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How to Use Test-Time Augmentation to Improve Model Performance for Image Classification
Author: Jason Brownlee
Data augmentation is a technique often used to improve performance and reduce generalization error when training neural network models for computer vision problems.
The image data augmentation technique can also be applied when making predictions with a fit model in order to allow the model to make predictions for multiple different versions of each image in the
test dataset. The predictions on the augmented images can be averaged, which can result in better predictive performance.
In this tutorial, you will discover test-time augmentation for improving the performance of models for image classification tasks.
After completing this tutorial, you will know:
• Test-time augmentation is the application of data augmentation techniques normally used during training when making predictions.
• How to implement test-time augmentation from scratch in Keras.
• How to use test-time augmentation to improve the performance of a convolutional neural network model on a standard image classification task.
Let’s get started.
Tutorial Overview
This tutorial is divided into five parts; they are:
1. Test-Time Augmentation
2. Test-Time Augmentation in Keras
3. Dataset and Baseline Model
4. Example of Test-Time Augmentation
5. How to Tune Test-Time Augmentation Configuration
Test-Time Augmentation
Data augmentation is an approach typically used during the training of the model that expands the training set with modified copies of samples from the training dataset.
Data augmentation is often performed with image data, where copies of images in the training dataset are created with some image manipulation techniques performed, such as zooms, flips, shifts, and
The artificially expanded training dataset can result in a more skillful model, as often the performance of deep learning models continues to scale in concert with the size of the training dataset.
In addition, the modified or augmented versions of the images in the training dataset assist the model in extracting and learning features in a way that is invariant to their position, lighting, and
Test-time augmentation, or TTA for short, is an application of data augmentation to the test dataset.
Specifically, it involves creating multiple augmented copies of each image in the test set, having the model make a prediction for each, then returning an ensemble of those predictions.
Augmentations are chosen to give the model the best opportunity for correctly classifying a given image, and the number of copies of an image for which a model must make a prediction is often small,
such as less than 10 or 20.
Often, a single simple test-time augmentation is performed, such as a shift, crop, or image flip.
In their 2015 paper that achieved then state-of-the-art results on the ILSVRC dataset titled “Very Deep Convolutional Networks for Large-Scale Image Recognition,” the authors use horizontal flip
test-time augmentation:
We also augment the test set by horizontal flipping of the images; the soft-max class posteriors of the original and flipped images are averaged to obtain the final scores for the image.
Similarly, in their 2015 paper on the inception architecture titled “Rethinking the Inception Architecture for Computer Vision,” the authors at Google use cropping test-time augmentation, which they
refer to as multi-crop evaluation.
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Test-Time Augmentation in Keras
Test-time augmentation is not provided natively in the Keras deep learning library but can be implemented easily.
The ImageDataGenerator class can be used to configure the choice of test-time augmentation. For example, the data generator below is configured for horizontal flip image data augmentation.
# configure image data augmentation
datagen = ImageDataGenerator(horizontal_flip=True)
The augmentation can then be applied to each sample in the test dataset separately.
First, the dimensions of the single image can be expanded from [rows][cols][channels] to [samples][rows][cols][channels], where the number of samples is one, for the single image. This transforms the
array for the image into an array of samples with one image.
# convert image into dataset
samples = expand_dims(image, 0)
Next, an iterator can be created for the sample, and the batch size can be used to specify the number of augmented images to generate, such as 10.
# prepare iterator
it = datagen.flow(samples, batch_size=10)
The iterator can then be passed to the predict_generator() function of the model in order to make a prediction. Specifically, a batch of 10 augmented images will be generated and the model will make
a prediction for each.
# make predictions for each augmented image
yhats = model.predict_generator(it, steps=10, verbose=0)
Finally, an ensemble prediction can be made. A prediction was made for each image, and each prediction contains a probability of the image belonging to each class, in the case of image multiclass
An ensemble prediction can be made using soft voting where the probabilities of each class are summed across the predictions and a class prediction is made by calculating the argmax() of the summed
predictions, returning the index or class number of the largest summed probability.
# sum across predictions
summed = numpy.sum(yhats, axis=0)
# argmax across classes
return argmax(summed)
We can tie these elements together into a function that will take a configured data generator, fit model, and single image, and will return a class prediction (integer) using test-time augmentation.
# make a prediction using test-time augmentation
def tta_prediction(datagen, model, image, n_examples):
# convert image into dataset
samples = expand_dims(image, 0)
# prepare iterator
it = datagen.flow(samples, batch_size=n_examples)
# make predictions for each augmented image
yhats = model.predict_generator(it, steps=n_examples, verbose=0)
# sum across predictions
summed = numpy.sum(yhats, axis=0)
# argmax across classes
return argmax(summed)
Now that we know how to make predictions in Keras using test-time augmentation, let’s work through an example to demonstrate the approach.
Dataset and Baseline Model
We can demonstrate test-time augmentation using a standard computer vision dataset and a convolutional neural network.
Before we can do that, we must select a dataset and a baseline model.
We will use the CIFAR-10 dataset, comprised of 60,000 32×32 pixel color photographs of objects from 10 classes, such as frogs, birds, cats, ships, etc. CIFAR-10 is a well-understood dataset and
widely used for benchmarking computer vision algorithms in the field of machine learning. The problem is “solved.” Top performance on the problem is achieved by deep learning convolutional neural
networks with a classification accuracy above 96% or 97% on the test dataset.
We will also use a convolutional neural network, or CNN, model that is capable of achieving good (better than random) results, but not state-of-the-art results, on the problem. This will be
sufficient to demonstrate the lift in performance that test-time augmentation can provide.
The CIFAR-10 dataset can be loaded easily via the Keras API by calling the cifar10.load_data() function, that returns a tuple with the training and test datasets split into input (images) and output
(class labels) components.
# load dataset
(trainX, trainY), (testX, testY) = load_data()
It is good practice to normalize the pixel values from the range 0-255 down to the range 0-1 prior to modeling. This ensures that the inputs are small and close to zero, and will, in turn, mean that
the weights of the model will be kept small, leading to faster and better learning.
# normalize pixel values
trainX = trainX.astype('float32') / 255
testX = testX.astype('float32') / 255
The class labels are integers and must be converted to a one hot encoding prior to modeling.
This can be achieved using the to_categorical() Keras utility function.
# one hot encode target values
trainY = to_categorical(trainY)
testY = to_categorical(testY)
We are now ready to define a model for this multi-class classification problem.
The model has a convolutional layer with 32 filter maps with a 3×3 kernel using the rectifier linear activation, “same” padding so the output is the same size as the input and the He weight
initialization. This is followed by a batch normalization layer and a max pooling layer.
This pattern is repeated with a convolutional, batch norm, and max pooling layer, although the number of filters is increased to 64. The output is then flattened before being interpreted by a dense
layer and finally provided to the output layer to make a prediction.
# define model
model = Sequential()
model.add(Conv2D(32, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform', input_shape=(32, 32, 3)))
model.add(MaxPooling2D((2, 2)))
model.add(Conv2D(64, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform'))
model.add(MaxPooling2D((2, 2)))
model.add(Dense(128, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(10, activation='softmax'))
The Adam variation of stochastic gradient descent is used to find the model weights.
The categorical cross entropy loss function is used, required for multi-class classification, and classification accuracy is monitored during training.
# compile model
model.compile(optimizer='adam', loss='categorical_crossentropy', metrics=['accuracy'])
The model is fit for three training epochs and a large batch size of 128 images is used.
# fit model
model.fit(trainX, trainY, epochs=3, batch_size=128)
Once fit, the model is evaluated on the test dataset.
# evaluate model
_, acc = model.evaluate(testX, testY, verbose=0)
The complete example is listed below and will easily run on the CPU in a few minutes.
# baseline cnn model for the cifar10 problem
from keras.datasets.cifar10 import load_data
from keras.utils import to_categorical
from keras.models import Sequential
from keras.layers import Conv2D
from keras.layers import MaxPooling2D
from keras.layers import Dense
from keras.layers import Flatten
from keras.layers import BatchNormalization
# load dataset
(trainX, trainY), (testX, testY) = load_data()
# normalize pixel values
trainX = trainX.astype('float32') / 255
testX = testX.astype('float32') / 255
# one hot encode target values
trainY = to_categorical(trainY)
testY = to_categorical(testY)
# define model
model = Sequential()
model.add(Conv2D(32, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform', input_shape=(32, 32, 3)))
model.add(MaxPooling2D((2, 2)))
model.add(Conv2D(64, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform'))
model.add(MaxPooling2D((2, 2)))
model.add(Dense(128, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(10, activation='softmax'))
# compile model
model.compile(optimizer='adam', loss='categorical_crossentropy', metrics=['accuracy'])
# fit model
history = model.fit(trainX, trainY, epochs=3, batch_size=128)
# evaluate model
_, acc = model.evaluate(testX, testY, verbose=0)
Running the example shows that the model is capable of learning the problem well and quickly.
A test set accuracy of about 66% is achieved, which is okay, but not terrific. The chosen model configuration has already started to overfit and could benefit from the use of regularization and
further tuning. Nevertheless, this provides a good starting point for demonstrating test-time augmentation.
Epoch 1/3
50000/50000 [==============================] - 64s 1ms/step - loss: 1.2135 - acc: 0.5766
Epoch 2/3
50000/50000 [==============================] - 63s 1ms/step - loss: 0.8498 - acc: 0.7035
Epoch 3/3
50000/50000 [==============================] - 63s 1ms/step - loss: 0.6799 - acc: 0.7632
Neural networks are stochastic algorithms and the same model fit on the same data multiple times may find a different set of weights and, in turn, have different performance each time.
In order to even out the estimate of model performance, we can change the example to re-run the fit and evaluation of the model multiple times and report the mean and standard deviation of the
distribution of scores on the test dataset.
First, we can define a function named load_dataset() that will load the CIFAR-10 dataset and prepare it for modeling.
# load and return the cifar10 dataset ready for modeling
def load_dataset():
# load dataset
(trainX, trainY), (testX, testY) = load_data()
# normalize pixel values
trainX = trainX.astype('float32') / 255
testX = testX.astype('float32') / 255
# one hot encode target values
trainY = to_categorical(trainY)
testY = to_categorical(testY)
return trainX, trainY, testX, testY
Next, we can define a function named define_model() that will define a model for the CIFAR-10 dataset, ready to be fit and then evaluated.
# define the cnn model for the cifar10 dataset
def define_model():
# define model
model = Sequential()
model.add(Conv2D(32, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform', input_shape=(32, 32, 3)))
model.add(MaxPooling2D((2, 2)))
model.add(Conv2D(64, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform'))
model.add(MaxPooling2D((2, 2)))
model.add(Dense(128, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(10, activation='softmax'))
# compile model
model.compile(optimizer='adam', loss='categorical_crossentropy', metrics=['accuracy'])
return model
Next, an evaluate_model() function is defined that will fit the defined model on the training dataset and then evaluate it on the test dataset, returning the estimated classification accuracy for the
# fit and evaluate a defined model
def evaluate_model(model, trainX, trainY, testX, testY):
# fit model
model.fit(trainX, trainY, epochs=3, batch_size=128, verbose=0)
# evaluate model
_, acc = model.evaluate(testX, testY, verbose=0)
return acc
Next, we can define a function with new behavior to repeatedly define, fit, and evaluate a new model and return the distribution of accuracy scores.
The repeated_evaluation() function below implements this, taking the dataset and using a default of 10 repeated evaluations.
# repeatedly evaluate model, return distribution of scores
def repeated_evaluation(trainX, trainY, testX, testY, repeats=10):
scores = list()
for _ in range(repeats):
# define model
model = define_model()
# fit and evaluate model
accuracy = evaluate_model(model, trainX, trainY, testX, testY)
# store score
print('> %.3f' % accuracy)
return scores
Finally, we can call the load_dataset() function to prepare the dataset, then repeated_evaluation() to get a distribution of accuracy scores that can be summarized by reporting the mean and standard
# load dataset
trainX, trainY, testX, testY = load_dataset()
# evaluate model
scores = repeated_evaluation(trainX, trainY, testX, testY)
# summarize result
print('Accuracy: %.3f (%.3f)' % (mean(scores), std(scores)))
Tying all of this together, the complete code example of repeatedly evaluating a CNN model on the MNIST dataset is listed below.
# baseline cnn model for the cifar10 problem, repeated evaluation
from numpy import mean
from numpy import std
from keras.datasets.cifar10 import load_data
from keras.utils import to_categorical
from keras.models import Sequential
from keras.layers import Conv2D
from keras.layers import MaxPooling2D
from keras.layers import Dense
from keras.layers import Flatten
from keras.layers import BatchNormalization
# load and return the cifar10 dataset ready for modeling
def load_dataset():
# load dataset
(trainX, trainY), (testX, testY) = load_data()
# normalize pixel values
trainX = trainX.astype('float32') / 255
testX = testX.astype('float32') / 255
# one hot encode target values
trainY = to_categorical(trainY)
testY = to_categorical(testY)
return trainX, trainY, testX, testY
# define the cnn model for the cifar10 dataset
def define_model():
# define model
model = Sequential()
model.add(Conv2D(32, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform', input_shape=(32, 32, 3)))
model.add(MaxPooling2D((2, 2)))
model.add(Conv2D(64, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform'))
model.add(MaxPooling2D((2, 2)))
model.add(Dense(128, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(10, activation='softmax'))
# compile model
model.compile(optimizer='adam', loss='categorical_crossentropy', metrics=['accuracy'])
return model
# fit and evaluate a defined model
def evaluate_model(model, trainX, trainY, testX, testY):
# fit model
model.fit(trainX, trainY, epochs=3, batch_size=128, verbose=0)
# evaluate model
_, acc = model.evaluate(testX, testY, verbose=0)
return acc
# repeatedly evaluate model, return distribution of scores
def repeated_evaluation(trainX, trainY, testX, testY, repeats=10):
scores = list()
for _ in range(repeats):
# define model
model = define_model()
# fit and evaluate model
accuracy = evaluate_model(model, trainX, trainY, testX, testY)
# store score
print('> %.3f' % accuracy)
return scores
# load dataset
trainX, trainY, testX, testY = load_dataset()
# evaluate model
scores = repeated_evaluation(trainX, trainY, testX, testY)
# summarize result
print('Accuracy: %.3f (%.3f)' % (mean(scores), std(scores)))
Running the example may take a while on modern CPU hardware and is much faster on GPU hardware.
The accuracy of the model is reported for each repeated evaluation and the final mean model performance is reported.
In this case, we can see that the mean accuracy of the chosen model configuration is about 68%, which is close to the estimate from a single model run.
> 0.690
> 0.662
> 0.698
> 0.681
> 0.686
> 0.680
> 0.697
> 0.696
> 0.689
> 0.679
Accuracy: 0.686 (0.010)
Now that we have developed a baseline model for a standard dataset, let’s look at updating the example to use test-time augmentation.
Example of Test-Time Augmentation
We can now update our repeated evaluation of the CNN model on CIFAR-10 to use test-time augmentation.
The tta_prediction() function developed in the section above on how to implement test-time augmentation in Keras can be used directly.
# make a prediction using test-time augmentation
def tta_prediction(datagen, model, image, n_examples):
# convert image into dataset
samples = expand_dims(image, 0)
# prepare iterator
it = datagen.flow(samples, batch_size=n_examples)
# make predictions for each augmented image
yhats = model.predict_generator(it, steps=n_examples, verbose=0)
# sum across predictions
summed = numpy.sum(yhats, axis=0)
# argmax across classes
return argmax(summed)
We can develop a function that will drive the test-time augmentation by defining the ImageDataGenerator configuration and call tta_prediction() for each image in the test dataset.
It is important to consider the types of image augmentations that may benefit a model fit on the CIFAR-10 dataset. Augmentations that cause minor modifications to the photographs might be useful.
This might include augmentations such as zooms, shifts, and horizontal flips.
In this example, we will only use horizontal flips.
# configure image data augmentation
datagen = ImageDataGenerator(horizontal_flip=True)
We will configure the image generator to create seven photos, from which the mean prediction for each example in the test set will be made.
The tta_evaluate_model() function below configures the ImageDataGenerator then enumerates the test dataset, making a class label prediction for each image in the test dataset. The accuracy is then
calculated by comparing the predicted class labels to the class labels in the test dataset. This requires that we reverse the one hot encoding performed in load_dataset() by using argmax().
# evaluate a model on a dataset using test-time augmentation
def tta_evaluate_model(model, testX, testY):
# configure image data augmentation
datagen = ImageDataGenerator(horizontal_flip=True)
# define the number of augmented images to generate per test set image
n_examples_per_image = 7
yhats = list()
for i in range(len(testX)):
# make augmented prediction
yhat = tta_prediction(datagen, model, testX[i], n_examples_per_image)
# store for evaluation
# calculate accuracy
testY_labels = argmax(testY, axis=1)
acc = accuracy_score(testY_labels, yhats)
return acc
The evaluate_model() function can then be updated to call tta_evaluate_model() in order to get model accuracy scores.
# fit and evaluate a defined model
def evaluate_model(model, trainX, trainY, testX, testY):
# fit model
model.fit(trainX, trainY, epochs=3, batch_size=128, verbose=0)
# evaluate model using tta
acc = tta_evaluate_model(model, testX, testY)
return acc
Tying all of this together, the complete example of the repeated evaluation of a CNN for CIFAR-10 with test-time augmentation is listed below.
# cnn model for the cifar10 problem with test-time augmentation
import numpy
from numpy import argmax
from numpy import mean
from numpy import std
from numpy import expand_dims
from sklearn.metrics import accuracy_score
from keras.datasets.cifar10 import load_data
from keras.utils import to_categorical
from keras.preprocessing.image import ImageDataGenerator
from keras.models import Sequential
from keras.layers import Conv2D
from keras.layers import MaxPooling2D
from keras.layers import Dense
from keras.layers import Flatten
from keras.layers import BatchNormalization
# load and return the cifar10 dataset ready for modeling
def load_dataset():
# load dataset
(trainX, trainY), (testX, testY) = load_data()
# normalize pixel values
trainX = trainX.astype('float32') / 255
testX = testX.astype('float32') / 255
# one hot encode target values
trainY = to_categorical(trainY)
testY = to_categorical(testY)
return trainX, trainY, testX, testY
# define the cnn model for the cifar10 dataset
def define_model():
# define model
model = Sequential()
model.add(Conv2D(32, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform', input_shape=(32, 32, 3)))
model.add(MaxPooling2D((2, 2)))
model.add(Conv2D(64, (3, 3), activation='relu', padding='same', kernel_initializer='he_uniform'))
model.add(MaxPooling2D((2, 2)))
model.add(Dense(128, activation='relu', kernel_initializer='he_uniform'))
model.add(Dense(10, activation='softmax'))
# compile model
model.compile(optimizer='adam', loss='categorical_crossentropy', metrics=['accuracy'])
return model
# make a prediction using test-time augmentation
def tta_prediction(datagen, model, image, n_examples):
# convert image into dataset
samples = expand_dims(image, 0)
# prepare iterator
it = datagen.flow(samples, batch_size=n_examples)
# make predictions for each augmented image
yhats = model.predict_generator(it, steps=n_examples, verbose=0)
# sum across predictions
summed = numpy.sum(yhats, axis=0)
# argmax across classes
return argmax(summed)
# evaluate a model on a dataset using test-time augmentation
def tta_evaluate_model(model, testX, testY):
# configure image data augmentation
datagen = ImageDataGenerator(horizontal_flip=True)
# define the number of augmented images to generate per test set image
n_examples_per_image = 7
yhats = list()
for i in range(len(testX)):
# make augmented prediction
yhat = tta_prediction(datagen, model, testX[i], n_examples_per_image)
# store for evaluation
# calculate accuracy
testY_labels = argmax(testY, axis=1)
acc = accuracy_score(testY_labels, yhats)
return acc
# fit and evaluate a defined model
def evaluate_model(model, trainX, trainY, testX, testY):
# fit model
model.fit(trainX, trainY, epochs=3, batch_size=128, verbose=0)
# evaluate model using tta
acc = tta_evaluate_model(model, testX, testY)
return acc
# repeatedly evaluate model, return distribution of scores
def repeated_evaluation(trainX, trainY, testX, testY, repeats=10):
scores = list()
for _ in range(repeats):
# define model
model = define_model()
# fit and evaluate model
accuracy = evaluate_model(model, trainX, trainY, testX, testY)
# store score
print('> %.3f' % accuracy)
return scores
# load dataset
trainX, trainY, testX, testY = load_dataset()
# evaluate model
scores = repeated_evaluation(trainX, trainY, testX, testY)
# summarize result
print('Accuracy: %.3f (%.3f)' % (mean(scores), std(scores)))
Running the example may take some time given the repeated evaluation and the slower manual test-time augmentation used to evaluate each model.
In this case, we can see a modest lift in performance from about 68.6% on the test set without test-time augmentation to about 69.8% accuracy on the test set with test-time augmentation.
> 0.719
> 0.716
> 0.709
> 0.694
> 0.690
> 0.694
> 0.680
> 0.676
> 0.702
> 0.704
Accuracy: 0.698 (0.013)
How to Tune Test-Time Augmentation Configuration
Choosing the augmentation configurations that give the biggest lift in model performance can be challenging.
Not only are there many augmentation methods to choose from and configuration options for each, but the time to fit and evaluate a model on a single set of configuration options can take a long time,
even if fit on a fast GPU.
Instead, I recommend fitting the model once and saving it to file. For example:
# save model
Then load the model from a separate file and evaluate different test-time augmentation schemes on a small validation dataset or small subset of the test set.
For example:
# load model
model = load_model('model.h5')
# evaluate model
datagen = ImageDataGenerator(...)
Once you find a set of augmentation options that give the biggest lift, you can then evaluate the model on the whole test set or trial a repeated evaluation experiment as above.
Test-time augmentation configuration not only includes the options for the ImageDataGenerator, but also the number of images generated from which the average prediction will be made for each example
in the test set.
I used this approach to choose the test-time augmentation in the previous section, discovering that seven examples worked better than three or five, and that random zooming and random shifts appeared
to decrease model accuracy.
Remember, if you also use image data augmentation for the training dataset and that augmentation uses a type of pixel scaling that involves calculating statistics on the dataset (e.g. you call
datagen.fit()), then those same statistics and pixel scaling techniques must also be used during test-time augmentation.
Further Reading
This section provides more resources on the topic if you are looking to go deeper.
In this tutorial, you discovered test-time augmentation for improving the performance of models for image classification tasks.
Specifically, you learned:
• Test-time augmentation is the application of data augmentation techniques normally used during training when making predictions.
• How to implement test-time augmentation from scratch in Keras.
• How to use test-time augmentation to improve the performance of a convolutional neural network model on a standard image classification task.
Do you have any questions?
Ask your questions in the comments below and I will do my best to answer.
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Radius Formula: Definition, Calculation, Solved Examples
In the world of geometry and mathematics, the concept of a circle holds great significance. One of the fundamental elements of a circle is the radius. Understanding the radius and its formula is
essential for solving various geometric problems and calculations. In this article, we will explore the definition of the radius, its formula, and provide step-by-step instructions on how to find the
radius of a circle in different scenarios. We will also examine solved examples to illustrate the practical application of the radius formula.
An Introduction to Radius Formula
Before diving into the specifics of the radius formula, let’s first understand the concept of radius. The radius of a circle is defined as a straight line segment that connects the center of the
circle to any point on its circumference. It is denoted by the letter ‘r’ and is crucial in determining the size and properties of a circle. The radius is half the length of the diameter, which is a
line segment passing through the center of the circle and connecting two points on its circumference.
What is Radius?
The radius of a circle is a fundamental measurement that plays a significant role in geometry and mathematics. It represents the distance from the center of the circle to any point on its
circumference. All radii of a circle are equal in length, as they connect the center to various points on the circumference. In simple terms, the radius defines the size of the circle and helps in
calculating other important parameters such as the diameter, circumference, and area.
What is the Radius Formula?
The radius formula allows us to calculate the length of the radius when other parameters of a circle are known. There are different formulas to find the radius based on the given information. Let’s
explore these formulas in detail.
Radius in Terms of Diameter
The diameter of a circle is the longest line segment that passes through the center and connects two points on the circumference. The radius is half the length of the diameter, and we can calculate
the radius using the following formula:
Radius = Diameter/2
In mathematical terms, the radius formula in terms of the diameter can be expressed as:
Radius = √(Diameter/2)
Radius in Terms of Circumference
The circumference of a circle is the distance around its outer boundary. It is calculated using the formula C = 2πr, where ‘C’ represents the circumference and ‘r’ represents the radius. By
rearranging the formula, we can find the radius when the circumference is known:
Radius = Circumference/2π
Radius in Terms of Area
The area of a circle is the amount of space enclosed within its boundaries. It is calculated using the formula A = πr², where ‘A’ represents the area and ‘r’ represents the radius. By rearranging the
formula, we can find the radius when the area is known:
Radius = √(Area/π)
How to Find the Radius of a Circle?
Now that we understand the different formulas to find the radius, let’s explore step-by-step instructions on how to find the radius of a circle in various scenarios.
How to Find the Radius of a Circle When a Diameter is Given?
Step 1: Identify the given value of the diameter.
Step 2: Apply the radius formula in terms of the diameter: Radius = Diameter/2.
Step 3: Divide the given diameter by 2 to find the radius.
Let’s solve an example to illustrate the process:
Example 1: Find the radius of a circle whose diameter is 24 inches long.
Step 1: Given diameter = 24 inches.
Step 2: Apply the radius formula: Radius = Diameter/2.
Step 3: Divide the diameter by 2: Radius = 24/2 = 12 inches.
Answer: The radius of the given circle is 12 inches.
How to Find the Radius of a Circle When Circumference is Given?
Step 1: Identify the given value of the circumference.
Step 2: Apply the radius formula in terms of the circumference: Radius = Circumference/2π.
Step 3: Divide the given circumference by 2π to find the radius.
Let’s solve an example to illustrate the process:
Example 2: Find the radius of a circle whose circumference is 4π units.
Step 1: Given circumference = 4π units.
Step 2: Apply the radius formula: Radius = Circumference/2π.
Step 3: Divide the circumference by 2π: Radius = 4π/2π = 2 units.
Answer: The radius of the given circle is 2 units.
How to Find the Radius of a Circle When Area is Given?
Step 1: Identify the given value of the area.
Step 2: Apply the radius formula in terms of the area: Radius = √(Area/π).
Step 3: Take the square root of the given area divided by π to find the radius.
Let’s solve an example to illustrate the process:
Example 3: Find the radius of a circle whose area is 4π units².
Step 1: Given area = 4π units².
Step 2: Apply the radius formula: Radius = √(Area/π).
Step 3: Take the square root of the area divided by π: Radius = √(4π/π) = 2 units.
Answer: The radius of the given circle is 2 units.
What is the Radius of the Circle that Passes Through Three Non-Collinear Points?
The radius of a circle that passes through three non-collinear points can be calculated using the following formula:
Radius = |O→X₁ - O→X₃| / (2 * Sin θ)
Here, O→X₁ and O→X₃ represent the vectors from the center of the circle to the respective points, and θ is the angle ∠X₁, X₂, X₃.
Radius of a Sphere
The concept of radius is not limited to circles alone. In three-dimensional geometry, a sphere also has a radius. The radius of a sphere is defined as the distance from the center of the sphere to
any point on its surface. The radius of a sphere is denoted by ‘r’ and is crucial in determining the size and properties of the sphere.
Differences between Radius and Diameter
The terms radius and diameter are often used interchangeably, but they have distinct meanings in geometry. Let’s explore the differences between radius and diameter.
• The radius is a straight line segment that connects the center of the circle or sphere to any point on its circumference or surface.
• The radius is half the length of the diameter.
• It is denoted by the letter ‘r’.
• The diameter is a straight line segment that passes through the center of the circle or sphere and connects two points on its circumference or surface.
• The diameter is twice the length of the radius.
• It is denoted by the letter ‘d’.
Radius Diameter
Represents the distance from the center of the circle or sphere to any point on its Represents the distance between two points on the circumference or surface of the circle or sphere, passing
circumference or surface. through the center.
Denoted by the letter ‘r’. Denoted by the letter ‘d’.
Half the length of the diameter. Twice the length of the radius.
Used to calculate other parameters such as circumference and area. Used to calculate other parameters such as circumference and area.
The radius formula is used to find the length of the radius. The diameter formula is used to find the length of the diameter.
Solved Examples on Radius Formula
Let’s explore a few solved examples to further enhance our understanding of the radius formula.
Example 1: Given the diameter of a circle is 10 cm, find the radius.
Step 1: Given diameter = 10 cm.
Step 2: Apply the radius formula: Radius = Diameter/2.
Step 3: Divide the diameter by 2: Radius = 10/2 = 5 cm.
Answer: The radius of the given circle is 5 cm.
Example 2: Find the radius of a circle whose circumference is 18π units.
Step 1: Given circumference = 18π units.
Step 2: Apply the radius formula: Radius = Circumference/2π.
Step 3: Divide the circumference by 2π: Radius = 18π/2π = 9 units.
Answer: The radius of the given circle is 9 units.
Example 3: Find the radius of a circle whose area is 25π units².
Step 1: Given area = 25π units².
Step 2: Apply the radius formula: Radius = √(Area/π).
Step 3: Take the square root of the area divided by π: Radius = √(25π/π) = 5 units.
Answer: The radius of the given circle is 5 units.
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4.4.8. (a) Prove that the set of all
4.4.8. (a) Prove that the set of all vectors orthogonal to a given subspace VC Rm forms a subspace. (b) Find a basis for the set of all vectors in R4 that are orthogonal to the subspace spanned by
(1, 2, 0, -1), (2,0,3,1)¹.
4.4.8. (a) Prove that the set of all vectors orthogonal to a given subspace VC Rm forms a subspace. (b) Find a basis for the set of all vectors in R4 that are orthogonal to the subspace spanned by
(1, 2, 0, -1), (2,0,3,1)¹.
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Transcribed Image Text:4.4.8. (a) Prove that the set of all vectors orthogonal to a given subspace VC Rm forms a subspace. (b) Find a basis for the set of all vectors in R4 that are orthogonal to the
subspace spanned by (1, 2, 0, -1), (2,0,3,1)¹.
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Nonlinear dynamic analysis of low-noise excavating system with two clearances
There are always clearances between the movement pairs in mechanical systems, which will deteriorate the dynamical performance of these systems when they become larger than the necessary size to
allow their relative movement between the movement parts. In this paper, a special linkage mechanism for excavation with two clearances was concerned, and the effects of size of the clearance and the
friction coefficient between the movement parts on the dynamical performance were investigated by modeling and numerical analysis. It shows that, with the increase of the friction coefficient and the
size of clearance, the trajectory of mass center of connecting rod does not change greatly, but the dynamic response, such as the angular velocity, the impact force, and the load torque, will sharply
increase, which may cause structure damage in many cases. Therefore, in the design and manufacture of this kind of mechanisms, we should find way to decrease the clearance and friction coefficient to
ensure the dynamical performance.
1. Introduction
Excavating instrument with low noise is one of the important equipments in civil engineering. In the machine, clearance is inevitable, and its size will affect the dynamic response and noise. Study
on clearance of machine is of essential significance. Many works on dynamic response of machine with clearance had been done by many scholars.
Different kinematic models with clearance are studied, assessing the actual configuration of a mechanism with clearance-affected pairs [1]. Imed [2] studied the dynamic behavior of a planar flexible
slider-crank mechanism with clearance, using a contact model based on the impact-function. Tsai and Lai [3], using the properties of reciprocal screws to determine the instantaneous configurations,
introduced a generalized method for error analysis of multi-loop mechanisms with joint clearance. Joint clearance was treated as a virtual link to simplify the study. Equivalent kinematical pair was
used to model the motion freedoms furnished by the joint clearances. Schwab [4] made a comparison between several continuous contact force models and an impact model. The results showed that, both
impact model and Hertzian contact force model could predict the dynamic response of mechanisms and machines having unlubricated revolute joint clearance including the peak values of the forces and
position and velocity deviations due to the clearance.
Flores [5] presented the dynamical analysis of mechanical systems considering joint with clearance and lubrication. The numerical results pointed that the existence of dry joint clearances caused
high peaks on the kinematic and characteristics of dynamic system due to contact-impact forces when compared to those obtained with lubricated model. Due to contact-impact forces when compared to
those obtained with lubricated model, the numerical results pointed that the existence of dry joint clearances caused high peaks on the characteristics of kinematic and dynamic system. In Hu’s
research [6], the support structure and the supports were considered flexible and the contribution from the flexible deformation to the clearance between the shaft and the support was included in the
shaft support interaction problem. The equation of final system was non-linear [7-8]. Flores [9] analyzed the dynamic characters of multi-body system with pair clearance. Rhee [10, 11] investigated
the response of a revolute joint in a four-bar mechanism with one clearance. The numerical results showed that nonlinearity depended on both the size of the clearance and the coefficient of friction
between the pin and bearing. Olyaei’s research results showed that the system may exhibit chaotic behavior under specific conditions [12].
In this paper, the nonlinear dynamic characters of excavating system with two clearances will be studied based on the references and research results. Our model will be established firstly, and the
dynamic equations are derived, then the response of excavating system with two clearances will be analyzed. The effects of the size of clearance and the coefficient of friction will be discussed
2. The dynamic model of excavating system with two clearances
The model of low-noise excavating system with two clearances is shown in Fig. 1. The machine has two parts, and the left and right part have similar structure. For simplification, only the right side
is considered. $OA$ is crank and $BC$ is cutter head. There are clearances at $A$, between the crank and connecting-rod, and $B$, between the connecting-rod and cutter head. ${l}_{1},{l}_{2},{l}_{3}$
are the length of crank, connecting-rod and cutter head, respectively, ${l}_{4}$ is the distance from fixed point $O$ to contact point when the cutter head is at the equilibrium position. ${m}_{1},
{m}_{2},{m}_{3}$ are the mass of crank, connecting-rod and cutter head, respectively. ${s}_{1},{s}_{2},{s}_{3}$ are the mass center of crank, connecting-rod and cutter head, respectively. ${J}_{s1}$
and ${J}_{1}$ are the rotational inertia of crank about fixed point $O$ and mass center, respectively, ${J}_{s2}$ and ${J}_{s3}$ are the rotational inertia of connecting-rod and cutter head about
mass center. The torque of crank is $T$, and the crushing torque between crushed body and cutter head is ${M}_{43}$, the torsion induced by the distortion of spring in tangential direction is ${M}_
{T}$. ${\theta }_{1},{\theta }_{2}$ are the angular of crank and connecting-rod, ${\theta }_{3}$ is angle of the engagement line of cutter head and crushed body, ${\theta }_{4}$ is the angle of the
centre line of cutter head. The curvature radius of crushed body is ${R}_{i}$. ${x}_{s2},{y}_{s2}$ are the projection of mass center ${s}_{2}$ of connecting rod in $x$ and $y$ directions, the
distance from shaft pin to shaft sleeve at $A$ is ${e}_{1}$, and ${e}_{2}$ at $B$.
Fig. 1Model of excavating system with two clearances
Choose ${\theta }_{1},{\theta }_{2},{\theta }_{3},{x}_{s2},{y}_{s2}$ as generalized coordinates, and $o$ as the origin of the coordinates, ${e}_{1x},{e}_{1y},{e}_{2x},{e}_{2y}$ are the projections of
distance between shaft pin and shaft sleeve in $x$ and $y$ direction. According to the geometrical relation, ${e}_{i}$ is expressed as follows:
$\text{Pair A}:\begin{array}{l}{e}_{1x}={x}_{s2}-{l}_{1}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{1}-{l}_{s2}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{2},\\ {e}_{1y}={y}_{s2}-{l}_{1}\mathrm{s}\mathrm
{i}\mathrm{n}{\theta }_{1}-{l}_{s2}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{2},\end{array}$
$\text{Pair B}:\begin{array}{l}{e}_{2x}={l}_{4}-{x}_{s2}-\left({l}_{2}-{l}_{s2}\right)\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{2}-\left(R-{l}_{3}\right)\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{3},\\
{e}_{2y}=\left(R-{l}_{3}\right)\left(1-\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{3}\right)-{y}_{s2}-\left({l}_{2}-{l}_{s2}\right)\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{2}.\end{array}$
When the distance ${e}_{i}$ between shaft pin and shaft sleeve is smaller than the clearance ${r}_{i}$, the shaft pin does not contact the sleeve, that is, the free state; otherwise, when ${e}_{i}$
is larger than the clearance ${r}_{i}$, the shaft pin contacts the sleeve and the system is in contact. The criterion is that:
${e}_{i}-{r}_{i}<0,\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(i=1,\text{\hspace{0.17em}\hspace{0.17em}}2\right),\text{the free state},$${e}_{i}-{r}_{i}>0,\text{\hspace{0.17em}\hspace
{0.17em}\hspace{0.17em}}\left(i=1,\text{\hspace{0.17em}\hspace{0.17em}}2\right),\text{the contact state}.$
The connected angle ${\alpha }_{i}$ can be determined by:
${\alpha }_{i}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{{e}_{iy}}{{e}_{ix}}\right),\left(i=1,\text{\hspace{0.17em}}2\right).$
At the contact point, the relative velocity of shaft pin relative to craft sleeve in normal and tangential direction is that:
${v}_{in}={\stackrel{˙}{e}}_{ix}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{i}+{\stackrel{˙}{e}}_{iy}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{i},\left(i=1,2\right),$${v}_{it}={\stackrel{˙}{e}}_{iy}\
mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{i}-{\stackrel{˙}{e}}_{ix}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{i}+\left({\stackrel{˙}{\theta }}_{i}-{\stackrel{˙}{\theta }}_{i+1}\right){R}_{i},\left(i=1,2\
The nonlinear contact force according to Hertz modeling is that [12, 13]:
${F}_{12}^{n}=\left\{\begin{array}{ll}{K}_{1}{\delta }_{1}^{1.5}+{C}_{1n}{v}_{1n},& {\delta }_{1}>0,\\ 0,& {\delta }_{1}\le 0,\end{array}\right\$
${F}_{12}^{t}=\left\{\begin{array}{ll}{f}_{1}{\sigma }_{1}{F}_{12}^{n}+{C}_{1t}{v}_{1t},& {\delta }_{1}>0,\\ 0,& {\delta }_{1}\le 0,\end{array}\right\$
${F}_{32}^{n}=\left\{\begin{array}{ll}{K}_{2}{\delta }_{2}^{1.5}+{C}_{2n}{v}_{2n},& {\delta }_{2}>0,\\ 0,& {\delta }_{2}\le 0,\end{array}\right\$
${F}_{32}^{t}=\left\{\begin{array}{ll}{f}_{2}{\sigma }_{2}{F}_{32}^{n}+{C}_{2t}{v}_{2t},& {\delta }_{2}>0,\\ 0,& {\delta }_{2}\le 0,\end{array}\right\$
${\sigma }_{1}=\left\{\begin{array}{ll}1,& {v}_{1t}\ge 0,\\ -1,& {v}_{1t}<0,\end{array}\right\$
${\sigma }_{2}=\left\{\begin{array}{ll}1,& {v}_{2t}\ge 0,\\ -1,& {v}_{2t}<0,\end{array}\right\$
where ${F}_{12}^{n},{F}_{12}^{t},{F}_{32}^{n},{F}_{32}^{t}$ are the contact forces at pair $A$ and $B$ in normal and tangential direction, respectively. ${K}_{1},{K}_{2},{C}_{1n},{C}_{2n},{C}_{1t},
{C}_{2t},{f}_{1},{f}_{2}$ are the stiffness coefficient, damping coefficient and friction coefficient in normal direction in normal and tangential direction, respectively.
The projections of contact force at pair $A$ in $x$ and $y$ direction are that:
${F}_{12x}={F}_{12}^{n}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{1}+{F}_{12}^{t}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{1},$${F}_{12y}={F}_{12}^{n}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{1}-{F}_{12}
^{t}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{1}.$
The projections of contact force at pair $B$ in $x$ and $y$ direction are that:
${F}_{32x}={F}_{32}^{n}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{2}+{F}_{32}^{t}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{2},$${F}_{32y}={F}_{32}^{n}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{2}-{F}_{32}
^{t}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{2}.$
Now the motion relationship, contact and force relationship are obtained.
The dynamical equations can be established according to d’Alembert’s principle. By substituting the expressions of clearance and forces in formula (1)-(13) into dynamical equations, the motion
equations are obtained. Motion equation of crank is as follows:
$-{F}_{21x}{l}_{1}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{1}+{F}_{21y}{l}_{1}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{1}+{m}_{1}{\stackrel{¨}{y}}_{s1}{l}_{s1}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_
{1}-{m}_{1}{\stackrel{¨}{x}}_{s1}{l}_{s1}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{1}-T-{R}_{1}\left({F}_{12x}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{1}-{F}_{12y}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha
}_{1}\right)-{J}_{s1}{\stackrel{¨}{\theta }}_{1}=0.$
The dynamic equations of connecting-rod are that:
${R}_{2}\left({F}_{32x}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{2}-{F}_{32y}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{2}\right)+{l}_{s2}\left({F}_{12x}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{2}-{F}_
{12y}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{2}\right)+\left({l}_{2}-{l}_{s2}\right)\left({F}_{32x}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{2}-{F}_{32y}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{2}\
right)+\left({R}_{1}+{e}_{1}\right)\left({F}_{12y}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{1}-{F}_{12x}\mathrm{s}\mathrm{i}\mathrm{n}{\alpha }_{1}\right)-{J}_{s2}{\stackrel{¨}{\theta }}_{2}=0.$
The dynamic equation of cutter head is that:
$\left\{{J}_{s3}-{m}_{3}\left[\left(R-{l}_{3}\right)\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{3}+{l}_{s3}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{4}\right]\left[{l}_{3}\left(\mathrm{s}\mathrm{i}\
mathrm{n}{\theta }_{3}+\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{3}\right)-{l}_{s3}\left(\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{4}+\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{4}\right)\right]\right\}{\
stackrel{¨}{\theta }}_{3}={m}_{3}{\stackrel{˙}{\theta }}_{3}^{2}\left[\left(R-{l}_{3}\right)\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{3}+{l}_{s3}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{4}\right]\left
[{l}_{3}\left(\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{3}-\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{3}\right)-{l}_{s3}\left(\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{4}-\mathrm{c}\mathrm{o}\mathrm{s}{\
theta }_{4}\right)\right]+\left({F}_{32x}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{3}-{F}_{32y}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{3}\right){l}_{3}+{F}_{32x}\left({R}_{2}\mathrm{s}\mathrm{i}\
mathrm{n}{\alpha }_{2}+{e}_{2y}\right)-{F}_{32y}\left({R}_{2}\mathrm{c}\mathrm{o}\mathrm{s}{\alpha }_{2}+{e}_{2x}\right)-{F}_{Ty}\left({l}_{3}\mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{3}-{l}_{T}\
mathrm{c}\mathrm{o}\mathrm{s}{\theta }_{4}\right)+{F}_{Tx}\left({l}_{3}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{3}-{l}_{T}\mathrm{s}\mathrm{i}\mathrm{n}{\theta }_{4}\right)+v{M}_{T}+u{M}_{43},$
$u=\left\{\begin{array}{ll}1,& {\theta }_{1}\in \left(0°+2k\pi ,180°+2k\pi \right),k\in {N}_{+},\\ -1,& {\theta }_{1}\in \left(180°+2k\pi ,360°+2k\pi \right),k\in {N}_{+}.\end{array}\right\$
Formula (19) is used to determine the direction of crushing torque ${M}_{43}$.
$v=\left\{\begin{array}{ll}1,& {\theta }_{3}\le \pi /2,\\ -1,& {\theta }_{3}>\pi /2.\end{array}\right\$
Formula (20) is used to determine the direction of torque ${M}_{T}$. The crushing torque ${M}_{43}$ can not be measured by experimental method, but it can be estimated through the transfer efficiency
and power. The low-noise excavating system is nonlinear dynamic system with 5-dof.
3. Results and discussions
According to the sample machine, the parameters of each part in excavating system are as follows: ${l}_{1}=20\text{mm}\text{,}$${l}_{2}=130\text{mm}\text{,}$${l}_{3}=365\text{mm}\text{,}$${l}_{T}=190
text{m}}^{2}\text{,}$${K}_{2}=3.91×{10}^{7}\text{N/}{\text{m}}^{2}\text{.}$ The numerical simulation results in different conditions are obtained.
When the crank rotate with speed ${\omega }_{1}=15.6\text{rad/s}$, and the model is Hertz contact modeling, the results are obtained when the friction coefficients in tangential direction is $f=0$
and $f=0.05$, shown in Figures 2 and 3.
It can be seen from Figures 2 and 3 that when the friction increases, the vibration amplitude of crank at mass center is increased, and the high frequency component will appear.
Fig. 2Trajectory of the mass center of connecting-rod with different friction coefficient
Fig. 3Angular curve of connecting-rod with different friction coefficient
Fig. 4Angular velocity curve of connecting-rod with different friction coefficient
The angular velocity curve of connecting-rod is also calculated, as shown in Figure 4. It can be seen from Figure 4 that, when $f=$ 0, the maximum angular velocity is $40$rad/s, but when $f=$ 0.05,
the maximum angular velocity is 90 rad/s. Therefore, the angular velocity increases with the increasing of friction coefficient. To reduce the vibration, lubrication is needed.
When the crank rotates with angular velocity ${\omega }_{1}=$ 15.6 rad/s, and the friction coefficient $f=$ 0.01, given the size of clearance as ${r}_{1}={r}_{2}=$ 0.01 mm and ${r}_{1}={r}_{2}=$ 0.5
mm, the responses are obtained, as shown in figures 5 and 6, respectively.
It can be seen from Figures 5 and 6 that, the trajectory of the mass center of connecting-rod is ellipse, when the size of clearance increases, the shape of trajectory do not change, but the curve
becomes non-smooth, the vibration is obvious.
