consumption per year. Monthly resetting and donating. No time discounting. -1.7% real Fedfunds borrowing rate. Geometric Brownian motion annual
real return 4.5% arithmetic, volatility 16.8%.
There are three additional lines in the above table that were computed using stochastic dynamic programming, and against which no bootstrap model comparison is available. They show a reasonable gain
in performance if the asset allocation is allowed to vary dynamically. A further gain in performance if the donation strategy is chosen to be optimal, as a function of portfolio size and time, rather
than 1/N. And a significant drop when a typical leveraged ETF performance lag exists.
Application of the model
Figure 13 presents the distribution of certainty equivalent values for the current time scenario, 1/N donating, and a 20 year time horizon. This plot incorporates the performance lag between the
leverage benchmark and an actual leveraged ETF. The associated summary statistics are presented in Table 6. It is easy to see the case for 2X leverage of effective altruism resources. It boosts both
the median and the mean donation amount. The case for 3X is harder to see as the mean is boosted, but the median falls. It is quite likely that you will end up with a smaller value than with 2X, but
there is a small chance it will be significantly larger boosting the mean. Despite this the case has been made since our preferences over different donation levels are captured by the concept of
utility, and effective altruists may not be mean donation amount maximizers, but should be mean utility maximizers. Beyond 3X, no case can be made. Both the mean and median fall.
Figure 13 - Projected distribution of real world donation certainty equivalents for the current time. Coefficient of relative risk aversion 2. Initial correlated assets $200,000. 20 year
non-stochastic life expectancy. $100,000 correlated consumption per year. Daily resetting and donating. 1/N annual donation rate. Performance lags: 1X 0.1%, 2X 1.4%, 3X 1.8%. No time discounting.
-1.4% Fedfunds borrowing rate. Projected S&P 500 real total return 4.5% arithmetic, volatility 16.8%.
asset allocation annual certainty equivalent donation amount
median mean
100% stocks $14,002 $14,901
200% stocks $15,035 $18,452
300% stocks $13,334 $19,528
400% stocks $9,388 $17,224
100% small cap value (no anomaly) $15,849 $17,384
100% small cap value (anomaly) $24,103 $25,658
Table 6. Summary statistics for previous figure. Plus 400% stocks with a performance lag of 2.2%. Plus 100% small cap value with a arithmetic mean real annual return of 6.5% without anomaly or 10.5%
with anomaly and real volatility of 22.2% and a 0.1% performance lag.
The gains from leverage are far smaller than those seen in the validation section, which saw a doubling or tripling of the certainty equivalent mean donation amount. There are two reasons for this.
First the results in the validation section were for 50 years rather than 20 years, giving more time for advantages to compound. Second the lag between idealized leverage performance and actual
leverage performance exerts a significant toll. 1.8% per year over 20 years is a 30% drag.
Also note the comparison against small cap value computed both on a purely risk adjusted basis (no anomaly), and as if there is a future 4% small and value performance anomaly (anomaly). Depending on
whether you believe there is an anomaly small cap value comes reasonably close to performing as well as leverage, or significantly exceeds the performance of leverage. This is quite different from
the results computed in Effective Altruism Asset Allocation where fixed and dynamically variable leverage both outperformed anomaly free small cap value by a wide margin. The likely reason for this
difference is the other paper assumes idealized leverage without the performance lag of leveraged ETFs seen in the real world.
Based on the previous two sections mathematically speaking a 3X leveraged ETF, such as UPRO, currently has a small advantage over 1X or 2X for effective altruism purposes, but it may be wise to use a
2X leveraged ETF, such as SSO, owing to the significantly reduced downside of doing so. Investing in the 2X ETF would probably make it slightly easier to sleep at night. That said, both options are
only just superior, or significantly inferior to investing in small cap value, depending on whether there are persistent small and value performance anomalies.
Important reasons the results contained here might not be valid include:
• The effective altruism specific recommendations depend on the shape of the cause specific utility function. I made a reasonable guess for this utility function, but it won't apply to all causes.
In particular, causes with a high discount rate are likely to eschew investing. Smarter-than-human AI risk mitigation a good example. The use of leverage will make little difference for such
causes. The causes that are best approximated by the chosen utility function have some donation amount that is independent of that made by knowledgeable effective altruists. Global poverty causes
are a good example.
• The analysis is based on the present. In particular it uses estimates of current projected index returns and the current real borrowing cost. At other times these would have different values
making leveraged ETFs more, or less, attractive.
• The available data for 3X leveraged ETFs only go back to 2008. Since then we have been in a very low interest rate environment. I use a model which incorporates borrowing costs, but without more
real world data to fully validate the model it isn't certain leveraged ETFs will operate as expected in a higher interest rate environment. The 2X leveraged ETF SSO is able to be validated back
to 2006.
• The results are based on the assumptions that any significant autocorrelated returns or non-constant volatility can be captured by looking at historical data in one month slices. To the extent
that these phenomena span a longer duration the results will be inaccurate.
Leveraged ETFs lag idealized benchmarks by 1.4% to 2.2% per year. A better understanding of the components of this lag is in order. Expense ratios and other fund expenses typically add up to around
0.95% per year. There is also the interest rate margin on the fund borrowing for the swap contract, which needs to be multiplied by the leverage factor. But with sufficient assets I would have
thought this could have been negotiated to a low value. Perhaps there are other cost factors of which I am unaware.
It would be informative to know if leveraged ETF swap contracts are pegged to the basic index, or the total return index. My analysis was based on the total return index less some lag. An alternative
analysis based on the basic index plus some margin is also possible. The iPath SFLA exchange traded note is indexed against the S&P 500 total return index, suggesting swap contracts indexed to the
total return index are possible.
Data was sourced from Fred, Yahoo Finance, and ProFunds. It was analyzed using a 700 line Python program called Leverage Analyze. For performance reasons it should normally be run using the PyPy
just-in-time Python compiler. Data was plotted using the leveraged_etfs.gnuplot GnuPlot script. The simple mathematical model was generated using the investing_math_model.py Python program and
plotted using the investing_math_model.gnuplot GnuPlot script.
© 2017 Gordon Irlam. Some rights reserved. This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. | {"url":"https://www.gordoni.com/effective-altruism/leveraged-etfs.html","timestamp":"2024-11-14T04:54:02Z","content_type":"text/html","content_length":"44249","record_id":"<urn:uuid:2a29ecd8-f3a2-4504-8ec1-ff8b5381610c>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00444.warc.gz"} |
Source code for torch.autograd
``torch.autograd`` provides classes and functions implementing automatic
differentiation of arbitrary scalar valued functions. It requires minimal
changes to the existing code - you only need to declare :class:`Tensor` s
for which gradients should be computed with the ``requires_grad=True`` keyword.
As of now, we only support autograd for floating point :class:`Tensor` types (
half, float, double and bfloat16) and complex :class:`Tensor` types (cfloat, cdouble).
import torch
import warnings
from torch.types import _TensorOrTensors
from typing import Any, Callable, List, Optional, Sequence, Tuple, Union
from .variable import Variable
from .function import Function, NestedIOFunction
from .gradcheck import gradcheck, gradgradcheck
from .grad_mode import no_grad, enable_grad, set_grad_enabled, inference_mode
from .anomaly_mode import detect_anomaly, set_detect_anomaly
from ..overrides import has_torch_function, handle_torch_function
from . import functional
from . import forward_ad
from . import graph
__all__ = ['Variable', 'Function', 'backward', 'grad_mode']
_OptionalTensor = Optional[torch.Tensor]
def _make_grads(outputs: Sequence[torch.Tensor], grads: Sequence[_OptionalTensor]) -> Tuple[_OptionalTensor, ...]:
new_grads: List[_OptionalTensor] = []
for out, grad in zip(outputs, grads):
if isinstance(grad, torch.Tensor):
if not out.shape == grad.shape:
raise RuntimeError("Mismatch in shape: grad_output["
+ str(grads.index(grad)) + "] has a shape of "
+ str(grad.shape) + " and output["
+ str(outputs.index(out)) + "] has a shape of "
+ str(out.shape) + ".")
if out.dtype.is_complex != grad.dtype.is_complex:
raise RuntimeError("For complex Tensors, both grad_output and output"
" are required to have the same dtype."