Fig. 5Trajectory of the mass center of connecting-rod with different clearance size
a)${r}_{1}={r}_{2}=$ 0.01 mm
b)${r}_{1}={r}_{2}=$ 0.5 mm
Fig. 6Angular curve of connecting-rod with different clearance size
a)${r}_{1}={r}_{2}=$ 0.01 mm
b)${r}_{1}={r}_{2}=$ 0.5 mm
Fig. 7Angular velocity curve of connecting-rod with different clearance size
a)${r}_{1}={r}_{2}=$ 0.01 mm
b)${r}_{1}={r}_{2}=$ 0.5 mm
The angular velocity of connecting-rod is also obtained, shown in Figure 7. It can be seen that, when the size of clearance becomes larger, the angular velocity will become larger, the maximum
velocity change from $20\text{rad/s}$ to $200\text{rad/s}$, and the impulse is more serious. The curves of load torque and contact force are shown in Figure 8.
Fig. 8Force curves with different angular velocity
The results in Figure 8 show that, when the size of clearance is large, there is very large contact and load torque in the system, the impact phenomena is obvious, which may cause noise and bring
breakage to system.
From the curves in Figure 9 we can see that, the existing of clearance will also affect the motion of system and decrease the precision of motion mechanism.
Through the calculation and analysis, it is obvious that, when the friction in clearance becomes larger, the vibration will be violent. In the design and manufacture, the precision should be improved
to reduce the clearance and friction; in the optimization of system structure, the clearance can not be omitted, and the two clearances should be considered.
Fig. 9The relation curves between velocity and acceleration
a) Curve between ${\omega }_{3}$ and ${\epsilon }_{3}$
b) Curve between ${\omega }_{2}$ and ${\epsilon }_{2}$
c) Curve between ${v}_{xs2}$ and ${a}_{xs2}$
d) Curve between ${v}_{ys2}$ and ${a}_{ys2}$
4. Conclusions
In this paper, the clearance is expressed with Hertz law, the dynamic response of excavating system with two clearances is analyzed with numerical simulation, and the effects of the size of clearance
and the coefficient of friction are discussed, respectively. The numerical results show that, when the friction coefficient and the size of clearance increase, the shape of trajectory of mass center
of connecting rod do not change, but the vibration is obvious, the angular velocity will become larger, which leads to the larger impact force and load torque, and brings breakage to system
structure. In the design and manufacture, the precision should be improved to reduce the clearance and friction; in the optimization of system structure, the clearance cannot be omitted, and the two
clearances should be considered.
• V. Parenti-Castelli, S. Venanzi Clearance influence analysis on mechanisms. Mechanism and Machine Theory, Vol. 40, Issue 12, 2005, p. 1316-1329.
• Imed Khemili, Lotfi Romdhane Dynamic analysis of a flexible slider–crank mechanism with clearance. European Journal of Mechanics A/Solids, Vol. 27, Issue 5, 2008, p. 882-898.
• Ming-June Tsai, Tien-Hsing Lai Accuracy analysis of a multi-loop linkage with joint clearances. Mechanism and Machine Theory, Vol. 43, Issue 9, 2008, p. 1141-1157.
• A. L. Schwab, J. P. Meijaard, P. Meijers A comparison of revolute joint clearance models in the dynamic analysis of rigid and elastic mechanical systems. Mechanism and Machine Theory, Vol. 37,
Issue 9, 2002, p. 895-913.
• P. Flores, J. Ambrosio, J. C. P. Claro, H. M. Lankarani, C. S. Koshy A study on dynamics of mechanical systems including joints with clearance and lubrication. Mechanism and Machine Theory, Vol.
41, Issue 3, 2006, p. 247-261.
• K. Hu, Z. P. Mourelatos, N. Vlahopoulos Computational analysis for dynamic response of a rotating shaft on flexible support structure with clearances. Journal of Sound and Vibration, Vol. 267,
Issue 1, 2003, p. 1-28.
• Eleonor D. Stoenescu, Dan B. Marghitu Dynamic analysis of a planar rigid-link mechanism with rotating slider joint and clearance. Journal of Sound and Vibration, Vol. 266, Issue 6, 2003, p.
• P. Metallidis, S. Natsiavas Vibration of a continuous system with clearance and motion constraints. International Journal of Non-Linear Mechanics, Vol. 35, 2000, p. 675-690.
• P. Flores, J. Ambrosio Revolute joints with clearance in multibody systems. Computers and Structures, Vol. 82, Issue 1, 2004, p. 1359-1369.
• Jungkeun Rhee, Adnan Akay Dynamic response of a revolute joint with clearance. Mechanism and Machine Theory, Vol. 31, Issue 1, 1996, p. 121-134.
• Chang Zongyu, Wang Yuxin, Zhang Ce Chaotic behavior in linkage with a clearance. Mechanical Science and Technology, Vol. 17, Issue 3, 1998, p. 245-348, (in Chinese).
• Ali Azimi Olyaei, Mohammad Reza Ghazavi Stabilizing slider-crank mechanism with clearance joints. Mechanism and Machine Theory, Vol. 53, Issue 7, 2012, p. 17-29.
About this article
05 September 2013
15 February 2014
excavating system
nonlinear vibration
numerical simulation
This work is supported by the National 863 Plans Project (Grant No. 2007AA04Z209), the National Natural Science Foundation of China (Grant No. 51009107), Tianjin Natural Science Foundation (Grant No.
13JCQNJC04200 and 13JCZDJC27100).
Copyright © 2014 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/14579","timestamp":"2024-11-11T16:41:21Z","content_type":"text/html","content_length":"156996","record_id":"<urn:uuid:8bda761f-5bb0-4f48-ac5a-51d4b994b51c>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00006.warc.gz"} |
Church overlap competition
I posted this a long time ago in our shared forum with Soap, but here it is again, and I figured I might as well post it here so everyone can choose to join in
Calculator advised. If you have boatloads of unreligious villages, this process is pretty much a waste of your time, so you can stop reading now.
Let's call a "space" on the map equal to the area taken up by a SINGLE VILLAGE.
Let c_1 be the number of level 1 churches you have, let c_2 be the number of level 2/first churches you have, and let c_3 be the number of level 3 churches you have. Let 'o' be equal to the total
overlap created by your churches such that for every space on the map covered by more than one church, o = (number of churches covering that space) - 1. So, if you have 3 churches overlapping on one
particular space, 'o' would be 2 higher as a result of that space. This is not a count of total overlapS, but total overlap. You need to look at the map and count individual spaces of overlap, and
add them up.
Let t = 48*c_1 + 112*c_2 + 196*c_3
Let r = t / (c_1 + c_2 + c_3)
Let x = (t - r) / o
Post x. The higher the x value, the more effectively your churches have been laid out in terms of overlap.
Obviously, there are other factors that go into placing churches, but this still sheds some light on planning and general thought put into church placement.
I get 55.5, which is pretty disappointing... It used to be above 90
Also, I would just like to say... 55.5 might be higher than many of your scores, but have no shame!
What are churches? I am suppose to have them? Crap...
The church is an indoctrination machine used to cover up pedophiles.
No wonder you like church worlds so much. :icon_wink:
Ok so I killed off a bunch of troops and put a lvl 1 church in every village. /amidoingitright?
Ok so I killed off a bunch of troops and put a lvl 1 church in every village. /amidoingitright?
DINGDINGDING! we have a winner!
ha ha...u people killed the thread before it even began...:icon_twisted:
anyways if its any consolation to Mills...i tried to calculate my "X"....but gave up while counting "O"...too troublesome...any easy way to count the Os ?
ha ha...u people killed the thread before it even began...:icon_twisted:
anyways if its any consolation to Mills...i tried to calculate my "X"....but gave up while counting "O"...too troublesome...any easy way to count the Os ?
Indeed I did, now why would I advertise this info? It was a good try though. Also as a side note, the math sucks and there wouldnt be anyway I would do it any how!
Warham said:
DINGDINGDING! we have a winner!
could you count my overlaps for me? i will pass you my account sit
Would be nice to turn this into a script of some sort that in overview page can tell. Could be really handy
I could do that, I suppose heh... Would probably take a pretty long time to run, but should be doable
I have a script written (for the second time, damn laptop batteries) to count o, if you want to help me test it, pm me. | {"url":"https://forum.tribalwars.net/index.php?threads/church-overlap-competition.243283/","timestamp":"2024-11-12T03:23:01Z","content_type":"text/html","content_length":"82258","record_id":"<urn:uuid:e6a6dcf9-6310-4b0c-8b79-fbae680aa32a>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00787.warc.gz"} |
A polynomial-time method to find the sparsest unobservable attacks in power networks
Power injection attacks that alter generation and loads at buses in power networks are studied. The system operator employs Phasor Measurement Units (PMUs) to detect such physical attacks, while
attackers devise attacks that are unobservable by such PMU networks. Unalterable buses, whose power injections cannot be changed, are also considered in our model. It is shown that, given the PMU
locations, the minimum sparsity of unobservable attacks has a simple form with probability one, namely, equation, where equation is defined as the vulnerable vertex connectivity of an augmented
graph. The constructive proof allows one to find the entire set of the sparsest unobservable attacks in polynomial time.
Publication series
Name Proceedings of the American Control Conference
Volume 2016-July
ISSN (Print) 0743-1619
Other 2016 American Control Conference, ACC 2016
Country/Territory United States
City Boston
Period 7/6/16 → 7/8/16
All Science Journal Classification (ASJC) codes
• Electrical and Electronic Engineering
Dive into the research topics of 'A polynomial-time method to find the sparsest unobservable attacks in power networks'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/a-polynomial-time-method-to-find-the-sparsest-unobservable-attack","timestamp":"2024-11-07T14:28:19Z","content_type":"text/html","content_length":"51344","record_id":"<urn:uuid:6e04778f-ac0a-48fa-b00c-4ea5f43aff48>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00121.warc.gz"} |
Number Types - Definition, Solved Examples, and FAQs
Just like different individuals of the same family live in different homes, different numbers belong to the same family but have different types. Over time, various patterns of ten digits have been
categorized into an array of number types. This framework of numbers varies from each other because of different properties and presentations.
That said, the types of numbers in maths are classified as per some purpose that they serve, property that they possess or fundamental rule that they follow.
What Are Numbers?
Those ten elegant digits, symbols, or numerals that we all learn early in life are the numbers. Numbers are algebraic in form and have a greatest influence in our lives in far more ways than we could
ever think of.
Numbers in Real Life
Ever wondered what our lives would be like in absence of these 10 digits and the innumerable array of other numbers that they can create? Numbers are everywhere in our birth dates, ages, height,
weight, addresses, phone numbers, credit card numbers, bank account numbers and a lot more.
Classification of Numbers
Numbers family can be classified in different categories. With that, we can also say that two or more types of numbers in maths can fall under one category. Refer to the image below for complete
understanding of classification of numbers:
(Image to be added soon)
Different Types of Numbers
There are various types of numbers in maths. Let’s discuss some of the following:
1. Natural Numbers - the set of numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 that we see and use in our routine life. The natural numbers are commonly known as positive integers and counting numbers.
2. Whole Numbers - the natural numbers plus (+) the zero (0).
3. Real Numbers - set of real numbers in addition to all the rational and irrational numbers and are represented by the letter R. It also includes all the numbers that can be written in the decimal
4. Fractional Numbers - any number expressible by the quotient of two numbers as in m/m, "m" greater than 1, in which "m" is the numerator and "n" is the denominator.
5. Irrational Numbers - any number that is unable to be expressed by an integer or the ratio of two integers. These numbers are expressible only as decimal fractions in which the digits are ongoing
with no repeating pattern. Examples of irrational numbers are √2 , √3.
6. Transcendental Numbers - any number unable to be the root of a polynomial equation with rational coefficients.
Quantum Numbers
Set of numbers used to define the energy and position of the electron in an atom are known as quantum numbers.
Types of Quantum Numbers
There are four quantum numbers that define the probable location of an electron in an atom which are as given:
• The Azimuthal Quantum Number denoted by symbol ‘l’
• The Magnetic Quantum Number denoted by symbol ‘ml’
• The Principal Quantum Number denoted by symbol ‘n’
• The Spin Projection Quantum Number denoted by symbol ‘ms’
Fun Facts
• Almost all of us whether mathematicians, scientists, doctors, engineers, manufacturers, cashiers or carpenters could not survive without numbers.
• Zero(0) as a number has the greatest value and importance.
Solved Examples
Find out the square root of -16? Write your answer in the form of imaginary number i.
Step 1: Write the number in terms of square root √ (-16)
Step 2: Separate out -1. √ (16 × -1)
Step 3: Move apart square roots.√ (16) × √ (-1)
Step 4: Solve and simplify the square root. 4 × √ (-1)
Step 5: Write in mathematical terms of i.4i
Sometimes you get an imaginary solution to the equations.
Example 2
Simplify and solve the equation: a2 + 2 = 0
Step 1: Take the constant term on other side of the linear expression: a2 = -2
Step 2: Take the square root on both sides of the equation √a2 = +√-2 or -√-2
Step 3: Solve and simplify: a = √ (2) × √ (-1)
a = +√2i or -√2i
Step 4: Double check the answers by substituting values in the initial equation and see if we obtain 0. a2 + 2
(+√2i) 2 + 2 = -2 + 2 = 0 [since i = √-1 and square of i will be -1]
(-√2i) 2 + 2 = -2 + 2 = 0 [since i = √-1 and square of i will be -1]
we would not be able to live without numbers in our lives. Interestingly, there exists an almost infinite array of number types and hidden wonders emanating from these acquainted symbols that we use
every day, the natural numbers.
FAQs on Number Types
1. What are imaginary numbers and its applications?
Just because their name is “imaginary” does not mean they are purposeless. They have several applications. In a few places, the imaginary number is also denoted by the letter ‘j’. One of the
substantial applications of imaginary numbers is its wide application in electric circuits. The computations of current and voltage are carried out with respect to imaginary numbers. Imaginary
numbers also have different applications in fields like systems, signals and fourier transform. These numbers are also frequently used in complex calculus calculations.
2. What are rational numbers and its applications?
Rational numbers are ones which can be mathematically expressed in the form of fraction. The term “rational” originates from the term, “ratio”, as rational numbers are the ratios of the 2 integers.
Rational numbers are denoted by the letter ‘Q’. For example, 0.9 is a rational number since this can be written as 9/10. Other examples of rational numbers are -2/5, 1/3, 79/80, 1.57, etc.
Assume a rational number x/y, where x and y are two integers. Here, the numerator x can be any integer (positive or negative), but the denominator y can never be 0, since the fraction is unspecified
then. Moreover, if y = 1, then the fraction will be an integer.
3. What are the types of rational numbers?
Rational Numbers can be in the form of whole numbers like (0, 1, 2, 3, 4, 5 ...), the integers (.-1, - 2, - 1, 0, 1, 2 ...), fractions (2/3) and repeating and terminating decimals. | {"url":"https://www.vedantu.com/maths/number-types","timestamp":"2024-11-02T14:23:12Z","content_type":"text/html","content_length":"261179","record_id":"<urn:uuid:dde3a67d-f5c3-4fe2-9096-53f28d3a547f>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00829.warc.gz"} |
to Yards
Rod to Yards Converter
β Switch toYards to Rod Converter
How to use this Rod to Yards Converter π €
Follow these steps to convert given length from the units of Rod to the units of Yards.
1. Enter the input Rod value in the text field.
2. The calculator converts the given Rod into Yards in realtime β using the conversion formula, and displays under the Yards label. You do not need to click any button. If the input changes, Yards
value is re-calculated, just like that.
3. You may copy the resulting Yards value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Rod to Yards?
The formula to convert given length from Rod to Yards is:
Length[(Yards)] = Length[(Rod)] / 0.1818181818098691
Substitute the given value of length in rod, i.e., Length[(Rod)] in the above formula and simplify the right-hand side value. The resulting value is the length in yards, i.e., Length[(Yards)].
Calculation will be done after you enter a valid input.
Consider that a boundary fence is 40 rods long.
Convert this length from rods to Yards.
The length in rod is:
Length[(Rod)] = 40
The formula to convert length from rod to yards is:
Length[(Yards)] = Length[(Rod)] / 0.1818181818098691
Substitute given weight Length[(Rod)] = 40 in the above formula.
Length[(Yards)] = 40 / 0.1818181818098691
Length[(Yards)] = 220
Final Answer:
Therefore, 40 rd is equal to 220 yd.
The length is 220 yd, in yards.
Consider that a farmer marks a field boundary using 25 rods.
Convert this distance from rods to Yards.
The length in rod is:
Length[(Rod)] = 25
The formula to convert length from rod to yards is:
Length[(Yards)] = Length[(Rod)] / 0.1818181818098691
Substitute given weight Length[(Rod)] = 25 in the above formula.
Length[(Yards)] = 25 / 0.1818181818098691
Length[(Yards)] = 137.5
Final Answer:
Therefore, 25 rd is equal to 137.5 yd.
The length is 137.5 yd, in yards.
Rod to Yards Conversion Table
The following table gives some of the most used conversions from Rod to Yards.
Rod (rd) Yards (yd)
0 rd 0 yd
1 rd 5.5 yd
2 rd 11 yd
3 rd 16.5 yd
4 rd 22 yd
5 rd 27.5 yd
6 rd 33 yd
7 rd 38.5 yd
8 rd 44 yd
9 rd 49.5 yd
10 rd 55 yd
20 rd 110 yd
50 rd 275 yd
100 rd 550 yd
1000 rd 5500 yd
10000 rd 55000 yd
100000 rd 550000 yd
A rod is a unit of length used in land measurement and surveying. One rod is equivalent to 16.5 feet or approximately 5.0292 meters.
The rod is defined as 16.5 feet, providing a measurement that is useful for various applications in land surveying, agriculture, and construction.
Rods are commonly used in tasks such as property measurement, plotting land, and agricultural practices. The unit provides a practical measurement for shorter distances and has historical
significance in land surveying.
A yard (symbol: yd) is a unit of length commonly used in the United States, the United Kingdom, and Canada. One yard is equal to 0.9144 meters.
The yard originated from various units used in medieval England. Its current definition is based on the international agreement of 1959, which standardized it to exactly 0.9144 meters.
Yards are often used to measure distances in sports fields, textiles, and land. Despite the global shift to the metric system, the yard remains in use in these countries.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Rod to Yards in Length?
The formula to convert Rod to Yards in Length is:
Rod / 0.1818181818098691
2. Is this tool free or paid?
This Length conversion tool, which converts Rod to Yards, is completely free to use.
3. How do I convert Length from Rod to Yards?
To convert Length from Rod to Yards, you can use the following formula:
Rod / 0.1818181818098691
For example, if you have a value in Rod, you substitute that value in place of Rod in the above formula, and solve the mathematical expression to get the equivalent value in Yards. | {"url":"https://convertonline.org/unit/?convert=rods-yards","timestamp":"2024-11-03T12:05:38Z","content_type":"text/html","content_length":"89327","record_id":"<urn:uuid:abcc9465-119e-452b-b7a9-c1c7155fbfe7>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00349.warc.gz"} |
Hoj James series stories-help me
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Welcome to Data Analysis and Interpretation - ppt video online download
1 Welcome to Data Analysis and Interpretation22-23 March 2011 Dick Schwanke
2 What will we be doing? Examining different mathematical /statistical analysis techniques Applying those techniques to our data. Definition of Statistics: the science of collecting, organizing,
summarizing, and analyzing information to draw conclusions or answer questions
3 Who is Data? What is data? Fact or Proposition used to draw a conclusion or make a decision Can be numerical Can be non-numerical
4 Definitions Population: The group to be studiedParameter is numerical summary of population It is all Greek to me Sample: The subset of the population Statistic is a numerical summary of a sample
When in Rome, do as the Romans do
5 Definitions Qualitative variable – classification of individuals based on some attribute or characteristic Quantitative variable – provide numerical measures of individuals Discrete Variable – has
either countable or finite number of possible values Continuous variable – has an infinite number of possible values
6 Some Administrative DetailsLet us gather some data Introductions Name / Job Function / Excel Experience / 1 fact Discuss these items with adjacent team member Class roster completed Schedule of
these two days Mix of lecture with problems Computer lab with Microsoft Excel 2007
7 Organizing and Summarizing Quantitative DataStep 1: Organize raw data into classes Step 2: Create tables for the data: frequency distribution relative frequency distribution cumulative frequency
distribution relative cumulative frequency distribution
8 Organizing and Summarizing Quantitative DataStep 3: Create graphs bar charts pie charts histograms frequency polygons ogives stem-and-leaf plots dot plots Step 4: Be cautious of misleading graphs
9 Organizing and Summarizing Quantitative DataOrganize Data Put into classes Start with lowest value Create Table Frequency distribution Relative frequency distribution, Cumulative frequency
distribution Relative cum. frequency distrib. Draw Graphic Displays Histogram Frequency polygon Ogive Stem & leaf plot Steps: Organize raw data into classes Create table with frequency distribution,
relative frequency distribution, cumulative frequency distribution, and relative cumulative frequency distribution Create graphical displays with histogram, frequency polygon, ogive, stem & leaf plot
Page &p of 6
10 Ways of Displaying Data - HistogramA graph using rectangles for each class of data, where the height of each rectangle is the frequency or relative frequency of the class Note 1 : width of each
rectangle is the same and rectangles touch each other Note 2: methods for discrete and for continuous data
11 More Ways of Displaying Data – Cumulative DistributionsCumulative frequency distribution: displays aggregate frequency of category i.e. total number of observations less than or equal to that
category Cumulative relative frequency distribution: displays the percentage (or proportion) of observations less than or equal to that category
12 More Ways of Displaying Data – Cumulative DistributionsAdditional notes: Works as a table or as a graph For continuous data display the total number of observations less than or equal to the upper
class limit of a class Class midpoint – determined by adding consecutive lower class limits then divide the result by 2
13 More Ways of Displaying Data – Frequency PolygonsConstruct with: class on horizontal axis frequency on vertical axis Plot a point above each class midpoint Draw straight lines between consecutive
points Page &p of 6
14 More Ways of Displaying Data – OgivesLike a frequency polygon except represents cumulative frequency or cumulative relative frequency for the class Construct with: upper class limits on horizontal
axis cumulative frequency (or cumulative relative frequency) on vertical axis
15 More Ways of Displaying Data – Stem and Leaf PlotStep 1: Select stem – the digit(s) to the left Leaf will be the rightmost digit Step 2: Write the stems in a vertical column in increasing order.
Draw vertical line to right of stems Step 3: Write each leaf to right of its stem Step 4: (re)Write leaves in ascending order
16 More Ways of Displaying Data – Stem and Leaf PlotOther notes about stem and leaf plots: Best used when data set is small Can use “split stem” method if data seems too bunched These sections give
us many “tools” for our “toolbox”, so that we may use the best one (the best graphical display) for our audience’s understanding of our point
17 Shapes of DistributionsUniform Symmetric (bell shaped) Skewed right (long tail to right) Slewed left (long tail to left)
18 Graphic Suggestions Title both axis, label includes unit of measureInclude data source when appropriate. Minimize white space in the graph, using available space to let the data stand out Avoid
clutter: pictures, excessive gridlines Avoid distortion. Never lie about the data Avoid three dimensional charts and graphs Let the data speak for themselves. | {"url":"http://slideplayer.com/slide/6237259/","timestamp":"2024-11-03T13:51:07Z","content_type":"text/html","content_length":"186040","record_id":"<urn:uuid:5ff88a33-44d6-47e7-b586-94b618f7707e>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00598.warc.gz"} |
Finite element trace theorems
The theorems on traces of functions from the Sobolev spaces play an important role in studying boundary value problems of mathematical physics. These theorems are commonly used for a priori estimates
of the stability with respect to boundary conditions. The trace theorems play also very important role to construct and to investigate effective domain decomposition methods. The main focus of this
talk is to study the case when norms of functions given in some domain dependent on parameters. Corresponding Sobolev spaces with the parameter dependent norms are generated, for instance, by
elliptic problems with disproportional anisotropic coefficients. The main goal is to introduce the parameter dependent norms of traces of functions on the boundary such that the corresponding
constants in the trace theorems are independent of the parameters. In the finite element case (finite element functions in the domain and finite element traces on the boundary), the corresponding
constants should be independent of the mesh step too. | {"url":"https://bulletin.iis.nsk.su/article/1398","timestamp":"2024-11-14T05:24:52Z","content_type":"text/html","content_length":"126697","record_id":"<urn:uuid:e7254f2f-20be-4af1-a170-9c83596273c5>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00873.warc.gz"} |
Minimum power energy spanners in wireless ad hoc networks
A power assignment is an assignment of transmission power to each of the nodes of a wireless network, so that the induced communication graph has some desired properties. The cost of a power
assignment is the sum of the powers. The energy of a transmission path from node u to node v is the sum of the squares of the distances between adjacent nodes along the path. For a constant t > 1, an
energy t-spanner is a graph G′, such that for any two nodes u and v, there exists a path from u to v in G′, whose energy is at most t times the energy of a minimum-energy path from u to v in the
complete Euclidean graph. In this paper, we study the problem of finding a power assignment, such that (i) its induced communication graph is a 'good' energy spanner, and (ii) its cost is 'low'. We
show that for any constant t > 1, one can find a power assignment, such that its induced communication graph is an energy t-spanner, and its cost is bounded by some constant times the cost of an
optimal power assignment (where the sole requirement is strong connectivity of the induced communication graph). This is a very significant improvement over the best current result due to Shpungin
and Segal [1], presented in last year's conference.
Original language English
Title of host publication 2010 Proceedings IEEE INFOCOM
State Published - 15 Jun 2010
Event IEEE INFOCOM 2010 - San Diego, CA, United States
Duration: 14 Mar 2010 → 19 Mar 2010
Publication series
Name Proceedings - IEEE INFOCOM
ISSN (Print) 0743-166X
Conference IEEE INFOCOM 2010
Country/Territory United States
City San Diego, CA
Period 14/03/10 → 19/03/10
ASJC Scopus subject areas
• General Computer Science
• Electrical and Electronic Engineering
Dive into the research topics of 'Minimum power energy spanners in wireless ad hoc networks'. Together they form a unique fingerprint. | {"url":"https://cris.bgu.ac.il/en/publications/minimum-power-energy-spanners-in-wireless-ad-hoc-networks-6","timestamp":"2024-11-14T14:56:42Z","content_type":"text/html","content_length":"58517","record_id":"<urn:uuid:216f1eb8-90f2-4f8b-a4d3-856ebef96c7f>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00332.warc.gz"} |
CDS & AFCAT 2 2024 Exam Maths Trigonometry Class 2
Trigonometry is an essential part of the mathematics syllabus for competitive exams like the Combined Defence Services (CDS) and Air Force Common Admission Test (AFCAT). A recent class dedicated to
this topic focused on key concepts such as heights and distances, line of sight, angle of elevation and depression, and the sine and cosine formulas. Additionally, the class included extensive
practice with multiple-choice questions (MCQs) to help reinforce these concepts. This article highlights the main areas discussed, providing a comprehensive understanding to aid students in their
exam preparation.
Key Topics in Trigonometry: Heights and Distances
Understanding the practical applications of trigonometry in solving problems related to heights and distances is crucial for competitive exams. The main topics covered in the class include:
1. Line of Sight
2. Angle of Elevation and Depression
Mastering these concepts enables students to tackle a variety of problems related to trigonometry efficiently.
Line of Sight
The line of sight is the straight line along which an observer looks at an object. It forms the basis for measuring angles of elevation and depression.
Angle of Elevation and Depression
The angle of elevation is the angle between the horizontal line and the line of sight when looking up at an object. Conversely, the angle of depression is the angle between the horizontal line and
the line of sight when looking down at an object.
Sine Formula
The sine formula is used to relate the sides of a triangle to the sine of its angles. This is particularly useful in non-right-angled triangles.
Cosine Formula
The cosine formula relates the lengths of the sides of a triangle to the cosine of one of its angles. It is useful in solving problems involving non-right-angled triangles.
Example MCQs and Solutions
To enhance understanding and application of these concepts, the class discussed several example MCQs. Here are some examples along with their solutions:
Example 1: Angle of Elevation
Question: A person standing 50 meters away from a building observes the top of the building at an angle of elevation of 45 degrees. What is the height of the building?
Solution: The height can be found using the tangent function (tangent of 45 degrees is 1).
Answer: Height of the building is 50 meters.
Example 2: Angle of Depression
Question: From the top of a 60-meter-high tower, the angle of depression to a point on the ground is 30 degrees. What is the horizontal distance from the tower to the point?
Solution: The distance can be found using the tangent function (tangent of 30 degrees).
Answer: Horizontal distance is approximately 103.92 meters.
Example 3: Sine Formula
Question: In a triangle, if one side is 8 units, the opposite angle is 30 degrees, and the hypotenuse is 16 units, is this consistent with the sine formula?
Solution: Verify using the sine formula.
Answer: Yes, it is consistent.
Strategies for Solving Trigonometry Problems
1. Understand the Problem: Carefully read the question to identify which trigonometric concept or formula to use.
2. Draw a Diagram: Visual representation of the problem often helps in understanding and solving it.
3. Memorize Key Formulas: Ensure you remember the key trigonometric formulas and identities.
4. Practice Regularly: Regular practice helps in reinforcing concepts and improving problem-solving speed.
5. Double-Check Calculations: Always verify your calculations to avoid simple errors.
Trigonometry, particularly the concepts of heights and distances, line of sight, angles of elevation and depression, and the sine and cosine formulas, is a vital topic for the CDS and AFCAT exams.
The recent class provided a comprehensive overview of these key areas, along with extensive practice of important MCQs to reinforce understanding.
Stay focused, keep practicing, and approach each problem with confidence. Good luck!
Leave Your Comment | {"url":"https://ssbcrackexams.com/cds-afcat-2-2024-exam-maths-trigonometry-class-2/","timestamp":"2024-11-04T17:25:27Z","content_type":"text/html","content_length":"339471","record_id":"<urn:uuid:565b2baa-4399-4b65-9d47-466d7546ffef>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00803.warc.gz"} |
Parameters vs. Statistics
Learning Objectives
• Describe the sampling distribution for sample proportions and use it to identify unusual (and more common) sample results.
• Distinguish between a sample statistic and a population parameter.
One of the goals of inference is to draw a conclusion about a population on the basis of a random sample from the population. Obviously, random samples vary, so we need to understand how much they
vary and how they relate to the population. Our ultimate goal is to create a probability model that describes the long-run behavior of sample measurements. We use this model to make inferences about
the population.
We begin our investigation with a simplified and artificial situation.
Proportions from Random Samples Vary
Imagine a small college with only 200 students, and suppose that 60% of these students are eligible for financial aid.
In this simplified situation, we can identify the population, the variable, and the population proportion.
• Population: 200 students at the college.
• Variable: Eligibility for financial aid is a categorical variable, so we use a proportion as a summary.
• Population proportion: 0.60 of the population is eligible for financial aid.
Note: Populations are usually much larger than 200 people. Also, in real situations, we do not know the population proportion. We are using a simplified situation to investigate how random samples
relate to the population. This is the first step in creating a probability model that will be useful in inference.
How accurate are random samples at predicting this population proportion of 0.60?
To answer this question, we randomly select 8 students and determine the proportion who are eligible for financial aid. We repeat this process several times. Here are the results for 3 random
Notice the following about these random samples:
• Each random sample came from a population in which the proportion eligible for financial aid is 0.60, but sample proportions vary. Each random sample has a different proportion who are eligible
for financial aid.
• Some sample proportions are larger than the population proportion of 0.60; some sample proportions are smaller than the population proportion.
• Some samples give good estimates of the population proportion. Some do not. In this case, 0.625 is a much better estimate than 0.375.
• A lot of variability occurs in these sample proportions. It is not surprising, therefore, that a sample of 8 students may give an inaccurate estimate for the proportion of those eligible for
financial aid in the population. It makes sense that small samples of only 8 students may not represent the population accurately. Later we investigate the effect of increasing the size of the
• The variability we see in proportions from random samples is due to chance.
Learn By Doing
In these activites, we use the following simulation to select a random sample of 8 students from the small college in the previous example. At the college, 60% of the students are eligible for
financial aid. For each sample, the simulation calculates the proportion in the sample who are eligible for financial aid. Repeat the sampling process many times to observe how the sample proportions
vary, then answer the questions.
Click here to open this simulation in its own window.
Means from Random Samples Vary
Now let’s consider a quantitative variable with this same population of 200 students at a small college. Let’s also suppose that the mean amount of financial aid received by students at the college
is $1,500.
In this simplified situation, we have
• Population: 200 students at the college.
• Variable: Financial aid amount ($) is a quantitative variable, so we use a mean as a summary.
• Population mean: $1,500.
How accurate are random samples at predicting this population mean of $1,500?
To answer this question, we randomly select 8 students and determine the mean amount of financial aid received by the students. We repeat this process several times. Here are the results for 3 random
Notice that observations we made earlier about sample proportions are true for sample means.
• Each random sample came from a population for which the mean amount of financial aid received by individual students is $1,500. But the sample means vary: Each random sample has a different mean.
• Some sample means are larger than the population mean of $1,500. Some sample means are smaller than the population mean.
• Some samples give better estimates of the population mean than others. For example, $1,325.00 is a much better estimate than $687.50.
• A lot of variability occurs in the sample means. It is not surprising, therefore, that a sample of 8 students may give an inaccurate estimate of the mean amount of financial aid received by the
population. Again, it makes sense that small samples of only 8 students may not represent the population accurately. We investigate the factors that affect the variability of means from random
samples in the module Inference for Means.
• The variability we see in the means from random samples is due to chance.
Before we continue our discussion of sampling variability, we introduce some vocabulary.
A parameter is a number that describes a population. A statistic is a number that we calculate from a sample.
Let’s use this new vocabulary to rephrase what we already know at this point:
• When we do inference, the parameter is not known because it is impossible or impractical to gather data from everyone in the population. (Note: In each example on this page, we assumed we knew
the parameter so that we could investigate how statistics relate to the parameter. This is the first step in creating a probability model. However, when we do inference, we use a statistic to
draw a conclusion about an unknown parameter.)
• We make an inference about the population parameter on the basis of a sample statistic.
• Statistics from samples vary.
In this course, if the variable is categorical, the parameter and the statistic are both proportions. If the variable is quantitative, the parameter and statistic are both means.
From our first example:
• Parameter: A population proportion. For this population of students at a small college, 0.60 are eligible for financial aid.
• Statistics: Sample proportions that vary. In the example, 0.75, 0.625, and 0.375 are all statistics that describe the proportion eligible for financial aid in a sample of 8 students.
From our second example:
• Parameter: A population mean. For this population of students at a small college, the mean amount of financial aid is $1,500.
• Statistics: Sample means that vary. In the example, $2,087.50, $1,325.00, and $687.50 are all statistics that describe the mean amount of financial aid received by a sample of 8 students.
We use different notation for parameters and statistics:
(Population) Parameter (Sample) Statistic
Proportion [latex]p[/latex] [latex]\stackrel{ˆ}{p}[/latex]
Mean [latex]\mu [/latex] [latex]\overline{x}[/latex]
Standard Deviation [latex]\sigma [/latex] [latex]s[/latex]
Sometimes we refer to the sample statistics as “p-hat” and “x-bar.”
Here we use this notation for the information from our examples.
For our first example:
• For the population of college students, p = 0.60.
• For the 3 random samples of 8 students, we have p-hats [latex]\stackrel{ˆ}{p}=0.75,\text{}\stackrel{ˆ}{p}=0.625,\text{}\stackrel{ˆ}{p}=0.375[/latex]
For our second example:
• For the population of college students, µ = $1,500.
• For the 3 random samples of 8 students, we have x-bars [latex]\stackrel{¯}{x}=\$2,087.50,\text{}\stackrel{¯}{x}=\$1,325.00,\text{}\stackrel{¯}{x}=\$687.50[/latex]
Important Comments about Notation
Many statistics packages and introductory statistics textbooks use the notation shown in the table. The notation for means and standard deviations is common in the field of statistics. However, you
will occasionally see other notation for proportions. In some statistical material, the Greek letter π represents the population proportion and p represents the sample proportion. This can be
particularly confusing because p is used in some statistical material for the population proportion and in other statistical material for a sample proportion. Whenever you work with symbols, always
be sure you understand what the symbol represents. You should be able to interpret the symbol from the context of the material. | {"url":"https://courses.lumenlearning.com/ivytech-wmopen-concepts-statistics/chapter/parameters-vs-statistics/","timestamp":"2024-11-07T14:18:29Z","content_type":"text/html","content_length":"59232","record_id":"<urn:uuid:c4b02454-ea26-428f-a5e9-f11caccccf51>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00604.warc.gz"} |
Response Level Probabilities for Linear Models
add_probs.lm {ciTools} R Documentation
Response Level Probabilities for Linear Models
This is the method add_probs uses if the model is of class lm. Probabilities are calculated parametrically, using a pivotal quantity.
## S3 method for class 'lm'
name = NULL,
yhatName = "pred",
comparison = "<",
log_response = FALSE,
df A data frame of new data.
fit An object of class lm. Predictions are made with this object.
q A real number. A quantile of the response distribution.
name NULL or a string. If NULL, probabilities automatically will be named by add_probs, otherwise, the probabilities will be named name in the returned data frame.
yhatName A character vector of length one. Names of the
comparison "<", or ">". If comparison = "<", then Pr(Y|x < q) is calculated for each observation in df. Otherwise, Pr(Y|x > q) is calculated.
log_response A logical. Default is FALSE. Set to TRUE if the model is log-linear: \log(Y) = X \beta + \epsilon.
... Additional arguments.
A dataframe, df, with predicted values and probabilities attached.
See Also
add_ci.lm for confidence intervals for lm objects, add_pi.lm for prediction intervals of lm objects, and add_quantile.lm for response quantiles of lm objects.
# Fit a linear model
fit <- lm(dist ~ speed, data = cars)
# Calculate the probability that a new dist will be less than 20,
# given the model.
add_probs(cars, fit, q = 20)
# Calculate the probability that a new dist will be greater than
# 30, given the model.
add_probs(cars, fit, q = 30, comparison = ">")
version 0.6.1 | {"url":"https://search.r-project.org/CRAN/refmans/ciTools/html/add_probs.lm.html","timestamp":"2024-11-11T03:54:31Z","content_type":"text/html","content_length":"4275","record_id":"<urn:uuid:bf31ea12-0b2d-45a1-a5a9-e1adebdb4097>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00025.warc.gz"} |
Phase with No Mass Gap in Non-Perturbatively Gauge-Fixed
THEORY OF THE MASS GAP IN QCD AND ITS VACUUM V. Gogokhia WIGNER RESEARCH CENTER FOR PHYSICS, HAS, BUDAPEST, HUNGARY
[email protected]
GGSWBS'12 (August 13-17) Batumi, Georgia 5-th "Georgian-German School and Workshop in Basic Science 0-0 Lagrangian of QCD LQCD = LYM + Lqg 1 L = ¡ F a F a¹º + L + L YM 4 ¹º g:f: gh: a a a abc b c 2
F¹º = @¹Aº ¡ @ºA¹ + gf A¹Aº; a = Nc ¡ 1;Nc = 3 j j j j j Lqg = iq¹®D®¯q¯ +q ¹®m0q¯; ®; ¯ = 1; 2; 3; j = 1; 2; 3; :::Nf j a a j D®¯q¯ = (±®¯@¹ ¡ ig(1=2)¸®¯A¹)γ¹q¯ (cov: der:) 0-1 Properties of the QCD
Lagrangian QCD or, equivalently Quantum ChromoDynamics 1). Unit coupling constant g 2). g » 1 while in QED (Quantum ElecroDynmics) it is g ¿ 1. 3). No mass scale parameter to which can be assigned a
physical meaning. Current quark mass m0 cannot be used, since the quark is a colored object QCD without quarks is called Yang-Mills, YM 0-2 The Ja®e-Witten (JW) theorem: Yang-Mills Existence And Mass
Gap: Prove that for any compact simple gauge group G, quantum Yang- Mills theory on R4 exists and has a mass gap ¢ > 0. (i). It must have a "mass gap". Every excitation of the vacuum has energy at
least ¢ (to explain why the nuclear force is strong but short-range). (ii). It must have "quark con¯nement" (why the physical particles are SU(3)-invariant). (iii). | {"url":"https://docslib.org/doc/854651/phase-with-no-mass-gap-in-non-perturbatively-gauge-fixed","timestamp":"2024-11-04T17:54:54Z","content_type":"text/html","content_length":"62978","record_id":"<urn:uuid:86de031e-bd30-46c0-a9ba-e850af609147>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00816.warc.gz"} |
Who Was Albert Einstein and Why Is Albert Einstein Regarded As the Father of Modern Physics?
One of the most startling discoveries in all scientific history was the amount of energy that is stored in matter.
For example, if all the energy that is stored in just one pound of coal could be converted into energy, it would produce the amount of electricity the entire world uses in one day.
This is the same principle that led to the development of the atomic bombs of World War II, bombs that destroyed entire cities.
The scientific equation that expresses this principle is the famous E = mc2, and the man who formulated the equation is probably the most famous scientist in history.
He changed the way we look at our universe more than any scientist ever had. He also tried very hard to make sure the world would survive his discoveries.
Albert Einstein was born in Ulm, Germany, in 1879. He was a very slow student and did not like playing games with the other kids.
He seemed to prefer to just sit and think. He did not pay much attention to the teachers and hated how they taught the students to learn by rote.
This all changed when he studied geometry in high school. He loved how statements needed proof and how problems were solved step-by-step using logic. He decided to devote his life to the study of
Einstein attended the famous Federal Polytechnic School in Zurich, Switzerland, and became an outstanding mathematics student. He was also now interested in physics and wanted to devote himself to
physics research full-time after graduation.
He decided that teaching physics would be the best way to do this.
Einstein graduated in 1905, but he could not obtain a teaching position anywhere. He was forced to accept a menial job as an examiner in a Swiss patent office. Luckily, Einstein had a lot of spare
time at his new job, enough to devise his theory of relativity, the most amazing scientific theory ever.
Einstein never conducted a laboratory experiment in his life. It was all done in his head as thought experiments, and he would then write down his mathematical formulas to prove his theories.
Albert Einstein is famous among the public and scientists alike for his equation E = mc2, his playful side made him popular as an “eccentric” scientific genius. | {"url":"https://zippyfacts.com/who-was-albert-einstein-and-why-is-albert-einstein-regarded-as-the-father-of-modern-physics/","timestamp":"2024-11-12T20:35:56Z","content_type":"text/html","content_length":"55634","record_id":"<urn:uuid:4af14b5a-73bf-445c-b51f-79f6c6ea04be>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00223.warc.gz"} |
JOB is CHARACTER*1
Specifies the type of backward transformation required:
= 'N', do nothing, return immediately;
= 'P', do backward transformation for permutation only;
= 'S', do backward transformation for scaling only;
= 'B', do backward transformations for both permutation and
JOB must be the same as the argument JOB supplied to CGEBAL.
SIDE is CHARACTER*1
= 'R': V contains right eigenvectors;
= 'L': V contains left eigenvectors.
N is INTEGER
The number of rows of the matrix V. N >= 0.
ILO is INTEGER
IHI is INTEGER
The integers ILO and IHI determined by CGEBAL.
1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
SCALE is REAL array, dimension (N)
Details of the permutation and scaling factors, as returned
by CGEBAL.
M is INTEGER
The number of columns of the matrix V. M >= 0.
V is COMPLEX array, dimension (LDV,M)
On entry, the matrix of right or left eigenvectors to be
transformed, as returned by CHSEIN or CTREVC.
On exit, V is overwritten by the transformed eigenvectors.
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,N).
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value. | {"url":"https://man.linuxreviews.org/man3/cgebak.f.3.html","timestamp":"2024-11-14T21:13:16Z","content_type":"text/html","content_length":"6207","record_id":"<urn:uuid:8e66c0e2-2fa9-465a-bd00-0cb5e70085f4>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00105.warc.gz"} |
Number Concepts – Troels Christensen
eric.ed.gov har udgivet: This report is a continuation of a study conducted at the University of Wisconsin during the spring of 1967. The previous study, Technical Report #38, succeeded in teaching
conservation of numerousness to small groups of kindergarten children, in a middle-class community. The purpose of the present study was to determine if the typical classroom teacher, in schools
differing in socio-economic levels, could successfully use the lessons developed in the previous study to effect conservation of numerousness with kindergarten children. Four questions were
considered – (1) can the typical classroom teacher teach the conservation lessons as successfully as a specially trained expert, (2) is the treatment of greater value for pupils from disadvantaged
backgrounds, (3) is the treatment of greater value for younger kindergarten children than for older… Continue Reading
Eric.ed.gov – Kodak Skills Enhancement Program Curriculum: Math for Manufacturing and Quality Control. Report No. AEP-93-01.
eric.ed.gov har udgivet: This teacher’s guide is intended for use in helping Kodak Corporation employees develop the basic mathematics skills required to perform the manufacturing and quality control
tasks expected of them. The following topics are covered in the first five modules: the four basic functions (adding, subtracting, multiplying, and dividing), calculations involving decimals,
percentages, positive and negative numbers, and fractions. The sixth module reviews the topics covered in the preceding modules and helps students transfer the mathematics skills presented to
applications on the shop floor. Each module includes some or all of the following: the module goal, an introduction, materials and guidelines for direct instruction, activities for use in guided
practice, materials for use in applied practice, activities to develop critical thinking strategies, and a pretest and posttest. Transparency… Continue Reading
Eric.ed.gov – Lower School Maths: Lesson Plans and Activities for Ages 7-9 Years. Series of Caribbean Volunteer Publications, No. 5.
eric.ed.gov har udgivet: This guide is a collection of ideas for mathematics activities which were assembled and tested by primary teachers. Activities are correlated to a mathematics curriculum for
ages 7-9 years. The activities supplement the teaching of basic numeracy and include topics such as the language of mathematics, matching numbers, tracing the numbers, number bonds, number rhymes,
number patterns, measurement, weight, money, shapes, and time. Each section of the core curriculum outline is accompanied by one or more activities. Worksheets for each activity are also provided.