" Mismatch in dtype: grad_output["
+ str(grads.index(grad)) + "] has a dtype of "
+ str(grad.dtype) + " and output["
+ str(outputs.index(out)) + "] has a dtype of "
+ str(out.dtype) + ".")
elif grad is None:
if out.requires_grad:
if out.numel() != 1:
raise RuntimeError("grad can be implicitly created only for scalar outputs")
new_grads.append(torch.ones_like(out, memory_format=torch.preserve_format))
raise TypeError("gradients can be either Tensors or None, but got " +
return tuple(new_grads)
def _tensor_or_tensors_to_tuple(tensors: Optional[_TensorOrTensors], length: int) -> Tuple[_OptionalTensor, ...]:
if tensors is None:
return (None, ) * length
if isinstance(tensors, torch.Tensor):
return (tensors, )
return tuple(tensors)
[docs]def backward(
tensors: _TensorOrTensors,
grad_tensors: Optional[_TensorOrTensors] = None,
retain_graph: Optional[bool] = None,
create_graph: bool = False,
grad_variables: Optional[_TensorOrTensors] = None,
inputs: Optional[_TensorOrTensors] = None,
) -> None:
r"""Computes the sum of gradients of given tensors with respect to graph
The graph is differentiated using the chain rule. If any of ``tensors``
are non-scalar (i.e. their data has more than one element) and require
gradient, then the Jacobian-vector product would be computed, in this
case the function additionally requires specifying ``grad_tensors``.
It should be a sequence of matching length, that contains the "vector"
in the Jacobian-vector product, usually the gradient of the differentiated
function w.r.t. corresponding tensors (``None`` is an acceptable value for
all tensors that don't need gradient tensors).
This function accumulates gradients in the leaves - you might need to zero
``.grad`` attributes or set them to ``None`` before calling it.
See :ref:`Default gradient layouts<default-grad-layouts>`
for details on the memory layout of accumulated gradients.
.. note::
Using this method with ``create_graph=True`` will create a reference cycle
between the parameter and its gradient which can cause a memory leak.
We recommend using ``autograd.grad`` when creating the graph to avoid this.
If you have to use this function, make sure to reset the ``.grad`` fields of your
parameters to ``None`` after use to break the cycle and avoid the leak.
.. note::
If you run any forward ops, create ``grad_tensors``, and/or call ``backward``
in a user-specified CUDA stream context, see
:ref:`Stream semantics of backward passes<bwd-cuda-stream-semantics>`.
.. note::
When ``inputs`` are provided and a given input is not a leaf,
the current implementation will call its grad_fn (even though it is not strictly needed to get this gradients).
It is an implementation detail on which the user should not rely.
See https://github.com/pytorch/pytorch/pull/60521#issuecomment-867061780 for more details.
tensors (Sequence[Tensor] or Tensor): Tensors of which the derivative will be
grad_tensors (Sequence[Tensor or None] or Tensor, optional): The "vector" in
the Jacobian-vector product, usually gradients w.r.t. each element of
corresponding tensors. None values can be specified for scalar Tensors or
ones that don't require grad. If a None value would be acceptable for all
grad_tensors, then this argument is optional.
retain_graph (bool, optional): If ``False``, the graph used to compute the grad
will be freed. Note that in nearly all cases setting this option to ``True``
is not needed and often can be worked around in a much more efficient
way. Defaults to the value of ``create_graph``.
create_graph (bool, optional): If ``True``, graph of the derivative will
be constructed, allowing to compute higher order derivative products.
Defaults to ``False``.
inputs (Sequence[Tensor] or Tensor, optional): Inputs w.r.t. which the gradient
be will accumulated into ``.grad``. All other Tensors will be ignored. If
not provided, the gradient is accumulated into all the leaf Tensors that
were used to compute the attr::tensors.
if grad_variables is not None:
warnings.warn("'grad_variables' is deprecated. Use 'grad_tensors' instead.")
if grad_tensors is None:
grad_tensors = grad_variables
raise RuntimeError("'grad_tensors' and 'grad_variables' (deprecated) "
"arguments both passed to backward(). Please only "
"use 'grad_tensors'.")
if inputs is not None and len(inputs) == 0:
raise RuntimeError("'inputs' argument to backward() cannot be empty.")
tensors = (tensors,) if isinstance(tensors, torch.Tensor) else tuple(tensors)
inputs = (inputs,) if isinstance(inputs, torch.Tensor) else
tuple(inputs) if inputs is not None else tuple()
grad_tensors_ = _tensor_or_tensors_to_tuple(grad_tensors, len(tensors))
grad_tensors_ = _make_grads(tensors, grad_tensors_)
if retain_graph is None:
retain_graph = create_graph
tensors, grad_tensors_, retain_graph, create_graph, inputs,
allow_unreachable=True, accumulate_grad=True) # allow_unreachable flag
[docs]def grad(
outputs: _TensorOrTensors,
inputs: _TensorOrTensors,
grad_outputs: Optional[_TensorOrTensors] = None,
retain_graph: Optional[bool] = None,
create_graph: bool = False,
only_inputs: bool = True,
allow_unused: bool = False
) -> Tuple[torch.Tensor, ...]:
r"""Computes and returns the sum of gradients of outputs with respect to
the inputs.
``grad_outputs`` should be a sequence of length matching ``output``
containing the "vector" in Jacobian-vector product, usually the pre-computed
gradients w.r.t. each of the outputs. If an output doesn't require_grad,
then the gradient can be ``None``).
.. note::
If you run any forward ops, create ``grad_outputs``, and/or call ``grad``
in a user-specified CUDA stream context, see
:ref:`Stream semantics of backward passes<bwd-cuda-stream-semantics>`.
.. note::
``only_inputs`` argument is deprecated and is ignored now (defaults to ``True``).
To accumulate gradient for other parts of the graph, please use
outputs (sequence of Tensor): outputs of the differentiated function.
inputs (sequence of Tensor): Inputs w.r.t. which the gradient will be
returned (and not accumulated into ``.grad``).
grad_outputs (sequence of Tensor): The "vector" in the Jacobian-vector product.
Usually gradients w.r.t. each output. None values can be specified for scalar
Tensors or ones that don't require grad. If a None value would be acceptable
for all grad_tensors, then this argument is optional. Default: None.
retain_graph (bool, optional): If ``False``, the graph used to compute the grad
will be freed. Note that in nearly all cases setting this option to ``True``
is not needed and often can be worked around in a much more efficient
way. Defaults to the value of ``create_graph``.
create_graph (bool, optional): If ``True``, graph of the derivative will
be constructed, allowing to compute higher order derivative products.