(DDR) Link til kilde
Eric.ed.gov – Math Olympics 2000/2002: Primary Mathematics Fun for Communities of Children and Adults.
eric.ed.gov har udgivet: The materials and ideas in this document are intended to be used by teachers in schools in order to bring adults and children together to experience good mathematics. It
features an evening of hands-on mathematics scheduled for one night a week for four weeks. Students and adults come together to share mathematics activities, solve problems, and have fun with
mathematics. The activities are intended for primary students. Sessions include measurement and geometry, number concepts and problem solving, estimation and calculators, and data collections and
probability. Descriptions of each session and a list of materials are presented. The description of each session includes directions for preparations to be done prior to the event, guidelines on how
to present each activity, and solutions to some of the explorations. Hard… Continue Reading
Eric.ed.gov – Math Olympics 2000/2002: Intermediate Mathematics Fun for Communities of Children and Adults.
eric.ed.gov har udgivet: The materials and ideas in this document are intended to be used by teachers in schools in order to bring adults and children together to experience good mathematics. It
features an evening of hands-on mathematics scheduled for one night a week for four weeks. Students and adults come together to share mathematics activities, solve problems, and have fun with
mathematics. The activities are intended for intermediate level students. Sessions include measurement and geometry, number concepts and problem solving, estimation and calculators, and data
collections and probability. Descriptions of each session and a list of materials are presented. The description of each session includes directions for preparations to be done prior to the event,
guidelines on how to present each activity, and solutions to some of the explorations.… Continue Reading
Eric.ed.gov – Using a Scientific Process for Curriculum Development and Formative Evaluation: Project FUSION
eric.ed.gov har udgivet: Given the vital importance of using a scientific approach for curriculum development, the authors employed a design experiment methodology (Brown, 1992; Shavelson et al.,
2003) to develop and evaluate, FUSION, a first grade mathematics intervention intended for students with or at-risk for mathematics disabilities. FUSION, funded through IES (Baker, Clarke, & Fien,
2008), targets students’ understanding of whole number concepts and skills and is being designed as a Tier 2 intervention for schools that use a multi-tiered service delivery model, such as Response
to Intervention (RtI). In developing this intervention, the authors have drawn extensively from the converging knowledge base of effective math instruction (Gersten et al., 2009; National Math
Advisory Panel, [NMAP] 2008) and the critical content areas of first grade mathematics recognized by national bodies… Continue Reading
Eric.ed.gov – Fun with Math: Real-Life Problem Solving for Grades 4-8.
eric.ed.gov har udgivet: This book was developed for teachers, youth group leaders, after-school child care providers, and parents, who may not have the time or the expertise to develop strategies
for preparing students to be effective problem solvers. The content is organized in a pyramid style to make it easy to locate and grasp the information provided. Information on effective strategies
for teaching general real-life problem solving is provided first. Similar information specific to real-life math problem solving follows. Together these two sections lay a foundation to prepare
teachers to successfully deliver the learning activities subsequently provided. The Learning Activities section is organized by strand as identified by the Ohio Mathematics Proficiency Outcomes. Each
section begins with an index of the activities included in that strand. Appendices provide additional details… Continue Reading
Eric.ed.gov – Dimensions of Math Avoidance among American Indian Elementary School Students. Final Report.
eric.ed.gov har udgivet: Using a cross-cultural perspective, researchers studied the “math avoidance syndrome,” which has reached crisis proportions among American Indians, at two elementary schools
on Utah’s Northern Ute Reservation and Wisconsin’s Oneida Indian Reservation in 1980. Researchers gathered data by observing math instruction at the schools and by interviewing parents, teachers,
tribal officials, and a group of students from third and fourth grade classrooms. They also discussed with tribal elders each tribe’s style of computation and problem solving. Results showed that,
contrary to widely held beliefs, neither degree of traditionality nor sex of student served as an accurate predictor of student math attainment or interest in math. Perceived conflicts between school
and home regarding function and purpose of education, social organization of math lessons, incompatibility of classroom management styles,… Continue Reading
Eric.ed.gov – GED as Project: Pathways to Passing the GED. Volume 2: Math.
eric.ed.gov har udgivet: This guide presents math-focused learning projects and accompanying inquiry activities to help students pass the math portion of the GED 2002. It is Volume 2 of a proposed
four-volume series; Volume 1 describing the concept of the GED as project is also available. Section 1 relates GED as project to the math portion of the GED and explains how inquiry activities use
Official GED Math Practice Test questions as stimuli and can serve as models for teacher-designed activities. It introduces the template for math inquiry activities, a series of steps and questions
that fulfill the learner-centered thinking and process this guide proposes. Section 2 is an introductory learning project that helps learners comprehend and internalize information about the GED,
“GED Math and You.” The two inquiry activities… Continue Reading
Eric.ed.gov – Cross Cultural Comparison of Rural Education Practice in China, Taiwan, and the United States
eric.ed.gov har udgivet: The purpose of this research is to compare the rural education practices of China, Taiwan, Canada and the United States. International comparisons of mathematics achievement
find that students in Asian countries outperform those from the USA. Excluded from these studies, however, are students from rural areas in China. This study compares the math abilities of 272
selectively chosen 5th grade students from rural, central China, 361 students from rural, northern Taiwan and 95 students from rural, central Pennsylvania. The test instrument was the same as used in
previous China vs. USA comparisons and focused on four subtopics: computation, number concepts, geometry and problem solving. The results showed that rural Chinese and Taiwanese students outperformed
similar American students in the area of mathematics achievement. The rural Chinese and… Continue Reading | {"url":"http://troelschristensen.dk/category/number-concepts/","timestamp":"2024-11-11T11:23:26Z","content_type":"text/html","content_length":"164903","record_id":"<urn:uuid:c47c22f8-386c-41b7-a218-5cf467a177d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00657.warc.gz"} |
Bob's Bookkeepers - How to Calculate Average Cost Inventory?
Thus, learning how to calculate average cost inventory is critical, as it’s an important component of key calculations, like COGS, with ripple effects throughout the income statement.
Below, we’ll explain what the average cost method for inventory is and how to calculate it, as well as discuss some of the drawbacks and advantages of using this approach over the others available.
What is the Average Cost Method for Inventory?
There are a few ways businesses can calculate the cost of their inventory, with one common option being the average cost method. Sometimes, this may be referred to as the weighted-average method.
As the name might suggest, when using the average cost method, companies take the total cost of all inventory purchased or produced over a certain period of time divided by the total number of items
in inventory. The resulting value is the average cost for each item in the inventory.
So, even if specific items were purchased at a higher or lower rate, this method applies the same uniform cost to all inventory.
When to Use Average Cost Inventory
As we mentioned above, the average cost method isn’t the only way to find the cost of inventory. There are two other options that companies commonly use:
• Last in, first out (LIFO)
• First in, first out (FIFO)
In general, the average cost method is seen as the simplest. With this method, you don’t have to worry about the timing of when certain items are delivered and sold, unlike with LIFO and FIFO.
The average cost method is practical in certain scenarios, including when:
• Inventory prices are relatively stable and consistent and do not fluctuate significantly over time
• A company wants a simple way to manage inventory
• The inventory consists of relatively homogeneous goods without a lot of variation between items
Companies have flexibility over which inventory valuation method they use. However, once they pick a method, they must use it consistently to support accurate financial reporting and remain compliant
with generally accepted accounting principles (GAAP).
How to Calculate Average Cost Inventory
Here is the formula you can use to calculate average cost inventory:
Average unit cost = Total cost of inventory / Total units in inventory
This is a relatively straightforward formula. However, there can be some nuances in calculating this value accurately, which we’ll explore in further detail below with a step-by-step example.
Average Cost Inventory Example
Learning how to calculate average cost inventory with the formula is only the first step toward understanding how to apply this concept in your business.
To better illustrate how to use the average cost method in the real world, we will now walk through a practical example of how this might look for an e-commerce store.
Company Background and Contex
Let’s use an example of an online retailer that sells pet products. They need to calculate the cost of their inventory to prepare their financial statements at the end of the period.
Order Summary
Over the past quarter, the e-commerce brand made multiple purchases of the same type of dog toy. The price of the product fluctuated over the quarter, so the cost of each purchase varied slightly.
Here is a breakdown of the purchases over the quarter:
• April 1: 50 dog toys at $6.50 each = $325 total
• May 6: 200 dog toys at $5.50 each = $1,100 total
• June 3: 125 dog toys at $5.75 each = $718.75 total
Initial Calculations
Using the above details, we need to make a few more simple calculations before landing on the average inventory cost.
Specifically, we still need to find the total cost of inventory and the total units, as shown in the above formula.
Total cost of inventory: $325 + $1,100 + $718.75 = $2,143.75
Total units of inventory: 50 + 200 + 125 = 375
So, throughout the quarter, the company purchased 375 dog toys, or total units of inventory, at a total cost of $2,143.75.
Average Inventory Cost Calculation
We now have all the necessary information to calculate the pet product company’s average inventory cost.
Average unit cost = $2,143.75 / 375
= $5.72
From our calculations, the average cost of the company’s inventory is $5.72 for the second quarter.
The company can use this value to account for the cost of goods sold based on the number of units sold over the period.
For example, if we know they sold 200 units during the quarter, we can calculate the COGS as:
COGS = 200 * $5.72
= $1,144
This value will be reported on the income statement, which is subtracted from total revenues to determine the company’s gross profit for the quarter.
Pros and Cons of Average Cost Inventory
After seeing how to calculate average cost inventory using a real-world example, it may be easier to understand why some companies are drawn to its simplicity.
However, there are some potential drawbacks of the average cost inventory method. Here is a quick view of the pros and cons of this method to round out our discussion:
• Easy to use, so it takes the team less time and resources to calculate
• Less prone to error since it’s easier to calculate
• Good for large volumes of homogenous goods, as companies don’t have to distinguish between the items to determine their value individually
• Can distort profits, as it doesn’t track the exact cost of each piece of inventory sold during the period
• It doesn’t track price fluctuations as accurately as LIFO or FIFO methods
Don’t Know How to Value Your Inventory? Leave it to the Experts
After reading through this guide, hopefully, you now have a better understanding of how to calculate average cost inventory.
Though this is typically seen as the simplest method for valuing inventory, it still requires diligent attention to the company’s purchases and careful calculations to ensure reporting accuracy and
If you’d rather not spend your valuable time poring over purchase orders so you can calculate your inventory costs, let the pros at Bob’s Bookkeepers handle it.
From basic bookkeeping service to fractional CFO support, our team can help you determine the best inventory valuation method for your company and industry, then keep your records up-to-date while
you focus on growing your business.
Contact us today to speak with our team about your company’s financial support needs. | {"url":"https://www.bobsbookkeepers.com/blog/learn-how-to-calculate-average-cost-inventory-and-see-it-used-in-a-real-world-example-to-understand-the-potential-pros-and-cons-of-this-method","timestamp":"2024-11-12T19:02:38Z","content_type":"text/html","content_length":"39213","record_id":"<urn:uuid:b4c175b3-0713-4fe9-be98-93c688eed422>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00094.warc.gz"} |
Drude polarizable force field
Next: MARTINI Residue-Based Coarse-Grain Forcefield Up: Force Field Parameters Previous: Water Models Contents Index
Drude polarizable force field
The Drude oscillator model represents induced electronic polarization by introducing an auxiliary particle attached to each polarizable atom via a zero-length harmonic spring. The advantage with the
Drude model is that it preserves the simple particle-particle Coulomb electrostatic interaction employed in nonpolarizable force fields, thus its implementation in NAMD is more straightforward than
alternative models for polarization. NAMD performs the integration of Drude oscillators by employing a novel dual Langevin thermostat to ``freeze'' the Drude oscillators while maintaining the normal
``warm'' degrees of freedom at the desired temperature [30]. Use of the Langevin thermostat enables better parallel scalability than the earlier reported implementation which made use of a dual
Nosé-Hoover thermostat acting on, and within, each nucleus-Drude pair [38]. Performance results show that the NAMD implementation of the Drude model maintains good parallel scalability with an
increase in computational cost by not more than twice that of using a nonpolarizable force field [30].
Excessive ``hyperpolarization'' of Drude oscillators can be prevented by two different schemes. The default ``hard wall'' option reflects elongated springs back towards the nucleus using a simple
collision model. Alternatively, the Drude oscillators can be supplemented by a flat-bottom quartic restraining potential (usually with a large force constant).
The Drude polarizable force field requires some extensions to the CHARMM force field. An anisotropic spring term is added to account for out-of-plane forces from a polarized atom and its covalently
bonded neighbor with two more covalently bonded neighbors (similar in structure to an improper bond). The screened Coulomb correction of Thole is calculated between pairs of Drude oscillators that
would otherwise be excluded from nonbonded interaction and optionally between non-excluded, nonbonded pairs of Drude oscillators that are within a prescribed cutoff distance [75,76]. Also included in
the Drude force field are the use of off-centered massless interaction sites, so called ``lone pairs'' (LPs), to avoid the limitations of centrosymmetric-based Coulomb interactions [26]. The
coordinate of each LP site is constructed based on three ``host'' atoms. The calculated forces on the massless LP must be transferred to the host atoms, preserving total force and torque. After an
integration step of velocities and positions, the position of the LP is updated based on the three host atoms, along with additional geometry parameters that give displacement and in-plane and
out-of-plane angles. See our research web page (http://www.ks.uiuc.edu/Research/Drude/) for additional details and parallel performance results.
No additional files are required by NAMD to use the Drude polarizable force field. However, it is presently beyond the capability of the psfgen tool to generate the PSF file needed to perform a
simulation using the Drude model. For now, CHARMM is needed to generate correct input files.
The CHARMM force field parameter files specific to the Drude model are required. The PDB file must also include the Drude particles (mass between 0.05 and 1.0) and the LPs (mass 0). The Drude
particles always immediately follow their parent atom. The PSF file augments the ``atom'' section with additional columns that include the ``Thole'' and ``alpha'' parameters for the screened Coulomb
interactions of Thole. The PSF file also requires additional sections that list the LPs, including their host atoms and geometry parameters, and list the anisotropic interaction terms, including
their parameters. A Drude-compatible PSF file is denoted by the keyword ``DRUDE'' given along the top line.
The NAMD logging to standard output is extended to provide additional temperature data on the cold and warm degrees of freedom. Four additional quantities are listed on the ETITLE and ENERGY lines:
gives the temperature for the warm center-of-mass degrees of freedom,
gives the temperature for the cold Drude oscillator degrees of freedom,
gives the average temperature (averaged since the previously reported temperature) for the warm center-of-mass degrees of freedom,
gives the average temperature (averaged since the previously reported temperature) for the cold Drude oscillator degrees of freedom.
The energies resulting from the Drude oscillators and the anisotropic interactions are summed into the BOND energy. The energies resulting from the LPs and the screened Coulomb interactions of Thole
are summed into the ELECT energy.
The Drude model should be used with the Langevin thermostat enabled (Langevin=on). Doing so permits the use of normal sized time steps (e.g., 1 fs). The Drude model is also compatible with constant
pressure simulation using the Langevin piston. Long-range electrostatics may be calculated using PME. The nonbonded exclusions should generally be set to use either the 1-3 exclusion policy (exclude=
1-3) or the scaled 1-4 exclusion policy (exclude=scaled1-4).
The Drude water model (SWM4-NDP) is a 5-site model with four charge sites and a negatively charged Drude particle [37], with the particles ordered in the input files as oxygen, Drude particle, LP,
hydrogen, hydrogen. The atoms in the water molecules should be constrained (rigidBonds=water), with use of the SETTLE algorithm recommended (useSettle=on). Explicitly setting the water model
(waterModel=swm4) is optional.
• drude
Acceptable Values: on or off
Default Value: off
Description: The integration uses a dual Langevin thermostat to freeze the Drude oscillators while maintaining the warm degrees of freedom at the desired temperature. Must also enable the
Langevin thermostat. If drude is set to on, then drudeTemp must also be defined.
• drudeTemp
Acceptable Values: non-negative decimal
Description: For stability, the Drude oscillators must be kept at a very cold termpature. Using a Langevin thermostat, it is possible to set this temperature to 0 K.
• drudeDamping
Acceptable Values: positive decimal
Description: The Langevin coupling coefficient to be applied to the Drude oscillators. If not given, drudeDamping is set to the value of langevinDamping, but values of as much as an order of
magnitude greater may be appropriate.
• drudeNbTholeCut
Acceptable Values: positive decimal
Default Value: 5.0
Description: If drudeNbTholeCut is non-zero, the screened Coulomb correction of Thole is also calculated for non-excluded, nonbonded pairs of Drude oscillators that are within this radius of
• drudeHardWall
Acceptable Values: on or off
Default Value: on
Description: Excessively elongated Drude oscillator bonds are avoided by reflective collisions induced at a fixed cutoff, drudeBondLen. A large number of such events is usually indicative of
unstable/unphysical dynamics and a simulation will stop if double the cutoff is exceeded.
• drudeBondLen
Acceptable Values: positive decimal
Default Value: 0.25
Description: If using drudeHardWall on, this is the distance at which collisions occur. Otherwise, this is the distance at which an additional quartic restraining potential is applied to each
Drude oscillator. In this latter case, a value of 0.2 Å (slightly smaller than default) is recommended.
Next: MARTINI Residue-Based Coarse-Grain Forcefield Up: Force Field Parameters Previous: Water Models Contents Index http://www.ks.uiuc.edu/Research/namd/ | {"url":"https://www.ks.uiuc.edu/Research/namd/3.0/ug/node27.html","timestamp":"2024-11-04T09:03:30Z","content_type":"text/html","content_length":"15128","record_id":"<urn:uuid:febc3d54-cc66-4bc4-b23b-3d60cd0f5538>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00878.warc.gz"} |
Mechanical Engineering and Math Test
Mechanical Engineering and Math Test
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The mechanical engineering and math test assesses knowledge of various engineering disciplines and the ability to solve mathematical problems.
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• Properly dimension electrical elements in an electric circuit.
• Understand the behavior of materials under different temperatures.
• Solve linear equations to calculate variables.
A good mechanical engineer needs a solid understanding of engineering physics, mathematics principles, and materials science in order to design, analyze, manufacture, and maintain mechanical systems
Sample public questions
Engineering Math
Pythagorean Theorem
Before purchasing seeds, a farmer needs to know the area of his land. Since one end of his property is currently unapproachable, he measured only three sides. This is his sketch with measurements:
What is the area of the land?
Mechanical Engineering
Materials Properties
A company is developing a temperature alarm, based on the schema below, that should close an electric circuit when the temperature rises above 150°C:
Linear coefficient of thermal expansion (x10^-6K^-1) Volumetric mass density (kg/m^3) Melting point (°C)
Brass 13 8600 1000
Copper 17 8940 1100
Iron 11.8 7870 1500
Mercury 181 13540 -39
Polystyrene 70 1060 100
Steel 12 7800 1600
Based on available materials and their properties, choose materials for the temperature alarm that require the smallest temperature difference to do the task.
For Material 1: ___
For Material 2: ___
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18 more premium Mechanical Engineering and Math questions
Construction Yard, KERS, Railway Track, Motor Speed, Classification Yard, Clutch Pedal, Attic, Handbarrow, Sensor Set, Chocolate Box, Bouquet of Roses, Overtime, Roulette, Car Velocity, Contest Idea,
Fraction Equation, Car Acceleration, Accelerometer.
Skills and topics tested
• Mechanical Engineering
• Materials Properties
• Volumetric Mass Density
• Angular Momentum
• Thermal Expansion
• Thermodynamics
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• Engineering Math
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• Independent Events
• Calculus
• Integration
• Nonlinear System
• Polynomial Function
• Arithmetic
• Fractions
• Derivation
• Function Extrema
• Matrix
• Matrix Addition
• Scalar Multiplication
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Density-functional study of undoped and doped <i>trans</i>-polyacetylene
We report a self-consistent linear-combination-of-Gaussian-orbitals study of the electronic states and ground-state geometry of an undoped and doped single, infinite chain of trans-polyacetylene
using the density-functional theory in the local-density approximation. We find a dimerized ground state for an undoped chain with a dimerization amplitude of about 0.01 Å, which is lower than the
experimental value of 0.023–0.03 Å. A pure Hartree calculation neglecting all exchange and correlation gives a much smaller dimerization amplitude of less than 0.005 Å. The local exchange-correlation
energy thus significantly favors the dimerization although its effect is not strong enough. In the calculations of the doped chains, the dopant ions were approximated by a uniform background charge.
We find that the undimerized state becomes energetically more favorable than any uniformly dimerized state at a critical doping level of about 0.04 (0.03) extra holes (electrons) per CH unit. The
band structures and total energies of polaron and soliton lattices at a higher doping level of 0.2 holes per CH unit are calculated and compared with those of the uniformly dimerized and undimerized
lattices, and possible models of the metallic state of trans-polyacetylene are discussed. According to our study, the bonds become increasingly similar with increasing doping. The undimerized chain
model seems to be a good approximation for the metallic state of trans-polyacetylene at high doping levels although the possibility for a marginal soliton lattice cannot be fully excluded.
Dive into the research topics of 'Density-functional study of undoped and doped trans-polyacetylene'. Together they form a unique fingerprint. | {"url":"https://cris.vtt.fi/en/publications/density-functional-study-of-undoped-and-doped-itransi-polyacetyle","timestamp":"2024-11-07T13:53:30Z","content_type":"text/html","content_length":"57065","record_id":"<urn:uuid:95841fb8-3770-4feb-8e11-b7375f5f03fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00459.warc.gz"} |
Title: Brahms 1
• Byzantine-Resilient Random Membership Sampling
Bortnikov, Gurevich, Keidar, Kliot, and Shraer
Edward (Eddie) Bortnikov
Maxim (Max) Gurevich
Idit Keidar
Alexander (Alex) Shraer
Gabriel (Gabi) Kliot
Why Random Node Sampling
• Gossip partners
• Random choices make gossip protocols work
• Unstructured overlay networks
• E.g., among super-peers
• Random links provide robustness, expansion
• Gathering statistics
• Probe random nodes
• Choosing cache locations
The Setting
• Many nodes n
• 10,000s, 100,000s, 1,000,000s,
• Come and go
• Churn
• Every joining node knows some others
• Connectivity
• Full network
• Like the Internet
• Byzantine failures
Byzantine Fault Tolerance (BFT)
• Faulty nodes (portion f)
• Arbitrary behavior bugs, intrusions, selfishness
• Choose f ids arbitrarily
• No CA, but no panacea for Cybil attacks
• May want to bias samples
• Isolate nodes, DoS nodes
• Promote themselves, bias statistics
Previous Work
• Benign gossip membership
• Small (logarithmic) views
• Robust to churn and benign failures
• Empirical study Lpbcast,Scamp,Cyclon,PSS
• Analytical study Allavena et al.
• Never proven uniform samples
• Spatial correlation among neighbors views PSS
• Byzantine-resilient gossip
• Full views MMR,MS,Fireflies,Drum,BAR
• Small views, some resilience SPSS
• We are not aware of any analytical work
Our Contributions
• Gossip-based BFT membership
• Linear portion f of Byzantine failures
• O(n1/3)-size partial views
• Correct nodes remain connected
• Mathematically analyzed, validated in simulations
• Random sampling
• Novel memory-efficient approach
• Converges to proven independent uniform samples
The view is not all bad
Better than benign gossip
1. Sampling - local component
2. Gossip - distributed component
Sampler Building Block
• Input data stream, one element at a time
• Bias some values appear more than others
• Used with stream of gossiped ids
• Output uniform random sample
• of unique elements seen thus far
• Independent of other Samplers
• One element at a time (converging)
Sampler Implementation
• Memory stores one element at a time
• Use random hash function h
• From min-wise independent family Broder et al.
• For each set X, and all ,
Keep id with smallest hash so far
Choose random hash function
Component S Sampling and Validation
id streamfrom gossip
using pings
Gossip Process
• Provides the stream of ids for S
• Needs to ensure connectivity
• Use a bag of tricks to overcome attacks
Gossip-Based Membership Primer
• Small (sub-linear) local view V
• V constantly changes - essential due to churn
• Typically, evolves in (unsynchronized) rounds
• Push send my id to some node in V
• Reinforce underrepresented nodes
• Pull retrieve view from some node in V
• Spread knowledge within the network
• Allavena et al. 05 both are essential
• Low probability for partitions and star topologies
Brahms Gossip Rounds
• Each round
• Send pushes, pulls to random nodes from V
• Wait to receive pulls, pushes
• Update S with all received ids
• (Sometimes) re-compute V
• Tricky! Beware of adversary attacks
Problem 1 Push Drowning
Push Alice
Push Bob
Push Mallory
Push Carol
Push Ed
Push Dana
Push MM
Push Malfoy
Trick 1 Rate-Limit Pushes
• Use limited messages to bound faulty pushes
• E.g., computational puzzles/virtual currency
• Faulty nodes can send portion p of them
• Views wont be all bad
Problem 2 Quick Isolation
Ha! Shes out! Now lets move on to the next guy!
Push Alice
Push Bob
Push Carol
Push Mallory
Push Ed
Push Dana
Push MM
Push Malfoy
Trick 2 Detection Recovery
• Do not re-compute V in rounds when too many
pushes are received
• Slows down isolation does not prevent it
Push Bob
Push Mallory
Hey! Im swamped! I better ignore all of em
Push MM
Push Malfoy
Trick 3 Balance Pulls Pushes
• Control contribution of push - aV ids versus
contribution of pull - ßV ids
• Parameters a, ß
• Pull-only ? eventually all faulty ids
• Pull from faulty nodes all faulty ids, from
correct nodes some faulty ids
• Push-only ? quick isolation of attacked node
• Push ensures system-wide not all bad ids
• Pull slows down (does not prevent) isolation
Trick 4 History Samples
• Attacker influences both push and pull
• Feedback ?V random ids from S
• Parameters a ß ? 1
• Attacker loses control - samples are eventually
perfectly uniform
Yoo-hoo, is there any good process out there?
View and Sample Maintenance
Pushed ids
Pulled ids
? V
View V
Key Property
• Samples take time to help
• Assume attack starts when samples are empty
• With appropriate parameters
• E.g.,
• Time to isolation gt time to convergence
Prove lower bound using tricks 1,2,3(not using
samples yet)
Prove upper bound until some good sample
persists forever
Self-healing from partitions
History Samples Rationale
• Judicious use essential
• Bootstrap, avoid slow convergence
• Deal with churn
• With a little bit of history samples (10) we
can cope with any adversary
• Amplification!
1. Sampling - mathematical analysis
2. Connectivity - analysis and simulation
3. Full system simulation
Connectivity ? Sampling
• Theorem If overlay remains connected
indefinitely, samples are eventually uniform
Sampling ? Connectivity Ever After
• Perfect sample of a sampler with hash h the id
with the lowest h(id) system-wide
• If correct, sticks once the sampler sees it
• Correct perfect sample ? self-healing from
partitions ever after
• We analyze PSP(t) probability of perfect sample
at time t
Convergence to 1st Perfect Sample
• n 1000
• f 0.2
• 40 unique ids in stream
• Analysis says
• For scalability, want small and constant
convergence time
• independent of system size, e.g., when
Connectivity Analysis 1 Balanced Attacks
• Attack all nodes the same
• Maximizes faulty ids in views system-wide
• in any single round
• If repeated, system converges to fixed point
ratio of faulty ids in views, which is lt 1 if
• ?0 (no history) and p lt 1/3 or
• History samples are used, any p
There are always good ids in views!
Fixed Point Analysis Push
Local view node i
Local view node 1
Time t
lost push
push from faulty node
Time t1
x(t) portion of faulty nodes in views at round
t portion of faulty pushes to correct nodes p
/ ( p ( 1 - p )( 1 - x(t) ) )
Fixed Point Analysis Pull
Local view node i
Local view node 1
Time t
pull from i faulty with probability x(t)
pull from faulty
Time t1
Ex(t1) ? p / (p (1 - p)(1 - x(t))) ?
( x(t) (1-x(t))?x(t) ) ?f
Faulty Ids in Fixed Point
Assumed perfect in analysis, real history in
With a few history samples, any portion of bad
nodes can be tolerated
Perfectly validated fixed pointsand convergence
Convergence to Fixed Point
Connectivity Analysis 2Targeted Attack
• Step 1 analysis without history samples
• Isolation in logarithmic time
• but not too fast, thanks to tricks 1,2,3
• Step 2 analysis of history sample convergence
• Time-to-perfect-sample lt Time-to-Isolation
• Step 3 putting it all together
• Empirical evaluation
• No isolation happens
Targeted Attack Step 1
• Q How fast (lower bound) can an attacker isolate
one node from the rest?
• Worst-case assumptions
• No use of history samples (? 0)
• Unrealistically strong adversary
• Observes the exact number of correct pushes and
complements it to aV
• Attacked node not represented initially
• Balanced attack on the rest of the system
Isolation w/out History Samples
Isolation time for V60
Depend on a,ß,p
Step 2 Sample Convergence
• n 1000
• p 0.2
• aß0.5, ?0
• 40 unique ids
Perfect sample in 2-3 rounds
Empirically verified
Step 3 Putting It All TogetherNo Isolation with
History Samples
Works well despite small PSP
Sample Convergence (Balanced)
Convergence twice as fast with
• O(n1/3)-size views
• Resist Byzantine failures of linear portion
• Converge to proven uniform samples
• Precise analysis of impact of failures
Balanced Attack Analysis (1)
• Assume (roughly) equal initial node degrees
• x(t) portion of faulty ids in correct node
views at time t
• Compute Ex(t1) as function of x(t), p, ?, ?, ?
• Result 1 Short-term Optimality
• Any non-balanced schedule imposes a smaller x(t)
in a single round
Balanced Attack Analysis (2)
• Result 2 Existence of Fixed Point X
• Ex(t1) x(t) X
• Analyze X (function of p, ?, ?, ?)
• Conditions for uniqueness
• For ??0.5, p lt 1/3, exists X lt 1
• The view is not entirely poisoned history
samples are not essential
• Result 3 Convergence to fixed point
• From any initial portion lt 1 of faulty ids
• From Hillam 1975 (sequence convergence)
User Comments (0) | {"url":"https://www.powershow.com/view4/738400-NzY3M/Brahms_powerpoint_ppt_presentation","timestamp":"2024-11-08T15:15:07Z","content_type":"application/xhtml+xml","content_length":"166136","record_id":"<urn:uuid:46287f03-38a9-492c-b17a-8801c0e73969>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00555.warc.gz"} |
How do you calculate external pressure in a pipe?
How do you calculate external pressure in a pipe?
Calculate Allowable External Pressure Pa if the value of factor B is determined.
1. Pa = 4B3(Dot)
2. Pa = 2AE3(Dot)
3. Pa = 2AE3(Dot)
4. Pa = 2AE3(Dot)
What is external design pressure?
External pressure can be caused in pressure vessels by a variety of conditions and circumstances. The design pressure may be less than atmospheric due to condensing gas or steam. Often vessels are
design for some amount of external pressure, to allow for steam cleaning and the effects of the condensing steam.
How do you create external pressure?
External pressure can be created three ways:
1. From a vacuum inside a vessel and atmospheric pressure outside.
2. From a pressure outside a vessel greater than atmospheric (typically from some types of jacket or a surrounding vessel)
How do you calculate the thickness of internal pressure in a pipe?
t = P * D / (2 * F *S * E)
1. t : Calculated Wall thickness (mm)
2. P : Design pressure for the pipeline (kPa)=78 bar-g=7800 KPa.
3. D : Outside diameter of pipe (mm)= 273.05 mm.
4. F : Design factor = 0.72.
5. S : Specified Minimum Yield Strength (MPa)=359870 KPa for the specified material.
6. E : Longitudinal joint factor = 1.0.
What is hoop stress formula?
For a cylindrical shell having diameter d and thickness t , the circumferential or hoop stress σh is given by the hoop stress equation: σh = p * d / (2 * t) where p is internal pressure.
What is hoop stress in thick cylinder?
Concept: In the case of thin cylinders, the hoop stress is determined by assuming it to be uniform across the thickness of the cylinder but in thick cylinders, the hoop stress is not uniform across
the thickness, it varies from a maximum value at the inner circumference to a minimum value at the outer circumference.
What happens if a thin pressure vessel is subjected to external pressure?
When a pressure vessel is subjected to a higher external than internal pressure, the above formulas are still valid. However, the stresses would now be negative since the wall is in compression
instead of tension. The hoop stress is twice as much as the longitudinal stress for the cylindrical pressure vessel.
Is external pressure the same as atmospheric pressure?
Yes, the external pressure is constant if it is related to atmospheric pressure. The internal pressure will give you a curved graph in the p-V diagram. The area between this curved graph and the
other graph in p-V diagram will give you the work that is done to the system.
What is external process?
External Processes (or) Exogenetic Processes The forces that act on the surface of the earth due to natural agents like running water, glacier, wind, waves, etc., are called External processes or
Exogenetic processes. These external processes tear the landscape down into relatively low elevated plains.
Is external pressure equal to internal pressure?
The internal pressure is either equal, or not equal, to the external pressure. If the work done by the system is equal in magnitude to the work done by the surroundings (which you confirmed was
true), the magnitude of the force applied by the system must equal the magnitude of the force applied by the surroundings.
What is internal pressure and external pressure?
Internal pressure is how much change in energy a system undergoes when it expands or contracts. External pressure is the amount of energy that is applied from the outside.
How do you design a vessel for external pressure?
The easiest way to design for external pressure is to make the shell thick enough to make the vessel stable with an acceptable factor of safety (pass code calculations). The length of the vessel used
in the calculations includes some of the head at each end. The calculations are found in ASME VIII-1 UG-28.
What is the design of pipe fittings under external pressure?
Design of pipe and pipe fittings under external pressure according to EN 13480-3 (2017) Pipes, Elbows, Mitre Bends and Reducers Interstiffener collapse The thickness of the pipe within the
unstiffened length L shall not be less than that determined by the following.
Can cylinders withstand high external pressure?
We can apply calculation criteria to these cylinders that are similar to those for cylinders under internal pressure; they can obviously withstand high external pressure. The equations of equilibrium
and congruence are the same, specifically (3.11) and (3.16), and we rewrite them below.
What is external pressure and how is it created?
External pressure can be created three ways: From a vacuum inside a vessel and atmospheric pressure outside From a pressure outside a vessel greater than atmospheric (typically from some types of
jacket or a surrounding vessel) From a combination of the first two – vacuum inside + pressure greater than atmospheric outside. | {"url":"https://gowanusballroom.com/how-do-you-calculate-external-pressure-in-a-pipe/","timestamp":"2024-11-08T11:46:49Z","content_type":"text/html","content_length":"53073","record_id":"<urn:uuid:d95b3304-d2b9-4434-96e6-2b3dbbbafbbb>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00869.warc.gz"} |
Fast ways to check if an element is in a vector of vector of elements
Hello ,
What would be the most efficient way to look for a number in a vector of vector of numbers?
Data: t = [ [1,2,3] , [4,5,6] , [1,10,12], [1,4,5] ]
Input: 1
Output: 1,3,4
I could do something like findall(map(x->1 in x, t)) but this is quite slow, and I’m pretty sure it’s not the best way to do this.
Make it a set, or give it some ordering.
Thanks, but could you also please tell me how this would help?
findall(v->in(1, v), t)
findall(1 in v for v in t)
1 Like
I’m interested in an answer, too.
The following didn’t speed up the searching:
t_set = map(Set, t)
ix = findall(map(x->1 in x, t_set))
t_sort = map(sort, t)
function fsn_loop(v, n)
# assume that the elements in v are sorted
hits = Int[]
for (i, v_l) in enumerate(v)
determined = false
for v_ll in v_l
if v_ll == n
push!(hits, i)
determined = true
if v_ll > n
determined = true
if determined
return hits
ix = fsn_loop(t_sort, 1)
For t having 1e5 elements, @btime gives around 1 ms, as does DNF’s solution.
My previous experience tells me that this should be the fastest:
But I must say I have seen a number of strange performance issues in v1.11, and now it benchmarks as slower than findall(map(x->1 in x, t)), which, frankly, makes no sense to me, since the latter
creates a redundant temporary array and also passes twice over memory
Hm, okay, thank you!
since the latter creates a redundant temporary array and also passes twice over memory
I thought there would be a better way, but I guess I’m better off sticking to findall(map(x->1 in x, t)) for now in that case.
So I have not understood the problem first. I think the big question is, how frequently you want to run this search. If once, the answer by @DNF is OK. if multiple times, you should build the index.
a = [[1,2,3],[4,5,6],[1,5,6]]
index = Dict{Int,Vector{Int}}()
for (i, jj) in enumerate(a)
for j in jj
push!(get!(index, j, Int[]), i)
julia> index[1]
2-element Vector{Int64}:
3 Likes
I would give this version a chance too
function find1s(a)
for i in eachindex(a)
if !isnothing(findfirst(==(1),a[i]))
2 Likes
My package SmallCollections.jl contains a vectorized version of in for suitable types (not yet in the published version).
EDIT: It’s also vectorized for SVector in StaticArrays.jl.
This might speed things up if the element vectors are “small” (say, up to 32 or 64 elements). For example, for
using SmallCollections, Chairmarks
using Base: Fix1
T = Int32
N = 3
t = [rand(T, N) for _ in 1:1_000_000]
x = T(1)
I get
julia> @b t findall(Fix1(in, $x), _) # analogous to OP
5.546 ms (5 allocs: 122.250 KiB)
julia> @b map(FixedVector{N}, t) findall(Fix1(in, $x), _)
881.295 μs (5 allocs: 122.250 KiB)
julia> @b map(SmallVector{N}, t) findall(Fix1(in, $x), _)
3.204 ms (5 allocs: 122.250 KiB)
FixedVector{N,T} is like SVector{N,T} from StaticArrays.jl. SmallVector{N,T} can hold up to N elements of type T.
To try it out, you can install the relevant branch via
pkg> add https://github.com/matthias314/SmallCollections.jl#fixedvector
EDIT: fast in for SmallVector is now implemented.
1 Like
could you check this ?
function f1(a,e)
for i in eachindex(a)
!isnothing(findfirst(==(e),a[i])) && (r[j+=1]=i)
1 Like
Was this for me? I get (with the same t and x as before)
julia> @b t f1(_, $x)
4.813 ms (3 allocs: 7.629 MiB)
julia> @b map(FixedVector{N}, t) f1(_, $x)
2.299 ms (3 allocs: 7.629 MiB)
Yes is for you.
I meant to do the proof by redefining the vector of vectors in the following way
T = Int32
N = 3
t = [T.(rand(1:10^5, N)) for _ in 1:1_000_000]
Here it is:
julia> @b f1($st, $x)
1.717 ms (3 allocs: 7.629 MiB)
julia> @b map(FixedVector{N}, st) findall(Fix1(in, $x), _)
902.658 μs (6 allocs: 122.531 KiB)
1 Like
f1 using findfirst -assuming that there is only one value being searched for or that it is enough to find the first one- would become more effective for vectors a little longer than 3.
Could you try for N=32?
As in previous post, just with N = 32:
julia> @b f1($st, $x)
14.917 ms (3 allocs: 7.629 MiB)
julia> @b map(FixedVector{N}, st) findall(Fix1(in, $x), _)
65.604 ms (7 allocs: 124.781 KiB)
julia> @b map(collect, st) findall(Fix1(in, $x), _)
26.547 ms (7 allocs: 124.781 KiB)
Now my version is much slower, even slower than findall with Vector. I don’t understand this because in is faster for FixedVector:
julia> w = st[1]; @b $x in $w
4.290 ns
julia> @b FixedVector{N}(w) $x in _
2.711 ns
julia> @b collect(w) $x in _
15.173 ns
Using findfirst looks slower:
julia> @b findfirst(==($x), $w) === nothing
13.547 ns
With in instead of findfirst, f1 becomes even faster (N = 32):
julia> @b f1($st, $x)
15.493 ms (3 allocs: 7.629 MiB)
julia> @b f1_in($st, $x)
6.463 ms (3 allocs: 7.629 MiB)
julia> @b map(FixedVector{N}, st) f1_in(_, $x)
6.782 ms (3 allocs: 7.629 MiB)
function f1_in(a,e)
for i in eachindex(a)
if e in a[i] # !isnothing(findfirst(==(e),a[i]))
The problem seems to be findall. With N = 32, T = Int32, x = T(1) and
t = [T.(rand(1:10^5, N)) for _ in 1:1_000_000]
st = map(SVector{N}, t)
(as before), I get
julia> @b findall(Fix1(in, $x), $st)
68.203 ms (7 allocs: 124.781 KiB)
julia> @b [i for (i, w) in enumerate($st) if $x in w]
6.780 ms (7 allocs: 7.562 KiB)
EDIT: Also
julia> t2 = map(SmallVector{N}, t);
julia> @b [i for (i, w) in enumerate($t2) if $x in w]
7.595 ms (7 allocs: 7.562 KiB)
1 Like
It seems that the implementation of some method of the function in is able to exploit the fact of having a static array better than findfirst can do.
it would be interesting to have a documentation of functions like these that at first glance seem to do the same thing (at least from a (high?) "logical" point of view), that explains what
algorithm (algorithms?) they use in the various cases and when one can be “convenient” compared to the other.
A curiosity of a similar kind comes to me from the fact that the use of enumerate that makes available both the index and the value of an array is slower than the following version where instead from
time to time, having only the index, you have to obtain the value of the array element (in this case in turn an array).
julia> T = Int32
julia> N = 32
julia> t = [T.(rand(1:10^5, N)) for _ in 1:1_000_000];
julia> st=SArray{Tuple{N}}.(t);
julia> x=T(1)
julia> @b [i for (i, w) in enumerate($st) if $x in w]
16.229 ms (6 allocs: 7.609 KiB)
julia> @b [i for i in eachindex($st) if $x in $st[i]]
7.199 ms (6 allocs: 7.609 KiB)
1 Like
This is because the implementation of in for SVector can be vectorized while that of findfirst (the default method for AbstractArray) cannot. However, findfirst for FixedVector and SmallVector is
vectorized (for suitable types), and in fact in for SmallVector is defined as
in(x, v::AbstractSmallVector) = findfirst(==(x), v) !== nothing
I don’t find it surprising that enumerate is slower than eachindex. With Julia 1.11.0, the difference is quite small on my machine:
julia> @b [i for (i, w) in enumerate($st) if $x in w]
6.792 ms (7 allocs: 7.562 KiB)
julia> @b [i for i in eachindex($st) if $x in $st[i]]
6.035 ms (7 allocs: 7.562 KiB) | {"url":"https://discourse.julialang.org/t/fast-ways-to-check-if-an-element-is-in-a-vector-of-vector-of-elements/121598","timestamp":"2024-11-05T04:10:50Z","content_type":"text/html","content_length":"57972","record_id":"<urn:uuid:7189813c-9e67-4c33-a1a2-ce3b5633ed49>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00721.warc.gz"} |
Applied Multivariate Statistics in R
30 k-Means Cluster Analysis
To consider ways of classifying sample units into non-hierarchical groups, including decisions about how many groups are desired and which criterion to use when doing so.
To begin visualizing patterns among sample units.
k-means cluster analysis is a non-hierarchical technique. It seeks to partition the sample units into k groups in a way that minimizes some criterion. Often, the criterion relates to the variance
between points and the centroid of the groups they are assigned to.
An essential part of a k-means cluster analysis, of course, is the decision of how many clusters to include in the solution. One way to approach this is to conduct multiple analyses for different
values of k and to then compare those analyses. For example, Everitt & Hothorn (2006) provide some nice code to compare the within-group sums of squares against k. Another approach is to use the
results of a hierarchical clustering method as the starting values for a k-means analysis. One of the older posts on Stack Overflow contains many examples of how to determine an appropriate number
of clusters to use in a k-means cluster analysis. Additional ideas about how to decide how many clusters to use are provided below.
The criterion in k-means clustering is generally to minimize the variance within groups, but how variance is calculated varies among algorithms. For example, kmeans() seeks to minimize within-group
sums of squared distances. Another function, pam(), seeks to minimize within-group sums of dissimilarities, and can be applied to any distance matrix. Both functions are described below.
k-means cluster analysis is an iterative process:
• Select k starting locations (centroids)
• Assign each observation to the group to which it is closest
• Calculate within-group sums of squares (or other criterion)
• Adjust coordinates of the k locations to reduce variance
• Re-assign observations to groups, re-assess variance
• Repeat process until stopping criterion is met
The iterative nature of this technique can be seen using the animation::kmeans.ani() function (see the examples in the help files) or in videos like this: https://www.youtube.com/watch?v=BVFG7fd1H30
. We will see a similar iterative process with NMDS.
k-means clustering may find a local (rather than global) minimum value due to differences in starting locations. A common solution to this is to use multiple starts, thus increasing the likelihood
of detecting a global minimum. This, again, is similar to the approach taken in NMDS.
As usual, we will illustrate these approaches with our oak dataset.
k-means Clustering in R
In R, k-means cluster analysis can be conducted using a variety of methods, including stats::kmeans(), cluster::pam(), cluster::clara(), and cluster::fanny().
The kmeans() function is available in the stats package. It seeks to identify the solution that minimizes the sum of squared (Euclidean) distances. Its usage is:
iter.max = 10,
nstart = 1,
algorithm = c("Hartigan-Wong", "Lloyd", "Forgy", "MacQueen"),
trace = FALSE
The key arguments are:
• x – data matrix. Note that a distance matrix is not allowed. Euclidean distances are assumed to be appropriate.
• centers – the number of clusters to be used. This is the ‘k’ in k-means.
• iter.max – maximum number of iterations allowed. Default is 10.
• nstart – number of random sets of centers to choose. Default is 1.
• algorithm – the algorithm used to derive the solution. I won’t pretend to understand what these algorithms are or how they differ. 🙂 The default is “Hartigan-Wong“.
This function returns an object of class ‘kmeans’ with the following values:
• cluster – a vector indicating which cluster each sample unit is assigned to. Note that the coding of clusters is arbitrary, and that different runs of the function may result in different
• centers – a matrix showing the mean value of each species in each sample unit.
• totss – the total sum of squares.
• withinss – the within-cluster sum of squares for each cluster.
• tot.withinss – the sum of the within-cluster sum of squares
• betweenss – the between-cluster sum of squares.
• size – the number of sample units in each cluster.
• iter – the number of iterations
An example for six groups:
k6 <- kmeans(x = Oak1, centers = 6, nstart = 100)
Note that we did not make any adjustments to our data but are using this here to illustrate the technique. Also, your results may differ depending on the number of starts (nstart) and the number of
iterations (iter.max). In addition, your groups may be identified by different integers. To see the group assignment of each sample unit:
Stand01 Stand02 Stand03 Stand04 Stand05 Stand06 Stand07
Stand08 Stand09 Stand10 Stand11 Stand12 Stand13 Stand14
Stand15 Stand16 Stand17 Stand18 Stand19 Stand20 Stand21
Stand22 Stand23 Stand24 Stand25 Stand26 Stand27 Stand28
Stand29 Stand30 Stand31 Stand32 Stand33 Stand34 Stand35
Stand36 Stand37 Stand38 Stand39 Stand40 Stand41 Stand42
Stand43 Stand44 Stand45 Stand46 Stand47
A more compact summary:
The kmeans() function permits a cluster analysis for one value of k. The cascadeKM() function, available in the vegan package, conducts multiple kmeans() analyses: one for each value of k ranging
from the smallest number of groups (inf.gr) to the largest number of groups (sup.gr):
cascadeKM(Oak1, inf.gr = 2, sup.gr = 8)
(output not displayed in notes)
The output can also be plotted:
plot(cascadeKM(Oak1, inf.gr = 2, sup.gr = 8))
K-means cluster analysis of oak plant community data for k=2 through k = 8 groups (vertical axis). Stands are shown on the horizontal axis. Within each row, colors distinguish different groups. On
the left is the Calinski criterion – larger values indicate more support for that group size.
It is important to recall that k-means solutions are not hierarchical – you can see evidence of this by looking at the colors in this graph.
The pam() function is available in the cluster package. The name is an abbreviation for ‘partitioning around medoids’ (a medoid is a representative example of a group – in this case, the sample unit
that is most similar to all other sample units in the cluster). It is more flexible than kmeans() because it allows the use of distance matrices produced from non-Euclidean measures. It also uses a
different criterion, identifying the solution that minimizes the sum of dissimilarities. Its usage is:
diss = inherits(x, "dist"),
metric = c("euclidean", "manhattan"),
medoids = if (is.numeric(nstart)) "random",
nstart = if (variant == "faster") 1 else NA,
stand = FALSE,
cluster.only = FALSE,
do.swap = TRUE,
keep.diss = !diss && !cluster.only && n < 100,
keep.data = !diss && !cluster.only,
variant = c("original", "o_1", "o_2", "f_3", "f_4", "f_5", "faster"),
pamonce = FALSE,
trace.lev = 0
There are quite a few arguments in this function, but the key ones are:
• x – data matrix or data frame, or distance matrix. Since this accepts a distance matrix as an input, it does not assume Euclidean distances as is the case with kmeans().
• k – the number of clusters to be used
• diss – logical; x is treated as a distance matrix if TRUE (default) and as a data matrix if FALSE.
• metric – name of distance measure to apply to data matrix or data frame. Options are “euclidean” and “manhattan“. Ignored if x is a distance matrix.
• medoids – if you want specific sample units to be the medoids, you can specify them with this argument. The default (NULL) results in the medoids being identified by the algorithm.
• stand – whether to normalize the data (i.e., subtract mean and divide by SD for each column). Default is to not do so (FALSE). Ignored if x is a dissimilarity matrix.
This function returns an object of class ‘pam’ that includes the following values:
• medoids – the identity of the sample units identified as ‘representative’ for each cluster.
• clustering – a vector indicating which cluster each sample unit is assigned to.
• silinfo – silhouette width information. The silhouette width of an observation compares the average dissimilarity between it and the other points in the cluster to which it is assigned and
between it and points in other clusters. A larger value indicates stronger separation or strong agreement with the cluster assignment.
An example for six groups:
k6.pam <- pam(x = Oak1.dist, k = 6)
Note that we called Oak1.dist, which is based on Bray-Curtis dissimilarities, directly since pam() cannot calculate this measure internally. Your groups may be identified by different integers. To
see the group assignment of each sample unit:
Stand01 Stand02 Stand03 Stand04 Stand05 Stand06 Stand07
Stand08 Stand09 Stand10 Stand11 Stand12 Stand13 Stand14
Stand15 Stand16 Stand17 Stand18 Stand19 Stand20 Stand21
Stand22 Stand23 Stand24 Stand25 Stand26 Stand27 Stand28
Stand29 Stand30 Stand31 Stand32 Stand33 Stand34 Stand35
Stand36 Stand37 Stand38 Stand39 Stand40 Stand41 Stand42
Stand43 Stand44 Stand45 Stand46 Stand47
The integers associated with each stand indicate the group to which it is assigned.
The help file about the object that is produced by this analysis (?pam.object) includes an example in which different numbers of clusters are tested systematically and the size with the smallest
average silhouette width is selected as the best number of clusters. Here, I’ve modified this code for our oak plant community dataset:
asw <- numeric(10)
for (k in 2:10) {
asw[k] <- pam(Oak1.dist, k = k) $ silinfo $ avg.width
cat("silhouette-optimal # of clusters:", which.max(asw), "\n")
silhouette-optimal # of clusters: 2
This algorithm suggests that there is most support for two clusters of sample units. We could re-run pam() with this specific number of clusters if desired.
Comparing k-means Analyses
Many of the techniques that we saw previously for comparing hierarchical cluster analyses are also relevant here. For example, we could generate a confusion matrix comparing the classifications from
the kmeans() and pam().
table(k6$cluster, k6.pam$clustering)
This can be a bit confusing to read because the row and column names are the integers 1:6 but the cells within the table are also numbers (i.e., number of stands in each group). Also, the number of
groups is the same in both analyses so it isn’t immediately apparent which analysis is shown as rows and which as columns.
Note that observations are more easily assigned to different groups in k-means analyses than they are in hierarchical analysis: in the latter, once two sample units are fused together they remain
fused through the rest of the analysis.
k-means cluster analysis is a technique for classifying sample units into k groups. This approach is not constrained by the decisions that might have been made earlier in a hierarchical analysis.
There is also no concern about other factors such as which group linkage method to use.
However, k-means cluster analysis does require that the user decide how many groups (k) to focus on. In addition, since the process is iterative and starts from random coordinates, it’s possible to
end up with different classifications from one run to another. Finally, functions such as kmeans() use a Euclidean criterion and thus are most appropriate for data expressed with that distance
measure. To apply this technique to a semimetric measure such as Bray-Curtis dissimilarity, some authors have suggested standardizing data using a Hellinger transformation: the square root of the
data divided by the row total. This standardization is available in decostand(), but I have not explored its utility.
Everitt, B.S., and T. Hothorn. 2006. A handbook of statistical analyses using R. Chapman & Hall/CRC, Boca Raton, LA. | {"url":"https://uw.pressbooks.pub/appliedmultivariatestatistics/chapter/k-means-cluster-analysis/","timestamp":"2024-11-06T08:01:35Z","content_type":"text/html","content_length":"89198","record_id":"<urn:uuid:b4d43410-0ee9-45c7-858a-990e379b99e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00464.warc.gz"} |
From Haskell to Racket
Table of contents
Basic Values
Let’s start by looking at something you know: Haskell. In Haskell, expressions can include literals for numbers, strings, booleans. Here we are using the Haskell’s GHCi which provides a
read-eval-print-loop (REPL) to type in examples and evaluate their results:
-- Haskell
> 8
> "haskell"
> True
> False
Note, evaluating values in GHCi gives the value back. Despite Haskell being a typed language, GHCi does not print the types of the expressions by default. We can use :set +t to require GHCi to print
the type of the expression it evaluates.
-- Haskell
> :set +t -- tell GHCi to automatically show types
> 8
it :: Num a => a
> "haskell"
it :: [Char]
> True
it :: Bool
> False
it :: Bool
GHCi now prints the type of the expression it evaluated as it :: Type. Here it refers to the last evaluated expression. Note how 8 is not printed as an Integer but as Num a => a. It means 8 has a
type a where a is an instance of the Num typeclass. The Num typeclass describes common properties of various kinds of number, such as for example. The native machine integer type Int, arbitrary-sized
integers Integer, and even floating point type Double are instances of Num.
The Racket REPL also operates similarly:
;; Racket
> 8
> "racket"
> #t
> #f
Racket only prints the value as it is an untyped language. The notation for booleans is slightly different, but both languages agree on numbers, strings, and booleans. The languages are essentially
the same so far.
Basic Operations
Haskell uses an infix notation for writing operations.
-- Haskell
> 1 + 2
it :: Num a => a
The order of operations follows the usual mathematical precendence rules (which you must memorize), or you can use parentheses to indicate grouping:
-- Haskell
> 1 + (2 * 2)
it :: Num a => a
> (1 + 2) * 2
it :: Num a => a
Extraneous parenthesis are fine:
-- Haskell
> (((1))) + ((2 * 2))
it :: Num a => a
Compared to many languages you may know, including Haskell, Racket employs a uniform, minimalistic concrete syntax based on the concept of parenthesized, prefix notation.
In this notation, parentheses play a much more central role. They are not optional and they signal the form of the expression.
Languages, like people, descend from their ancestors and inherit some of their properties. In the case of notation, Racket inherits the Lisp (and Scheme) notation for programs. It takes a bit of
getting used to, but once aclimated, the notation should feel lightweight and consistent; there is verry little to memorize when it comes to syntax.
So in Racket, we would write:
;; Racket
> (+ 1 (* 2 2))
> (* (+ 1 2) 2)
Note that there are no precendence rules for addition and multiplication: the form of the expression makes it unambiguous.
Parenthesis indicate function applications, so adding extraneous parens means something different than in Haskell:
;; Racket
> (1)
application: not a procedure;
expected a procedure that can be applied to arguments
given: 1
Haskell also has a notation for writing functions:
-- Haskell
> \x y -> x + y
<interactive>:18:1: error:
* No instance for (Show (Integer -> Integer -> Integer))
arising from a use of 'print'
(maybe you haven't applied a function to enough arguments?)
* In a stmt of an interactive GHCi command: print it
The REPL prints the values of the expression it evaluated. Once functions are evaluated, their source cannot be printed. However, we can still ask GHCi to print the type of the expression by
preceeding it with a :t.
-- Haskell
> :t \x y -> x + y
(\x y -> x + y) :: Num a => a -> a -> a
This make an anonymous function that consumes two numbers and produces their sum.
To apply it, we can write it justapoxed with arguments:
-- Haskell
> (\x y -> x + y) 3 4
it :: Num a => a
Note that in Haskell, every function is a function of exactly one argument. Therefore \x y -> x + y is actuallty shorthand for \x -> \y -> x + y.
Applying such a function to fewer than 2 arguments will do a partial function application, which will produce a function that take the remaining arguments:
-- Haskell
> :t (\x y -> x + y) 3
(\x y -> x + y) 3 :: Num a => a -> a
To encode functions that must always be given two arguments, a tuple can be used:
-- Haskell
> :t \(x, y) -> x + y
\(x, y) -> x + y :: Num a => (a, a) -> a
To apply such a function, it must be given a pair of integers:
-- Haskell
> (\(x, y) -> x + y)(3, 4)
it :: Num a => a
The use of (x, y) here in the function parameters is actually a pattern. This can be understood as shorthand for:
-- Haskell
> :t \p -> case p of (x, y) -> x + y
\p -> case p of (x, y) -> x + y :: Num a => (a, a) -> a
So even this function is actually taking a single argument (which must be a pair of numbers).
Racket has a similar notation for writing functions:
;; Racket
> (λ (x) (λ (y) (+ x y)))
You can also write this without the fancy λ by spelling it lambda:
;; Racket
> (lambda (x) (lambda (y) (+ x y)))
(In DrRacket, to insert a λ press Cmd + )
To apply it, it must be written in parens, juxtaposed with arguments:
;; Racket
> (((λ (x) (λ (y) (+ x y))) 3) 4)
Functions in Racket do not always consume a single argument. They can consume 0, 1, or more arguments.
;; Racket
> (λ (x y) (+ x y))
This is not a shorthand for the function above it; rather it is a function that expects two arguments:
;; Racket
> ((λ (x y) (+ x y)) 3 4)
Applying a function to the wrong number of arguments will result in an error (and not perform partial function application):
;; Racket
> ((λ (x y) (+ x y)) 3)
arity mismatch;
the expected number of arguments does not match the given number
expected: 2
given: 1
In Haskell, variables can be defined with let:
> let x = 3
x :: Num a => a
> let y = 4
y :: Num a => a
> x + y
it :: Num a => a
> :{
let fact = \n -> case n of
0 -> 1
n -> n * (fact (n - 1))
fact :: (Eq a, Num a) => a -> a
> fact 5
it :: (Eq a, Num a) => a
The :{ and :} marks the start and end of a multi-line definition in the REPL.
In Racket, variables are defined with the define form:
> (define x 3)
> (define y 4)
> (+ x y)
> (define fact
(λ (n)
(match n
[0 1]
[n (* n (fact (- n 1)))])))
> (fact 5)
In Haskell, function definitions can be written as:
let fact n = case n of
0 -> 1
n -> n * (fact (n - 1))
This is just a shorthand for the definition written above in terms of \.
Similarly in Racket, function definitions can be written as:
> (define (fact n)
(match n
[0 1]
[n (* n (fact (- n 1)))]))
which is shorthand for the definition above using λ.
Notice both Haskell and Racket have pattern matching forms, which are quite useful for writing function in terms of a number of “cases.” More on this in a minute.
Haskell has a built-in list datatype. The empty list is written [] and : is an operation for “consing” an element on to a list. So to build a list with three integer elements, 1, 2, and 3, you’d
> 1 : 2 : 3 : []
it :: Num a => [a]
Racket has a built-in list datatype. The empty list is written '() and cons is an operation for consing an element on to a list. To build the same list, you’d write:
> (cons 1 (cons 2 (cons 3 '())))
'(1 2 3)
The notation (list 1 2 3) is shorthand for the above.
There is a slight difference here. For one, Haskell lists must be homogeneous. You can have a list of strings or a list of numbers, but you can’t have a list of strings and numbers.
> ["a", 3]
<interactive>:45:7: error:
* No instance for (Num [Char]) arising from the literal '3'
* In the expression: 3
In the expression: ["a", 3]
In an equation for 'it': it = ["a", 3]
In Racket, there is no such restriction:
Also, in Racket, cons plays the role of both tupling (making pairs) and making lists (making a pair of an element and another list).
So in Haskell, you could make a pair ("a", 3). In Racket, you’d write (cons "a" 3). Note this is a pair and not a proper list. In Haskell, tuples and lists are disjoint things. In Racket, lists and
tuples (pairs) are made out of the same stuff.
This can be confusing the first time you encounter it, so let’s go over it a bit more.
In Racket (or any Lisp), cons plays the role of both the pair constructor and the list constructor. Non-empty lists are a subset of pairs: they are pairs whose second component is a list (either the
empty list or another pair whose second component is a list, etc.).
You can make pairs out of any kind of element and you can make lists out of any kind of elements. We can precisely define these sets as:
;; type ListofAny =
;; | '()
;; | (cons Any ListofAny)
;; type PairofAny =
;; | (cons Any Any)
Or, to give more useful parameterized definitions:
;; type (Listof A) =
;; | '()
;; | (cons A (Listof A))
;; type (Pairof A B) =
;; | (cons A B)
The functions first and rest operate on non-empty lists, producing the first element of the list and the tail of the list, respectively.
> (first (cons 3 (cons 4 '())))
> (rest (cons 3 (cons 4 '())))
These function will produce errors if given something that is a pair but not a list:
> (first (cons 3 4))
first: contract violation
expected: (and/c list? (not/c empty?))
given: '(3 . 4)
> (rest (cons 3 4))
rest: contract violation
expected: (and/c list? (not/c empty?))
given: '(3 . 4)
On the other hand, the functions car and cdr access the left and right components of a pair (the names are admittedly awful and an artifact of Lisp history):
> (car (cons 3 4))
> (cdr (cons 3 4))
When given pairs that are also lists, they behave just like first and rest:
> (car (cons 3 (cons 4 '())))
> (cdr (cons 3 (cons 4 '())))
Pattern Matching
Haskell has a very nice pattern matching for letting you express case analysis and decomposition in a concise way.
Each pattern maching expression has a sub-expression that produce a value to be matched against and a number of clauses. Each clause has a pattern and an expression. The pattern potentially consists
of data constructors, variables, and literals. If the value matches the first pattern, meaning the value and the template match up on constructors and literals, then the variables are bound to the
correspond parts of the value, and the right-hand side expression is evaluated. If the value doesn’t match, the next pattern is tried, and so on. It’s an error if none of the patterns match.
So for example, we can write a function that recognize even digits as:
> :{
let evenDigit n = case n of 0 -> True
2 -> True
4 -> True
6 -> True
8 -> True
_ -> False
evenDigit :: (Eq a, Num a) => a -> Bool
The patterns here, save the last one, are just integer literals. If n is the same as any of these integers, the value True is produced. The last case uses a “wildcard,” which matches anything and
produces False.
Here’s an example that matches a tuple, binding each part of the tuple to a name and then using those names to construct a different tuple:
> let swap p = case p of (x, y) -> (y, x)
swap :: (b, a) -> (a, b)
Here the pattern uses a data constructor (the tuple constructor). It matches any value that is made with the same constructor.
Here is a recursive function for computing the sum of a list of numbers, defined with pattern matching:
> :{
let addNums xs = case xs of [] -> 0
x:xs -> x + (addNums xs)
addNums :: Num p => [p] -> p
> addNums [4,5,6]
it :: Num p => p
We can do the same in Racket:
> (define (even-digit n)
(match n
[0 #t]
[2 #t]
[4 #t]
[6 #t]
[8 #t]
[_ #f]))
> (define (swap p)
(match p
[(cons x y) (cons y x)]))
> (define (sum xs)
(match xs
['() 0]
[(cons x xs)
(+ x (sum xs))]))
> (sum (list 4 5 6))
Haskell has the ability to declare new datatypes. For example, we can define type for binary trees of numbers:
data BinaryTree = Leaf
| Node Integer BinaryTree BinaryTree
This declares a new type, named BinaryTree. There are two variants of the BinaryTree type, each with their own constructor: Leaf and Node. The Leaf constructor takes no arguments, so just writing
Leaf creates a (empty) binary tree:
> :t Leaf
Leaf :: BinaryTree
The Node constructor takes three arguments: an integer and two binary trees. Applying the constructor to a tuple of three things, makes a (non-empty) binary tree:
> :t Node 3 Leaf Leaf
Node 3 Leaf Leaf :: BinaryTree
Binary trees are an example of a recursive datatype, since one of the variants contains binary trees. This means we can build up arbitrarily large binary trees by nesting nodes within nodes:
> :t Node 3 (Node 4 Leaf Leaf) (Node 7 Leaf Leaf)
Node 3 (Node 4 Leaf Leaf) (Node 7 Leaf Leaf) :: BinaryTree
Pattern matching is used to do case analysis and deconstruct values. So for example, a function that determines if a binary tree is empty can be written as:
> :{
let btEmpty bt = case bt of Leaf -> True
Node _ _ _ -> False
btEmpty :: BinaryTree -> Bool
> btEmpty Leaf
it :: Bool
> btEmpty (Node 4 Leaf Leaf)
it :: Bool
The patterns use the constructor names to discriminate on which constructor was used for a given binary tree. The use of the wildcard here is just saying it doesn’t matter what’s inside a node; if
you’re a node, you’re not empty.
Recursive functions work similarly, but use variables inside patterns to bind names to the binary trees contained inside a node:
> :{
let btHeight bt = case bt of Leaf -> 0
Node _ l r -> 1 + (max (btHeight l) (btHeight r))
btHeight :: (Num p, Ord p) => BinaryTree -> p
> btHeight Leaf
it :: (Num p, Ord p) => p
> btHeight (Node 4 (Node 2 Leaf Leaf) Leaf)
it :: (Num p, Ord p) => p
We do something very similar in Racket using structures. A structure type is like a (single) variant of a data type in Haskell: it’s a way of combining several things into one new kind of value.
> (struct leaf ())
> (struct node (i left right))
This declares two new kinds of values: leaf structures and node structures. For each, we get a constructor, which is a function named after the structure type. The leaf constructor takes no
arguments. The node constructor takes 3 arguments.
> (leaf)
> (node 5 (leaf) (leaf))
(node 5 (leaf) (leaf))
> (node 3 (node 2 (leaf) (leaf)) (leaf))
(node 3 (node 2 (leaf) (leaf)) (leaf))
There is no type system in Racket, but we can conceptually still define what we mean in a comment. Just like in Haskell, we can use pattern matching to discriminate and deconstruct:
;; type BinaryTree = (leaf | (node Integer BinaryTree BinaryTree))
> (define (bt-empty? bt)
(match bt
[(leaf) #t]
[(node _ _ _) #f]))
> (bt-empty? (leaf))
> (bt-empty? (node 5 (leaf) (leaf)))
> (define (bt-height bt)
(match bt
[(leaf) 0]
[(node _ left right)
(+ 1 (max (bt-height left)
(bt-height right)))]))
> (bt-height (leaf))
> (bt-height (node 4 (node 2 (leaf) (leaf)) (leaf)))
One of the built-in datatypes we will use often in Racket is that of a symbol. A symbol is just an atomic peice of data. A symbol is written using the quote notation (quote symbol-name), which is
abbreviated 'symbol-name. What’s allowable as a symbol name follows the same rules as what’s allowable as a Racket identifier.
Symbols don’t have a whole lot of operations. The main thing you do with symbols is tell them apart from eachother:
> (equal? 'fred 'fred)
> (equal? 'fred 'wilma)
It is possible to convert between symbols and strings:
> (symbol->string 'fred)
> (string->symbol "fred")
There’s also a convient function that produces a symbol that is guaranteed to have not been used so far each time you call it:
> (gensym)
> (gensym)
> (gensym)
They can be used to define “enum” like datatypes:
; type Flintstone = 'fred | 'wilma | 'pebbles
You can use pattern matching to match symbols:
> (define (flintstone? x)
(match x
['fred #t]
['wilma #t]
['pebbles #t]
[_ #f]))
> (flintstone? 'fred)
> (flintstone? 'barney)
There’s really not a precise analog to symbols in Haskell.
Quote, quasiquote, and unquote
One of the distinguishing features of languages in the Lisp family (such as Scheme and Racket) is the quote operator and its closely related cousins quasiquote, unquote, and unquote-splicing.
Let’s start with quote.
The “tick” character 'd is used as a shorthand for (quote d).
You’ve already seen it show up with symbols: 'x is the symbol x. It also shows up in the notation for the empty list: '().
But you can also write quote around non-empty lists like '(x y z). This makes a list of symbols. It is equivalent to saying (list 'x 'y 'z).
In fact, you can nest lists within the quoted list: '((x) y (q r)). This is equivalent to (list (list 'x) 'y (list 'q 'r)).
Here’s another: '(() (()) ((()))). This is equivalent to
(list '() (list '()) (list (list '())))
So, anything you can write with quoted lists, you can write without quoted lists by pushing the quote inward until reaching a symbol or an empty set of parenthesis.
You can also put strings, booleans, and numbers inside of a quote. As you push the quote inward, it simply disappears when reaching a string, boolean or number. So '5 is just 5. Likewise '#t is #t
and '"Fred" is "Fred".
You can also write pairs with quote, which uses the . notation for separating the left and right part of the pair. For example, '(1 . 2) is equivalent to (cons 1 2). If you write something like '(1 2
3 . 4), what you are in effect saying is (cons 1 (cons 2 (cons 3 4))), an improper list that ends in 4.
In essence, quote is a shorthand for conveniently constructing data and is a very concise notation for writing down ad-hoc data. It serves much the same purpose as formats like JSON and XML, except
there’s even less noise.
To summarize, with quote, you can construct
• strings
• booleans
• numbers
• symbols
• and… pairs (or lists) of those things (including this one)
The kind of things you can construct with the quote form are often called s-expressions, short for symbolic expressions.
We can give a type definition for s-expressions:
; type S-Expr =
; | String
; | Boolean
; | Number
; | Symbol
; | (Listof S-Expr)
The reason for this name is because anything you can write down as an expression, you can write down inside a quote to obtain a data representation of that expression. You can render an expression as
a symbolic representation of itself.
For example, (+ 1 2) is an expression. When run, it applies the function bound to the variable + to the arguments 1 and 2 and produces 3. On the other hand: '(+ 1 2) constructs a peice of data,
namely, a list of three elements. The first element is the symbol +, the second element is 2, the third element is 3.
We will be using (subsets of) s-expressions extensively as our data representation of AST and IR expressions, so it’s important to gain a level of fluency with them now.
Once you understand quote, moving on to quasiquote, unquote, and unquote-splicing are pretty straight-forward.
Let’s start with quasiquote. The “backtick” character `d is used as a shorthand for (quasiquote d) and the “comma” character ,e is shorthand for (unquote e). The (quasiquote d) form means the same
thing as (quote d), with the exception that if (unquote e) appears anywhere inside d, then the expression e is evaluated and it’s value will be used in place of (unquote e).
This gives us the ability to “escape” out of a quoted peice of data and go back to expression mode.
If we think of quasiquote like quote in terms of “pushing in” then the rules are exactly the same except that when a quasiquote is pushed up next to an unquote, the two “cancel out.” So `,e is just
For example, `(+ 1 ,(+ 1 1)) is equivalent to (list '+ 1 (+ 1 1)), which is equivalent to (list '+ 1 2).
So if quote signals us to stop interpreting things as expressions, but instead as data, quasiquote signals us to stop interpreting things as expression, but instead as data.. unless we encounter a
unquote, in which case you go back to interpreting things as expressions.
The last remaining peice is unquote-splicing, which is abbreviated with “comma-at”: ,@e means (unquote-splicing e). The unquote-splicing form is like unquote in that if it occurs within a quasiquote,
it means we switch back in to expression mode. The difference is the expression must produce a list (or pair) and the elements of that list (or pair) are spliced in to the outer data.
So for example, `(+ 1 ,@(map add1 '(2 3))) is equivalent to (cons '+ (cons 1 (map add1 (list 2 3)))), which is equivalent to (list '+ 1 3 4), or '(+ 1 3 4).
If the expression inside the unquote-splicing produces something other than a pair, an error is signalled.
Poetry of s-expressions
The use of structures lets us program in a style very similar to idiomatic Haskell programming. For each variant data type, we can define a structure type for each variant and use pattern matching to
process such values.
However, we are going to frequently employ a different idiom for programming with recursive variants which doesn’t rely on structures, but rather uses symbols in place of constructors and lists in
place of fields.
Let’s revisit the binary tree example, using this style.
Notice that leaf structure is a kind of atomic data. It doesn’t contain anything and its only real purpose is to be distinguishable from node structures. On the other hand a node structure needs to
be distinguishable from leafs, but also contain 3 peices of data within it.
We can formulate definition of binary trees using only symbols and lists as:
;; type BinaryTree = 'leaf | (list 'node Integer BinaryTree BinaryTree)
So the following are binary trees:
> 'leaf
> (list 'node 3 'leaf 'leaf)
'(node 3 leaf leaf)
> (list 'node 3
(list 'node 7 'leaf 'leaf)
(list 'node 9 'leaf 'leaf))
'(node 3 (node 7 leaf leaf) (node 9 leaf leaf))
This formulation has the added benefit that we write binary trees as s-expressions:
> 'leaf
> '(node 3 leaf leaf)
'(node 3 leaf leaf)
> '(node 3
(node 7 leaf leaf)
(node 9 leaf leaf))
'(node 3 (node 7 leaf leaf) (node 9 leaf leaf))
We re-write our functions to match this new datatype definition:
> (define (bt-empty? bt)
(match bt
['leaf #t]
[(cons 'node _) #f]))
> (bt-empty? 'leaf)
> (bt-empty? '(node 3
(node 7 leaf leaf)
(node 9 leaf leaf)))
> (define (bt-height bt)
(match bt
['leaf 0]
[(list 'node _ left right)
(+ 1 (max (bt-height left)
(bt-height right)))]))
> (bt-height 'leaf)
> (bt-height '(node 3
(node 7 leaf leaf)
(node 9 leaf leaf)))
We even can use quasiquote notation in patterns to write more concise definitions:
> (define (bt-empty? bt)
(match bt
[`leaf #t]
[`(node . ,_) #f]))
> (bt-empty? 'leaf)
> (bt-empty? '(node 3
(node 7 leaf leaf)
(node 9 leaf leaf)))
> (define (bt-height bt)
(match bt
[`leaf 0]
[`(node ,_ ,left ,right)
(+ 1 (max (bt-height left)
(bt-height right)))]))
> (bt-height 'leaf)
> (bt-height '(node 3
(node 7 leaf leaf)
(node 9 leaf leaf)))
Moreover, we can embrace quasiquotation at the type-level and write:
; type BinaryTree = `leaf | `(node ,Integer ,BinaryTree ,BinaryTree)
Testing, modules, submodules
We will take testing seriously in this class. Primarily this will take the form of unit tests, for which we will use the rackunit library. To use the library, you must require it.
Here is a simple example:
> (require rackunit)
> (check-equal? (add1 4) 5)
> (check-equal? (* 2 3) 7)
name: check-equal?
location: eval:76:0
actual: 6
expected: 7
The check-equal? function takes two arguments (and an optional third for a message to display should the test fail) and checks that the first argument produces something that is equal? to the
expected outcome given as the second argument.
There are many other forms of checks and utilities for building up larger test suites, but check-equal? will get us a long way.
As a matter of coding style, we will place tests nearby the function they are testing and locate them within their own module. Let’s talk about modules for a minute.
In Racket, a module is the basic unit of code organization. Every file is a module whose name is derived from the filename, but you can also write modules without saving them in a file. For example:
> (module bt racket
(provide bt-height)
(define (bt-height bt)
(match bt
[`leaf 0]
[`(node ,_ ,left ,right)
(+ 1 (max (bt-height left)
(bt-height right)))])))
This declares a module named bt. It provides a single value named bt-height.
We can require the module from the REPL to gain access to the modules provided values:
> (require 'bt)
> (bt-height 'leaf)
We could have also used the #lang racket shorthand for (module bt racket ...) and saved this in a file called bt.rkt. To import from a file in the current directory, you’d write (require "bt.rkt").
But this doesn’t work well in REPL.
For the most part we will organize our programs into single module files using the #lang racket shorthand. But we will place tests within a “sub”-module, i.e. a module nested inside of the module
that contains the code it tests. We will use a special form called module+ which declares a submodule that has access to the enclosing module. Moreover, repeated uses of module+ will add content to
the submodule. By convention, we will name the testing submodule test.
So here’s a second version of the bt module with unit tests included (and more code). Note the use of all-defined-out to provide everything:
> (module bt2 racket
; provides everything defined in module
(provide (all-defined-out))
(module+ test
(require rackunit))
(define (bt-empty? bt)
(match bt
['leaf #t]
[(cons 'node _) #f]))
(module+ test
(check-equal? (bt-empty? 'leaf) #t)
(check-equal? (bt-empty? '(node 3
(node 7 leaf leaf)
(node 9 leaf leaf)))
(define (bt-height bt)
(match bt
[`leaf 0]
[`(node ,_ ,left ,right)
(+ 1 (max (bt-height left)
(bt-height right)))]))
(module+ test
(check-equal? (bt-height 'leaf) 0)
; intentionally wrong test:
(check-equal? (bt-height '(node 3 leaf leaf)) 2)))
Requiring this module with make bt-height, but it will not run the tests:
Running the tests only happens when the test submodule is required:
> (require (submod 'bt2 test))
name: check-equal?
location: eval:80:0
actual: 1
expected: 2
Putting it all together, we can write the following code and save it in a file called bt.rkt. (You can click the tiny clipboard icon on top-right to copy it.)
#lang racket
(provide (all-defined-out))
(module+ test
(require rackunit))
;; type Bt =
;; | `leaf
;; | `(node ,Integer ,Bt ,Bt)
;; Bt -> Boolean
;; Is the binary tree empty?
(define (bt-empty? bt)
(match bt
['leaf #t]
[(cons 'node _) #f]))
(module+ test
(check-equal? (bt-empty? 'leaf) #t)
(check-equal? (bt-empty? '(node 3
(node 7 leaf leaf)
(node 9 leaf leaf)))
;; Bt -> Natural
;; Compute the height of a binary tree
(define (bt-height bt)
(match bt
[`leaf 0]
[`(node ,_ ,left ,right)
(+ 1 (max (bt-height left)
(bt-height right)))]))
(module+ test
(check-equal? (bt-height 'leaf) 0)
(check-equal? (bt-height '(node 3 leaf leaf)) 1)
(check-equal? (bt-height '(node 2 leaf (node 1 leaf leaf)))
This code follows a coding style that we will use in this course:
• it’s organized in a module,
• data type definitions occur at the top of the file,
• it uses a test submodule to group unit tests,
• tests occur immediately after the functions they test,
• functions are annotated with type signatures and short purpose statements, and
• indentation follows standard conventions (which DrRacket can apply for you).
From the command line, you can run a module’s tests using the Racket command line testing tool raco test:
$ raco test bt.rkt
raco test: (submod "bt.rkt" test)
5 tests passed
Or simply give a directory name and test everything within that directory:
$ raco test .
raco test: (submod "./bt.rkt" test)
5 tests passed
These notes are adapted from CMSC430 at UMD. | {"url":"https://sankhs.com/eecs662/notes/02-hs2rkt/","timestamp":"2024-11-09T20:21:39Z","content_type":"text/html","content_length":"120658","record_id":"<urn:uuid:c8e9cd9a-d6b4-417d-a4e2-efed4c461909>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00677.warc.gz"} |
direct: DIviding RECTangles Algorithm for Global Optimization in nloptr: R Interface to NLopt
DIRECT is a deterministic search algorithm based on systematic division of the search domain into smaller and smaller hyperrectangles. The DIRECT_L makes the algorithm more biased towards local
search (more efficient for functions without too many minima).
direct( fn, lower, upper, scaled = TRUE, original = FALSE, nl.info = FALSE, control = list(), ... ) directL( fn, lower, upper, randomized = FALSE, original = FALSE, nl.info = FALSE, control = list(),
... )
fn objective function that is to be minimized.
lower, upper lower and upper bound constraints.
scaled logical; shall the hypercube be scaled before starting.
original logical; whether to use the original implementation by Gablonsky – the performance is mostly similar.
nl.info logical; shall the original NLopt info been shown.
control list of options, see nl.opts for help.
... additional arguments passed to the function.
randomized logical; shall some randomization be used to decide which dimension to halve next in the case of near-ties.
logical; whether to use the original implementation by Gablonsky – the performance is mostly similar.
logical; shall some randomization be used to decide which dimension to halve next in the case of near-ties.
The DIRECT and DIRECT-L algorithms start by rescaling the bound constraints to a hypercube, which gives all dimensions equal weight in the search procedure. If your dimensions do not have equal
weight, e.g. if you have a “long and skinny” search space and your function varies at about the same speed in all directions, it may be better to use unscaled variant of the DIRECT algorithm.
The algorithms only handle finite bound constraints which must be provided. The original versions may include some support for arbitrary nonlinear inequality, but this has not been tested.
The original versions do not have randomized or unscaled variants, so these options will be disregarded for these versions.
par the optimal solution found so far.
value the function value corresponding to par.
iter number of (outer) iterations, see maxeval.
convergence integer code indicating successful completion (> 0) or a possible error number (< 0).
message character string produced by NLopt and giving additional information.
integer code indicating successful completion (> 0) or a possible error number (< 0).
D. R. Jones, C. D. Perttunen, and B. E. Stuckmann, “Lipschitzian optimization without the Lipschitz constant,” J. Optimization Theory and Applications, vol. 79, p. 157 (1993).
J. M. Gablonsky and C. T. Kelley, “A locally-biased form of the DIRECT algorithm," J. Global Optimization, vol. 21 (1), p. 27-37 (2001).
The dfoptim package will provide a pure R version of this algorithm.
### Minimize the Hartmann6 function hartmann6 <- function(x) { a <- c(1.0, 1.2, 3.0, 3.2) A <- matrix(c(10.0, 0.05, 3.0, 17.0, 3.0, 10.0, 3.5, 8.0, 17.0, 17.0, 1.7, 0.05, 3.5, 0.1, 10.0, 10.0, 1.7,
8.0, 17.0, 0.1, 8.0, 14.0, 8.0, 14.0), nrow=4, ncol=6) B <- matrix(c(.1312,.2329,.2348,.4047, .1696,.4135,.1451,.8828, .5569,.8307,.3522,.8732, .0124,.3736,.2883,.5743, .8283,.1004,.3047,.1091,
.5886,.9991,.6650,.0381), nrow=4, ncol=6) fun <- 0 for (i in 1:4) { fun <- fun - a[i] * exp(-sum(A[i,] * (x - B[i,]) ^ 2)) } fun } S <- directL(hartmann6, rep(0, 6), rep(1, 6), nl.info = TRUE,
control = list(xtol_rel = 1e-8, maxeval = 1000)) ## Number of Iterations....: 1000 ## Termination conditions: stopval: -Inf ## xtol_rel: 1e-08, maxeval: 1000, ftol_rel: 0, ftol_abs: 0 ## Number of
inequality constraints: 0 ## Number of equality constraints: 0 ## Current value of objective function: -3.32236800687327 ## Current value of controls: ## 0.2016884 0.1500025 0.4768667 0.2753391
0.311648 0.6572931
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/nloptr/man/direct.html","timestamp":"2024-11-06T20:40:04Z","content_type":"text/html","content_length":"30566","record_id":"<urn:uuid:436f7dda-9e34-4a5c-8e46-35cc01598d4f>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00627.warc.gz"} |
Divide and conquer - ASCR Discovery
Imagine that the only way to appreciate the beauty of a Ming vase was to shatter it into a thousand pieces, examine the shape and color of each piece, then glue it back together – here’s the hard
part – so it’s impossible to detect the seams.
This is much like the challenge that faced physicist Lin-Wang Wang and his colleagues at Lawrence Berkeley National Laboratory when they set out to develop physics algorithms to model the
nanomaterials equivalent of a Ming vase. The next generation of materials for solar cells and a wide range of electronic components will depend on such solutions.
Wang and his LBNL team – Byounghak Lee, Hongzhang Shan, Zhengji Zhao, Juan Meza, Erich Strohmaier and David Bailey – needed new algorithms to simulate the electronic structure of semiconductor
nanomaterials. It is a class of systems so intricate and complex that physicists continue to puzzle over the materials’ unique properties. Because nanomaterials are so tiny – approaching electron
wavelengths – they behave very differently from the same material in bulk forms.
Those who study nanostructures are interested mainly in the location and energy level of electrons in the system – information that determines a material’s properties. Unlike uniform systems such as
graphite and diamond, Wang’s structures can’t be represented by just a few atoms. Because it is a coordinated system, any attempt to understand the materials’ properties must simulate the system as a
whole, Wang says.
A reliable method called density functional theory (DFT) allows physicists to simulate the electronic properties of materials. But DFT calculations are time-consuming and any system with more than
1,000 atoms quickly overwhelms computing resources. The challenge for Wang became to find a way of retaining DFT’s accuracy while performing calculations with tens of thousands of atoms.
Petascale computing, which can distribute calculations over thousands of processor cores, presented a new way to solve large systems like nanostructure calculations. But even at the petascale,
calculating a system with tens of thousands of atoms will be extremely challenging because the computational cost of the conventional DFT method scales as the third power of the size of the system.
Thus, when a nanostructure size increases 10 times, computing power must increase 1,000 times. Making up that power gap would take the hardware community more than a decade of development. Wang had
to find a way to make computational cost scale linearly to the size of nanostructure systems. He settled on a method he calls “divide and conquer,” and it worked like a charm.
It fact, it worked so well that the LBNL team recently won the coveted Gordon Bell Prize, sponsored by the Association for Computing Machinery (ACM), for special achievement in high-performance
“The linear scaling method suddenly makes a big class of problems amenable to simulation,” Wang says.
Oxygen atoms stud the material like raisins in a bowl of oatmeal.
Wang calls his set of new algorithms LS3DF, for linear scaling three-dimensional fragment method. The method’s key lies in the nifty mathematical sleight of hand Wang devised to eliminate evidence of
the numerical seams created when the material is divided up and then reassembled.
The program divides the system into numerical squares. It then subdivides the squares into even smaller overlapping squares and rectangles. Once divided, the problem can be easily split up among the
thousands of processors available on petascale computing systems. After the calculations are complete the program reassembles the pieces, then performs Wang’s critical step.
“If you do this cleverly, and we have a formula to do this, then the surface of the larger pieces cancels out the surface of the smaller pieces so there will be one core piece left, which just
corresponds to the interior of the original system,” Wang says.
This core piece, with artificial seams removed, provides crucial information about the system’s quantum mechanical state, which can then be used to complete the total energy calculation. The program
then uses this information to calculate the electrostatic energy of whole system, a much less resource-intensive calculation. The two energy results combine to generate the structure’s total energy
and potential.
Being able to remove the artifacts created by dividing up the problem allows large complex systems to be treated like a series of small problems, which is ideal for distributed petascale computing.
The mathematical breakthrough was only the first step to achieving efficient scaling on massively parallel computing systems. Bailey, Shan and Strohmaier of the DOE Office of Science’s Scientific
Discovery through Advanced Computing (SciDAC) Performance Engineering Research Institute (PERI) analyzed the algorithms to identify potential performance improvements.
The analysis revealed major roadblocks to achieving peak performance on parallel machines. Many of the programming glitches dealt with how four subroutines communicate and share information. For
example, the computational team suggested storing data in memory, rather than storing data on disk, which requires a much more cumbersome and time-consuming method of communication. This change
greatly improved scalability and overall performance.
The team performed a series of small test simulations to compare LS3DF’s results to the best DFT programs, such as PARATEC and PEtot. The algorithm produced results that approach the accuracy of DFT.
In addition, based on how long it takes PEtot to complete the simulation, the team calculated that LS3DF would be faster with any system larger than 600 atoms.
Armed with a program optimized to parallel computing, Wang and his colleagues chose a real-world problem for a proof of concept. They used LS3DF to test whether a proposed new solar cell material
could generate enough electricity to be cost effective. The zinc-telluride-oxygen alloy, first proposed by LBNL, is an oxide that could potentially be less expensive to produce than current solar
cell materials. The catch is that similar zinc-telluride semiconductors have proven too inefficient to be practical for solar cells.
The new material included 3 percent oxygen to provide what the materials scientists hope will be a boost to its energy efficiency. The oxygen acts like a stepping stone, allowing electrons excited by
solar photons to span an expanse, or “band gap” in the middle of the zinc-telluride, that would otherwise be too wide for many of them to jump. This extra step could increase the material’s energy
efficiency from 30 percent to 60 percent.
The oxygen atoms stud the material like raisins in a bowl of oatmeal, but because there are so few of them and they are randomly scattered, simulating a small portion would miss the raisins. They had
to simulate the whole bowl with tens of thousands of atoms. That’s where LS3DF came in.
The team ran the model on Franklin, the 36,864-core Cray XT4 at LBNL’s National Energy Research Scientific Computing Center, achieving 135 teraflops. The initial model, which simulated a 13,824-atom
zinc-telluride-oxygen alloy, ran 400 times faster than a similar direct DFT calculation, assuming it were even possible to run such a large problem using the more cumbersome algorithm. In reality,
Wang estimated, it would have taken four to six weeks to achieve the same result as LS3DF did in two to three hours.
The initial simulation demonstrated that adding oxygen did in fact provide the atomic stepping stone the researchers hoped it would but with slightly lower efficiency than the best theoretical
“We found that if the percentage of oxygen is small – about 3 percent – there are separate states in the middle of the band gap,” Wang says. “So that indicates this material might be able to be used
to generate solar energy.”
Since then, the researchers have run the same problem to test LS3DF’s efficiency on Intrepid, the IBM Blue Gene/P at Argonne National Laboratory. The code achieved 224 teraflops on 163,840 cores – 40
percent of the machine’s peak speed. In a later run on Jaguar, Oak Ridge National Laboratory’s Cray XT5, the code achieved 442 teraflops on 147,456 cores – 33 percent of peak. The ease of moving the
program to a new computer system was good news for the research team because they had recently been granted time on Jaguar under DOE’s Innovative and Novel Computational Impact on Theory and
Experiment (INCITE) program.
That grant will allow Wang and his team to explore the nature and behavior of new solar cell materials and other nanostructures as never before.
Wang reels off a few of the big questions. “Within an electron field, how do the electrons move? Are they trapped by the surface states or do they move more freely? How do they couple with the photon
energy? How does the vibration of the atoms change the electron transport in a nanosystem? All of these questions are not well understood.”
Trial and error can’t address the dynamics of these complex nanostructure systems, Wang says. “To reach that understanding, simulation does play and will continue to play a very important role.”
Understanding the electronic properties of new semiconductor materials may allow researchers to predict how a new nanostructure will behave before actually spending the time and money to make the
material. In the case of solar cells, the ability of semiconductor nanomaterials to generate and carry electricity could ease dependence on fossil fuels and generate clean energy, Wang says.
He also is interested in using LS3DF to study the properties of quantum rods and dots. Scientists are already using quantum dots to track the movements of molecules inside living cells. Physicians
are enlisting the dots to track tumors, assisting in diagnosis and treatment of cancers. Quantum rods also could be used to create nano-sized electronic devices, and semiconductor materials can
transport electricity.
The ability to predict the behavior of new nanomaterials, Wang says, will speed the development of new materials and applications that we can now only dream about. | {"url":"https://ascr-discovery.org/2009/01/divide-and-conquer/amp/","timestamp":"2024-11-14T04:47:49Z","content_type":"text/html","content_length":"116937","record_id":"<urn:uuid:dd6d30c9-ee38-43e9-a9a9-f1b644a00460>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00638.warc.gz"} |
Lesson 13
The Volume of a Cylinder
13.1: A Circle's Dimensions (10 minutes)
The purpose of this warm-up is for students to review how to compute the area of a circle. This idea should have been carefully developed in grade 7. This warm-up gives students an opportunity to
revisit this idea in preparation for finding the volume of a cylinder later in the lesson.