Default: ``False``.
allow_unused (bool, optional): If ``False``, specifying inputs that were not
used when computing outputs (and therefore their grad is always zero)
is an error. Defaults to ``False``.
outputs = (outputs,) if isinstance(outputs, torch.Tensor) else tuple(outputs)
inputs = (inputs,) if isinstance(inputs, torch.Tensor) else tuple(inputs)
overridable_args = outputs + inputs
if has_torch_function(overridable_args):
return handle_torch_function(
if not only_inputs:
warnings.warn("only_inputs argument is deprecated and is ignored now "
"(defaults to True). To accumulate gradient for other "
"parts of the graph, please use torch.autograd.backward.")
grad_outputs_ = _tensor_or_tensors_to_tuple(grad_outputs, len(outputs))
grad_outputs_ = _make_grads(outputs, grad_outputs_)
if retain_graph is None:
retain_graph = create_graph
return Variable._execution_engine.run_backward(
outputs, grad_outputs_, retain_graph, create_graph,
inputs, allow_unused, accumulate_grad=False)
# This function applies in case of gradient checkpointing for memory
# optimization. Currently, gradient checkpointing is supported only if the
# execution engine is invoked through torch.autograd.backward() and its
# inputs argument is not passed. It is not supported for torch.autograd.grad().
# This is because if inputs are specified, the gradient won't be calculated for
# anything else e.g. model parameters like weights, bias etc.
# This function returns whether the checkpointing is valid i.e. torch.autograd.backward
# or not i.e. torch.autograd.grad. The implementation works by maintaining a thread
# local variable in torch/csrc/autograd/engine.cpp which looks at the NodeTask
# in the stack and before a NodeTask is executed in evaluate_function, it
# checks for whether reentrant backwards is imperative or not.
# See https://github.com/pytorch/pytorch/pull/4594 for more discussion/context
def _is_checkpoint_valid():
return Variable._execution_engine.is_checkpoint_valid()
def variable(*args, **kwargs):
warnings.warn("torch.autograd.variable(...) is deprecated, use torch.tensor(...) instead")
return torch.tensor(*args, **kwargs)
if not torch._C._autograd_init():
raise RuntimeError("autograd initialization failed")
# Import all native method/classes
from torch._C._autograd import (DeviceType, ProfilerActivity, ProfilerState, ProfilerConfig, ProfilerEvent,
_enable_profiler_legacy, _disable_profiler_legacy, _profiler_enabled,
_enable_record_function, _set_empty_test_observer, kineto_available,
_supported_activities, _add_metadata_json, SavedTensor,
_register_saved_tensors_default_hooks, _reset_saved_tensors_default_hooks)
from torch._C._autograd import (_ProfilerResult, _KinetoEvent,
_prepare_profiler, _enable_profiler, _disable_profiler)
from . import profiler | {"url":"https://pytorch.org/docs/1.10/_modules/torch/autograd.html","timestamp":"2024-11-12T19:00:24Z","content_type":"text/html","content_length":"68799","record_id":"<urn:uuid:352fcd40-c6f0-4b14-9d72-d61b5f44c9ee>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00883.warc.gz"} |
Understanding Mathematical Functions: How To Find The Average Rate Of
Mathematical functions are a fundamental concept in algebra and calculus, representing a relationship between input and output values. Understanding functions is crucial for analyzing and predicting
various phenomena in the natural world. One of the key aspects of functions is finding the average rate of change, which plays a vital role in numerous real-world applications such as physics,
economics, and engineering. In this blog post, we will delve into the concept of mathematical functions and explore the importance of finding the average rate of change in practical scenarios.
Key Takeaways
• Mathematical functions represent a relationship between input and output values, and are crucial for analyzing and predicting various phenomena in the natural world.
• Finding the average rate of change is important in real-world applications such as physics, economics, and engineering.
• Understanding mathematical functions involves knowing the definition and examples of common types such as linear, quadratic, and exponential functions.
• Finding the average rate of change requires understanding the formula and step-by-step process, as well as practicing with example problems.
• The importance of average rate of change lies in its practical use in analyzing trends and its connection to slope in graphing.
Understanding Mathematical Functions
Mathematical functions are fundamental to understanding and analyzing various phenomena in the world. They provide a way to describe and model real-world situations using mathematical expressions. In
this chapter, we will delve into the definition of a mathematical function and explore examples of common mathematical functions.
A. Definition of a mathematical function
A mathematical function is a relation between a set of inputs (the domain) and a set of outputs (the range), such that each input is related to exactly one output. This means that for every value of
the input, there is a unique value of the output.
Examples of common mathematical functions
• Linear function: A linear function is a function that can be graphically represented as a straight line. It has the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept.
• Quadratic function: A quadratic function is a function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. It produces a parabolic curve when graphed.
• Exponential function: An exponential function is a function of the form f(x) = a^x, where a is a positive constant. It grows or decays exponentially as x increases or decreases.
Average Rate of Change
When working with mathematical functions, understanding the average rate of change is crucial for analyzing the behavior of the function over a given interval. The average rate of change measures how
the output of a function changes on average as the input changes over a specific interval.
Definition of average rate of change
The average rate of change of a function f(x) over an interval [a, b] is defined as the change in the function's output divided by the change in the input over that interval. In other words, it
represents the average slope of the function over that interval.
Formula for finding the average rate of change of a function
The formula for finding the average rate of change of a function f(x) over an interval [a, b] is given by:
Average Rate of Change (ARC) = (f(b) - f(a)) / (b - a)
• ARC is the average rate of change of the function f(x) over the interval [a, b]
• f(b) is the value of the function at the upper bound of the interval
• f(a) is the value of the function at the lower bound of the interval
• b is the upper bound of the interval
• a is the lower bound of the interval
Finding the Average Rate of Change of a Function
Understanding how to find the average rate of change of a function is an essential skill in mathematics and is often used in various fields such as physics, economics, and engineering. The average
rate of change of a function represents the average rate at which the function's output value changes with respect to its input value over a specific interval. Here, we will discuss the step-by-step
process for finding the average rate of change of a function and provide example problems to demonstrate the process.
Step-by-step process for finding the average rate of change
• Select two points on the function: To find the average rate of change of a function over a specific interval, you will need to select two distinct points on the function within that interval.
These two points will act as the starting and ending points for the calculation.
• Calculate the change in the function's output: Determine the change in the function's output value between the two selected points by subtracting the output value at the starting point from the
output value at the ending point.
• Calculate the change in the function's input: Similarly, calculate the change in the function's input value between the two selected points by subtracting the input value at the starting point
from the input value at the ending point.
• Find the average rate of change: Divide the change in the function's output by the change in the function's input to obtain the average rate of change of the function over the specified interval.
Example problems to demonstrate the process
Let's work through a couple of example problems to demonstrate the process of finding the average rate of change of a function.
• Example 1: Consider the function f(x) = 2x + 3. Find the average rate of change of the function over the interval [1, 3].
• Example 2: Now, let's consider the function g(x) = x^2 - 4. Determine the average rate of change of the function over the interval [2, 5].
By following the step-by-step process and working through example problems, you can gain a better understanding of how to find the average rate of change of a function and its practical applications
in real-world scenarios.
Real-World Applications
Mathematical functions are not just abstract concepts used in mathematical theory. They have real-world applications in various fields, including physics and engineering. One important concept in
understanding mathematical functions is the average rate of change, which is used in analyzing real-world problems.
A. How average rate of change is used in physics and engineering
• Physics
In physics, the average rate of change is used to analyze the motion of objects. For example, when studying the velocity of an object over a specific time interval, the average rate of change of
velocity can provide insights into the object's acceleration and overall motion.
• Engineering
In engineering, the concept of average rate of change is crucial for analyzing the performance of systems and processes. Engineers use this concept to understand how variables such as
temperature, pressure, and flow rates change over time, helping them make informed decisions in designing and optimizing systems.
B. Examples of real-world problems that involve finding average rate of change
• Financial Analysis
Financial analysts often use average rate of change to analyze the growth or decline of investments, market trends, and economic indicators. By calculating the average rate of change, they can
make predictions and strategic decisions based on the observed trends.
• Population Studies
In demographic studies, researchers use the average rate of change to analyze population growth or decline over specific time periods. This information is crucial for urban planning, resource
allocation, and policy-making.