Students begin the activity with a whole-class discussion in which they identify important features of a circle including its radius and diameter. They use this information and the formula for the
area of the circle to choose expressions from a list that are equivalent to the area of the circle. In the final question, students are given the area of the circle and find the corresponding
As students are working, monitor for students who can explain why \(16\pi\), \(\pi 4^2\), and “approximately 50” square units represent the area of the circle.
Display the diagram from the task statement for all to see and ask students:
• “Name a segment that is a radius of circle A.” (AC, AD, and AB or those with the letters reversed are all radii.) Review the meaning of the radius of a circle.
• “What do we call a segment like BC, with endpoints on the circle that contains the center of the circle?” (A diameter.) Review the meaning of a diameter of a circle.
• “What is the length of segment \(AB\)?” (4 units.) Review the fact that all radii of a circle have the same length.
Give students 3 minutes of quiet work time and follow with a whole-class discussion.
Student Facing
Here is a circle. Points \(A\), \(B\), \(C\), and \(D\) are drawn, as well as Segments \(AD\) and \(BC\).
1. What is the area of the circle, in square units? Select all that apply.
1. \(4\pi\)
2. \(\pi 8\)
3. \(16\pi\)
4. \(\pi 4^2\)
5. approximately 25
6. approximately 50
2. If the area of a circle is \(49\pi\) square units, what is its radius? Explain your reasoning.
Anticipated Misconceptions
If students struggle to recall how to find the area of a circle, encourage them to look up a formula using any available resources.
Activity Synthesis
The purpose of this discussion is to make sure students remember that the area of a circle can be found by squaring its radius and multiplying by \(\pi\).
Select previously identified students to share answers to the first question and explain why each of the solutions represents the area of the circle. If not brought up during the discussion, tell
students that sometimes it is better to express an area measurement in terms of \(\pi\). Other times it may be better to use an approximation of \(\pi\), like 3.14, to represent the area measurement
in decimal form. In this unit, we will often express our answers in terms of \(\pi\).
Display in a prominent place for all to see for the next several lessons: Let \(A\) be the area of a circle of radius \(r\), then \(A=\pi r^2\).
13.2: Circular Volumes (15 minutes)
The purpose of this activity is for students to connect their previous knowledge of the volume of rectangular prisms to their understanding of the volume of cylinders. From previous work, students
should know that the volume of rectangular prisms is found by multiplying the area of the base by the height. Here we expand upon that to compute the volume of a cylinder.
Students start by calculating the volume of a rectangular prism. Then they extrapolate from that to calculate the volume of a cylinder given the area of its base and its height. If some
students don’t know they should multiply the area of the base by its height, then they are prompted to connect prisms and cylinders to make a reasonable guess.
We want students to conjecture that the volume of a cylinder is the area of its base multiplied by its height.
Arrange students in groups of 2. Remind students that a rectangular prism has a base that is a rectangle, and a cylinder has a base that is a circle. It may have been some time since students have
thought about the meaning of a result of a volume computation. Consider showing students a rectangular prism built from 48 snap cubes with the same dimensions as Figure A. It may help them to see
that one layer is made of 16 cubes. Give students 3–5 minutes of quiet work time followed by a partner discussion. During their discussion, partners compare the volumes they found for the cylinders.
If they guessed the volumes, partners explain their reasoning to one another. Follow with a whole-class discussion.
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide connecting cubes for students to create rectangular
prisms and calculate the volume of. Ask students to represent the volume of the rectangular prism in an equation and connect the base, width, and height to the 3-D model.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
What is the volume of each figure, in cubic units? Even if you aren’t sure, make a reasonable guess.
1. Figure A: A rectangular prism whose base has an area of 16 square units and whose height is 3 units.
2. Figure B: A cylinder whose base has an area of 16\(\pi\) square units and whose height is 1 unit.
3. Figure C: A cylinder whose base has an area of 16\(\pi\) square units and whose height is 3 units.
Student Facing
Are you ready for more?
│prism │prism │prism │cylinder │
│base: square│base: hexagon │base: octagon │base: circle│
Here are solids that are related by a common measurement. In each of these solids, the distance from the center of the base to the furthest edge of the base is 1 unit, and the height of the solid is
5 units. Use 3.14 as an approximation for \(\pi\) to solve these problems.
1. Find the area of the square base and the circular base.
2. Use these areas to compute the volumes of the rectangular prism and the cylinder. How do they compare?
3. Without doing any calculations, list the figures from smallest to largest by volume. Use the images and your knowledge of polygons to explain your reasoning.
4. The area of the hexagon is approximately 2.6 square units, and the area of the octagon is approximately 2.83 square units. Use these areas to compute the volumes of the prisms with the hexagon
and octagon bases. How does this match your explanation to the previous question?
Activity Synthesis
Highlight the important features of cylinders and their definitions: the radius of the cylinder is the radius of the circle that forms its base; the height of a cylinder is the length between its
circular top and bottom; a cylinder of height 1 can be thought of as a “layer” in a cylinder with height \(h\). To highlight the connection between finding the area of a rectangular prism and finding
the area of a cylinder, ask:
• “How are prisms and cylinders different?” (A prism has a base that is a polygon, and a cylinder has a base that is a circle.)
• “How are prisms and cylinders the same?” (The volume of cylinders and prisms is found by multiplying the area of the base by the height. \(V=Bh\))
• “How do you find the area of the base, \(B\), of a cylinder?” (\(B=\pi r^2\)).
Writing, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise a response to the question, “What is similar and different about
cylinders and rectangular prisms?” Ask each student to meet with 2–3 other partners in a row for feedback. Provide student with prompts for feedback that will help students strengthen their ideas and
clarify their language (e.g., “Can you give an example?”, “Why do you think that?”, “Can you say more about ...?”, etc.). Students can borrow ideas and language from each partner to strengthen their
final explanation.
Design Principle(s): Optimize output (for explanation)
13.3: A Cylinder's Dimensions (10 minutes)
Optional activity
In this optional activity, students use colored pencils (or pens or highlighters) to label the radius and height on different pictures of cylinders. Then they sketch their own cylinders and label the
radius and heights of those. The purpose of this activity is for students to practice identifying the radius and height of various cylinders, some of which are in context.
This activity can also be abbreviated if students demonstrate prior understanding of how to draw or label cylinders and only need a brief refresh.
Distribute colored pencils. Give students 1–2 minutes of quiet work time followed by a whole-class discussion.
Student Facing
1. For cylinders A–D, sketch a radius and the height. Label the radius with an \(r\) and the height with an \(h\).
2. Earlier you learned how to sketch a cylinder. Sketch cylinders for E and F and label each one’s radius and height.
Anticipated Misconceptions
In diagrams D and E, some students might mistake the diameter for the height of the cylinder. Some students may mark a height taller than the cylinder due to the tilt of the figure.
Activity Synthesis
Select students to share where they marked the radius and height and the cylinders sketched for images E and F. Discuss examples of other cylinders students see in real life.
Speaking, Listening: MLR3 Clarify, Critique, Correct. Present an incorrect drawing that reflects a possible misunderstanding from the class about the height of a cylinder. For example, draw a
cylinder with the diameter incorrectly placed at its height and the height at its diameter. Prompt students to identify the error, (e.g., ask, “Do you agree with the representation? Why or why not?
”), correct the representation, and write feedback to the author explaining the error. This helps students develop an understanding and make sense of identifying the heights of differently-oriented
Design Principle(s): Support sense-making; Maximize meta-awareness
13.4: A Cylinder's Volume (10 minutes)
The purpose of this activity is to give students opportunities to find the volumes of some cylinders. By finding the area of the base before finding the volume, students are encouraged to compute the
volume by multiplying the area of its base by its height. This way of thinking about volume might be more intuitive for students than the formula \(V=\pi r^2 h\). Notice students who plug the radius
into a formula for the volume and students who find the area of the base and multiply that by the cylinder’s height.
The second problem of this activity focuses on exploring cylinders in a context. Generally, the volume of a container is the amount of space inside, but in this context, that also signifies the
amount of material that fits into the space. Students are not asked to find the area of the base of the silo. Notice students that find the area before computing the volume and those that use the
volume formula and solve for it directly. Notice at what step in the computations students approximate \(\pi\).
When working with problems in a given context, it is sometimes convenient or practical to use an approximation of \(\pi\). An example of this is given in the question regarding the volume of a grain
silo and interpretations of the answer.
Provide access to colored pencils to shade the cylinder’s base. Give students 5–6 minutes of quiet work time followed by a whole-class discussion.
Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
1. Here is a cylinder with height 4 units and diameter 10 units.
1. Shade the cylinder’s base.
2. What is the area of the cylinder’s base? Express your answer in terms of \(\pi\).
3. What is the volume of this cylinder? Express your answer in terms of \(\pi\).
2. A silo is a cylindrical container that is used on farms to hold large amounts of goods, such as grain. On a particular farm, a silo has a height of 18 feet and diameter of 6 feet. Make a sketch
of this silo and label its height and radius. How many cubic feet of grain can this silo hold? Use 3.14 as an approximation for \(\pi\).
Student Facing
Are you ready for more?
One way to construct a cylinder is to take a rectangle (for example, a piece of paper), curl two opposite edges together, and glue them in place.
Which would give the cylinder with the greater volume: Gluing the two dashed edges together, or gluing the two solid edges together?
Anticipated Misconceptions
Students might use the silo’s diameter instead of the radius to find the volume. Remind them of the formula for area of a circle and the discussion about diameter and radius earlier in this lesson.
Activity Synthesis
The goal of this discussion is to ensure students understand how to use the area of the cylinder’s base to calculate its volume. Consider asking the following questions:
• “How does knowing the area of a circular base help determine the volume of a cylinder?” (The volume is this area multiplied by the height of the cylinder.)
• “If the cylinder were on its side, how do you know which measurements to use for the volume?” (Since we need area of the base first, the radius or diameter of the circle will always be the
measurement used for \(r\), and the height is always the distance between the bases, or the measurement perpendicular to the bases. It doesn’t matter which direction the cylinder is turned.)
• “When is it better to use approximations of pi instead of leaving it exact?” (Approximating pi helps us interpret an answer that has pi as a factor, like the area of a circular region or the
volume of a cylindrical container.)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each response that is shared, ask students to restate and/or revoice what they heard using mathematical
language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately
able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement.This will help students produce and make sense of the language needed to
communicate about different strategies for calculating volume of cylinders.
Design Principle(s): Support sense-making; Optimize output (for explanation)
Lesson Synthesis
Make a display that includes the formula for a cylinder’s volume, \(V=B h\), along with a labeled diagram of a cylinder. This display should be kept posted in the classroom for the remaining lessons
within this unit.
Previously, students computed the volume of prisms by multiplying the area of the base by the prism’s height. To help students summarize the ideas in this lesson, ask “How is finding the volume of a
cylinder like finding the volume of a prism?” and give students 2–3 minutes to write a response. Encourage students to use diagrams to show their thinking. Invite students to share their ideas,
displaying any diagrams for all to see.
13.5: Cool-down - Liquid Volume (5 minutes)
Student Facing
We can find the volume of a cylinder with radius \(r\) and height \(h\) using two ideas we've seen before:
• The volume of a rectangular prism is a result of multiplying the area of its base by its height.
• The base of the cylinder is a circle with radius \(r\), so the base area is \(\pi r^2\).
Remember that \(\pi\) is the number we get when we divide the circumference of any circle by its diameter. The value of \(\pi\) is approximately 3.14.
Just like a rectangular prism, the volume of a cylinder is the area of the base times the height. For example, take a cylinder whose radius is 2 cm and whose height is 5 cm.
The base has an area of \(4\pi\) cm^2 (since \(\pi\boldcdot 2^2=4\pi\)), so the volume is \(20\pi\) cm^3 (since \(4\pi \boldcdot 5 = 20\pi\)). Using 3.14 as an approximation for \(\pi\), we can say
that the volume of the cylinder is approximately 62.8 cm^3.
In general, the base of a cylinder with radius \(r\) units has area \(\pi r^2\) square units. If the height is \(h\) units, then the volume \(V\) in cubic units is \(\displaystyle V=\pi r^2h\) | {"url":"https://curriculum.illustrativemathematics.org/MS/teachers/3/5/13/index.html","timestamp":"2024-11-07T21:57:26Z","content_type":"text/html","content_length":"123284","record_id":"<urn:uuid:2aa6d89b-ed62-42b2-bbd1-c7f2695688a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00291.warc.gz"} |
What helps to research the mobile advertising market?
How to understand whether mobile advertising is effective: what you need to monitor February 27, 2023
What indicators affect the effectiveness of mobile advertising
The mobile advertising market occupies a very important niche along with the mobile app market. As they develop, so does mobile marketing. To assess its effectiveness, there are many metrics that
help determine engagement, conversion rates, and other criteria for the effectiveness of mobile advertising.
It is essential to determine the percentage of detected and blocked fraudulent app downloads in relation to their total number. The formula to help calculate this metric is expressed as: the number
of fraudulent installations refers to the total number of inorganic installations.
A metric such as ARPU, or average user revenue, demonstrates how much income an average user brings in on average. It takes into account data on impressions and interactions with ads, subscriptions,
and paid downloads. The formula for calculating the metric involves the ratio of revenue for the period to the number of users who have interacted with the advertising message over the specified time
The average revenue per paying user (ARPPU) is a metric that shows the approximate revenue received from one paying user over a certain period. To calculate ARPPU, you need to divide the revenue
generated by the application by the total number of users who pay internally.
The average session is also an important metric. It is a metric that allows determining the average time people spend in the app per session. With the help of this metric, it is possible to determine
how many users are engaged and how many are less active. To calculate the metric, you need to divide the number of sessions by the number of active users.
To determine the profitability of a particular advertising campaign, use the Return on Ad Spend (ROAS) metric by dividing revenue from users by the amount of money spent on marketing.
The Click to Install (CTI) metric allows you to determine the ratio of users who clicked on an ad and took the targeted action of installing a mobile app after viewing it. To calculate the index you
need to divide the total number of downloads by the number of clicks.
CPI metric clearly shows the price that the advertiser pays for the installation of the application. To obtain the value, you need to divide the advertising costs by the number of installations for a
certain period of advertising campaigns.
CPA shows how much a target user action costs the company. Another useful metric for marketers is CPM – the actual cost per thousand impressions. It is used to determine the effectiveness of an
advertising campaign in terms of coverage of the target audience.
Lifetime Value (LTV) determines the revenue per user who installed the app over a certain period. The formula by which LTV is determined is expressed as revenue generated since a certain installation
date/number of users who installed the app on that date.
Cost per click (PPC) shows the cost per click on an ad. The formula for calculating this metric is ad spend/number of clicks.
Retention Rate (RR) measures how many people started using the app again in a given period of time. The formula for RR is the number of active users during the date interval after installation/the
number of people who launched the app for the first time during the same interval.
A metric such as repeat purchase frequency (RPR) shows how many people have made multiple purchases in more than one session, which demonstrates a higher LTV. Calculated as follows: the number of
purchases from existing users divided by the total number of purchases.
Return on experience (ROX) determines how successful the customer’s experience of interacting with a specific channel of the company’s promotion has been. The formula: benefit (revenue)/value of
experience (software, services, labor) x 100%.
Finally, remarketing conversion value determines the percentage of remarketing conversions compared to all marketing conversions. The formula is the number of remarketing conversions/number of
marketing conversions.
Marketers can use these formulas to simplify the evaluation of the performance of marketing campaigns during marketing strategy or hypothesis testing. | {"url":"https://ckitsourcing.com/mobile-advertising/","timestamp":"2024-11-12T23:22:38Z","content_type":"text/html","content_length":"121837","record_id":"<urn:uuid:4ad0ce3f-1abe-46c3-9e69-46b80904ec25>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00213.warc.gz"} |
McGraw Hill Math Grade 3 Chapter 8 Lesson 4 Answer Key More Fractions on a Number Line
Excel in your academics by accessing McGraw Hill Math Grade 3 Answer Key PDF Chapter 8 Lesson 4 More Fractions on a Number Line existing for free of cost.
McGraw-Hill Math Grade 3 Answer Key Chapter 8 Lesson 4 More Fractions on a Number Line
Write the missing fractions in the box.
Question 1.
The missing fraction in the number line is 1/4.
Question 2.
The missing fraction in the number line is 2/3
Question 3.
The missing fraction in the number line is 2/6 and 5/6.
Question 4.
This fraction is on a number line divided into eighths. It appears four number places after 0. What fraction is it?
This fraction is on a number line divided into eighths. It appears four number places after 0.
Leave a Comment
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In these notes we give some examples of the interaction of mathematics with experiments and numerical simulations on the search for singularities.
We consider the dynamics of an interface given by two incompressible fluids with different characteristics evolving by Darcy’s law. This scenario is known as the Muskat problem, being in 2D
mathematically analogous to the two-phase Hele-Shaw cell. The purpose of this paper is to outline recent results on local existence, weak solutions, maximum principles and global existence.
The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an ${L}^{2}
\left(ℝ\right)$ maximum principle, in the form of a new “log” conservation law which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique
strong solutions if the initial data is smaller than an explicitly computable constant, for instance ${∥f∥}_{1}\le 1/5$. Previous results of this... | {"url":"https://eudml.org/search/page?q=sc.general*op.AND*l_0*c_0author_0eq%253A1.C%25C3%25B3rdoba%252C+Diego&qt=SEARCH","timestamp":"2024-11-12T07:17:07Z","content_type":"application/xhtml+xml","content_length":"63562","record_id":"<urn:uuid:ef036f92-b3c6-4fe9-9667-ff2ed7dcc1e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00207.warc.gz"} |
Transient structural solution and derived quantities
A TransientStructuralResults object contains the displacement, velocity, and acceleration in a form convenient for plotting and postprocessing.
Displacement, velocity, and acceleration are reported for the nodes of the triangular or tetrahedral mesh generated by generateMesh. The displacement, velocity, and acceleration values at the nodes
appear as FEStruct objects in the Displacement, Velocity, and Acceleration properties. The properties of these objects contain the components of the displacement, velocity, and acceleration at the
nodal locations.
To evaluate the stress, strain, von Mises stress, principal stress, and principal strain at the nodal locations, use evaluateStress, evaluateStrain, evaluateVonMisesStress, evaluatePrincipalStress,
and evaluatePrincipalStrain, respectively.
To evaluate the reaction forces on a specified boundary, use evaluateReaction.
To interpolate the displacement, velocity, acceleration, stress, strain, and von Mises stress to a custom grid, such as the one specified by meshgrid, use interpolateDisplacement, interpolateVelocity
, interpolateAcceleration, interpolateStress, interpolateStrain, and interpolateVonMisesStress, respectively.
Solve a dynamic linear elasticity problem by using the solve function. This function returns a transient structural solution as a TransientStructuralResults object.
Displacement — Displacement values at nodes
FEStruct object
This property is read-only.
Displacement values at the nodes, returned as an FEStruct object. The properties of this object contain components of displacement at nodal locations.
Velocity — Velocity values at nodes
FEStruct object
This property is read-only.
Velocity values at the nodes, returned as an FEStruct object. The properties of this object contain components of velocity at nodal locations.
Acceleration — Acceleration values at nodes
FEStruct object
This property is read-only.
Acceleration values at the nodes, returned as an FEStruct object. The properties of this object contain components of acceleration at nodal locations.
SolutionTimes — Solution times
real vector
This property is read-only.
Solution times, returned as a real vector. SolutionTimes is the same as the tlist input to solve.
Data Types: double
Mesh — Finite element mesh
FEMesh object
This property is read-only.
Finite element mesh, returned as a FEMesh object.
Object Functions
evaluateStress Evaluate stress for dynamic structural analysis problem
evaluateStrain Evaluate strain for dynamic structural analysis problem
evaluateVonMisesStress Evaluate von Mises stress for dynamic structural analysis problem
evaluateReaction Evaluate reaction forces on boundary
evaluatePrincipalStress Evaluate principal stress at nodal locations
evaluatePrincipalStrain Evaluate principal strain at nodal locations
filterByIndex Access transient results for specified time steps
interpolateDisplacement Interpolate displacement at arbitrary spatial locations
interpolateVelocity Interpolate velocity at arbitrary spatial locations for all time or frequency steps for dynamic structural problem
interpolateAcceleration Interpolate acceleration at arbitrary spatial locations for all time or frequency steps for dynamic structural problem
interpolateStress Interpolate stress at arbitrary spatial locations
interpolateStrain Interpolate strain at arbitrary spatial locations
interpolateVonMisesStress Interpolate von Mises stress at arbitrary spatial locations
Solve Transient Structural Problem
Solve for the transient response of a thin 3-D plate under a harmonic load at the center.
Create a geometry of a thin 3-D plate and plot it.
gm = multicuboid([5,0.05],[5,0.05],0.01);
Zoom in to see the face labels on the small plate at the center.
axis([-0.2 0.2 -0.2 0.2 -0.1 0.1])
Create an femodel object for transient structural analysis and include the geometry.
model = femodel(AnalysisType="structuralTransient", ...
Specify Young's modulus, Poisson's ratio, and the mass density of the material.
model.MaterialProperties = ...
materialProperties(YoungsModulus=210E9, ...
PoissonsRatio=0.3, ...
Specify that all faces on the periphery of the thin 3-D plate are fixed boundaries.
model.FaceBC(5:8) = faceBC(Constraint="fixed");
Apply a sinusoidal pressure load on the small face at the center of the plate.
First, define a sinusoidal load function, sinusoidalLoad, to model a harmonic load. This function accepts the load magnitude (amplitude), the location and state structure arrays, frequency, and
phase. Because the function depends on time, it must return a matrix of NaN of the correct size when state.time is NaN. Solvers check whether a problem is nonlinear or time-dependent by passing NaN
state values and looking for returned NaN values.
function Tn = sinusoidalLoad(load,location,state,Frequency,Phase)
if isnan(state.time)
Tn = NaN*(location.nx);
if isa(load,"function_handle")
load = load(location,state);
load = load(:);
% Transient model excited with harmonic load
Tn = load.*sin(Frequency.*state.time + Phase);
Now, apply a sinusoidal pressure load on face 12 by using the sinusoidalLoad function.
Pressure = 5e7;
Frequency = 25;
Phase = 0;
pressurePulse = @(location,state) ...
model.FaceLoad(12) = faceLoad(Pressure=pressurePulse);
Generate a mesh with linear elements.
model = generateMesh(model,GeometricOrder="linear",Hmax=0.2);
Specify zero initial displacement and velocity.
model.CellIC = cellIC(Displacement=[0;0;0],Velocity=[0;0;0]);
Solve the model.
tlist = linspace(0,1,300);
R = solve(model,tlist);
The solver finds the values of the displacement, velocity, and acceleration at the nodal locations. To access these values, use R.Displacement, R.Velocity, and so on. The displacement, velocity, and
acceleration values are returned as FEStruct objects with the properties representing their components. Note that properties of an FEStruct object are read-only.
ans =
FEStruct with properties:
ux: [2217x300 double]
uy: [2217x300 double]
uz: [2217x300 double]
Magnitude: [2217x300 double]
ans =
FEStruct with properties:
vx: [2217x300 double]
vy: [2217x300 double]
vz: [2217x300 double]
Magnitude: [2217x300 double]
ans =
FEStruct with properties:
ax: [2217x300 double]
ay: [2217x300 double]
az: [2217x300 double]
Magnitude: [2217x300 double]
Version History
Introduced in R2018a | {"url":"https://ch.mathworks.com/help/pde/ug/pde.transientstructuralresults.html","timestamp":"2024-11-03T04:14:00Z","content_type":"text/html","content_length":"98247","record_id":"<urn:uuid:e18eca28-4888-40df-9227-db366dd71692>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00489.warc.gz"} |
, Wintersemester 201
course description and syllabus
lecture notes
These notes were updated regularly during the semester and also include the notes from the previous semester's Topologie I course. I have since revised them to incorporate many of the homework
problems and correct some errors pointed out by the students.
(last update: 12.04.2021)
Ankündigungen / Announcements
Somebody miscalculated the fundamental group.
• 14.02.2019: The final lecture (on 15.02.2019) has been cancelled due to the BVG strike. The planned contents of that lecture will be added to the (Hermannplatz, 8.06.2017)
notes soon, but will not be considered relevant for final exams.
• 5.02.2019: The graded take-home midterms will be returned on Friday, 8.02.2019.
• 27.11.2018: By popular demand, Felix has written up a solution for Problem Set 4 #4 (on the long exact sequence of a mapping torus) which is now
posted below.
• 31.10.2018: Starting next week (7.11.2018), the Wednesday lectures will start at 11:00 sharp instead of 15 minutes after the hour. This way they can
end earlier so that there is more time for lunch before the Übung.
previous announcements (no longer relevant)
Übungsblätter / Problem sets
Problem sets will normally be handed out on Wednesdays and discussed in the Übung on the following Wednesday. They will not be collected or graded, with
the exception of the take-home midterm, a two-week assignment midway through the semester.
Chris Wendl's homepage | {"url":"http://www.mathematik.hu-berlin.de/~wendl/Winter2018/Topologie2/","timestamp":"2024-11-09T22:40:07Z","content_type":"text/html","content_length":"4350","record_id":"<urn:uuid:5ef5c9e7-1737-4477-9bef-3bb2f24ee112>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00046.warc.gz"} |
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Math Problem Analysis
Mathematical Concepts
Quadratic Equations
Completing the Square
Completing the Square formula: (x - h)^2 = k
Quadratic equation: ax^2 + bx + c = 0
Zero Product Property
Properties of Square Roots
Suitable Grade Level
Grade 9-10 | {"url":"https://math.bot/q/find-zeros-fx--5x2-4x-1-completing-square-FMOFQv0p","timestamp":"2024-11-01T22:17:55Z","content_type":"text/html","content_length":"87235","record_id":"<urn:uuid:b5d5f3e7-5956-4d65-83de-d184e86fa36e>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00259.warc.gz"} |
How to break down colour variable in sjPlot::plot_model into equally-sized bins | Pablo Bernabeu
How to break down colour variable in sjPlot::plot_model into equally-sized bins
Whereas the direction of main effects can be interpreted from the sign of the estimate, the interpretation of interaction effects often requires plots. This task is facilitated by the R package
sjPlot. For instance, using the plot_model function, I plotted the interaction between two continuous variables.
#> Loading required package: Matrix
#> Learn more about sjPlot with 'browseVignettes("sjPlot")'.
# Create data partially based on code by Ben Bolker
# from https://stackoverflow.com/a/38296264/7050882
spin = runif(800, 1, 24)
trait = rep(1:40, each = 20)
ID = rep(1:80, each = 10)
testdata <- data.frame(spin, trait, ID)
testdata$fatigue <-
testdata$spin * testdata$trait /
rnorm(800, mean = 6, sd = 2)
# Model
fit = lmer(fatigue ~ spin * trait + (1|ID),
data = testdata, REML = TRUE)
#> boundary (singular) fit: see help('isSingular')
plot_model(fit, type = 'pred', terms = c('spin', 'trait'))
#> Warning: Ignoring unknown parameters: linewidth
^Created on 2023-06-24 with reprex v2.0.2
However, I needed an extra feature, as sjPlot by default breaks down the colour (fill) variable into few levels that do not include the minimum or the maximum values in my variable. What I would like
to do is to stratify the colour variable into equally-sized levels that include the minimum and the maximum values.
Furthermore, in the legend, I would also like to display the number of levels of a grouping variable (ID) that are contained in each level of the colour variable.
Below is a solution using custom functions called deciles_interaction_plot and sextiles_interaction_plot.
#> Loading required package: Matrix
# Create data partially based on code by Ben Bolker
# from https://stackoverflow.com/a/38296264/7050882
spin = runif(800, 1, 24)
trait = rep(1:40, each = 20)
ID = rep(1:80, each = 10)
testdata <- data.frame(spin, trait, ID)
testdata$fatigue <-
testdata$spin * testdata$trait /
rnorm(800, mean = 6, sd = 2)
# Model
fit = lmer(fatigue ~ spin * trait + (1|ID),
data = testdata, REML = TRUE)
#> boundary (singular) fit: see help('isSingular')
# plot_model(fit, type = 'pred', terms = c('spin', 'trait'))
# Binning the colour variable into ten levels (deciles)
# Read in function from GitHub
model = fit,
x = 'spin',
fill = 'trait',
fill_nesting_factor = 'ID'
#> Loading required package: dplyr
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#> filter, lag
#> The following objects are masked from 'package:base':
#> intersect, setdiff, setequal, union
#> Loading required package: RColorBrewer
#> Loading required package: ggtext
#> Loading required package: Cairo
#> Warning in RColorBrewer::brewer.pal(n, pal): n too large, allowed maximum for palette Set1 is 9
#> Returning the palette you asked for with that many colors
#> Warning: Ignoring unknown parameters: linewidth
#> Scale for 'y' is already present. Adding another scale for 'y', which will
#> replace the existing scale.
#> Scale for 'colour' is already present. Adding another scale for 'colour',
#> which will replace the existing scale.
# If you wanted or needed to make six levels (sextiles) instead
# of ten, you could use the function sextiles_interaction_plot.
# Read in function from GitHub
model = fit,
x = 'spin',
fill = 'trait',
fill_nesting_factor = 'ID'
#> Warning: Ignoring unknown parameters: linewidth
#> Scale for 'y' is already present. Adding another scale for 'y', which will
#> replace the existing scale.
#> Scale for 'colour' is already present. Adding another scale for 'colour',
#> which will replace the existing scale.
^Created on 2023-06-24 with reprex v2.0.2 | {"url":"https://pablobernabeu.github.io/2023/how-to-break-down-colour-variable-in-sjplot-plot-model-into-equally-sized-bins/","timestamp":"2024-11-14T14:56:43Z","content_type":"text/html","content_length":"24114","record_id":"<urn:uuid:181e8ee6-6836-4f80-b873-e5a503f61708>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00815.warc.gz"} |
TRIANGLE's Properties | Types | Area | Perimeter | Similarity and Congruence of triangle
Nowadays, we study various types of shapes which is useful in day to day life. These shapes are used in building elevation, flooring, fashion designing, and various arts.
In this article, we are discussing triangle and it's various topics.
Here is the content of our mini-guide on the triangle which will help you as a beginner.
So first we start with the definition of a triangle and discuss parts of a triangle.
1. What is a triangle?
Triangle is a simple closed shape which is made only from three line segments.
Triangle is a polygon of three sides.
Triangle is a simple closed shape with three vertices.
2. Parts of a triangle
There are various parts of a triangle-
A side is a line segment by which we make a triangle.
Vertex is a point where two sides join together.
An angle is an exposer between two sides.
An angle that is inside the triangle is known as an interior angle.
An angle that is outside the triangle is known as an exterior angle.
Median is a line segment that divides a triangle into two equal parts.
Height is a perpendicular line segment that shows the elevation of a triangle.
Sometimes median and height are same in a triangle.
Triangle has three sides, three vertices, three interior angles.
We explained triangles & components of triangles now we need to explain the types of triangles.
3. Types of triangles
There are various types of triangles. Triangles may be classified based on their sides & angles.
First, we discuss types of triangles based on their sides.
Types of triangles based on sides-
Equilateral triangle
Triangle whose all sides and angles are equal (60°) is known as an equilateral triangle.
Isosceles triangle
Triangle whose any two sides and any two angles are equal is known as an isosceles triangle.
Scalene triangle
Triangle whose all sides are different in length is known as a scalene triangle.
Types of triangles based on angles-
Right-angled triangle
Triangle whose any one angle is 90° and the other two angles are less than 90° is known as a right-angled triangle.
Acute angled triangle
Triangle whose all angles are less than 90° is known as an acute-angled triangle.
Obtuse angled triangle
Triangle whose any one angle is more than 90° and the other two angles are less than 90° is known as an obtuse-angled triangle.
Generally, we learn above all kinds of the triangle.
But when a triangle contains two properties(based on their sides & angles) then we must study the below triangle.
Types of triangles based on both angles and sides-
Acute-angled equilateral triangle
When a triangle contains both properties in which all angles are equal(60°, 60°, 60°) or less than 90° and all sides are equal is known as an acute-angled equilateral triangle.
Right-angled isosceles triangle
When a triangle contains one right angle(90°) and the other two angles(45°, 45°) or sides are equal is known as a right-angled isosceles triangle. This triangle includes 90°, 45°, 45° angles.
Obtuse-angled scalene triangle
When a triangle contains one angle more than 90° and all sides are different in length is known as the obtuse-angled scalene triangle.
Above all these are the classifications of the triangle which we generally study.
Now we are going to talk about the properties of a triangle. By the properties of the triangle, we may solve various problems.
4. Properties of triangle
There are several properties of the triangle which helps to solve the questions.
Let's discuss it one by one-
Pythagoras theorem (Pythagoras property of right triangle)
It states - In a right-angled triangle square of the longest side is equal to the sum of the square of the other two sides.
It means the square of the hypotenuse(H) is equal to the sum of the square of perpendicular(P) and base(B).
H² = P² + B²
When we have given any two sides in the right-angled triangle then we may find the remaining side by Pythagoras theorem.
Angle sum property of triangle
∠A + ∠B + ∠C = 180°
When we have any two interior angles in a triangle and we want to find the remaining interior angle of a triangle then we may use this property.
Exterior angle property of triangle
x° + y° = Exterior angle
Sum of two sides of the triangle
It states the sum of any two sides of a triangle is always greater than its third side.
By the above property, we can find any exterior angle of the triangle.
And we know exterior angle so we may find interior angles.
Now it's turn to know about the area of triangle.
5. Area of triangle
Area of triangle is the region surrounded by sides of a triangle.
We have several formulas to find the area of a triangle. Let's discuss-
Area of a right-angled triangle(Right triangle)
ar∆ = 1/2×Base×Height
This formula is very useful if we have a base and height of any right triangle.
Area of an equilateral triangle
ar∆ = √3/4×side²
Heron's Formula
Heron's formula helps to find the area of all types of triangles. If we know all the sides of the triangle.
ar∆ = √{s(s-a)(s-b)(s-c)}
s(semi-perimeter) = (a+b+c)/2
a, b and c = sides of the triangle.
6. Perimeter of triangle
The perimeter of a triangle is the length of its boundary.
The formula for the perimeter of a triangle = The sum of the length of all sides.
3×side of a triangle (for equilateral triangle)
By the above formulas, we may find the perimeter and any unknown sides of the triangle.
7. Congruence of a triangle
When all sides and angles of a triangle are equal to the corresponding sides and angles of another triangle then both triangles are congruent to each other.
There are various rules for the congruence of the triangle-
SSS congruence rule (Side-Side-Side)
If all sides of a triangle are equal to the corresponding sides of another triangle then both triangles are congruent by the Side-Side-Side (SSS) congruence rule.
ASA congruence rule (Angle-Side-Angle)
When any two angles and a side of a triangle are equal to the corresponding angles and side of another triangle so both the triangles are congruent by the Angle-Side-Angle (ASA) congruence rule.
SAS congruence rule (Side-Angle-Side)
When any two sides and an angle of a triangle are equal to the corresponding sides and angle of another triangle so both the triangles are congruent by the Side-Angle-Side (SAS) congruence rule.
RHS congruence rule (Right angle-Hypotenuse-Side)
When one side and hypotenuse of a right-angled triangle are equal to the corresponding side and hypotenuse of another right-angled triangle then both the triangles are congruent by Right
Angle-Hypotenuse-Side (RHS) congruence rule.
8. Similarity of a triangle
When the corresponding sides of two triangles are in proportion or equal and corresponding angles are equal then they are similar to each other.
Questions / Answers
1.) How triangle are congruent?
When any two triangles satisfy SSS, SAS, ASA and RHS congruence rule then both triangles are congruent to each other.
2.) How triangles are similar?
When all interior angles of two triangles are equal and their corresponding sides are in proportion so they are similar to each other.
3.) What triangle is both scalene and right?
Triangle whose all sides are different in length and anyone interior angle is 90°. So it will be both a scalene and right triangle.
4.) Which triangle is a 30°-60°-90° triangle?
A right-angled triangle is known as 30°-60°-90° triangle because it has one angle is 90° and the other two angles are less than 90°.
5.) Which triangle is both scalene and acute?
Triangle whose all sides are different in length and all angles are less than 90°. So it will be both a scalene and acute triangle.
Finally, we completed our mini-guide on the triangle which covers the introduction of triangle, properties.
Please comment us for any query related to this article.... | {"url":"https://www.tirlaacademy.com/2021/01/triangles-properties-types-area.html","timestamp":"2024-11-03T00:28:28Z","content_type":"application/xhtml+xml","content_length":"352978","record_id":"<urn:uuid:6c7e0398-f7ec-42bf-9ee4-bda6bcdc9fab>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00647.warc.gz"} |
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Printable Time Tables 112 Activity Shelter
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Multiplication Table 112
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Printable Multiplication Times Table 1 12 Times Tables Worksheets
Web this multiplication table 1 to 12 is consist of 12 rows with a respective operation of multiplication, which is very beneficial to learn the basic multiplication of 1 to 12. Get a free printable
multiplication chart pdf for your class! Web free printable multiplication charts (times tables) available in pdf format. You can use it as a reminder or..
You will find below complete or. Web free printable multiplication charts (times tables) available in pdf format. Web here is the printable multiplication chart (pdf) from the 1 time table up to the
12 times table, it's a free resource. Web this multiplication table 1 to 12 is consist of 12 rows with a respective operation of multiplication, which is very beneficial to learn the basic
multiplication of 1 to 12. Use these colorful multiplication tables to help your child build confidence while mastering the. You can use it as a reminder or. Get a free printable multiplication chart
pdf for your class!
You Can Use It As A Reminder Or.
You will find below complete or. Web free printable multiplication charts (times tables) available in pdf format. Use these colorful multiplication tables to help your child build confidence while
mastering the. Get a free printable multiplication chart pdf for your class!
Web This Multiplication Table 1 To 12 Is Consist Of 12 Rows With A Respective Operation Of Multiplication, Which Is Very Beneficial To Learn The Basic Multiplication Of 1 To 12.
Web here is the printable multiplication chart (pdf) from the 1 time table up to the 12 times table, it's a free resource.
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Which Argument's is/are strong as per given statement.
Statement-Should there be a complete ban on strike by government employees in India ?
I. Yes, this is the only way to teach discipline to the employees.
II. No, this deprives the citizens of their democratic rights.
1. if only argument I is strong
2. if only argument II is strong
3. if either I or II is strong
4. if both I & II are strong
Rahul saves 10% of his total salary. Next year, he increase his expenses by 20%, but his percentage of savings remain the same What is the percenatage increase in his salary next year?
1. 10%
2. 16.66%
3. 20%
4. 40%
Anil said to Anand, "that boy playing with the football is the younger of the two brothers of the daughter of my fathe~s wife". How is the boy playing football related to Anil?
1. father
2. brother
3. son in law
4. uncle
Efficiency of X is 20% less than Y to do a certain task. If X alone can complete a piece of work in $$7\frac{1}{2}$$ hours, then Y alone cant do it in:
1. $$4\frac{1}{2}$$ hours
2. $$5$$ hours
3. $$7\frac{1}{2}$$ hours
4. $$6$$ hours
If the cost price of 10 articles is equal to the selling price of 7 articles, then the gain or loss percent is:
1. $$51$$% gain
2. $$42\frac{6}{7}$$% gain
3. $$35$$% loss
4. $$42\frac{6}{7}$$% loss
An outlet pipe can empty a cistern in 3 hours. In what time will it empty 2/3 rd of the cistern?
1. 2 hours
2. 3 hours
3. 4 hours
4. 5 hours
X takes 2 hours more than Y to walk d km, but if X doubles his speed, then he can make it in 1 hours less than Y. How much time does Y require for walking d km?
1. $$3$$ hours
2. $$4$$ hours
3. $$\frac{d}{2}$$ hours
4. $$\frac{2d}{3}$$ hours
If x% of a is the same as y% of b, then z% of b is
1. xy/z % of a
2. yz/x % of a
3. xz/y % of a
4. xyz %
which option is correct mirror image of figure(x)?
1. a
2. b
3. C
4. d
Tank is fitted with two taps X and Y. In how much time will the tank be full if both the taps are opened together? Which of the followmgs statements is/are required to answer this question
(A) X is 50% more efficient than Y
(B) X alone takes 16 hours to fill the tank.
(C) Y alone takes 24 hours to fill the tank.
Choose the correct answer from the options given below·
1. (A) and (C) only
2. (A) and (B) only
3. (B) and (C) only.
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Journal club: Estimating the dimensionality of neuronal representations during cognitive tasks
It's a bit of a cliche that the best papers are the ones that raise more questions than they answer (in fact, many papers seem to answer hardly anything at all on close inspection and it doesn't mean
they're great). But I think this might be one of those papers for which the cliche holds true in a positive sense. Rigotti and colleagues (2013, Nature) reported a really intriguing re-analysis of
some single-unit data from macaque PFC. The central idea here is to attempt to estimate the dimensionality of the neuronal representation, and to connect this to task performance. This sounds
abstract, but I think the strength of the paper lies in how the authors frame dimensionality in terms of linear separability.
The basic idea goes like this: If we represent neuronal firing rates in some task with a n by c matrix where n represents cells and c the unique conditions, the most task-related dimensions that a
neural representation can encode would equal to c. Ordinarily, you could take the rank of the matrix (assuming n=>c) to test how many dimensions are present. The rank will be less than c if some of
the conditions are linear combinations of each other. The catch is that neuroscientific data is noisy, which inflates the dimensionality all the way up to n in practically all cases. So how do you
estimate the dimensionality in the presence of noise?
Rigotti's solution is to approach the problem indirectly via linear separability. One way to think of a representation's dimensionality is that it's related to the number of ways in which you can
bisect the space with a discriminant. Imagine arbitrarily splitting the conditions into two classes, and using a standard linear discriminant analysis to find a hyperplane that separates the two
classes. If the matrix is full rank, this is always possible for all arbitrary splits of the conditions. So the number of successful discriminants (there's 2^c) is related to the rank of the matrix.
This is useful because we can now deal with the noise by cross-validating the discriminant. So the number of successful cross-validated discriminants (and by successful, we mean accuracy over some
threshold) provides a noise-corrected measure of the dimensionality of the underlying representation.
The most convincing evidence in the paper is in Fig 5, of which two panels appear below. (a) shows that the estimated dimensionality is lower for correct trials than for error trials. By contrast,
decoding of the stimulus cue is similar for these trial types (b), which makes two points: first that it's not that the monkey simply fell asleep on the error trials because this stimulus distinction
is present in the responses. Second, and less intuitively, this one arguably task-relevant dimension does not distinguish correct and error trials, while the total count over many discriminants does,
even though a good number of these splits would have very little behavioural relevance. This is puzzling.
A final few notes on this:
• The paper has a strong spin on the topic of 'non-linear mixed selectivity', by which the authors simply mean that a neuronal code based on tuning to single dimensions or linear combinations
thereof cannot support the kind of high dimensionality they observe here. Lots of analyses in the paper focus on removing linear selectivity and characterising it separately in different ways to
support the case that non-linear tunings are essential for this. I don't think this point is as new or as controversial as it is presented in the manuscript.
• The authors' dimensionality estimation approach is neat for this application because it has a natural link to neuronal readout - part of the popularity of linear classifiers stems from the
intuitive cartoon of the weights vector as the synaptic weights on some downstream representation. In this sense, a higher-dimensional representation seems more suited to flexible behaviour
because a downstream region would be able to make a large number of distinctions by changing the weights. But there are of course many other ways to estimate the rank of noisy data and one
wonders how this approach compares to methods used in other fields, where the classifier intuition is less appealing but the problem potentially very similar.
• If PFC really furnishes such high-dimensional representations (note that all stimulus dimensions are present in the population code according to Fig 5A above), why are some distinctions
behaviourally more difficult than others? Presumably monkeys would find it much harder to learn an XOR-like stimulus-response mapping than a simple feature mapping, which doesn't seem to follow
if the code were this high-dimensional.
Rigotti, M., Barak, O., Warden, M. R., Wang, X.-J., Daw, N. D., Miller, E. K., & Fusi, S. (2013). The importance of mixed selectivity in complex cognitive tasks. Nature, 497(7451), 585–90. http://
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Reproductive Number – bdocs rambling
Chart by Michael Levitt illustrating his Gompertz function curve fitting methodology
I have been wondering for a while how to characterise the difference in approaches to Coronavirus modelling of cases and deaths, between “curve-fitting” equations and the SIR differential equations
approach I have been using (originally developed in Alex de Visscher’s paper this year, which included code and data for other countries such as Italy and Iran) which I have adapted for the UK.
Part of my uncertainty has its roots in being a very much lapsed mathematician, and part is because although I have used modelling tools before, and worked in some difficult area of mathematical
physics, such as General Relativity and Cosmology, epidemiology is a new application area for me, with a wealth of practitioners and research history behind it.
Curve-fitting charts such as the Sigmoid and Gompertz curves, all members of a family of curves known as logistics or Richards functions, to the Coronavirus cases or deaths numbers as practised,
notably, by Prof. Michael Levitt and his Stanford University team has had success in predicting the situation in China, and is being applied in other localities too.
Michael’s team have now worked out an efficient way of reducing the predictive aspect of the Gompertz function and its curves to a straight line predictor of reported data based on a version of the
Gompertz function, a much more efficient use of computer time than some other approaches.
The SIR model approach, setting up an series of related differential equations (something I am more used to in other settings) that describe postulated mechanisms and rates of virus transmission in
the human population (hence called “mechanistic” modelling), looks beneath the surface presentation of the epidemic cases and deaths numbers and time series charts, to model the growth (or otherwise)
of the epidemic based on postulated characteristics of viral transmission and behaviour.
Research literature
In researching the literature, I have become familiar with some names that crop up or frequently in this area over the years.