• Environmental Science
Environmental scientists use the average rate of change to study changes in ecological systems, such as the rate of deforestation, species extinction, and climate change. This information helps
in understanding the impact of human activities on the environment and in formulating conservation strategies.
Importance of Average Rate of Change
Understanding the concept of average rate of change is crucial in the field of mathematics and various other disciplines. It allows us to analyze the trends in a function and make predictions about
its behavior. Additionally, it helps in understanding the relationship between different variables and their rates of change over a specific interval. Let's delve deeper into the importance of
average rate of change in mathematical functions.
A. How understanding average rate of change helps in analyzing trends
Calculating the average rate of change of a function over a given interval provides insight into the overall trend of the function. By analyzing how the function changes over time, we can identify
whether it is increasing, decreasing, or remaining constant. This information is essential for making informed decisions in various scenarios, such as economic forecasting, scientific research, and
engineering applications.
For instance, in economics, understanding the average rate of change helps in predicting market trends and assessing the growth or decline of a company's performance over time. In physics, it assists
in analyzing the motion of objects and determining their velocity or acceleration. Therefore, grasping the concept of average rate of change is valuable for interpreting the behavior of functions and
drawing meaningful conclusions from the data.
B. Connection between average rate of change and slope in graphing
The average rate of change of a function is closely related to the concept of slope in graphing. When we calculate the average rate of change over an interval, we are essentially finding the slope of
the secant line that connects two points on the function. This connection is fundamental in understanding the steepness or direction of a function's graph and how it changes over time.
By determining the average rate of change, we can visualize the function's behavior graphically and comprehend its rate of ascent or descent. In essence, the slope of the function's graph corresponds
to its average rate of change, making it a valuable tool for interpreting the overall trend and direction of the function.
Understanding mathematical functions and how to find the average rate of change is crucial for anyone studying mathematics or pursuing a career in a related field. It not only helps in understanding
the behavior of a function but also provides valuable insights into real-world applications. I strongly encourage you to practice finding average rate of change to improve your math skills and gain a
deeper understanding of mathematical functions.
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Neural network
Visualization of a simple neural network for educational purposes.
What is this?
This is implementation of neural network with back-propagation. There aren't any special tricks, it's as simple neural network as it gets.
Cost function
The cost is defined as \(C = \frac{1}{2 \times sampleCnt}\sum^{sampleCnt}_{m=1}(\sum^{outputSize}_{n=1}(neruon_n-target_n)^2)\). In words: Error is defined as \((value - target)^2\). To get error of
neural network for one training sample, you simply add errors of all output neurons. The total cost is then defined as average error of all training samples.
Forward propagation
Let's say that the value of connection is the connection's weight (how wide it is) times the first connected neuron. To calculate the value of some neuron you add the values of all incoming
connections and apply the sigmoid function to that sum. Other activation functions are possible, but I have not implemented them yet.
Back propagation
The cost function defined above is a function dependend on weights of connections in the same way as \(f(x, y) = x^2 + y^2\) is dependend on x and y. In the beginning, the weights are random. Let's
say x = 5 and y = 3. The cost at this point would be 25 + 9 = 34, which we want to get to 0. Now we take the derivate with respect to each of these weights, which tells us how to adjust the wieghts
to minimize the function. \(\frac{\partial f(x, y)}{\partial x} = 2x\), \(\frac{\partial f(x, y)}{\partial y} = 2y\). Now that we have the derivatives, we know the "direction" in which to change the
weights. \(x_{new} = x_{old} - rate \times 2x = 5 - 0.1 \times 2 \times 5 = 4\) and that's a little bit closer to the desired 0 result of f(x, y). The rate is necessary to avoid stepping over the
In practice is the computation of the derivatives is a little bit harder, but all you need to know is the chain rule. I highly recommend 3blue1brown's series and this paper for better understanding.