Focusing on some familiar and frequently recurring names, rather than more recent practitioners, might lead me to fall into “The Trouble with Physics” trap (the tendency, highlighted by Lee Smolin in
his book of that name, exhibited by some University professors to recruit research staff (“in their own image”) who are working in the mainstream, rather than outliers whose work might be seen as
off-the-wall, and less worthy in some sense.)
In this regard, Michael Levitt‘s new work in the curve-fitting approach to the Coronavirus problem might be seen by others who have been working in the field for a long time as on the periphery
(despite his Nobel Prize in Computational Biology and Stanford University position as Professor of Structural Biology).
His results (broadly forecasting, very early on, using his curve-fitting methods (he has used Sigmoid curves before, prior to the current Gompertz curves), a much lower incidence of the virus going
forward, successfully so in the case of China) are in direct contrast to that of some some teams working as advisers to Governments, who have, in some cases, all around the world, applied fairly
severe lockdowns for a period of several months in most cases.
In particular the work of the Imperial College Covid response team, and also the London School of Hygiene and Tropical Medicine have been at the forefront of advice to the UK Government.
Some Governments have taken a different approach (Sweden stands out in Europe in this regard, for several reasons).
I am keen to understand the differences, or otherwise, in such approaches.
Twitter and publishing
Michael chooses to publish his work on Twitter (owing to a glitch (at least for a time) with his Stanford University laboratory‘s own publishing process. There are many useful links there to his
My own succession of blog posts (all more narrowly focused on the UK) have been automatically published to Twitter (a setting I use in WordPress) and also, more actively, shared by me on my FaceBook
But I stopped using Twitter routinely a long while ago (after 8000+ posts) because, in my view, it is a limited communication medium (despite its reach), not allowing much room for nuanced posts. It
attracts extremism at worst, conspiracy theorists to some extent, and, as with a lot of published media, many people who choose on a “confirmation bias” approach to read only what they think they
might agree with.
One has only to look at the thread of responses to Michael’s Twitter links to his forecasting results and opinions to see examples of all kinds of Twitter users: some genuinely academic and/or
thoughtful; some criticising the lack of published forecasting methods, despite frequent posts, although they have now appeared as a preprint here; many advising to watch out (often in extreme terms)
for “big brother” government when governments ask or require their populations to take precautions of various kinds; and others simply handclapping, because they think that the message is that this
all might go away without much action on their part, some of them actively calling for resistance even to some of the most trivial precautionary requests.
One of the recent papers I have found useful in marshalling my thoughts on methodologies is this 2016 one by Gustavo Chowell, and it finally led me to calibrate the differences in principle between
the SIR differential equation approach I have been using (but a 7-compartment model, not just three) and the curve-fitting approach.
I had been thinking of analogies to illustrate the differences (which I will come to later), but this 2016 Chowell paper, in particular, encapsulated the technical differences for me, and I summarise
that below. The Sergio Alonso paper also covers this ground.
Categorization of modelling approaches
Gerard Chowell’s 2016 paper summarises modelling approaches as follows.
Phenomenological models
A dictionary definition – “Phenomenology is the philosophical study of observed unusual people or events as they appear without any further study or explanation.”
Chowell states that phenomenological approaches for modelling disease spread are particularly suitable when significant uncertainty clouds the epidemiology of an infectious disease, including the
potential contribution of multiple transmission pathways.
In these situations, phenomenological models provide a starting point for generating early estimates of the transmission potential and generating short-term forecasts of epidemic trajectory and
predictions of the final epidemic size.
Such methods include curve fitting, as used by Michael Levitt, where an equation (represented by a curve on a time-incidence graph (say) for the virus outbreak), with sufficient degrees of freedom,
is used to replicate the shape of the observed data with the chosen equation and its parameters. Sigmoid and Gompertz functions (types of “logistics” or Richards functions) have been used for such
fitting – they produce the familiar “S”-shaped curves we see for epidemics. The starting growth rate, the intermediate phase (with its inflection point) and the slowing down of the epidemic, all
represented by that S-curve, can be fitted with the equation’s parametric choices (usually three or four).
Chart by Michael Levitt illustrating his Gompertz function curve fitting methodology
A feature that some epidemic outbreaks share is that growth of the epidemic is not fully exponential, but is “sub-exponential” for a variety of reasons, and Chowell states that:
“Previous work has shown that sub-exponential growth dynamics was a common phenomenon across a range of pathogens, as illustrated by empirical data on the first 3-5 generations of epidemics of
influenza, Ebola, foot-and-mouth disease, HIV/AIDS, plague, measles and smallpox.”
Choices of appropriate parameters for the fitting function can allow such sub-exponential behaviour to be reflected in the chosen function’s fit to the reported data, and it turns out that the
Gompertz function is more suitable for this than the Sigmoid function, as Michael Levitt states in his recent paper.
Once a curve-fit to reported data to date is achieved, the curve can be used to make forecasts about future case numbers.
Mechanistic and statistical models
Chowell states that “several mechanisms have been put forward to explain the sub-exponential epidemic growth patterns evidenced from infectious disease outbreak data. These include spatially
constrained contact structures shaped by the epidemiological characteristics of the disease (i.e., airborne vs. close contact transmission model), the rapid onset of population behavior changes, and
the potential role of individual heterogeneity in susceptibility and infectivity.“
He goes on to say that “although attractive to provide a quantitative description of growth profiles, the generalized growth model (described earlier) is a phenomenological approach, and hence cannot
be used to evaluate which of the proposed mechanisms might be responsible for the empirical patterns.
Explicit mechanisms can be incorporated into mathematical models for infectious disease transmission, however, and tested in a formal way. Identification and analysis of the impacts of these factors
can lead ultimately to the development of more effective and targeted control strategies. Thus, although the phenomenological approaches above can tell us a lot about the nature of epidemic patterns
early in an outbreak, when used in conjunction with well-posed mechanistic models, researchers can learn not only what the patterns are, but why they might be occurring.“
On the Imperial College team’s planning website, they state that their forecasting models (they have several for different purposes, for just these reasons I guess) fall variously into the
“Mechanistic” and “Statistical” categories, as follows.
Imperial College models use a combination of mechanistic and statistical approaches.
“Mechanistic model: Explicitly accounts for the underlying mechanisms of diseases transmission and attempt to identify the drivers of transmissibility. Rely on more assumptions about the disease
“Statistical model: Do not explicitly model the mechanism of transmission. Infer trends in either transmissibility or deaths from patterns in the data. Rely on fewer assumptions about the disease
“Mechanistic models can provide nuanced insights into severity and transmission but require specification of parameters – all of which have underlying uncertainty. Statistical models typically have
fewer parameters. Uncertainty is therefore easier to propagate in these models. However, they cannot then inform questions about underlying mechanisms of spread and severity.“
So Imperial College’s “statistical” description matches more to Chowell’s description of a phenomenological approach, although may not involve curve-fitting per se.
The SIR modelling framework, employing differential equations to represent postulated relationships and transitions between Susceptible, Infected and Recovered parts of the population (at its most
simple) falls into this Mechanistic model category.
Chowell makes the following useful remarks about SIR style models.
“The SIR model and derivatives is the framework of choice to capture population-level processes. The basic SIR model, like many other epidemiological models, begins with an assumption that
individuals form a single large population and that they all mix randomly with one another. This assumption leads to early exponential growth dynamics in the absence of control interventions and
susceptible depletion and greatly simplifies mathematical analysis (note, though, that other assumptions and models can also result in exponential growth).
The SIR model is often not a realistic representation of the human behavior driving an epidemic, however. Even in very large populations, individuals do not mix randomly with one another—they have
more interactions with family members, friends, and coworkers than with people they do not know.
This issue becomes especially important when considering the spread of infectious diseases across a geographic space, because geographic separation inherently results in nonrandom interactions, with
more frequent contact between individuals who are located near each other than between those who are further apart.
It is important to realize, however, that there are many other dimensions besides geographic space that lead to nonrandom interactions among individuals. For example, populations can be structured
into age, ethnic, religious, kin, or risk groups. These dimensions are, however, aspects of some sort of space (e.g., behavioral, demographic, or social space), and they can almost always be modeled
in similar fashion to geographic space“.
Here we begin to see the difference I was trying to identify between the curve-fitting approach and my forecasting method. At one level, one could argue that curve-fitting and SIR-type modelling
amount to the same thing – choosing parameters that make the theorised data model fit the reported data.
But, whether it produces better or worse results, or with more work rather than less, SIR modelling seeks to understand and represent the underlying virus incubation period, infectivity,
transmissibility, duration and related characteristics such as recovery and immunity (for how long, or not at all) – the why and how, not just the what.
The (nonlinear) differential equations are then solved numerically (rather than analytically with exact functions) and there does have to be some fitting to the initial known data for the outbreak
(i.e. the history up to the point the forecast is being done) to calibrate the model with relevant infection rates, disease duration and recovery timescales (and death rates).
This makes it look similar in some ways to choosing appropriate parameters for any function (Sigmoid, Gompertz or General Logistics function (often three or four parameters)).
But the curve-fitting approach is reproducing an observed growth pattern (one might say top-down, or focused on outputs), whereas the SIR approach is setting virological and other behavioural
parameters to seek to explain the way the epidemic behaves (bottom-up, or focused on inputs).
Metapopulation spatial models
Chowell makes reference to population-level models, formulations that are used for the vast majority of population based models that consider the spatial spread of human infectious diseases and that
address important public health concerns rather than theoretical model behaviour. These are beyond my scope, but could potentially address concerns about indirect impacts of the Covid-19 pandemic.
a) Cross-coupled metapopulation models
These models, which have been used since the 1940s, do not model the process that brings individuals from different groups into contact with one another; rather, they incorporate a contact matrix
that represents the strength or sum total of those contacts between groups only. This contact matrix is sometimes referred to as the WAIFW, or “who acquires infection from whom” matrix.
In the simplest cross-coupled models, the elements of this matrix represent both the influence of interactions between any two sub-populations and the risk of transmission as a consequence of those
interactions; often, however, the transmission parameter is considered separately. An SIR style set of differential equations is used to model the nature, extent and rates of the interactions between
b) Mobility metapopulation models
These models incorporate into their structure a matrix to represent the interaction between different groups, but they are mechanistically oriented and do this by considering the actual process by
which such interactions occur. Transmission of the pathogen occurs within sub-populations, but the composition of those sub-populations explicitly includes not only residents of the sub-population,
but visitors from other groups.
One type of model uses a “gravity” approach for inter-population interactions, where contact rates are proportional to group size and inversely proportional to the distance between them.
Another type described by Chowell uses a “radiation” approach, which uses population data relating to home locations, and to job locations and characteristics, to theorise “travel to work” patterns,
calculated using attractors that such job locations offer, influencing workers’ choices and resulting travel and contact patterns.
Transportation and mobile phone data can be used to populate such spatially oriented models. Again SIR-style differential equations are used to represent the assumptions in the model about between
whom, and how the pandemic spreads.
Summary of model types
We see that there is a range of modelling methods, successively requiring more detailed data, but which seek increasingly to represent the mechanisms (hence “mechanistic” modelling) by which the
virus might spread.
We can see the key difference between curve-fitting (what I called a surface level technique earlier) and the successively more complex models that seek to work from assumed underlying causations of
infection spread.
An analogy (picking up on the word “surface” I have used here) might refer to explaining how waves in the sea behave. We are all aware that out at sea, wave behaviour is perceived more as a “swell”,
somewhat long wavelength waves, sometimes of great height, compared with shorter, choppier wave behaviour closer to shore.
I’m not here talking about breaking waves – a whole separate theory is needed for those – René Thom‘s Catastrophe Theory – but continuous waves.
A curve fitting approach might well find a very good fit using trigonometric sine waves to represent the wavelength and height of the surface waves, even recognising that they can be encoded by depth
of the ocean, but it would need an understanding of hydrodynamics, as described, for example, by Bernoulli’s Equation, to represent how and why the wavelength and wave height (and speed*) changes
depending on the depth of the water (and some other characteristics).
(*PS remember that the water moves, pretty much, up and down, in an elliptical path for any fluid “particle”, not in the direction of travel of the observed (largely transverse) wave. The horizontal
motion and speed of the wave is, in a sense, an illusion.)
Concluding comments
There is a range of modelling methods, successively requiring more detailed data, from phenomenological (statistical and curve-fitting) methods, to those which seek increasingly to represent the
mechanisms (hence “mechanistic”) by which the virus might spread.
We see the difference between curve-fitting and the successively more complex models that build a model from assumed underlying interactions, and causations of infection spread between parts of the
I do intend to cover the mathematics of curve fitting, but wanted first to be sure that the context is clear, and how it relates to what I have done already.
Models requiring detailed data about travel patterns are beyond my scope, but it is as well to set into context what IS feasible.
Setting an understanding of curve-fitting into the context of my own modelling was a necessary first step. More will follow.
I have found several papers very helpful on comparing modelling methods, embracing the Gompertz (and other) curve-fitting approaches, including Michaels Levitt’s own recent June 30th one, which
explains his methods quite clearly.
Gerard Chowell’s 2016 paper on Mathematical model types September 2016
The Coronavirus Chronologies – Michael Levitt, 13th March 2020
COVID-19 Virus Epidemiological Model Alex de Visscher, Concordia University, Quebec, 22nd March 2020
Empiric model for short-time prediction of Covid-19 spreading , Sergio Alonso et al, Spain, 19th May 2020
Universality in Covid-19 spread in view of the Gompertz function Akira Ohnishi et al, Kyoto University) 22nd June 2020
Predicting the trajectory of any Covid-19 epidemic from the best straight line – Michael Levitt et al 30th June 2020
New confirmed cases of coronavirus, 7-day rolling average, USA vs Europe
A couple of interesting articles on the Coronavirus pandemic came to my attention this week; a recent one in National Geographic on June 26th, highlighting a startling comparison, between the USA’s
cases history, and recent spike in case numbers, with the equivalent European data, referring to an older National Geographic article, from March, by Cathleen O’Grady, referencing a specific chart
based on work from the Imperial College Covid-19 Response team.
I noticed, and was interested in that reference following a recent interaction I had with that team, regarding their influential March 16th paper. It prompted more thought about “herd immunity” from
Covid-19 in the UK.
Meanwhile, my own forecasting model is still tracking published data quite well, although over the last couple of weeks I think the published rate of deaths is slightly above other forecasts as well
as my own.
The USA
The recent National Geographic article from June 26th, by Nsikan Akpan, is a review of the current situation in the USA with regard to the recent increased number of new confirmed Coronavirus cases.
A remarkable chart at the start of that article immediately took my attention:
7 day average cases from the US Census Bureau chart, NY Times / National Geographic
The thrust of the article concerned recommendations on public attitudes, activities and behaviour in order to reduce the transmission of the virus. Even cases per 100,000 people, the case rate, is
worse and growing in the USA.
7 day average cases per 100,000 people from the US Census Bureau chart, NY Times / National Geographic
A link between this dire situation and my discussion below about herd immunity is provided by a reported statement in The Times by Dr Anthony Fauci, Director of the National Institute of Allergy and
Infectious Diseases, and one of the lead members of the Trump Administration’s White House Coronavirus Task Force, addressing the Covid-19 pandemic in the United States.
Reported Dr Fauci quotation by the Times newspaper 30th June 2020
If the take-up of the vaccine were 70%, and it were 70% effective, this would result in roughly 50% herd immunity (0.7 x 0.7 = 0.49).
If the innate characteristics of the the SARS-CoV-2 virus don’t change (with regard to infectivity and duration), and there is no other human-to-human infection resistance to the infection not yet
understood that might limit its transmission (there has been some debate about this latter point, but this blog author is not a virologist) then 50% is unlikely to be a sufficient level of population
My remarks later about the relative safety of vaccination (eg MMR) compared with the relevant diseases themselves (Rubella, Mumps and Measles in that case) might not be supported by the anti-Vaxxers
in the US (one of whose leading lights is the disgraced British doctor, Andrew Wakefield).
This is just one more complication the USA will have in dealing with the Coronavirus crisis. It is one, at least, that in the UK we won’t face to anything like the same degree when the time comes.
The UK, and implications of the Imperial College modelling
That article is an interesting read, but my point here isn’t really about the USA (worrying though that is), but about a reference the article makes to some work in the UK, at Imperial College,
regarding the effectiveness of various interventions that have been or might be made, in different combinations, work reported in the National Geographic back on March 20th, a pivotal time in the
UK’s battle against the virus, and in the UK’s decision making process.
This chart reminded me of some queries I had made about the much-referenced paper by Neil Ferguson and his team at Imperial College, published on March 16th, that seemed (with others, such as the
London School of Hygiene and Infectious Diseases) to have persuaded the UK Government towards a new approach in dealing with the pandemic, in mid to late March.
Possible intervention strategies in the fight against Coronavirus
The thrust of this National Geographic article, by Cathleen O’Grady, was that we will need “herd immunity” at some stage, even if the Imperial College paper of March 16th (and other SAGE Committee
advice, including from the Scientific Pandemic Influenza Group on Modelling (SPI-M)) had persuaded the Government to enforce several social distancing measures, and by March 23rd, a combination of
measures known as UK “lockdown”, apparently abandoning the herd immunity approach.
The UK Government said that herd immunity had never been a strategy, even though it had been mentioned several times, in the Government daily public/press briefings, by Sir Patrick Vallance (UK Chief
Scientific Adviser (CSA)) and Prof Chris Whitty (UK Chief Medical Officer (CMO)), the co-chairs of SAGE.
The particular part of the 16th March Imperial College paper I had queried with them a couple of weeks ago was this table, usefully colour coded (by them) to allow the relative effectiveness of the
potential intervention measures in different combinations to be assessed visually.
PC=school and university closure, CI=home isolation of cases, HQ=household quarantine, SD=large-scale general population social distancing, SDOL70=social distancing of those over 70 years for 4
months (a month more than other interventions)
Why was it, I wondered, that in this chart (on the very last page of the paper, and referenced within it) the effectiveness of the three measures “CI_HQ_SD” in combination (home isolation of cases,
household quarantine & large-scale general population social distancing) taken together (orange and yellow colour coding), was LESS than the effectiveness of either CI_HQ or CI_SD taken as a pair of
interventions (mainly yellow and green colour coding)?
The explanation for this was along the following lines.
It’s a dynamical phenomenon. Remember mitigation is a set of temporary measures. The best you can do, if measures are temporary, is go from the “final size” of the unmitigated epidemic to a size
which just gives herd immunity.
If interventions are “too” effective during the mitigation period (like CI_HQ_SD), they reduce transmission to the extent that herd immunity isn’t reached when they are lifted, leading to a
substantial second wave. Put another way, there is an optimal effectiveness of mitigation interventions which is <100%.
That is CI_HQ_SDOL70 for the range of mitigation measures looked at in the report (mainly a green shaded column in the table above).
While, for suppression, one wants the most effective set of interventions possible.
All of this is predicated on people gaining immunity, of course. If immunity isn’t relatively long-lived (>1 year), mitigation becomes an (even) worse policy option.
Herd Immunity
The impact of very effective lockdown on immunity in subsequent phases of lockdown relaxation was something I hadn’t included in my own (single phase) modelling. My model can only (at the moment)
deal with one lockdown event, with a single-figure, averaged intervention effectiveness percentage starting at that point. Prior data is used to fit the model. It has served well so far, until the
point (we have now reached) at which lockdown relaxations need to be modelled.
But in my outlook, potentially, to modelling lockdown relaxation, and the potential for a second (or multiple) wave(s), I had still been thinking only of higher % intervention effectiveness being
better, without taking into account that negative feedback to the herd immunity characteristic, in any subsequent more relaxed phase, other than through the effect of the changing comparative
compartment sizes in the SIR-style model differential equations.
I covered the 3-compartment SIR model in my blog post on April 8th, which links to my more technical derivation here, and more complex models (such as the Alex de Visscher 7-compartment model I use
in modified form, and that I described on April 14th) that are based on this mathematical model methodology.
In that respect, the ability for the epidemic to reproduce, at a given time “t” depends on the relative sizes of the infected (I) vs. the susceptible (S, uninfected) compartments. If the R
(recovered) compartment members don’t return to the S compartment (which would require a SIRS model, reflecting waning immunity, and transitions from R back to the S compartment) then the ability of
the virus to find new victims is reducing as more people are infected. I discussed some of these variations in my post here on March 31st.
My method might have been to reduce the % intervention effectiveness from time to time (reflecting the partial relaxation of some lockdown measures, as Governments are now doing) and reimpose it to a
higher % effectiveness if and when the R[t] (the calculated R value at some time t into the epidemic) began to get out of control. For example, I might relax lockdown effectiveness from 90% to 70%
when R[t] reached R[t]<0.7, and increase again to 90% when R[t] reached R[t]>1.2.
This was partly owing to the way the model is structured, and partly to the lack of disaggregated data I would have available to me for populating anything more sophisticated. Even then, the
mathematics (differential equations) of the cyclical modelling was going to be a challenge.
In the Imperial College paper, which does model the potential for cyclical peaks (see below), the “trigger” that is used to switch on and off the various intervention measures doesn’t relate to R[t],
but to the required ICU bed occupancy. As discussed above, the intervention effectiveness measures are a much more finely drawn range of options, with their overall effectiveness differing both
individually and in different combinations. This is illustrated in the paper (a slide presented in the April 17th Cambridge Conversation I reported in my blog article on Model Refinement on April
What is being said here is that if we assume a temporary intervention, to be followed by a relaxation in (some of) the measures, the state in which the population is left with regard to immunity at
the point of change is an important by-product to be taken into account in selecting the (combination of) the measures taken, meaning that the optimal intervention for the medium/long term future
isn’t necessarily the highest % effectiveness measure or combined set of measures today.
The phrase “herd immunity” has been an ugly one, and the public and press winced somewhat (as I did) when it was first used by Sir Patrick Vallance; but it is the standard term for what is often the
objective in population infection situations, and the National Geographic articles are a useful reminder of that, to me at least.
The arithmetic of herd immunity, the R number and the doubling period
I covered the relevance and derivation of the R[0] reproduction number in my post on SIR (Susceptible-Infected-Recovered) models on April 8th.
In the National Geographic paper by Cathleen O’Grady, a useful rule of thumb was implied, regarding the relationship between the herd immunity percentage required to control the growth of the
epidemic, and the much-quoted R[0] reproduction number, interpreted sometimes as the number of people (in the susceptible population) one infected person infects on average at a given phase of the
epidemic. When R[t] reaches one or less, at a given time t into the epidemic, so that one person is infecting one or fewer people, on average, the epidemic is regarded as having stalled and to be
under control.
Herd immunity and R[0]
One example given was measles, which was stated to have a possible starting R[0] value of 18, in which case almost everyone in the population needs to act as a buffer between an infected person and a
new potential host. Thus, if the starting R[0] number is to be reduced from 18 to R[t]<=1, measles needs A VERY high rate of herd immunity – around 17/18ths, or ~95%, of people needing to be immune
(non-susceptible). For measles, this is usually achieved by vaccine, not by dynamic disease growth. (Dr Fauci had mentioned over 95% success rate in the US previously for measles in the reported
quotation above).
Similarly, if Covid-19, as seems to be the case, has a lower starting infection rate (R[0] number) than measles, nearer to between 2 and 3 (2.5, say (although this is probably less than it was in the
UK during March; 3-4 might be nearer, given the epidemic case doubling times we were seeing at the beginning*), then the National Geographic article says that herd immunity should be achieved when
around 60 percent of the population becomes immune to Covid-19. The required herd immunity H% is given by H% = (1 – (1/2.5))*100% ~= 60%.
Whatever the real Covid-19 innate infectivity, or reproduction number R[0] (but assuming R[0]>1 so that we are in an epidemic situation), the required herd immunity H% is given by:
H%=(1-(1/R[0]))*100% (1)
(*I had noted that 80% was referenced by Prof. Chris Whitty (CMO) as loose talk, in an early UK daily briefing, when herd immunity was first mentioned, going on to mention 60% as more reasonable (my
words). 80% herd immunity would correspond to R[0]=5 in the formula above.)
R[0] and the Doubling time
As a reminder, I covered the topic of the cases doubling time T[D ]here; and showed how it is related to R[0] by the formula;
R[0]=d(log[e]2)/T[D ] (2)
where d is the disease duration in days.
Thus, as I said in that paper, for a doubling period T[D ]of 3 days, say, and a disease duration d of 2 weeks, we would have R[0]=14×0.7/3=3.266.
If the doubling period were 4 days, then we would have R[0]=14×0.7/4=2.45.
As late as April 2nd, Matt Hancock (UK secretary of State for Health) was saying that the doubling period was between 3 and 4 days (although either 3 or 4 days each leads to quite different outcomes
in an exponential growth situation) as I reported in my article on 3rd April. The Johns Hopkins comparative charts around that time were showing the UK doubling period for cases as a little under 3
days (see my March 24th article on this topic, where the following chart is shown.)
In my blog post of 31st March, I reported a BBC article on the epidemic, where the doubling period for cases was shown as 3 days, but for deaths it was between 2 and 3 days ) (a Johns Hopkins
University chart).
Doubling time and Herd Immunity
Doubling time, T[D](t) and the reproduction number, R[t] can be measured at any time t during the epidemic, and their measured values will depend on any interventions in place at the time, including
various versions of social distancing. Once any social distancing reduces or stops, then these measured values are likely to change – T[D] downwards and R[t] upwards – as the virus finds it
relatively easier to find victims.
Assuming no pharmacological interventions (e.g. vaccination) at such time t, the growth of the epidemic at that point will depend on its underlying R[0] and duration d (innate characteristics of the
virus, if it hasn’t mutated**) and the prevailing immunity in the population – herd immunity.
(**Mutation of the virus would be a concern. See this recent paper (not peer reviewed)
The doubling period T[D](t) might, therefore, have become higher after a phase of interventions, and correspondingly R[t] < R[0], leading to some lockdown relaxation; but with any such interventions
reduced or removed, the subsequent disease growth rate will depend on the interactions between the disease’s innate infectivity, its duration in any infected person, and how many uninfected people it
can find – i.e. those without the herd immunity at that time.
These factors will determine the doubling time as this next phase develops, and bearing these dynamics in mind, it is interesting to see how all three of these factors – T[D](t), R[t] and H(t) –
might be related (remembering the time dependence – we might be at time t, and not necessarily at the outset of the epidemic, time zero).
Eliminating R from the two equations (1) and (2) above, we can find:
H=1-T[D]/d(log[e]2) (3)
So for doubling period T[D]=3 days, and disease duration d=14 days, H=0.7; i.e. the required herd immunity H% is 70% for control of the epidemic. (In this case, incidentally, remember from equation
(2) that R[0]=14×0.7/3=3.266.)
(Presumably this might be why Dr Fauci would settle for a 70-75% effective vaccine (the H% number), but that would assume 100% take-up, or, if less than 100%, additional immunity acquired by people
who have recovered from the infection. But that acquired immunity, if it exists (I’m guessing it probably would) is of unknown duration. So many unknowns!)
For this example with 14 day infection period d, and exploring the reverse implications by requiring R[t] to tend to 1 (so postulating in this way (somewhat mathematically pathologically) that the
epidemic has stalled at time t) and expressing equation (2) as:
T[D] (t)= d(log[e]2)/R[t] (4)
then we see that T[D](t)= 14*log[e](2) ~= 10 days, at this time t, for R[t]~=1.
Thus a sufficiently long doubling period, with the necessary minimum doubling period depending on the disease duration d (14 days in this case), will be equivalent to the R[t] value being low enough
for the growth of the epidemic to be controlled – i.e. R[t] <=1 – so that one person infects one or less people on average.
Confirming this, equation (3) tells us, for the parameters in this (somewhat mathematically pathological) example, that with T[D](t)=10 and d=14,
H(t) = 1 – (10/14*log[e](2)) ~= 1-1 ~= 0, at this time t.
In this situation, the herd immunity H(t) (at this time t) required is notionally zero, as we are not in epidemic conditions (R[t]~=1). This is not to say that the epidemic cannot restart – it simply
means that if these conditions are maintained, with R[t] reducing to 1, and the doubling period being correspondingly long enough, possibly achieved through social distancing (temporarily), across
whole or part of the population (which might be hard to sustain) then we are controlling the epidemic.
It is when the interventions are reduced, or removed altogether that the sufficiency of % herd immunity in the population will be tested, as we saw from the Imperial College answer to my question
earlier. As they say in their paper:
“Once interventions are relaxed (in the example in Figure 3, from September onwards), infections begin to rise, resulting in a predicted peak epidemic later in the year. The more successful a
strategy is at temporary suppression, the larger the later epidemic is predicted to be in the absence of vaccination, due to lesser build-up of herd immunity.“
Herd immunity summary
Usually herd immunity is achieved through vaccination (eg the MMR vaccination for Rubella, Mumps and Measles). It involves less risk than the symptoms and possible side-effects of the disease itself
(for some diseases at least, if not for chicken-pox, for which I can recall parents hosting chick-pox parties to get it over and done with!)
The issue, of course, with Covid-19, is that no-one knows yet if such a vaccine can be developed, if it would be safe for humans, if it would work at scale, for how long it might confer immunity, and
what the take-up would be.
Until a vaccine is developed, and until the duration of any CoVid-19 immunity (of recovered patients) is known, this route remains unavailable.
Hence, as the National Geographic article says, there is continued focus on social distancing, as an effective part of even a somewhat relaxed lockdown, to control transmission of the virus.
Is there an uptick in the UK?
All of the above context serves as a (lengthy) introduction to why I am monitoring the published figures at the moment, as the UK has been (informally as well as formally) relaxing some aspects of it
lockdown, imposed on March 23rd, but with gradual changes since about the end of May, both in the public’s response and in some of the Government interventions.
My own forecasting model (based on the Alex de Visscher MatLab code, and my variations, implemented in the free Octave version of the MatLab code-base) is still tracking published data quite well,
although over the last couple of weeks I think the published rate of deaths is slightly above other forecasts, as well as my own.
Worldometers forecast
The Worldometers forecast is showing higher forecast deaths in the UK than when I reported before – 47924 now vs. 43,962 when I last posted on this topic on June 11th:
Worldometers UK deaths forecast based on Current projection scenario by Oct 1, 2020
My forecasts
The equivalent forecast from my own model still stands at 44,367 for September 30th, as can be seen from the charts below; but because we are still near the weekend, when the UK reported numbers are
always lower, owing to data collection and reporting issues, I shall wait a day or two before updating my model to fit.
But having been watching this carefully for a few weeks, I do think that some unconscious public relaxation of social distancing in the fairer UK weather (in parks, on demonstrations and at beaches,
as reported in the press since at least early June) might have something to do with a) case numbers, and b) subsequent numbers of deaths not falling at the expected rate. Here are two of my own
charts that illustrate the situation.
In the first chart, we see the reported and modelled deaths to Sunday 28th June; this chart shows clearly that since the end of May, the reported deaths begin to exceed the model prediction, which
had been quite accurate (even slightly pessimistic) up to that time.
Model vs. reported deaths, linear scale, to June 28th 2020
In the next chart, I show the outlook to September 30th (comparable date to the Worldometers chart above) showing the plateau in deaths at 44,367 (cumulative curve on the log scale). In the daily
plots, we can see clearly the significant scatter (largely caused by weekly variations in reporting at weekends) but with the daily deaths forecast to drop to very low numbers by the end of
Model vs. reported deaths, log scale, cumulative and daily, to Sep 30th 2020
I will update this forecast in a day or two, once this last weekend’s variations in UK reporting are corrected.
Reported Deaths vs. Modelled deaths, both cumulative and daily,
This post covers the current status of my UK Coronavirus (SARS-CoV-2) model, stating the June 2nd position, and comparing with an update on June 3rd, reworking my UK SARS-CoV-2 model with 83.5%
intervention effectiveness (down from 84%), which reduces the transmission rate to 16.5% of its pre-intervention value (instead of 16%), prior to the 23rd March lockdown.
This may not seem a big change, but as I have said before, small changes early on have quite large effects later. I did this because I see some signs of growth in the reported numbers, over the last
few days, which, if it continues, would be a little concerning.
I sensed some urgency in the June 3rd Government update, on the part of the CMO, Chris Whitty (who spoke at much greater length than usual) and the CSA, Sir Patrick Vallance, to highlight the
continuing risk, even though the UK Government is seeking to relax some parts of the lockdown.
They also mentioned more than once that the significant “R” reproductive number, although less than 1, was close to 1, and again I thought they were keen to emphasise this. The scientific and medical
concern and emphasis was pretty clear.
These changes are in the context of quite a bit of debate around the science between key protagonists, and I begin with the background to the modelling and data analysis approaches.
Curve fitting and forecasting approaches
Curve-fitting approach
I have been doing more homework on Prof. Michael Levitt’s Twitter feed, where he publishes much of his latest work on Coronavirus. There’s a lot to digest (some of which I have already reported, such
as his EuroMOMO work) and I see more methodology to explore, and also lots of third party input to the stream, including Twitter posts from Prof. Sir David Spiegelhalter, who also publishes on Medium
I DO use Twitter, although a lot less nowadays than I used to (8.5k tweets over a few years, but not at such high rate lately); much less is social nowadays, and more is highlighting of my https://
www.briansutton.uk/ blog entries.
Core to that work are Michael’s curve fitting methods, in particular regarding the Gompertz cumulative distribution function and the Change Ratio / Sigmoid curve references that Michael describes.
Other functions are also available(!), such as The Richard’s function.
This curve-fitting work looks at an entity’s published data regarding cases and deaths (China, the Rest of the World and other individual countries were some important entities that Michael has
analysed) and attempts to fit a postulated mathematical function to the data, first to enable a good fit, and then for projections into the future to be made.
This has worked well, most notably in Michael’s work in forecasting, in early February, the situation in China at the end of March. I reported this on March 24th when the remarkable accuracy of that
forecast was reported in the press:
The Times coverage on March 24th of Michael Levitt’s accurate forecast for China
Forecasting approach
Approaching the problem from a slightly different perspective, my model (based on a model developed by Prof. Alex de Visscher at Concordia University) is a forecasting model, with my own parameters
and settings, and UK data, and is currently matching death rate data for the UK, on the basis of Government reported “all settings” deaths.
The model is calibrated to fit known data as closely as possible (using key parameters such as those describing virus transmission rate and incubation period, and then solves the Differential
Equations, describing the behaviour of the virus, to arrive at a predictive model for the future. No mathematical equation is assumed for the charts and curve shapes; their behaviour is constructed
bottom-up from the known data, postulated parameters, starting conditions and differential equations.
The model solves the differential equations that represent an assumed relationship between “compartments” of people, including, but not necessarily limited to Susceptible (so far unaffected),
Infected and Recovered people in the overall population.
I had previously explored such a generic SIR model, (with just three such compartments) using a code based on the Galbraith solution to the relevant Differential Equations. My following post article
on the Reproductive number R0 was set in the context of the SIR (Susceptible-Infected-Recovered) model, but my current model is based on Alex’s 7 Compartment model, allowing for graduations of
sickness and multiple compartment transition routes (although NOT with reinfection).
SEIR models allow for an Exposed but not Infected phase, and SEIRS models add a loss of immunity to Recovered people, returning them eventually to the Susceptible compartment. There are many such
options – I discussed some in one of my first articles on SIR modelling, and then later on in the derivation of the SIR model, mentioning a reference to learn more.
Although, as Michael has said, the slowing of growth of SARS-CoV-2 might be because it finds it hard to locate further victims, I should have thought that this was already described in the
Differential Equations for SIR related models, and that the compartment links in the model (should) take into account the effect of, for example, social distancing (via the effectiveness % parameter
in my model). I will look at this further.
The June 2nd UK reported and modelled data
Here are my model output charts exactly up to, June 2nd, as of the UK Government briefing that day, and they show (apart from the last few days over the weekend) a very close fit to reported death
data**. The charts are presented as a sequence of slides:
These charts all represent the same UK deaths data, but presented in slightly different ways – linear and log y-axes; cumulative and daily numbers; and to date, as well as the long term outlook. The
current long term outlook of 42,550 deaths in the UK is within error limits of the the Worldometers linked forecast of 44,389, presented at https://covid19.healthdata.org/united-kingdom, but is not
modelled on it.
**I suspected that my 84% effectiveness of intervention would need to be reduced a few points (c. 83.5%) to reflect a little uptick in the UK reported numbers in these charts, but I waited until
midweek, to let the weekend under-reporting work through. See the update below**.
I will also be interested to see if that slight uptick we are seeing on the death rate in the linear axis charts is a consequence of an earlier increase in cases. I don’t think it will be because of
the very recent and partial lockdown relaxations, as the incubation period of the SARS-CoV-2 virus means that we would not see the effects in the deaths number for a couple of weeks at the earliest.
I suppose, anecdotally, we may feel that UK public response to lockdown might itself have relaxed a little over the last two or three weeks, and might well have had an effect.
The periodic scatter of the reported daily death numbers around the model numbers is because of the reguar weekend drop in numbers. Reporting is always delayed over weekends, with the ground caught
up over the Monday and Tuesday, typically – just as for 1st and 2nd June here.
A few numbers are often reported for previous days at other times too, when the data wasn’t available at the time, and so the specific daily totals are typically not precisely and only deaths on that
particular day.
The cumulative charts tend to mask these daily variations as the cumulative numbers dominate small daily differences. This applies to the following updated charts too.
**June 3rd update for 83.5% intervention effectiveness
I have reworked the model for 83.5% intervention effectiveness, which reduces the transmission rate to 16.5% of its starting value, prior to 23rd March lockdown. Here is the equivalent slide set, as
of 3rd June, one day later, and included in this post to make comparisons easier:
These charts reflect the June 3rd reported deaths at 39,728 and daily deaths on 3rd June of 359. The model long-term prediction is 44,397 deaths in this scenario, almost exactly the Worldometer
forecast illustrated above.
We also see the June 3rd reported and modelled cumulative numbers matching, but we will have to watch the growth rate.
Concluding remarks
I’m not as concerned to model cases data as accurately, because the reported numbers are somewhat uncertain, collected as they are in different ways by four Home Countries, and by many different
regions and entities in the UK, with somewhat different definitions.
My next steps, as I said, are to look at the Sigmoid and data fitting charts Michael uses, and compare the same method to my model generated charts.
*NB The UK Office for National Statistics (ONS) has been working on the Excess Deaths measure, amongst other data, including deaths where Covid-19 is mentioned on the death certificate, not requiring
a positive Covid-19 test as the Government numbers do.
As of 2nd June, the Government announced 39369 deaths in its standard “all settings” – Hospitals, Community AND Care homes (with a Covid-19 test diagnosis) but the ONS are mentioning 62,000 Excess
Deaths today. A little while ago, on the 19th May, the ONS figure was 55,000 Excess Deaths, compared with 35,341 for the “all settings” UK Government number. I reported that in my blog post https://
www.briansutton.uk/?p=2302 in my EuroMOMO data analysis post.
But none of the ways of counting deaths is without its issues. As the King’s Fund says on their website, “In addition to its direct impact on overall mortality, there are concerns that the Covid-19
pandemic may have had other adverse consequences, causing an increase in deaths from other serious conditions such as heart disease and cancer.
“This is because the number of excess deaths when compared with previous years is greater than the number of deaths attributed to Covid-19. The concerns stem, in part, from the fall in numbers of
people seeking health care from GPs, accident and emergency and other health care services for other conditions.
“Some of the unexplained excess could also reflect under-recording of Covid-19 in official statistics, for example, if doctors record other causes of death such as major chronic diseases, and not
Covid-19. The full impact on overall and excess mortality of Covid-19 deaths, and the wider impact of the pandemic on deaths from other conditions, will only become clearer when a longer time series
of data is available.”
Owing to the serendipity of a contemporary and friend of mine at King’s College London, Andrew Ennis, wishing one of HIS contemporaries in Physics, Michael Levitt, a happy birthday on 9th May, and
mentioning me and my Coronavirus modelling attempts in passing, I am benefiting from another perspective on Coronavirus from Michael Levitt.
The difference is that Prof. Michael Levitt is a Nobel laureate in 2013 in computational biosciences…and I’m not! I’m not a Fields Medal winner either (there is no Nobel Prize for Mathematics, the
Fields Medal being an equivalently prestigious accolade for mathematicians). Michael is Professor of Structural Biology at the Stanford School of Medicine.
I did win the Drew Medal for Mathematics in my day, but that’s another (lesser) story!
Michael has turned his attention, since the beginning of 2020, to the Coronavirus pandemic, and had kindly sent me a number of references to his work, and to his other recent work in the field.
I had already referred to Michael in an earlier blog post of mine, following a Times report of his amazingly accurate forecast of the limits to the epidemic in China (in which he was taking a
particular interest).
Report of Michael Levitt’s forecast for the China outbreak
I felt it would be useful to report on the most recent of the links Michael sent me regarding his work, the interview given to Freddie Sayers of UnHerd at https://unherd.com/thepost/
nobel-prize-winning-scientist-the-covid-19-epidemic-was-never-exponential/ reported on May 2nd. I have added some extracts from UnHerd’s coverage of this interview, but it’s better to watch the
Michael’s interview with UnHerd
As UnHerd’s report says, “With a purely statistical perspective, he has been playing close attention to the Covid-19 pandemic since January, when most of us were not even aware of it. He first spoke
out in early February, when through analysing the numbers of cases and deaths in Hubei province he predicted with remarkable accuracy that the epidemic in that province would top out at around 3,250
“His observation is a simple one: that in outbreak after outbreak of this disease, a similar mathematical pattern is observable regardless of government interventions. After around a two week
exponential growth of cases (and, subsequently, deaths) some kind of break kicks in, and growth starts slowing down. The curve quickly becomes ‘sub-exponential’.
UnHerd reports that he takes specific issue with the Neil Ferguson paper, that along with some others, was of huge influence with the UK Government (amongst others) in taking drastic action, moving
away from a ‘herd immunity” approach to a lockdown approach to suppress infection transmission.
“In a footnote to a table it said, assuming exponential growth of 15% for six days. Now I had looked at China and had never seen exponential growth that wasn’t decaying rapidly.
“The explanation for this flattening that we are used to is that social distancing and lockdowns have slowed the curve, but he is unconvinced. As he put it to me, in the subsequent examples to China
of South Korea, Iran and Italy, ‘the beginning of the epidemics showed a slowing down and it was very hard for me to believe that those three countries could practise social distancing as well as
China.’ He believes that both some degree of prior immunity and large numbers of asymptomatic cases are important factors.
“He disagrees with Sir David Spiegelhalter’s calculations that the totem is around one additional year of excess deaths, while (by adjusting to match the effects seen on the quarantined Diamond
Princess cruise ship, and also in Wuhan, China) he calculates that it is more like one month of excess death that is need before the virus peters out.
“He believes the much-discussed R0 is a faulty number, as it is meaningless without the time infectious alongside.” I discussed R0 and its derivation in my article about the SIR model and R0.
Interestingly, Prof Alex Visscher, whose original model I have been adapting for the UK, also calibrated his thinking, in part, by considering the effect of the Coronavirus on the captive, closed
community on the Diamond Princess, as I reported in my Model Update on Coronavirus on May 8th.
The UnHerd article finishes with this quote: “I think this is another foul-up on the part of the baby boomers. I am a real baby boomer — I was born in 1947, I am almost 73 years old — but I think
we’ve really screwed up. We’ve caused pollution, we’ve allowed the world’s population to increase threefold in my lifetime, we’ve caused the problems of global warming and now we’ve left your
generation with a real mess in order to save a relatively small number of very old people.”
I suppose, as a direct contemporary, that I should apologise too.
There’s a lot more at the UnHerd site, but better to hear it directly from Michael in the video.
Deaths reported vs. model log graph
Introduction and summary
This is a brief update to my UK model predictions in the light of a week’s published data regarding Covid-19 cases and deaths in all settings – hospitals, care homes and the community – rather than
just hospitals and the community, as previously.
In order to get the best fit between the model and the published data, I have had to reduce the effectiveness of interventions (lockdown, social distancing, home working etc) from 85% last week ( in
my post immediately following the Government change of reporting basis) to 84.1% at present.
This reflects the fact that care homes, new to the numbers, seem to influence the critical R0 number upwards on average, and it might be that R0 is between .7 and .9, which is uncomfortably near to
1. It is already higher in hospitals than in the community, but the care home figures in the last week have increased R0 on average. See my post on the SIR model and importance of R0 to review the
meaning of R0.
Predicted cases are now at 2.8 million (not reflecting the published data, but an estimate of the underlying real cases) with fatalities at 42,000.
Possible model upgrades
The Government have said that they are to sample people randomly in different settings (hospital, care homes and the community), and regionally, better to understand how the transmission rate, and
the influence on the R0 reproductive number, differs in those settings, and also in different parts of the UK.
Ideally a model would forecast the pandemic growth on the basis of these individually, and then aggregate them, and I’m sure the Government advisers will be doing that. As for my model, I am
adjusting overall parameters for the whole population on an average basis at this point.
Another model upgrade which has already been made by academics at Imperial College and at Harvard is to explore the cyclical behaviour of partial relaxations of the different lockdown components, to
model the response of the pandemic to these (a probable increase in growth to some extent) and then a re-tightening of lockdown measures to cope with that, followed by another fall in transmission
rates; and then repeating this loop into 2021 and 2022, showing a cyclical behaviour of the pandemic (excluding any pharmaceutical (e.g. vaccine and medicinal) measures). I covered this in my
previous article on exit strategy.
This explains Government reluctance to promise any significant easing of lockdown in any specific timescales.
Current predictions
My UK model (based on the work of Prof. Alex Visscher at Concordia University in Montreal for other countries) is calibrated on the most accurate published data up to the lockdown date, March 23rd,
which is the data on daily deaths in the UK.
Once that fit of the model to the known data has been achieved, by adjusting the assumed transmission rates, the data for deaths after lockdown – the intervention – is matched by adjusting parameters
reflecting the assumed effectiveness of the intervention measures.
Data on cases is not so accurate by a long way, and examples from “captive” communities indicate that deaths vs. cases run at about 1.5% (e.g. the Diamond Princess cruise ship data).
The Italy experience also plays into this relationship between deaths and actual (as opposed to published) case numbers – it is thought that a) only a single figure percentage of people ever get
tested (8% was Alex’s figure), and b) again in Italy, the death rate was probably higher than 1.5% because their health service couldn’t cope for a while, with insufficient ICU provision.
In the model, allowing for that 8%, a factor of 12.5 is applied to public total and active cases data, to reflect the likely under-reporting of case data, since there are relatively few tests.
In the model, once the fit to known data (particularly deaths to date) is made as close as possible, then the model is run over whatever timescale is desired, to look at its predictions for cases and
deaths – at present a short-term forecast to June 2020, and a longer term outlook well into 2021, by when outcomes in the model have stabilised.
Model charts for deaths
The model fit is accurate to date, and the outlook prediction is for 42,000 deaths
The fit of the model here can be managed well, post lockdown, by adjusting the percentage effectiveness of the intervention measure, and this is currently set at 84.1%. This model predicts fatalities
in the UK at 42,000. They are reported currently (8th May 2020) at 31241.
Model charts for cases
The outcome prediction is for 2.8 million cases in the UK when the pandemic stabilises
As we can see here, the fit for cases isn’t as good, but the uncertainty in case number reporting accuracy, owing to the low level of testing, and the variable experience from other countries such as
Italy, means that this is an innately less reliable basis for forecasting. The model prediction for the outcome of UK case numbers is 2.8 million.
If testing, tracking and tracing is launched effectively in the UK, then this would enable a better basis for predictions for case numbers than we currently have.
I’m certainly not at a concluding stage yet. A more complex model is probably necessary to predict the situation, once variations to the current lockdown measures begin to happen, likely over the
coming month or two in the first instance.
Models are being developed and released by research groups, such as that being developed by the RAMP initiative at https://epcced.github.io/ramp/
Academics from many institutions are involved, and I will take a look at the models being released to see if they address the two points I mentioned here: the variability of R0 across settings and
geography, and the cyclical behaviour of the pandemic in response to lockdown variations.
At the least, perhaps, my current model might be enhanced to allow a time-dependent interv_success variable, instead of a constant lockdown effectiveness representation.
Change of reporting basis
The UK Government yesterday changed the reporting basis for Coronavirus numbers, retrospectively (since 6th March 2020) adding in deaths in the Care Home and and other settings, and also modifying
the “Active Cases” to match, and so I have adjusted my model to match.
This historic information is more easily found on the Worldometer site; apart from current day numbers, it is harder to find the tabular data on the UK.gov site, and I guess Worldometers have a
reliable web services feed from most national reporting web pages.
The increase in daily and cumulative deaths over the period contrasts with a slight reduction in daily active case numbers over the period.
Understanding the variations in epidemic parameters
With more resources, it would make sense to model different settings separately, and then combine them. If (as it is) the reproduction number R[0]<1 for the community, the population at large
(although varying by location, environment etc), but higher in hospitals, and even higher in Care Homes, then these scenarios would have different transmission rates in the model, different
effectiveness of counter-measures, and differences in several other parameters of the model(s). Today the CSA (Sir Patrick Vallance) stated that indeed, there is to be a randomised survey of people
in different places (geographically) and situations (travel, work etc) to work out where the R-value is in different parts of the population.
But I have continued with the means at my disposal (the excellent basis for modelling in Alex Visscher’s paper that I have been using for some time).
Ultimately, as I said I my article at https://www.briansutton.uk/?p=1595, a multi-phase model will be needed (as per Imperial College and Harvard models illustrated here:-
Repeated peaks with no pharmaceutical intervention
and I am sure that it is the Imperial College version of this (by Neil Ferguson and his team) that will be to the forefront in that advice. The models looks at variations in policy regarding
different aspects of the lockdown interventions, and the response of the epidemic to them. This leads to the cyclicity illustrated above.
Model adjustments
In my model, the rate of deaths is the most accurately available data, (even though the basis for reporting it has just changed) and the model fit is based on that. I have incorporated that reporting
update into the model.
Up to lockdown (March 23rd in the UK, day 51), an infection transmission rate k[11] (rate of infection of previously uninfected people by those in the infected compartment) and a correction factor
are used to get this fit for the model as close as possible prior to the intervention date. For example, k[11] can be adjusted, as part of a combination of infection rates; k[12] from sick (S)
people, k[13] from seriously sick (SS) people and k[14] from recovering (B, better) people to the uninfected community (U). All of those sub-rates could be adjusted in the model, and taken together
define the overall rate of transition of people from from Unifected to Infected.
After lockdown, the various interventions – social distancing, school and large event closures, restaurant and pub closures and all the rest – are represented by an intervention effectiveness
percentage, and this is modified (as an average across all those settings I mentioned before) to get the fit of the model after the lockdown measures as close as possible to the reported data, up to
the current date.
I had been using an intervention effectiveness of 90% latterly, as the UK community response to the Government’s advice has been pretty good.
But with the UK Government move to include data from other settings (particularly the Care Home setting) I have had to reduce that overall percentage to 85% (having modelled several options from 80%
upwards) to match the increased reported historic death rate.
It is, of course, more realistic to include all settings in the reported numbers, and in fact my model was predicting on that basis at the start. Now we have a few more weeks of data, and all the
reported data, not just some of it, I am more confident that my original forecast for 39,000 deaths in the UK (for this single phase outlook) is currently a better estimate than the update I made a
week or so ago (with 90% intervention effectiveness) to 29,000 deaths in the Model Refinement article referred to above, when I was trying to fit just hospital deaths (having no other reference point
at that time).
Here are the charts for 85% intervention effectiveness; two for the long term outlook, into 2021, and two up until today’s date (with yesterday’s data):
Forecasts for UK deaths long term, and up to end April 2020
Another output would be for UK cases, and I’ll just summarise with these charts for all cases up until June 2020 (where the modelled case numbers just begin to level off in the model):-
All UK cases until June 30th 2020, linear and log charts
As we can see, the fit here isn’t as good, but this also reflects the fact that the data is less certain than for deaths, and is collected in many different ways across the UK, in the four home
countries, and in the conurbations, counties and councils that input to the figures. I will probably have to adjust the model again within a few days, but the outlook, long term, of the model is for
2.6 million cases of all types. We’ll see…
Outlook beyond the Lockdown – again
I’m modest about my forecasts, but the methodology shows me the kind of advice the Government will be getting. The behavioural “science” part of the advice (not in the model) taking the public
“tiredness” into account, was the reason for starting partial lockdown later, wasn’t it?
It would be more of the same if we pause the wrong aspects of lockdown too early for these reasons. Somehow the public have to “get” the rate of infection point into their heads, and that you can be
infecting people before you have symptoms yourself. The presentation of the R number in today’s Government update might help that awareness. My article on R[0] refers
Neil Ferguson of Imperial College was publishing papers at least as far back as 2006 on the mitigation of flu epidemics by such lockdown means, modelling very similar non-pharmaceutical methods of
controlling infection rates – social distancing, school closures, no public meetings and all the rest. Here is the 2006 paper, just one of 188 publications over the years by Ferguson and his team.
The following material is very recent, and, of course, focused on the current pandemic. https://www.imperial.ac.uk/…/Imperial-College-COVID19…
All countries would have been aware of this from the thinking around MERS, SARS and other outbreaks. We have a LOT of prepared models to fall back on.
As other commentators have said, Neil Ferguson has HUGE influence with the UK Government. I’m not sure how quickly UK scientists themselves were off the mark (as well as Government). We have moved
from “herd immunity” and “flattening the curve” as mantras, to controlling the rate of infection by the measures we currently have in place, the type of lockdown that other countries were already
using (in Europe, Italy did that two weeks before we did, although Government is saying that we did it earlier in the life of the epidemic here in the UK).
One or two advisory scientists have broken ranks (John Edmunds reported at https://www.reuters.com/…/special-report-johnson… ) on this to say that the various Committees should have been faster with
their firm advice to Govenment. Who knows?
But what is clear from the public pronouncements is that Governments are now VERY aware of the issue of further peaks in the epidemic, and I would be very surprised to see rapid or significant change
in the lockdown (it already allows some freedoms here in the UK, not there in some other countries, for example exercise outings once a day). I wouldn’t welcome being even more socially distanced
than others, as a fit 70+ year-old person, through the policy of shielding, but if it has to be done, so be it.
Solving log(x) = R(x-1) for a family of R values
In the recent daily UK Government presentations, the R0 Reproductive Number has been mentioned a few times, and with good reason. Its value is as a commonly accepted measure of the propensity of an
infectious disease outbreak to become an epidemic.
It turns out to be a relatively simple number to define, although working back from current data to calculate it is awkward if you don’t have the data. That’s my next task, from public data.
If R0 is below 1, then the epidemic will reduce and disappear. If it is greater than 1, then the epidemic grows.
The UK Government and its health advisers have made a few statements about it, and I covered these in an earlier post.
This is a more technical post, just to present a derivation of R0, and its consequences. I have used a few research papers to help with this, but the focus, brevity(!) and mistakes are mine (although
I have corrected some in the sources).
See it at The SIR model and R0 | {"url":"https://www.briansutton.uk/?cat=105","timestamp":"2024-11-13T18:36:11Z","content_type":"text/html","content_length":"431225","record_id":"<urn:uuid:95a4baf1-c277-40a0-8cab-cfc62f73115a>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00351.warc.gz"} |
Random vs. Non-Random Walk Theory in the Financial Markets - Forex Training Group
The book has been revised many times, and there is a new addition as recently as 2007. Malkiel’s random walk theory is based on the notion that returns produced by stocks are unpredictable and random
and therefore, a portfolio manager cannot produce consistent returns that will outperform the broader market. The book states that using different types of analysis will only lead to
underperformance, as there is no way to predict prices over the long term.
His conclusion, to his random walk model, is that an investor is better off purchasing an index fund that replicates the returns of the broader market, and use a buy and hold strategy. This is the
essence of Malkiel’s random walk hypothesis.
The Random walk theory asserts that stock price returns are efficient because all currently available information is reflected in the present price of a security and that movements are based purely
on traders’ sentiment which cannot be measured consistently.
When new information becomes available, the price of a security will quickly adjust and immediately reflect the new information. Since new information is random and unpredictable, there is market
randomness, and therefore the returns associated with prices are unpredictable, creating a random market.
Efficient Market Hypothesis
The Random Walk theory is predicated on the notion that the market is efficient, and that when new information becomes available to traders, they will react in a way to change the price to reflect
new information. This theory has some issues as not every market participant has the same motivation.
For example, a corporate treasurer and a hedge fund manager will have some different motivations as to when they should transact. While a hedge fund manager might shy away from a period when a
security price is tumbling, a corporate treasurer might be looking to use a large price drop to initiate a buyback program.
A corporate treasurer will also use derivative securities in different ways. For example, if a stock price is declining rapidly and a company has a buyback program, a corporate treasurer might use a
technique where he is selling puts below the market to enhance the program by receiving a premium.
In this situation, if a treasurer sells put options below the market, he can receive premium regardless of whether the strike price of the put option is reached. This type of motivation will alter
the way efficient market theory occurs, since the corporate treasurer, sees the market differently than a trader or portfolio manager.
Additionally, the time horizon used by traders can alter the efficiency of a market. Investors who are looking to hold stock for the long term will behave differently than those who are attempting to
day trade a security. For example, if you are dollar cost averaging, where you purchase a stock as it declines, your goal is different than the trader who is looking to capture small moves on both
long and short trades.
Algorithms Enhance Efficient Market Theory
The markets have changed considerably since the last version of a Random Walk Down Wall Street was written in 2007. Today, algorithms are a large part of short-term movements in nearly every capital
market. An algorithm is a computer program that looks for changes in information and immediately reacts by buying and selling securities. These securities could be stocks, currencies pairs, bonds or
even commodities.
High-frequency algorithmic trading strategies use computer algorithms that are trading thousands of times a day trying to influence the market as well as capture inefficiencies. High-frequency
traders made their first foray into the equity markets. New regulation allowed electronic exchanges to compete with one another, which left the door open for high-frequency traders to step in and
search for discrepancies in prices.
Today, algorithms use data that is accumulated from many different sources. Algorithms are scanning websites and the Twitter universe looking for keywords to determine how they should transact. A
simple term such as the “Feder Reserve Rate Hike” could set off a cascade of transactions, which could result in volatile market movements. Many of the recent flash crashes have been generated by
algorithms that quickly buy and sell securities and create a snowball effect when new market information becomes available.
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Algorithms also alter the distribution of stock returns. In general, the returns reflected in the capital markets are not normally distributed. What does this mean? For example, if you were to
measure the weight of 100 school children and plot the distribution you would likely see a classic bell shaped curve. The most recurring weight would be the middle and the remaining weights of these
children would be distributed on either side. Approximately 68 % would fall within 1-standard deviation of the middle and 95% would fall within 2-standard deviations.
There have been numerous studies that have found that returns of securities are not normally distributed and these returns have fat tails. This means that there will be a high number of returns that
are outside the normal distribution. Some might be lower and many might be higher.
Since algorithms are designed to take advantage of new information, their rapid reaction to new information generates returns that are not normally distributed. They are trained to do nothing when
there is no new information, providing little liquidity, but to erupt when there is new information generating volatile market conditions.
Non-Random Walk Theory
While the theory that Malkiel provides has merit, to the extent that he can make the argument that prices are random, there have been many portfolio managers who have outperformed the broader
markets. This means that a buy and hold methodology is not the best way to achieve risk-adjusted returns. As an example, over the past 20-years, Berkshire Hathaway has experienced 613% return on
capital compared to the S&P 500 index (excluding dividend) which has a return profile of 190%.
There have also been several papers and articles that have been written to counter the arguments made by Burton Malkiel, asserting that there is a non-random market. There is a collection of articles
called “A Non-Random Walk Down Wall Street” which offers evidence that the price of a stock offers valuable information.
The empirical data that was used was a series of econometric models that tested the randomness of prices. The non-random walk was composed by Andrew Lo, who is a non-random proponent, with a
conclusion that there are many techniques that can be used to beat the major averages, but the question remains for how long can these methodologies be successful. Lo said, “The more creativity you
bring to the investment process, the more rewarding it will be. The only way to maintain ongoing success, however, is to constantly innovate.” His thought process on beating the markets over the long
term was to change your methodology to adjust to market conditions constantly.
Test for Market Randomness
There are several tests that can be performed to determine if a data series is random. For example, a RUNS, test named after Abraham Wald and Jacob Wolfowitz, is a statistical methodology that
evaluates the randomness of a two or more-time series.
The runs test can determine if trends exist in a market and how often they occur. The null hypothesis is assumed, which means that there is no dependence and no trend that exists, and the populations
are identical in nature. The runs tests ranks the values and either proves the null hypothesis or suggests a trend.
Regression Analysis
Another way to determine if a variable is dependent on another variable is to run a regression analysis. A regression formula designates an independent and dependent variable as well as an R-squared,
which describes how dependent one variable is on another.
The simplest regression analysis uses a predictor variable and a response variable. The data points are reported using a least-squared method. If there are outliers in the data series that are
suspect, resistant methods can be used to fit the model. An R-squared of 1 means the dependent variable moves in tandem with the independent variable.
Correlation Analysis
Another technique that is used to determine the non-random nature of securities is correlation analysis. Correlation is like regression in that you are using multiple time series to determine if the
returns move in tandem.
The analysis evaluates the returns of one time series relative to another and provides you with a correlation coefficient between 1 and -1. A correlation coefficient of 1 means that the returns of
the 2-time series move together in lockstep. A correlation coefficient of -1 means that the returns of the 2-time series move in the opposite directions. When you are evaluating the relationship, it
is important to analyze the returns as opposed to the price.
Although correlation does not imply that the movement of one security is dependent on another security it does show that the movements of the two securities are related to one another.
The higher the correlation coefficient, the more closely the performance of the two assets are to one another. Correlation coefficients of 70 or negative 70 mean that the assets have a significant
correlation or negative correlation.
An example of how to use correlation is to find an asset that might influence another asset. For example, a country like Canada has a significant number of oil companies that employ millions of
people. The Canadian economy relies heavily on these companies and these companies rely heavily on the price of oil to ensure profitability. When the price of oil falls dramatically, as it did during
the first half of 2015, the economies in countries like Canada face significant headwinds.
A correlation analysis can be performed over many different periods. You can perform correlation analysis over one long period, such as 1-year or over rolling periods. The number that you see over a
1-year period will incorporate the aggregate correlation period, but will not show you the nuances of how the correlation changes over specific time horizons. For example, the USD/CAD might have a
correlation coefficient of -0.80 over a year but might range between -1 and -0.20 during different 20-day timeframes over the course of a year.
Technical analysis is widely used to determine the future direction of a security. There has been some empirical work done that suggests that technical analysis can be used to outperform the broader
Many technicians believe that they can predict future price movements using historical data points. For example, technical analysts believe that all the available information is currently in the
price of a security.
With this as a backdrop, you can only determine the future price by using studies or patterns, as past price action forecasts future price movements. At the very least, technical analysis can be used
as a self-fulfilling prophecy.
If many people use technical analysis to determine the future price movements, it is important for you to understand technical analysis so you know what other might be thinking. In the following
section, we will discuss some basic types of technical tools that traders utilize to predict future movement.
Demand and Support Levels
The values of a security is based on changes to the supply and demand for that security. When an investor thinks that the price of a security is cheap relative to the market’s expectations, they will
buy the security in the hopes that it will climb in value. As demand for the stock increases, the price will find a pivot point where further declines will be unattainable. This is considered
There are many ways you can use technical analysis to determine support. Many traders use trend lines which connect swing lows to determine trend support. Upward sloping trend lines that connect
higher lows are a very popular way to find support levels in a bull market.
Supply and Resistance Levels
Resistance is the opposite of support. It is an area of supply that reflects market price action where prices have a difficult time moving higher. Pent up supply builds up at resistance. Like support
there are several ways you can determine resistance levels using technical analysis. You can use trend lines which connect swing highs, or you can use a horizontal trend line that also connects pivot
Moving Averages
Another technical methodology that is often employed to determine the future direction of a security is using moving averages to smooth the price action and help describe a path. A moving average is
an average of a specific number of days. When the next price is recorded, the first price is dropped from the calculation.
For example, if you are calculating the 10-day moving average of a security, you would average the first 10-days. On the 11^th day, you would drop the first day which would generate a new data point.
Moving Average Crossover
Moving averages can help you determine the future direction by using the popular crossover methodology. The moving average crossover helps you determine if there is a new emerging trend in the
security you are trading. If you are looking for a change over a short-term timeframe, then it is best for you to use short-term moving averages.
One of the more popular settings is the 5-day moving average cross above or below the 20-day moving average. This encapsulates a 1-week period and a 1-month period and is very helpful at catching
short-term trends. If you are looking for a longer period, you might consider the 20-day and 50-day moving averages.
A long-term moving average crossover such as the 50-day moving average and the 200-day moving average, is very popular and known as either the “golden cross” on a cross above, and the “death cross”
on the cross below. The moving average crossover is a robust way to define a trend.
As a technical analyst, I do not believe that the markets are random, and it is quite apparent that there are individuals that have track records that are better than the broader markets over a long
period. Warren Buffet’s performance, has easily beat the S&P 500 index by 423% over the past 20-years. While Buffet uses a fundamental approach to picking companies, there are numerous examples of
successful traders, who use statistical models, as well as, technical analysis to create robust and consistent returns.
There are also several statistical tools such as the runs test, regression and correlation, that show that there is dependence, and correlation between assets. The premise that all the available
information is currently priced into a security has merit, and it is also clear that there is a new paradigm, where algorithms trade information that is ubiquitous, including information that is
available on social media including Facebook and Twitter.
At the end of the day, even though the Random vs Non-Random proponents will continue their debate, I expect that the informed technical analyst who has a positive expectancy strategy will continue to
outperform the market. | {"url":"https://forextraininggroup.com/random-vs-non-random-walk-theory-financial-markets/","timestamp":"2024-11-06T23:10:38Z","content_type":"text/html","content_length":"380805","record_id":"<urn:uuid:9f858902-e139-4620-b84b-ee91b4cef48b>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00377.warc.gz"} |
Power Automate Split Function And Arrays
There are a variety of Power Automate string functions of string functions that we can use in our MS flows. In this tutorial, we’ll find out the usage and importance of another complicated string
function – the Power Automate split function.
The split function simply returns an array of items based on the provided string and a delimiter.
How The Power Automate Split Function Works
We need to define its parameters (text and delimiter) for this function to properly work.
To explain it further, let’s say our text is “the apple falls far from the tree”. Then the delimiter is just a simple space.
By using the split function, we’ll have an array that contains all the words within the text/string that we specified. An array is a collection of items.
Using The Power Automate Split Function
Let’s now have a sample use case to understand why we would want to do this. We’ll add another input in this sample flow that I previously created. Let’s set that as Input 4 then type in “Please
enter the message that you want to cascade”.
Then, let’s use the split function on our Message text. Under the Expressions tab, choose the split function.
For the text parameter, choose the Input 4 variable.
For the separator, we’re going to use a space. Then click the OK button.
Lastly, click Save.
Testing The Flow
Let’s now see what happens. Click Test.
Choose the I’ll perform the trigger action option. Then click Test.
Enter any data for the other inputs. As for Input 4, just type “the apple falls far from the tree”. Then click Run flow.
Click Done.
Upon checking Slack, the returned output is exactly what we thought it would be.
This is now sending us an array and not the actual string.
Usage Of Arrays
Let’s now discuss some things that you can do with an array. First, let’s add a new step.
Then click Control.
After that, choose the Apply to each control.
Under the Expressions tab, choose the split function.
Then select the text itself, which is the Input 4 variable.
Make sure to type the separator which is a space. Then click OK.
After that, let’s add another action.
Search and select the Slack connector.
Then choose the Post message action.
Let’s post a message to the same Slack channel (discord).
For the Message text, instead of posting Input 4 or the array, we’re going to post the Current item.
By using the Apply to each control, it’s first getting the array of strings via the split function. Then, for each item in the array, it’s posting that current item. So let’s see what that looks
like. Click Save.
Click Test.
We’ll just use the previous run since no inputs have changed. Then click Test.
Upon checking our Slack, we’ll see that it successfully displayed each of the items from the array.
By doing this, we can perform actions on the array items separately.
***** Related Links *****
Power Automate String Functions: Substring And IndexOf
Triggers In Power Automate Flows
Power Automate Expressions – An Introduction
To sum up, we’re able to discuss the usage of the Power Automate split function in our flows. It breaks down strings into an array of strings using the defined delimiter. The delimiter serves as the
separator. In our example, we used space. People usually use commas, but you can certainly have different delimiters for each purpose.
We can also iterate through the array of items and then perform certain actions on each of them. We usually do that by using the Apply to each control. This technique is definitely useful when
working on a collection of data.
All the best,
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Guide to Intermediate DAX Functions for Power BI
Optimizing DAX: Performance Tips for Power BI Reports
DAX Table Functions Deep Dive | {"url":"https://blog.enterprisedna.co/power-automate-split-function-and-arrays/page/2/?et_blog","timestamp":"2024-11-13T11:07:37Z","content_type":"text/html","content_length":"187791","record_id":"<urn:uuid:6147ef7e-2842-47f1-ad26-f73b6c89c6e6>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00009.warc.gz"} |
What do the Fresnel equations represent?
Fresnel's equations describe the reflection and transmission of electromagnetic waves at an interface between two media. It turns out that these equations can be used in quasistatics and even
statics, for example to straightforwardly calculate magnetic forces between a permanent magnet and a bulk medium.
How to Calculate Fresnel's Law of Reflection?
Fresnel's Law of Reflection calculator uses Reflection Loss = (Refractive Index of Medium 2-Refractive Index of Medium 1)^2/(Refractive Index of Medium 2+Refractive Index of Medium 1)^2 to calculate
the Reflection Loss, Fresnel's Law of Reflection formula is defined as the Fresnel equations give the ratio of the reflected wave's electric field to the incident wave's electric field, and the ratio
of the transmitted wave's electric field to the incident wave's electric field, for each of two components of polarization. Reflection Loss is denoted by r[λ] symbol.
How to calculate Fresnel's Law of Reflection using this online calculator? To use this online calculator for Fresnel's Law of Reflection, enter Refractive Index of Medium 2 (n[2]) & Refractive Index
of Medium 1 (n[1]) and hit the calculate button. Here is how the Fresnel's Law of Reflection calculation can be explained with given input values -> 0.043199 = (1.54-1.01)^2/(1.54+1.01)^2. | {"url":"https://www.calculatoratoz.com/en/fresnels-law-of-reflection-calculator/Calc-38278","timestamp":"2024-11-02T21:24:37Z","content_type":"application/xhtml+xml","content_length":"119514","record_id":"<urn:uuid:5dc859d7-0ef0-44d8-9bda-315db9f8b9c4>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00088.warc.gz"} |
The Standard Template Library (STL) [STL94], now part of the C++ Standard Library [C++98], is a generic container and algorithm library. Typically STL algorithms operate on container elements via
function objects. These function objects are passed as arguments to the algorithms.
Any C++ construct that can be called with the function call syntax is a function object. The STL contains predefined function objects for some common cases (such as plus, less and not1). As an
example, one possible implementation for the standard plus template is:
template <class T>
struct plus : public binary_function<T, T, T> {
T operator()(const T& i, const T& j) const {
return i + j;
The base class binary_function<T, T, T> contains typedefs for the argument and return types of the function object, which are needed to make the function object adaptable.
In addition to the basic function object classes, such as the one above, the STL contains binder templates for creating a unary function object from an adaptable binary function object by fixing one
of the arguments to a constant value. For example, instead of having to explicitly write a function object class like:
class plus_1 {
int _i;
plus_1(const int& i) : _i(i) {}
int operator()(const int& j) { return _i + j; }
the equivalent functionality can be achieved with the plus template and one of the binder templates (bind1st). E.g., the following two expressions create function objects with identical
functionalities; when invoked, both return the result of adding 1 to the argument of the function object:
bind1st(plus<int>(), 1)
The subexpression plus<int>() in the latter line is a binary function object which computes the sum of two integers, and bind1st invokes this function object partially binding the first argument to
1. As an example of using the above function object, the following code adds 1 to each element of some container a and outputs the results into the standard output stream cout.
transform(a.begin(), a.end(), ostream_iterator<int>(cout),
bind1st(plus<int>(), 1));
To make the binder templates more generally applicable, the STL contains adaptors for making pointers or references to functions, and pointers to member functions, adaptable. Finally, some STL
implementations contain function composition operations as extensions to the standard [SGI02].
All these tools aim at one goal: to make it possible to specify unnamed functions in a call of an STL algorithm, in other words, to pass code fragments as an argument to a function. However, this
goal is attained only partially. The simple example above shows that the definition of unnamed functions with the standard tools is cumbersome. Complex expressions involving functors, adaptors,
binders and function composition operations tend to be difficult to comprehend. In addition to this, there are significant restrictions in applying the standard tools. E.g. the standard binders allow
only one argument of a binary function to be bound; there are no binders for 3-ary, 4-ary etc. functions.
The Boost Lambda Library provides solutions for the problems described above:
• Unnamed functions can be created easily with an intuitive syntax. The above example can be written as:
transform(a.begin(), a.end(), ostream_iterator<int>(cout),
1 + _1);
or even more intuitively:
for_each(a.begin(), a.end(), cout << (1 + _1));
• Most of the restrictions in argument binding are removed, arbitrary arguments of practically any C++ function can be bound.
• Separate function composition operations are not needed, as function composition is supported implicitly.
Lambda expression are common in functional programming languages. Their syntax varies between languages (and between different forms of lambda calculus), but the basic form of a lambda expressions
lambda x[1] ... x[n].e
A lambda expression defines an unnamed function and consists of:
• the parameters of this function: x[1] ... x[n].
• the expression e which computes the value of the function in terms of the parameters x[1] ... x[n].
A simple example of a lambda expression is
lambda x y.x+y
Applying the lambda function means substituting the formal parameters with the actual arguments:
(lambda x y.x+y) 2 3 = 2 + 3 = 5
In the C++ version of lambda expressions the lambda x[1] ... x[n] part is missing and the formal parameters have predefined names. In the current version of the library, there are three such
predefined formal parameters, called placeholders: _1, _2 and _3. They refer to the first, second and third argument of the function defined by the lambda expression. For example, the C++ version of
the definition
lambda x y.x+y
_1 + _2
Hence, there is no syntactic keyword for C++ lambda expressions. The use of a placeholder as an operand implies that the operator invocation is a lambda expression. However, this is true only for
operator invocations. Lambda expressions containing function calls, control structures, casts etc. require special syntactic constructs. Most importantly, function calls need to be wrapped inside a
bind function. As an example, consider the lambda expression:
lambda x y.foo(x,y)
Rather than foo(_1, _2), the C++ counterpart for this expression is:
bind(foo, _1, _2)
We refer to this type of C++ lambda expressions as bind expressions.
A lambda expression defines a C++ function object, hence function application syntax is like calling any other function object, for instance: (_1 + _2)(i, j).
A bind expression is in effect a partial function application. In partial function application, some of the arguments of a function are bound to fixed values. The result is another function, with
possibly fewer arguments. When called with the unbound arguments, this new function invokes the original function with the merged argument list of bound and unbound arguments.
A lambda expression defines a function. A C++ lambda expression concretely constructs a function object, a functor, when evaluated. We use the name lambda functor to refer to such a function object.
Hence, in the terminology adopted here, the result of evaluating a lambda expression is a lambda functor. | {"url":"https://www.boost.org/doc/libs/1_73_0/doc/html/lambda/s03.html","timestamp":"2024-11-03T15:41:11Z","content_type":"text/html","content_length":"14909","record_id":"<urn:uuid:b6175af0-bffb-4dde-bc01-08cf032b6f55>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00573.warc.gz"} |
Digital Math Resources
Direct Variation: Circumference vs. Diameter
A simple application of linear functions involves the ratio of the circumference of a circle to its diameter. Students are familiar with the irrational number π, but many may not know where the
number comes from.
Even students who are familiar with the ratio of circumference to diameter may not realize this represents a linear function. Use function notation to show that circumference (C) is a function of
diameter (d).
C = πd
In this form, connect it to the linear function
y = mx
The graph of this equation is a line that crosses the origin. The value of m is the slope, or steepness, of the line.
Then write this version of the equation:
y = πx
Ask students what π represents. Have them connect the notion of slope, constant of proportionality, and the ratio of circumference to diameter. The idea that π represents a slope may come as a
Next, have students calculate the value for π with a hands-on activity. Before doing that review the formula for calculating slope.
m = (y2 - y1)/(x2 - x1)
In the case of direct variation, the graph of the line crosses the origin, whose coordinates are (0, 0). This means that for a direct variation, the slope can be found with one set of coordinates.
For example, plug in (0, 0) into the slope formula and you get
m = y/x
Now, to the hands-on activity. Take jar lids of different sizes and string. Have students loop the string around the lid and cut the length of string. Then have them do the same for the diameter.
Have them create a data table of x-y coordinates, where x is the diameter and y is the circumference. Use a spreadsheet to graph the points. The graph will be a scatterplot of a line. If you are
using a graphing calculator, find the line of best fit and look at the value for the slope.
For a self-paced student activity on linear functions that focuses on Hooke's Law, take a look at our Classroom Module, linked here: https://www.media4math.com/classroom/browse-modules/
Or take a look at our other resources on Hooke's Law: https://www.media4math.com/library/search?keys=circumference+vs+diameter&type=All | {"url":"https://www.media4math.com/BlogPost--CircumferenceVsDiameter","timestamp":"2024-11-07T14:13:49Z","content_type":"text/html","content_length":"262041","record_id":"<urn:uuid:d9269243-1c45-4b49-8d5f-d093aa581519>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00590.warc.gz"} |
Calculated tags
Calculated tags have a formula that uses the values of one or more existing tags to compute the result. Both live and historical calculations are provided. For the calculation of historical values,
data records of all parameters must be available.
Create calculated tag
In tree, a new calculated tag can be created in the Calculated values node via the icon.
First, a data type (numeric or logical) must be selected. Note that the data type cannot be changed after it has been saved for the first time.
Then the formula can be entered. Tags can be added via drag & drop or by entering the name. Tip: Via Ctrl+space a selection of all available tags and functions appears.
Formula operators
In addition to the standard mathematical functions, such as addition (+), subtraction (-), multiplication (*) and division (/), the following operations are offered:
Pow("num", "exp") Returns a specified number num raised to the specified power exp.
Sqrt("num") Returns the square root of a specified number num.
Abs("num") Returns the absolute value of a specified number num.
If("exp","t","f") Returns t if the conditions exp is true, otherwise returns f.
Min("num", "num") Returns the smaller of two numbers.
Max("num", "num") Returns the larger of two specified numbers.
Log("num", "base") Returns the logarithm of a specified number num in a specified base base.
Cos("num") Returns the cosine of the specified angle num.
Sin("num") Returns the sine of the specified angle num.
Tan("num") Returns the tangent of the specified angle num.
Logical conditions can be used in AND/OR notation as well as in &&/|| notation.
Comparison operators are <, >, as well as <= and >=.
The operator for modulo is %.
Constant numbers are specified in international format, i.e. with a dot (.) as decimal separator.
Historical analysis
The historical analysis of a calculated tags only works if the history of all parameters exists. For calculation, all time series are loaded and the formula is applied to each time/value combination.
Special case: Logical tags
If a logical tag is used in a numerical formula, the operating hours are used. Example: EnginOn * 1.5. With this simple formula the energy of a machine can be calculated. With this simple formula the
energy of a machine can be calculated. The number of hours the machine is running, is multiplied by the power (here 1.5kW). When calculating the live value, either 0kW (0 * 1.5 if EnginOn=false) or
1.5kW (1 * 1.5 if EnginOn=true) is returned.
Counter and consumption values
When using counter and consumption tags, power/throughput is always used (see counter concept). | {"url":"https://www.anyviz.io/help/en/tags-calc","timestamp":"2024-11-03T04:20:55Z","content_type":"text/html","content_length":"12462","record_id":"<urn:uuid:6ef677f0-722d-420d-a680-540e0a19f159>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00852.warc.gz"} |
Excel Formula: Count Non-Empty Cells
In this tutorial, we will learn how to count non-empty cells in Excel using the COUNTA function. The COUNTA function is used to count the number of non-empty cells in a given range. This can be
useful for data analysis and spreadsheet management. We will provide a step-by-step explanation of the formula and provide examples to illustrate its usage.
To count non-empty cells in a range, we can use the following formula:
This formula counts each cell in the range P4:P200 that contains any data, including text, numbers, and formulas. The result of the formula is the total count of cells in the range that contain any
For example, if we have the following data in the range P4:P200:
| P |
| |
| 4 |
| 7 |
| |
| 2 |
| 5 |
| 9 |
| |
| 1 |
| 3 |
The formula =COUNTA(P4:P200) would return the value 7, which is the count of cells in the range P4:P200 that contain any data. By using the COUNTA function, we can easily count non-empty cells in
Excel and perform various data analysis tasks.
In this tutorial, we have learned how to count non-empty cells in Excel using the COUNTA function. This can be useful for various data analysis tasks and spreadsheet management. The COUNTA function
counts each cell in a given range that contains any data, including text, numbers, and formulas. We have provided a step-by-step explanation of the formula and illustrated its usage with examples.
Now you can apply this knowledge to your own Excel projects and enhance your data analysis skills.
An Excel formula
Formula Explanation
This formula uses the COUNTA function to count each cell that contains any data in the range P4:P200.
Step-by-step explanation
1. The COUNTA function is used to count the number of non-empty cells in a range.
2. In this case, the range P4:P200 is specified as the argument for the COUNTA function.
3. The COUNTA function counts each cell in the range that contains any data, including text, numbers, and formulas.
4. The result of the formula is the total count of cells in the range P4:P200 that contain any data.
For example, if we have the following data in the range P4:P200:
| P |
| |
| 4 |
| 7 |
| |
| 2 |
| 5 |
| 9 |
| |
| 1 |
| 3 |
The formula =COUNTA(P4:P200) would return the value 7, which is the count of cells in the range P4:P200 that contain any data. | {"url":"https://codepal.ai/excel-formula-generator/query/bm0u1tum/excel-formula-count-non-empty-cells","timestamp":"2024-11-09T04:12:04Z","content_type":"text/html","content_length":"90359","record_id":"<urn:uuid:c6904475-35f0-4443-bd43-006eacc7f7f6>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00645.warc.gz"} |
The Polynomial Path
A Beginner's Guide to PIOPs
The following research builds on Arantxa Zapico’s recent PROGCRYPTO presentation on Why You Should Care About Polynomials and this research paper on Interactive Oracle Proofs by Eli Ben-Sasson,
Alessandro Chiesa, and Nicholas Spooner.
This article is a general overview. As the field of zero-knowledge proofs and cryptographic protocols continues to evolve at a rapid pace, it is crucial to investigate and understand emerging
technologies from various perspectives.
This article represents one such exploration, aiming to demystify PIOPs and showcase their potential in reshaping the landscape of privacy-preserving computations.
The concept of an efficient proof system, whereby a computationally limited verifier can be convinced of the correctness of an arbitrary complex computation, is fundamental to complexity theory and
cryptography. At a high level, a proof system, the mathematical bedrock, allows one entity (the prover) to convince another entity (the verifier) that some statement is true, where the verifier has
bounded resources and is also pivotal in ensuring the security and functionality of various digital protocols, from blockchain transactions to secure multiparty computations.
The essence of efficiency in these proof systems cannot be overstated; it directly translates to lower computational costs and broader adoption, making secure digital interactions both more
accessible and sustainable, but at scale.
Historically, this quest for optimization has been advanced through the development of interactive proofs (IPs) and probabilistically checkable proofs (PCPs), each marking significant strides in our
ability to conduct verifications with minimal resources.
• IPs introduced back-and-forth communication between prover and verifier, evolving past static proofs.
• PCPs demonstrated verifying statements probabilistically by checking small portions of proofs, pioneering "sublinear" approaches.
These laid foundations for reconciling rigorous verification with real-world efficiency constraints. But for even wider applications, a new generation of proof systems would be needed. Enter
Interactive Oracle Proofs (IOPs).
Proof Systems in the Real World
In many digital systems today, one party needs to convince another that certain statements are true or computations are correct, without disclosing private underlying data. For example, a client
wanting to store genetic data may need to verify that a cloud server correctly ran analytics over it without revealing the actual health data. Or a customer may need to validate that an online seller
securely handled a sensitive transaction without sharing details.
Interactive oracle proofs emerged as frameworks that allowed this convincing to happen through a back-and-forth query and response flow between the prover and verifier. The prover computationally
encodes the private data into an "oracle" representation that hides the raw contents. By asking the prover to perform certain operations over this encoded data and return the results, the verifier
becomes incrementally convinced without ever seeing the underlying inputs.
How IOPs Work
1. Prover encodes private sensitive data into an "oracle" - a cryptographic commitment that conceals raw contents
2. Verifier sends challenge or query about the hidden data to prover
3. Prover performs operations on oracle-encoded data without decrypting it
4. Prover sends back proof that computation was done properly
5. Verifier checks proof is formatted correctly
6. Steps 2-5 repeat in interactive back-and-forth until verifier convinced of truth
Key Features
• Computation integrity verified without exposing raw data
• Incrementally builds confidence through conversation
• Queries and responses efficient for prover and verifier
The interactivity and encryption makes IOPs uniquely useful for web3 platforms dealing with sensitive or confidential data. By keeping data supplies and computations private while allowing
validation, IOPs overcome transparency vs. privacy tradeoffs.
Interactive Oracle Proofs Meet Polynomials
The introduced interactive oracle proof paradigm offers a more practical path to verification over encrypted data. However, the techniques for actually implementing the "oracles" representing secret
computations can significantly impact efficiency and flexibility.
This brings us to polynomials - algebraic structures with useful computational properties. By encoding sensitive data into polynomial coefficients, we can evaluate, transform, and reason about
encrypted information without ever decrypting actual contents.
Welcome to PIOPs
Polynomial interactive oracle proofs (PIOPs) specifically utilize polynomial commitment schemes for realizing IOPs efficiently. They represent the secrets within multivariate polynomial equations,
with each term consisting of variable exponents, coefficients encoding data, and modular arithmetic tying it all together.
Polynomials are a very powerful tool for constructing efficient zero-knowledge proof systems. Specifically, there is a framework called polynomial interactive oracle proofs (PIOPs) that enables a
prover to convince a verifier that certain statements are true about some data set or computation, without revealing the actual underlying data or computation.
The key reason polynomials are so useful for this is they provide a way to take a very large data set and "encode" it by representing it as the coefficient vector of a polynomial. So you could have a
data set with 2^20 elements, construct a polynomial with degree 2^20-1 that has each of those data elements as coefficients. This polynomial could be enormous, but has a useful feature - if you
evaluate the polynomial at any given point, you get just a single numeric value out. So it takes that huge data set and "compresses" it down to one number.
This means the prover can send that evaluated polynomial value (just one number) to the verifier as a commitment that encodes all of the underlying data. Then, without ever revealing the data, the
prover can prove certain statements are true about the data by manipulating the polynomial - doing operations like multiplying polynomials, evaluating them, etc. - and proving statements about the
resulting polynomials using PIOPs. The verifier only sees the polynomial values, not the actual data.
This polynomial formulation enables indirect operations over concealed datasets along with zero knowledge proofs that computations were performed properly. We can multiply encodings, evaluate
commitments, and prove statements about encrypted contents through algebraic rules.
So in a PIOP, the prover commits confidential data into a polynomial representation that obscures the raw underlying elements. The verifier then issues challenges for demonstrating relationships over
this encoded information. As long as mathematical rules are followed, the verifier gets convinced without accessing the source secrets.
The Role of Kate Commitment Scheme (KZG) in PIOPs
Kate commitment scheme (KZG) plays a crucial role in enabling efficient and secure proof systems. KZG is a polynomial commitment scheme, which means it allows us to commit to a polynomial and later
prove that the polynomial evaluates to specific values at certain points.
Imagine you have a large vector of data, like a list of transactions or a database. Instead of dealing with the entire vector directly, you can encode this data into a polynomial. Each element of the
vector becomes a coefficient of the polynomial. This is where KZG comes in handy.
In KZG, the prover commits to the polynomial by using powers of a secret element in a mathematical group. This secret element is like a special key that allows the prover and verifier to work with
the polynomial without revealing the actual data. The prover sends the commitment to the verifier, which is much smaller than the original vector.
Later, when the prover wants to prove something about the data, they can use the committed polynomial. For example, the prover might want to show that the polynomial evaluates to a specific value at
a particular point. To do this, they create a proof using the KZG scheme, which involves some mathematical operations on the polynomial and the secret element.
The beauty of KZG is that it allows for efficient proofs and aggregation. Instead of proving each element of the vector separately, the prover can create a single proof that covers multiple elements.
This makes the proof much smaller and faster to verify.
KZG is used within Polynomial IOPs (PIOPs), which are interactive proof systems based on polynomials. In a PIOP, the prover and verifier communicate using polynomials. The prover sends polynomials,
and the verifier checks the polynomials by querying them at specific points. KZG enables the prover to create concise and convincing proofs within this framework.
One interesting aspect of KZG is the Lagrange basis. By encoding the data using the Lagrange basis, the prover can efficiently prove the values of individual elements in the vector. Each polynomial
in the Lagrange basis is designed to evaluate to 1 at its corresponding element and 0 at all other elements. This property makes it easy to prove specific elements without revealing the entire
PIOPs: Building Blocks for zkSNARKs
Zero-Knowledge Succinct Non-Interactive Arguments of Knowledge (zkSNARKs) are a powerful cryptographic tool that allows one party to prove to another that a statement is true without revealing any
additional information. PIOPs play a crucial role in constructing efficient zkSNARKs by providing the underlying framework for encoding and manipulating data using polynomials.
In a zkSNARK, the prover wants to convince the verifier that they possess knowledge of a secret value that satisfies a certain computational statement, without revealing the secret itself. PIOPs
enable this by allowing the prover to encode the secret value and the computational statement into polynomial representations.
The prover then uses the properties of polynomials to generate a succinct proof that the encoded secret satisfies the encoded computational statement. This proof is constructed using polynomial
commitments and zero-knowledge protocols, ensuring that the verifier can verify the proof's validity without learning any information about the secret.
By leveraging the efficiency and security of PIOPs, zkSNARKs can achieve remarkable properties:
1. Succinctness: The proofs are very short and can be verified quickly, even for complex computations.
2. Non-interactivity: The proof can be generated offline and verified later without further interaction between the prover and verifier.
3. Zero-knowledge: The proof reveals nothing about the secret beyond the validity of the computational statement.
The combination of PIOPs and zkSNARKs opens up a wide range of possibilities for privacy-preserving applications, such as confidential transactions, anonymous credentials, and verifiable computation
on encrypted data.
Let's dive into a very simplified mathematical example of how Polynomial IOPs work in the context of SNARKs. We'll use a simple arithmetic circuit to illustrate the process.
Suppose we have the following arithmetic circuit: (a + b) * (c - d) = e
We want to prove that we know the values of a, b, c, and d that satisfy this equation without revealing the actual values.
Step 1: Encode the values as polynomials First, we encode the values a, b, c, and d as coefficients of polynomials. Let's say:
• a = 3, represented by the polynomial A(x) = 3
• b = 1, represented by the polynomial B(x) = 1
• c = 4, represented by the polynomial C(x) = 4
• d = 2, represented by the polynomial D(x) = 2
Step 2: Construct the constraint polynomial We create a constraint polynomial that represents the arithmetic circuit. In this case, the constraint polynomial would be: CP(x) = (A(x) + B(x)) * (C(x) -
Substituting the values: CP(x) = (3 + 1) * (4 - 2) CP(x) = 4 * 2 CP(x) = 8
Step 3: Commit to the polynomials The prover commits to the polynomials A(x), B(x), C(x), and D(x) using a polynomial commitment scheme like KZG (Kate-Zaverucha-Goldberg). The commitments are
cryptographic values that hide the actual polynomials but allow for later evaluation and verification.