If you have enjoyed this project, you might also like my tool for creating gif memes and Vocabuo - an app for learning vocabulary. | {"url":"https://nnplayground.com/","timestamp":"2024-11-07T18:42:22Z","content_type":"text/html","content_length":"10195","record_id":"<urn:uuid:e1bf087f-85cb-4e46-82e1-22926cb05d7d>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00313.warc.gz"} |
IMARGUMENT() Formula in Google Sheets
The IMARGUMENT function returns the angle (also known as the argument or \theta) of the given complex number in radians.
Common Questions about the IMARGUMENT Formula
1. What does the IMARGUMENT formula do?
2. What arguments can be passed to the IMARGUMENT formula?
3. How do I use the IMARGUMENT formula?
How to Use the IMARGUMENT Formula Appropriately
1. Enter the data into Google Sheets correctly.
2. Use the syntax of the IMARGUMENT formula correctly.
3. Specify appropriate arguments for the IMARGUMENT formula.
4. Calculate the IMARGUMENT formula correctly.
How the IMARGUMENT Formula Can be Commonly Mistyped
1. Typing in ""IMARGUEMENT"" instead of ""IMARGUMENT"".
2. Accidentally adding a space between the ""I"" and the ""M"" in the formula.
3. Omitting the parentheses when uisng the formula.
Common Ways the IMARGUMENT Formula is Used Inappropriately
1. Passing inappropriate arguments to the IMARGUMENT formula.
2. Failing to specify the required arguments for the IMARGUMENT formula.
3. Entering information that is inappropriate for use with the IMARGUMENT formula.
Common Pitfalls When Using the IMARGUMENT Formula
1. Not entering or using the proper syntax for the formula.
2. Failing to use the appropriate arguments or specifying incorrect arguments for the formula.
3. Calculating results using the wrong type of data.
Common Mistakes When Using the IMARGUMENT Formula
1. Reversing the order of the arguments in the formula.
2. Overwriting the data used by the formula.
3. Using incorrect logical tests to evaluate results.
Common Misconceptions People Might Have with the IMARGUMENT Formula
1. Thinking that the IMARGUMENT formula is the same as the
2. Assuming the IMARGUMENT formula can be used for all kinds of data.
3. Believing that the IMARGUMENT formula can be used to calculate more than one cell at a time. | {"url":"https://www.bettersheets.co/formulas/imargument","timestamp":"2024-11-08T18:23:14Z","content_type":"text/html","content_length":"31649","record_id":"<urn:uuid:98894bb8-2662-4429-997c-b8ef77191ee4>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00799.warc.gz"} |
Strings C++ | HackerRank Solutions
Strings C++
Problem Statement :
C++ provides a nice alternative data type to manipulate strings, and the data type is conveniently called string. Some of its widely used features are the following:
string a = "abc";
int len = a.size();
Concatenate two strings:
string a = "abc";
string b = "def";
string c = a + b; // c = "abcdef".
Accessing i th element:
string s = "abc";
char c0 = s[0]; // c0 = 'a'
char c1 = s[1]; // c1 = 'b'
char c2 = s[2]; // c2 = 'c'
s[0] = 'z'; // s = "zbc"
P.S.: We will use cin/cout to read/write a string.
Input Format
You are given two strings, a and b , separated by a new line. Each string will consist of lower case Latin characters ('a'-'z').
Output Format
In the first line print two space-separated integers, representing the length of a and b respectively.
In the second line print the string produced by concatenating a and b(a + b).
In the third line print two strings separated by a space, a' and b' . a' and b' are the same as a and b, respectively, except that their first characters are swapped.
Solution :
Solution in C :
#include <iostream>
#include <string>
using namespace std;
int main() {
string a, b;
cin >> a >> b;
cout << a.length() << " " << b.length() << endl;
cout << a << b << endl;
swap(a[0], b[0]);
cout << a << " " << b << endl;
return 0;
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