Step 4: Prove the constraint polynomial evaluates to the expected value The prover needs to convince the verifier that the constraint polynomial CP(x) evaluates to the expected value (8 in this case)
without revealing the actual values of a, b, c, and d.
To do this, the prover generates a PIOP proof:
1. The prover evaluates the polynomials A(x), B(x), C(x), and D(x) at a random point z chosen by the verifier.
2. The prover computes the evaluation of CP(x) at the same point z using the evaluations of A(z), B(z), C(z), and D(z).
3. The prover sends the evaluations A(z), B(z), C(z), D(z), and CP(z) to the verifier, along with a proof that these evaluations are consistent with the polynomial commitments.
Step 5: Verify the PIOP proof The verifier checks the PIOP proof:
1. The verifier verifies that the evaluations A(z), B(z), C(z), and D(z) are consistent with the polynomial commitments.
2. The verifier computes the expected evaluation of CP(z) using the received evaluations and checks that it matches the claimed value (8 in this case).
If the PIOP proof verifies successfully, the verifier is convinced that the prover knows the values a, b, c, and d that satisfy the arithmetic circuit, without learning the actual values.
This is a simplified example, but it illustrates the key steps involved in using PIOPs within a SNARK proof system.
In practice, the arithmetic circuits can be much more complex, and the polynomials are typically of higher degrees. The PIOP proof ensures that the prover can convince the verifier of the
satisfiability of the arithmetic circuit while preserving the privacy of the input values.
Demystifying PIOPs
At the heart of PIOPs lie two key concepts:
• Polynomial Encodings as Locked Boxes
• A Conversational Verification Dance
Polynomial Encodings as Locked Boxes
A useful analogy is to think of polynomial encodings as special locked boxes that allow restricted operations without ever unlocking the contents.
Specifically, the prover starts by encoding sensitive raw data (e.g financial records, medical data, etc.) numerically. This long string of numbers then becomes the coefficient vector of a
multivariate polynomial equation.
We can visualize this polynomial commitment scheme as taking the sensitive data and sealing it within a cryptographic box secured by mathematical locks. The box conceals all actual values, showing
only the outer shell.
However, the box allows selectively applying certain keys to reshape and transform its exterior in provable ways, indirectly operating on the contents:
• Special keys can stretch/compress the box by evaluating the polynomial encoding at points
• Other keys can verify modular rotations orOperations that rearrange its composition
• Some keys attach cryptographic proofs of valid transformations
So by asking for the right kind of manipulations and validating the keys used follow mathematical rules, we can gain confidence the concealed raw data has been handled properly without ever directly
accessing it. The polynomial encodings act as data containers we can compute over while keeping contents private.
1. Sensitive Raw Data:
□ At the beginning of the process, we have some sensitive information that we want to protect. This could be things like financial records or medical data.
□ We don't want anyone to be able to see this data directly because it's private and confidential.
2. Encode Data as Polynomial Coefficients:
□ To keep the sensitive data safe, we need to transform it into a different form that hides the original information.
□ In this step, we take the raw data and convert it into a list of numbers called polynomial coefficients.
□ Think of it like putting the data through a special machine that scrambles it up into a secret code.
3. Coefficient Vector:
□ After encoding the data, we end up with a list of numbers known as the coefficient vector.
□ In the image, the coefficient vector is shown as (3, 1, 4, 1, 5, 9, ...).
□ Each number in this vector represents a piece of the encoded data.
4. Polynomial Encoding (Locked Box):
□ Now, we take the coefficient vector and place it inside a special "locked box" called the Polynomial Encoding.
□ This locked box is like a secure container that keeps the encoded data hidden from anyone who doesn't have the right keys to open it.
□ The box conceals the original sensitive data, so even if someone gets their hands on the box, they can't see what's inside.
5. Privacy:
□ The locked box (Polynomial Encoding) is designed to protect the privacy of the sensitive data.
□ As shown in the image, there's an arrow pointing from the locked box to the "Privacy" section.
□ This means that the raw data remains hidden and inaccessible to anyone who doesn't have permission to unlock the box.
6. Apply Special Key (Evaluation Key):
□ Now, let's say we want to perform some operations or calculations on the encoded data without revealing the actual information.
□ To do this, we use a special key called the Evaluation Key.
□ In the image, there's an arrow pointing from the locked box to the "Apply Special Key" step.
□ The evaluation key allows us to interact with the locked box in specific ways without actually opening it or seeing the raw data inside.
7. Evaluate Polynomial at a Point:
□ One of the things we can do with the evaluation key is evaluate the polynomial (the encoded data) at a particular point.
□ This means we plug in a specific value and calculate the result based on the encoded data inside the locked box.
□ It's like asking the locked box a question and getting an answer without ever seeing the original data.
8. Evaluation Result (Single Number):
□ When we evaluate the polynomial at a point, we get a single number as the result.
□ This number is the outcome of the calculation we performed on the encoded data.
□ We can use this result for further computations or comparisons without ever knowing the actual sensitive information hidden inside the locked box.
So, in summary, this process allows us to take sensitive data, encode it into a special form (polynomial coefficients), and lock it up in a secure box (Polynomial Encoding). We can then use a special
key (Evaluation Key) to perform calculations on the encoded data without ever seeing the original information. This way, we can work with the data while still protecting its privacy.
A Conversational Verification Dance
Rather than a static proof, the verification process happens interactively through a back-and-forth "dance" between the prover and verifier.
The verifier issues a series of polynomial operation challenges tailored to confirm computations over the concealed data were done correctly. The prover manipulates the polynomial box in response to
each challenge, sending back the transformed encodings plus proofs of proper execution.
As long as these responses methodically follow valid algebraic rules, the verifier becomes incrementally convinced of correct handling of the raw data. By steering this conversation and seeing keys
used properly unlock intermediate proof states, assurance builds interactively.
So PIOPs bridge interactivity with polynomial cryptography to facilitate a dialogue rooted in mathematical formalisms yet flexible enough for practical applications needing to verify computations
over sensitive data. The dialogue grants confidence while encryption protects underlying secrets.
In this diagram:
1. The verification dance begins with the "Start of Verification Dance" note, indicating the commencement of the interactive process between the Verifier and the Prover.
2. The verification dance proceeds with a series of interactions:
□ The Verifier issues the first polynomial operation challenge (Challenge 1) to the Prover, requesting a specific manipulation of the polynomial encoding of the sensitive data.
□ The Prover receives Challenge 1, manipulates the polynomial accordingly without revealing the underlying data, and generates Proof 1 to demonstrate the correct execution of the requested
□ The Prover sends Proof 1 back to the Verifier.
□ The Verifier receives Proof 1 and verifies its correctness. If the proof is valid, the Verifier's confidence in the Prover's claims begins to build.
□ The Verifier issues the second polynomial operation challenge (Challenge 2) to the Prover, requesting another specific manipulation of the polynomial encoding.
□ The Prover receives Challenge 2, manipulates the polynomial accordingly, generates Proof 2, and sends it back to the Verifier.
□ The Verifier receives Proof 2 and verifies its correctness. If the proof is valid, the Verifier's confidence in the Prover's claims increases further.
3. The diagram also explores an optional section where the Verifier can issue further challenges to the Prover if necessary. The Prover responds with subsequent proofs, and the Verifier verifies
each new proof. This process can continue until the Verifier is satisfied or until a predetermined number of challenges is reached.
4. After all the challenges have been issued and the corresponding proofs have been verified, the verification dance concludes with the "End of Verification Dance" note.
5. If all the proofs provided by the Prover are valid and the Verifier's confidence reaches a sufficient level, the Verifier sends a message to the Prover indicating that the verification has
The Prover manipulates the polynomial encoding and provides proofs, while the Verifier checks the responses and builds confidence incrementally. The incremental building of the Verifier's confidence
through multiple challenges and proofs is a key aspect of the PIOP verification process.
Comparing PIOPs
As we've seen, PIOPs offer a powerful framework for enabling efficient and secure verification of computations over encrypted data. But how do they compare to other well-known proof systems? Let's
take a closer look at the key properties of Interactive Proofs (IPs), Probabilistically Checkable Proofs (PCPs), zkSNARKs, and PIOPs to understand their unique strengths and differences.
While IPs and PCPs laid the groundwork for interactive and efficient verifications, zkSNARKs and PIOPs have built upon these foundations to offer solutions that are both efficient and
privacy-preserving. zkSNARKs excel in providing succinct and non-interactive zero-knowledge proofs, making them especially suitable for blockchain applications. PIOPs, on the other hand, balance the
benefits of interactive proofs with the efficiency and privacy offered by polynomial commitments, making them versatile for a wide range of cryptographic applications. Each of these proof systems has
its place in the cryptographic landscape, chosen based on the specific requirements of efficiency, scalability, privacy, and trust assumptions of the application at hand.
Throughout this article, we've explored the fascinating journey of PIOPs, starting from their foundations in Interactive Proofs and Probabilistically Checkable Proofs. We've seen how PIOPs leverage
the magic of polynomial encodings to create "locked boxes" that allow us to perform computations on encrypted data without revealing the underlying secrets. The conversational verification dance
between the prover and verifier showcases the elegance of PIOPs, as they incrementally build confidence in the correctness of the computations through a series of challenges and responses.
But the real excitement lies in the practical applications of PIOPs. As we've discovered, PIOPs serve as the building blocks for the groundbreaking Zero-Knowledge Succinct Non-Interactive Arguments
of Knowledge (zkSNARKs). By enabling efficient and succinct zero-knowledge proofs, PIOPs open up a world of possibilities for privacy-preserving protocols, from confidential transactions and
anonymous credentials to verifiable computations on encrypted data.
Of course, PIOPs are not the only players in the cryptographic proof system arena. Our comparison with Interactive Proofs, Probabilistically Checkable Proofs, and zkSNARKs highlights the unique
strengths and trade-offs of each approach. While PIOPs may not always be the perfect fit for every scenario, their versatility and balance between efficiency, scalability, and privacy make them a
compelling choice for a wide range of applications.
As we look ahead, the future of PIOPs and their role in shaping the landscape of privacy-enhancing technologies is truly exciting.
Find L2IV at l2iterative.com and on Twitter @l2iterative
Author: Arhat Bhagwatkar, Research Analyst, L2IV (@0xArhat)
Disclaimer: This content is provided for informational purposes only and should not be relied upon as legal, business, investment, or tax advice. You should consult your own advisors as to those
matters. References to any securities or digital assets are for illustrative purposes only and do not constitute an investment recommendation or offer to provide investment advisory services. | {"url":"https://l2ivresearch.substack.com/p/the-polynomial-path","timestamp":"2024-11-14T02:08:23Z","content_type":"text/html","content_length":"220770","record_id":"<urn:uuid:88aa03eb-e580-4636-9144-fe61d33292c2>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00479.warc.gz"} |
Strategic Thinking: Form, Content, and Context
This is about the difficulty of pinning down which strategic thinking is correct and which is not.
Croesus. Photograph by Marco Prins
“If one kilo of rice costs one million pesos, then two kilos of rice costs two million pesos.” Everyone agrees.
Notice that while one agrees with this statement quoted, no one agrees that one kilo of rice costs one million pesos. As everyone observes, there is no store of any kind anywhere that sells one kilo
of rice for one million pesos.
What everyone affirms in the statement quoted is the form of the statement taken apart from its content. The form, meaning the arithmetic involved. If one is to one, then two is to two.
The issue of form is not the same as the issue of content.
If I am going to Spain, then I will need a passport.
Therefore, I will need a passport.
If I am going to eat, then I will need food.
Therefore, I will need food.
If I am going swimming, then I will be wet.
Therefore, I will be wet.
All of these arguments have the same form but not the same content. The first is about travel to Spain. The second is about eating. The third is about swimming.
When one communicates, three dimensions are involved with the medium of communication. The first is form. The second is content. The third is context — that which Croesus should have known.
Logos is the Greek term for “word.”
Apparently the Ancient Greeks have thought words are, somehow, names of forms — “forms” themselves being synonymous with “ideas.” Thus the derivative of the term “logos” which is “logoi,” meaning
form. Ergo, logic being the study of forms.
Therefore, Socrates is mortal.
The study of forms of arguments like this has led Aristotle to invent the discipline of logic. Nowadays, however, logic takes on a mathematical form. Thus, Mathematical Logic that is sometimes
referred to as Modern Symbolic Logic — as distinguished from Aristotelian that is sometimes referred to as Classic Logic.
Similar to catching some fish and declaring the fish caught as either juvenile or adult, when one communicates, what is communicated can either as acceptable or not. That is how one considers which
form, content or context is acceptable or not.
Form is measured in mathematical terms, ergo the expression “mathematical logic.”
Content is measured in terms of observation, meaning the use of one’s senses. This is the domain of science so far as generalisations are concerned.
The measure of context, unlike the other two dimensions already adumbrated, is not as easily pinned down.
One observes a table. Observed is that a pen is on the table. In addition, a phone is on the table. In addition, a laptop is on the table. More. The many truths about the table can be taken
altogether as the whole truth about the table.
Part of the whole truth about the table is that it is in a room.
Observed of the room is that it has a ceiling fan. In addition, there is a swivel chair in the room. In addition, there is a person in the room. The many truths about the room can be taken altogether
as the whole truth about the the room.
The whole truth about the table is part of the whole truth about the room. The whole truth about the room is part of the whole truth about Metro Manila. The whole truth about Metro Manila is part of
the whole truth about the Philippines, the whole truth about the globe, the whole truth about our universe.
What is indicated here is that nothing, whether some object, some event, or some generalisation, among others, can be taken in isolation. This leaves everyone with an infinite array of levels of
emergence from which anyone can choose from by whatever standard. That is what context is all about.
The most that one can say about context is that there is some harmony or coherence that binds everything into some single unit. That is all of it.
This is the catch. What may be coherently “music” for some may be coherently “noise” for others.
More. Competing perspectives of individuals can, at times, become incommensurable. It can happen that the standard one uses to identify what is correct and what is not is internal to one's mental
model in a way that comparison of the competing mental models themselves becomes untenable. This is what happens, for example, when it comes to the alternative geometries one can work with —
Euclidean geometry being only one among many.
Strategic planning involves analysis of context. Analysis of context involves an infinite array of levels of emergence from which anyone can choose from by some standard. The trick involves how to
identify the standard which which one chooses the standard — standard about standards.
Discoveries about our universe by science are made every now and then. This means invention of tools that can be used to better observe events take place every now and then. This means new data are
generated and new perspectives are developed with which one can view old data.
What can be “true” at one time may not be “true” at another time. Ergo the use of the term “theory” rather than the term “truth” to refer to the principles of science. Thus, theories are expected
either to be revised or rejected in favor of better theories as science develops, as time goes by. What this means is that even the standards of science are underdetermined.
With the infinite array of constantly changing perspectives to choose from, choice of which to use, one way of the other, becomes extremely difficult.
“If Croesus goes to war, he will destroy a great empire.” Croesus is happy thinking that he is about to destroy his adversary’s empire. Croesus goes to war and, in the process, he destroys his own
Easy to illustrate when something goes wrong. That is not the issue, however. The issue is with what is anyone to measure the correct context with. So long as this issue is unresolved, strategic
thinking is going to be both interesting as well as frustrating. | {"url":"https://www.pinoytoolbox.org/post/strategic-thinking-form-content-and-context","timestamp":"2024-11-10T14:40:28Z","content_type":"text/html","content_length":"1039047","record_id":"<urn:uuid:298d609b-0c3d-426d-a508-e9c2a979b997>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00719.warc.gz"} |
01 - S
Woburn Challenge 2001 - Suicidal
Thursday night — Blind Date
Ah, it's the end of frosh week and love is in the air. New couples are forming everywhere and everyone is happy. Hi, I'm Roger Lodge. Let's meet our couple, shall we? Simon is a Computer Engineering
frosh who has a "fetish" for precision. His favourite books are "Advanced Number theory" and "The Hermit, The Hospital and The Place: baby names for mathematicians". Simone is an electrical
engineering frosh who likes sparks to fly on the first date. She likes to mix psychology with physics and is currently reading "Jung on Young" and "Freud and the solenoid". Yikes.
Simon and Simone are at dinner. When their drinks (ie. beer) arrive, it is in a large glass that holds 2N litres (the glass is full). In addition, the waiter has been kind enough to supply 2 other
glasses: a small one that holds N-K litres and a medium one holding N+K litres, where 0 < K < N.
These 2 glasses are empty. Simon, being very precise, wants to pour exactly the same amount of beer for him and his date (so that there are exactly N litres left in the large and medium sized
glasses). However, the only measuring devices he has are the 3 glasses.
Since the jugs are not calibrated, the only way that he can know how much volume he is pouring is by emptying a jug fully or filling another jug fully. Dr. Love says that Simon hasn't been to the gym
in a while (he's been too busy reading early 21st century literature), and so he finds the jugs heavy and wants to minimize lifting a glass and pouring it. Plus, he doesn't want to go pouring back
and forth like an idiot, if it isn't even possible to split the beer evenly.
N K (0 < K < N < 4000; -1 -1 denotes end of input)
Josh, another Waterloo keener, has solved this program for 0 < K < N < 64000. Simon doesn't want to lose his reputation as the class nerd, so he's desperate to repeat this feat. If you can help him,
you'll get twice as many points.
P (P = the minimum number of pours will fit within a longint; output “infinity” if it isn't possible to split the beer evenly)
Sample Input
-1 -1
Sample Output
All Submissions
Best Solutions
Point Value: 20 (partial)
Time Limit: 2.00s
Memory Limit: 32M
Added: Dec 05, 2008
Languages Allowed:
C++03, PAS, C, HASK, ASM, RUBY, PYTH2, JAVA, PHP, SCM, CAML, PERL, C#, C++11, PYTH3
My solution runs within time and memory constraints(after realizing that solution wasn't actually O(n^2 or n^3) time and memory), but gives WA for the bonus, and not the regular cases.
My solution doesn't seem to have any mistakes. It covers all of the cases (small->medium, small->large, 4 others, assert() show it is completely filling or emptying at least one pitcher after each pour, as stated) and fits within memory, time constraints.
assert() shows amount of beer does not become negative or exceed capacity of cups, so cases seem to be handled correctly.
There is no reason for integer overflow to happen for n< 64000, since separate integers are used for each cup.
--This leaves me with no idea of what could be wrong, except the judge, or some ridiculously tiny detail I overlooked
My solution runs within time and memory constraints(after realizing that solution wasn't actually O(n^2 or n^3) time and memory), but gives WA for the bonus, and not the regular cases.
My solution doesn't seem to have any mistakes. It covers all of the cases (small->medium, small->large, 4 others, assert() show it is completely filling or emptying at least one pitcher after each
pour, as stated) and fits within memory, time constraints.
assert() shows amount of beer does not become negative or exceed capacity of cups, so cases seem to be handled correctly.
There is no reason for integer overflow to happen for n< 64000, since separate integers are used for each cup.
--This leaves me with no idea of what could be wrong, except the judge, or some ridiculously tiny detail I overlooked
Well, I don't know what to say - Hanson agrees with the output that we had for this problem, and he's Hanson.
You only get 1 of the 6 sub-cases wrong (the 3rd one, which is close to the limit), in which you output a fairly large number instead of infinity.
I don't know which of you is right, but I wouldn't worry too much about it if I were you.
Well, I don't know what to say - Hanson agrees with the output that we had for this problem, and he's Hanson.
You only get 1 of the 6 sub-cases wrong (the 3rd one, which is close to the limit), in which you output a fairly large number instead of infinity.
I don't know which of you is right, but I wouldn't worry too much about it if I were you.
However, I do notice that Hanson's solution is different than yours, and it most definitely uses a small-enough amount of time and memory, while yours seems a bit uncertain. Plus, it's Hanson.
However, I do notice that Hanson's solution is different than yours, and it most definitely uses a small-enough amount of time and memory, while yours seems a bit uncertain. Plus, it's Hanson.
??? did you even look at the third test case when you said that?
The third could easily be solved by hand, unlike the others
In the third line of second test case(sorry about getting it on my own),
there are two large numbers n, k, where k = n-1
That means there is one jug with n-k = n-(n-1) = 1 unit of capacity, and another jug with n+k = n+(n-1) = 2n - 1 units of capacity
Since there is a jug with a capacity of one, IT MUST BE POSSIBLE to pour n units
So there is no way it could be infinity
One possible solution involves pouring 1 from L->S and S->M n times. This is rather obvious, and takes 2n steps, but is not optimal
Another possible solution involves pouring as much as possible from larger to medium (size n+(n-1)) then pouring 1 from M->S, then S->L (n-1) times.
This takes 1 + 2*(n-1) = 2n-1 steps, and is optimal. This is the answer my program gets by simulation in my solution
There must be a bug in hanson's code if he says its infinity. (it is a border case, after all)
Please correct the judge for that case and test most recent 10/20 submission. Or did you not mean the third, but some other one
*S(size 1) represents small
M (size 2n-1) represents medium
L (size 2n ) represents large
??? did you even look at the third test case when you said that?
The third could easily be solved by hand, unlike the others
In the third line of second test case(sorry about getting it on my own),
there are two large numbers n, k, where k = n-1
That means there is one jug with n-k = n-(n-1) = 1 unit of capacity, and another jug with n+k = n+(n-1) = 2n - 1 units of capacity
Since there is a jug with a capacity of one, IT MUST BE POSSIBLE to pour n units
So there is no way it could be infinity
One possible solution involves pouring 1 from L->S and S->M n times. This is rather obvious, and takes 2n steps, but is not optimal
Another possible solution involves pouring as much as possible from larger to medium (size n+(n-1)) then pouring 1 from M->S, then S->L (n-1) times.
This takes 1 + 2*(n-1) = 2n-1 steps, and is optimal. This is the answer my program gets by simulation in my solution
There must be a bug in hanson's code if he says its infinity. (it is a border case, after all)
Please correct the judge for that case and test most recent 10/20 submission. Or did you not mean the third, but some other one
*S(size 1) represents small
M (size 2n-1) represents medium
L (size 2n ) represents large
I agree with you, Hanson must be wrong somehow. It's fixed.
I agree with you, Hanson must be wrong somehow. It's fixed.
It appears you'll get half the points if you don't do the bonus.
It appears you'll get half the points if you don't do the bonus. | {"url":"https://wcipeg.com/problem/wc01Sp6","timestamp":"2024-11-13T21:40:30Z","content_type":"text/html","content_length":"23921","record_id":"<urn:uuid:5ffbf5d2-93b0-42e7-9ef3-cbe38373c893>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00141.warc.gz"} |
Mitigating Legislative Risk in Retirement: The Critical Role of IRMAA and IRA Strategies
It’s Legislative Risk Awareness Month, and today I want to share one of the clearest examples of Legislative Risk many seniors face. And it’s an example that seems to be everywhere these days.
I’m talking about IRMAA.
IRMAA, of course, is the Income-Related Monthly Adjustment Amount (IRMAA), a fee Medicare beneficiaries at certain income thresholds pay for their Part B and Part D premiums.
When the IRMAA adjustment rules were originally passed, they created IRMAA brackets and associated fees and introduced an annual process for adjusting these fees going forward.
While we don’t know the exact amount IRMAA fees will rise each year going forward, the existence of those annual increases is already law. Historically, these increases have ranged from around 4.73%
to 8.02% annually.
Why is this a relevant topic for Legislative Risk Awareness Month?
Medicare and Social Security expenses make up a majority of the mandatory spending in our nation’s annual budget. In fact, the majority of all revenue generated by our country is made up of these two
major expenditures.
And the age demographics of our country will only compound this problem.
IRMAA increases are inevitable. Clients can’t control them; only the government can.
But, we can control the impact of our IRAs and 401ks on the IRMAA calculation each year. If we can establish a baseline income throughout retirement from non-IRA assets, we can then look at the
incremental impact on IRMAA from the addition of IRA distributions.
It comes down to a few assumptions:
• How long will you live?
• How many tiers of IRMAA do you move by the incremental IRA income?
• Are you married?
You can then apply the new combined income to the tiered IRMAA tables for Parts B and D.
When it comes to IRMAA fees, the “IRMAA Increments” are all equal. If you jump from one bracket to the next higher bracket, the additional premium is equal for each tier. The same goes for paying
IRMAA as a single person or married. This means one jump in the bracket is equal to one more year of paying IRMAA, and it is also equal to paying IRMAA for a second person (spouse).
So, you can define an IRMAA Increment as an equal, additional amount of IRMAA premium paid per year of paying IRMAA, plus additional tiers of IRMAA from higher income in a given year, per person
(either 1 for a single or 2 for a spouse).
Suppose we can simplify this down to a very simple analysis of IRMAA Increments. In that case, we can quickly identify, based on realistic, reasonable assumptions, whether it makes sense to take IRA
distributions over your lifetime compared to converting money over a few years to a tax-free instrument like a Roth IRA or Cash Value Life Insurance.
Let’s explore deeper through an example.
Here’s a 65-year-old couple with a $1M IRA.
If they ask, “How will this IRA impact my IRMAA calculations?” or better yet, “Can I limit the impact of my IRA on my IRMAA calculations?” Here's how we could help them answer it.
Let’s assume they have a baseline non-IRA income of $150,000, and the IRA distributions will be added to that.
As for the $1M IRA, let’s consider two different scenarios.
Scenario 1 assumes a strategy of keeping the IRA and using it for income. With a blended growth rate of 5%, net of fees, you could take about $80,000 a year from the IRA and make it last to life
expectancy (20 years).
Or, the second scenario considers converting that IRA to a tax-free vehicle, like a Roth IRA. I’ll go with a very aggressive 2-year conversion pattern. That means $500,000 will be added to the
baseline income for the next two years. But it also means that income is off the books when it comes to IRMAA calculations in a shorter amount of time.
Now we’ll just add up the additional IRMAA increments for each scenario.
Scenario 1:
Years – 20
Tiers – 1 (Meaning the added $80k jumps the IRMAA calculation by one bracket tier)
People – 2 (Assumes both live for 20 years)
My quick 3rd-grade multiplication tables tell me this is 40 IRMAA Increments.
Scenario 2:
Years – 2 (Remember, we’re not looking at total IRMAA, only the years the IRA impacts IRMAA)
Tiers – 4 (This is the big fear. Large distributions from IRAs drive up income for IRMAA calculations)
People – 2
Another quick calculation tells me this scenario only carries 16 IRMAA Increments.
In fourth grade, I learned greater than/less than. 40 is definitely greater than 16.
Some very simple analysis tells me that converting to tax-free is well worth the up-front “cost” of conversion.
Oh, and did someone mention that the cost of IRMAA is going up? Every year? That’s right. Add on a 4-8% increase to that IRMAA Increment and suddenly conversion looks even better. Because by
executing a Roth Conversion now, they are paying their IRMAA Increments at their lowest values!
Let’s look at the actual current value of an IRMAA Increment. The difference in tiers of the part B calculation is $104.80. And the difference for part D is $20.50. So that’s $125.30 per month, which
equates to $1503.60.
$1503.60 per Tier. Per Year. Per Person.
And that’s at today’s prices. The lowest percentage increase since IRMAA’s inception in 2007 was a growth of 4.73%. So using that lowest fee increase going forward, the cost would grow to $2385.95 in
10 years, and $3789.26 in the final 20^th year of the example.
And that’s assuming the smallest increase we’ve seen. Those increases are not always that small.
I understand that every client has their own unique set of circumstances. And there are multiple factors that come into play when considering the possible conversion of tax-deferred money to tax-free
But when it comes to IRMAA, the case is pretty cut and dry. If your IRMAA Increments are higher by staying in the IRA, then a conversion must at least be considered. And quickly.
There is an incredibly high level of legislative risk around increases to IRMAA. After all, those increases have already been codified into law.
Helping our clients minimize IRMAA as part of their conversion strategies can be an important part of helping protect our clients from legislative risk in retirement. (If you would like a deeper dive
into IRMAA, watch our most recent study group here.) | {"url":"https://blog.stonewoodfinancial.com/irmaa-legislative-risk","timestamp":"2024-11-12T11:49:38Z","content_type":"text/html","content_length":"85279","record_id":"<urn:uuid:4e89ea53-f8fd-42dd-b95a-0c08131a6704>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00101.warc.gz"} |
Finite Markov Chains : Summary
This book is an awesome resource for understanding finite Markov chains. The book is written in Pre-Latex era and hence one has to struggle to follow the notation. Is the Struggle worth it? IT IS, if
you are looking at developing a Matrix perspective towards “Discrete Stochastic Process”. IT IS NOT, if you are looking at quick and dirty formulae. The first version of the book appeared in 1960 and
second version in 1976. I wasn’t even born when these books came out –:) . However like many things in life, things that age have their own charm. The book serves as a precursor to understanding
Markov chain Monte Carlo method which is an essential tool for any quant.
**Chapter 1: Pre-requisites:
**Most of the stuff can be quick read in this chapter. What I found interesting was the mention of set theory concepts such as equivalence relation, weak ordering relation, minimal elements etc. One
must patiently go over these stuff as they find immense application in classifying Markov chains. An attempt is made to describe a simple stochastic process and readers might find this example
appealing before venturing in to complicated stochastic processes. I understood Cesaro-Summability and Euler Summability clearly in this context , even though I had quick read them in “Understanding
Analysis”- Abbott.
Chapter 2: Basic Concepts of Markov Chains
The chapter starts off by defining a Markov process which essentially is a process where essentially previous outcome is all that is needed to predict the probability of future outcome. Markov chain
is a finite Markov process where the transition probability from step i to step j does not depend on the trail number. One of the equivalent ways to state a Markov chain is that , given the present,
the past and future are completely independent of each other. To deviate in to philosophy here a bit, I think we should lead our lives also like a Markov process. Whatever be the past, given the
present, we must work in the present, be in the NOW so that future becomes independent of the past. Ok, enough of this philosophy crap J , Let me get back to the contents of the chapter.
What’s the difference between Markov Process and Markov Chain ?
Well, Markov chain is a type of Markov process. Apart from that, one of the biggest differences arises from the fact that a Reverse Markov process is also Markov process but a reverse Markov chain
need not be a Markov chain. The transition probability matrix of a reversed Markov chain may well depend on the trail number and hence does not qualify to be a Markov chain. This nifty difference is
sometimes not highlighted clearly most of the times in other books.
The author gives a set of examples representing Markov chains to get an idea of various finite Markov chains one can come across. One of the reasons I got interested in Markov chains is that they are
an excellent computational tool for classic probability problems. At the same time, they serve as an ideal platform to understand stochastic processes. I used to struggle to work with problems
involving absorbing barriers, reflecting barriers in a simple random walk. Setting up a recurrence relation and solving using PDE approach or some tricky method never gave me a tool to understand the
stuff. Markov chains on the other hand actually give a fantastic way to compute these using matrices. I love when any problem is formulated using matrices. It makes everything so pleasant for a
problem solver.
Classification of states in a Markov Chain:
The author uses set theory concepts such as partial ordering, equivalence classes induced by partial ordering, minimal elements of the partial ordering of equivalence classes etc to classify the
states of a Markov Chain. Well, I was aware that “Relation” can be used to define equivalence classes that partition a set. But beyond that, my understanding of “Partial Ordering”, “Minimal Elements”
etc was shallow. Hence I took a detour for a few hours and went through “Naive Set Theory” – Paul Halmos. I managed just enough to understand stuff about Markov Chain.
You start off with a relation – iRj, you can reach state i from state j, meaning it is a type of relation and you can then extend this to another relation ( iRj and jRi ) meaning you can reach i from
j and j from i . So, you start with a weak ordering relation, and extend it form equivalence classes. When I first came across these concepts, I found some resistance in understanding these ideas.
Why should I care if you start with a weak relation and you can extend that weak relation so that you can form equivalence classes?
Well, all these concepts make tremendous sense when you look at a Markov Chain. Firstly,the weak ordering relation can be extended to”communication” relation to first classify the states in to
equivalence classes. Subsequently, you can order these states and the minimal elements of these ordered states form ergodic states. Cutting the fancy stuff around the word ,”minimal”, all it means is
that if you take any element x of a set U, if iRx holds then xRi holds too. All those elements i form an ergodic set. One must understand the reason for this ordering and the rationale behind
ordering. Once you order stuff , you have the transition matrix as follows
This pattern emerges once you order the states accordingly. The first thing that strikes you about the above matrix is that you can higher powers of this matrix retains the individual transition
matrices on the diagonal and thus one can individually analyze these matrices without worrying about the entire chain.
The chapter then goes in to giving labels to various kinds of Finite Markov Chains. There are two broad categories of Markov Chain. Firstly the chains which do not have transition states. Secondly,
the states have transition states.
Chains without Transition States are also called chains with single ergodic set (ergodic chain). They can be further divided in to:
• Regular Markov chains symbolize the transition matrix where each state can be reached from any other state. Thus all the elements of the transition matrix will eventually become positive.
• If the chain has a period d then such a chain is called cyclic Markov chain.
Chains with transition states can be further divided in to
• Absorbing chain where the ergodic states are unit sets
• All ergodic sets are regular. This means if you take an ergodic set, then you can reach any state from any state in that ergodic set
• All ergodic sets are cyclic. This means that the states in the ergodic state are cyclical
• Both cyclical and regular ergodic states
As can be seen from the above illustration, almost all the categories can be expressed using simple random walk. Given various states, what are the kinds of questions that one would be interested in?
Regular Markov Chain Questions
• If a chain starts in Si, what is the probability after n steps that it will be in Sj?
• Can we predict the average number of steps that the process will be in Sj? Does it depend on where the process starts?
• We may wish to consider the process as it goes from Si to Sj. What is the mean and variance of the number of states passed? What is the probability that process passes through Sk?
• Study a subset of states and observe the process only when it is in these states
Transient States Questions
• Probability of entering a given ergodic set , starting from Si
• Mean and Variance of the number of times that the process is in Si, before entering an ergodic set, and how this number depends on the starting position ?
• Mean and variance of the number of steps needed before entering an ergodic set starting at Si?
• The mean number of states passed before entering an ergodic set , starting at Si
The book is sequenced in a way that the above questions are answered systematically. Absorbing Chains, Regular Markov Chains and Ergodic Chains are covered as specific chapters.
Chapter 3: Absorbing Markov Chains
The book looks at Absorbing Markov chain by segregating transient states from Persistent states. It then analyzes Transient State matrix and derives basic formulae in terms of matrix operations. The
highlight of this book is “Fundamental Matrix Approach”. For any Markov chain, the book tries to zero on to a specific matrix which is fundamental to all the calculations relating to the Markov chain
Fundamental Matrix for Absorbing Markov Chains is derived:
Solutions in Matrix form are furnished for the following:
• Total # of times that it is in various states
• Probability of absorption
• Total time before it gets absorbed
• Variance of total time before it gets absorbed.
From the examples give, all of them have high variance for the mean estimates. Is this a general case ? I mean are all the estimates of Markov chain characterized by high variance? I don’t know. Will
find out someday.
Chapter 4: Regular Markov Chains
Regular Markov Chains are explored in this chapter using Matrices. The limiting properties of Regular Markov chain are explored where the chain settles down in to a constant row matrix after n
iterations. This limiting markov chain is then used to state “Law of Large Numbers” . Most of us would have heard of “Law of Large Numbers” for independent trials where the statements become weak law
or strong law based on whether the convergence is almost sure or convergence is in probability. This law from a Markov chain is radically different as it removes , “iid” clause from the law of large
numbers and thus massively expanding its scope. This form of law was worked out by A.A.Markov in 1907. In the 100 odd years that have passed since then, Markov chains are now practically used in tons
of applications.
• The book’s USP is Fundamental Matrix and hence one sees the Fundamental Matrix derived for these Regular Markov chains as well, which takes the form
• Mean of first passage times for various states
• Variance of first passage times for various states
• Correlation for the above events
The chapter then talks about Central Limit theorems for Markov chains. It merely states the theorem and defers proving the same. CLT for Markov chains connects the (average number of visits to a
particular state, its limiting value) WITH Standard Normal Distribution.
The preface in the book mentions one of the reasons for the second edition. In the above formulae, Fixed weight vector was used to obtain the Fundamental Matrix. However there is an alternative
method where any constant row vector can be used to obtain Z. The appendix of the book mentions the paper where pseudo-inverses are used.
Chapter 5: Ergodic Markov Chains
What if there is cyclicality in chains with transient states ? This chapter delves in to the math behind such chains. Condition for a markov chain to be considered as reversible is also mentioned. A
Markov chain being reversible or not, has a great impact from a simulation perspective.The book then talks about further extensions to the above mentioned chains. Basically it involves stuff where a
transient chain is made in to an persistent chain forcefully to compute some interesting stuff and vice versa.
In the concluding chapter, the book shows various application of Markov chains like
• Random walks
• Ehrenfest Model
• Application to Genetics
• Learning Theory
• Mobility Theory
• Open Leontif Model
By no means these examples should make you think that Markov chain is used as Toy examples. The research and amount of stuff that has been done on Markov chains for the last 100 years is so massive
that it has found applications in a wide variety of fields from sports betting to
Take away:
Persi Diaconis (the iconoclastic Stanford Professor) remarks “To someone working in my part of the world, asking about applications of Markov chain Monte Carlo (MCMC) is a little like asking about
applications of the quadratic formula. The results are really used in every aspect of scientific inquiry.”
Indeed, Markov Chain Monte Carlo methods have revolutionized the field of statistics. My takeaway from the book is not so much so about simulation as it is about understanding Martingale Objects.
While formulating a Martingale, If you are able to visualize vaguely the discreet Markov Chain in the background, I bet your understanding will be much better. | {"url":"https://www.rksmusings.com/2011/05/28/finite-markov-chains-summary/","timestamp":"2024-11-08T17:07:11Z","content_type":"text/html","content_length":"22811","record_id":"<urn:uuid:90d92e5d-d5aa-419b-89da-d47a01c703f3>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00736.warc.gz"} |
has max currency value (1..1)
This property specifies the UPPER BOUND of the amount of money for a price RANGE per unit, shipping charges, or payment charges. The currency and other relevant details are attached to the respective
gr:PriceSpecification etc.
For a gr:UnitPriceSpecification, this is the UPPER BOUND for the price for one unit or bundle (as specified in the unit of measurement of the unit price specification) of the respective
gr:ProductOrService. For a gr:DeliveryChargeSpecification or a gr:PaymentChargeSpecification, it is the UPPER BOUND of the price per delivery or payment.
Using gr:hasCurrencyValue sets the upper and lower bounds to the same given value, i.e., x gr:hasCurrencyValue y implies x gr:hasMinCurrencyValue y, x gr:hasMaxCurrencyValue y.
type DatatypeProperty
This property specifies the UPPER BOUND of the amount of money for a price RANGE per unit, shipping charges, or payment charges. The currency and other relevant details are attached to
the respective gr:PriceSpecification etc.
For a gr:UnitPriceSpecification, this is the UPPER BOUND for the price for one unit or bundle (as specified in the unit of measurement of the unit price specification) of the respective
comment gr:ProductOrService. For a gr:DeliveryChargeSpecification or a gr:PaymentChargeSpecification, it is the UPPER BOUND of the price per delivery or payment.
Using gr:hasCurrencyValue sets the upper and lower bounds to the same given value, i.e., x ...
domain Price specification
isDefinedBy GoodRelations Ontology
label has max currency value (1..1)
range xsd:float | {"url":"https://data.ox.ac.uk/doc/?uri=http%3A%2F%2Fpurl.org%2Fgoodrelations%2Fv1%23hasMaxCurrencyValue&format=html","timestamp":"2024-11-09T05:58:38Z","content_type":"text/html","content_length":"10913","record_id":"<urn:uuid:5b3f67c3-b01c-4742-8e34-e2feca52d41d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00809.warc.gz"} |
Duality and form factors in the thermally deformed two-dimensional tricritical Ising model
Axel Cortés Cubero, Robert M. Konik, Máté Lencsés, Giuseppe Mussardo, Gabor Takács
SciPost Phys. 12, 162 (2022) · published 16 May 2022
• doi: 10.21468/SciPostPhys.12.5.162
The thermal deformation of the critical point action of the 2D tricritical Ising model gives rise to an exact scattering theory with seven massive excitations based on the exceptional $E_7$ Lie
algebra. The high and low temperature phases of this model are related by duality. This duality guarantees that the leading and sub-leading magnetisation operators, $\sigma(x)$ and $\sigma'(x)$, in
either phase are accompanied by associated disorder operators, $\mu(x)$ and $\mu'(x)$. Working specifically in the high temperature phase, we write down the sets of bootstrap equations for these four
operators. For $\sigma(x)$ and $\sigma'(x)$, the equations are identical in form and are parameterised by the values of the one-particle form factors of the two lightest $\mathbb{Z}_2$ odd particles.
Similarly, the equations for $\mu(x)$ and $\mu'(x)$ have identical form and are parameterised by two elementary form factors. Using the clustering property, we show that these four sets of solutions
are eventually not independent; instead, the parameters of the solutions for $\sigma(x)/\sigma'(x)$ are fixed in terms of those for $\mu(x)/\mu'(x)$. We use the truncated conformal space approach to
confirm numerically the derived expressions of the matrix elements as well as the validity of the $\Delta$-sum rule as applied to the off-critical correlators. We employ the derived form factors of
the order and disorder operators to compute the exact dynamical structure factors of the theory, a set of quantities with a rich spectroscopy which may be directly tested in future inelastic neutron
or Raman scattering experiments.
Authors / Affiliations: mappings to Contributors and Organizations
See all Organizations.
Funders for the research work leading to this publication | {"url":"https://scipost.org/SciPostPhys.12.5.162","timestamp":"2024-11-11T20:41:44Z","content_type":"text/html","content_length":"40420","record_id":"<urn:uuid:d19f0ab8-8748-42b0-970a-edd44003974e>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00106.warc.gz"} |
How Many Spades Are In a Deck of Cards? (52 Card Standard Deck)
How many spades are in a deck of 52 cards?
If you are new to a deck of cards or card magic, you might not know, how many Spades are in a deck of 52 cards? This post will answer that and some other related questions you might have as well.
In this post, you’ll learn how many Spades are in a deck of cards, what all the Spades cards are, and you will be able to learn how playing cards work.
How Many Spades are in a Deck of 52 Cards?
The exact number of Spade cards in a deck of cards is 13. This is because there are four different suits that make up a complete deck. They are Hearts, Club, Diamonds, and Spades.
Each of these suits has a total of 13 cards each. These are one of each of the number cards and face cards, Ace through King. These are the Ace, Two, Three, Four, Five, Six, Seven, Eight, Nine,
Ten, Jack, Queen, and King. (You can learn more about how a deck works at A Deck of Playing Cards Explained)
There are 13 spades in a deck of 52 cards
There are 13 Spades in a deck of 52 cards.
Understanding Playing Cards
Here are two videos that teach how playing cards work.
Understanding Playing Cards
An Introduction to Playing Cards
Video Link: How Many Spades are in a Deck of 52 Cards?
What Are All the Spade Cards in a Deck?
The 13 Spade cards in a deck of 52 cards are as follows:
1. Ace of Spades
2. Two of Spades
3. Three of Spades
4. Four of Spades
5. Five of Spades
6. Six of Spades
7. Seven of Spades
8. Eight of Spades
9. Nine of Spades
10. Ten of Spades
11. Jack of Spades
12. Queen of Spades
13. King of Spades
How Many Ace of Spades are in a Deck of Cards?
There is one Ace of Spades in a deck of cards. This is because there are 4 ace cards total in a deck. Each of the four suits has its own ace.
There are 13 Spades in a deck and 1 Ace of Spades
How Many King of Spades are in a Deck of Cards?
There are 4 suits in a deck. Each deck gets one of the 13 value cards which are Ace through King. So, the Spades get one King. This means there is one King of Spades in a deck of cards.
(Learn more about Kings at How Many Kings are in a Deck of Cards?)
How Many Jack of Spades are in a Deck of Cards?
There are 4 Jacks in a deck of 52 cards. Each of the 4 suits has one Jack each. So there is only one Jack of Spades in a deck of cards.
How Many Spades Appear on the Ace of Spades?
On most playing cards there are actually 3 spades that appear on the Ace of Spades. Although there are always the same number of pips (pictures of the suit) as the value of the card in the center of
a card, there are also the two pips that are in the corners underneath the number or letter that represents the playing card.
In the case of the Ace of Spades, there is one Spade pip in the center of the card, but then there are two more spades underneath the A letter that appears in the corner. If you look at the image
above of the Ace of Spades you will see what I mean.
This means there are actually 3 spades that appear on the Ace of Spades.
How Many Black Spades are in a Deck of Cards?
Since all the Spades are already part of the black cards, which include all the spades and clubs, there are 13 black Spades in a deck of cards.
What is the Probability of Getting a Spade from a Deck of 52 Cards?
The probability of getting a spade from a deck of 52 cards is 1/4 or 25%.
This is because there are 13 spades cards in a deck of 52 cards. So there are 13 chances out of 52 to get a Spade card, which is 13/52 or 1/4.
Find out more about card probability at Playing Card Probability Explained.
How Many Black 3s are in a Deck of 52 Cards?
There are two black 3s in a deck of 52 cards. This is because there are two black suits, the Spades and the Clubs. Each suit has one of each value card, so each of these suits has one 3.
So, the two black 3s are the 3 of Clubs and the 3 of Spades.
Is there a 12 in a Deck of Cards?
There is no 12 in a deck of cards. Instead, after the number 10, the cards become face cards instead of the numbers 11, 12, and 13. Instead of the number 12, the Queen is used in the same place
that a 12 would be.
Why is There a Joker in a Card Deck?
Here, you can find out more about the Joker card and why it appears in a deck.
Related Questions
We have seen that a standard deck of cards is composed of exactly 13 spades. These are the 13 values Ace through King, with each value only being used once per suit.
The magician started magic as a kid and has learned from some of the greats. He loves to share his knowledge with others and help out with the subtleties he’s learned along the way.
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Ected with this operate are presented Kifunensine Technical Information inside the last section. two. Methodology | calpaininhibitor.com
Ected with this operate are presented Kifunensine Technical Information inside the last section. two. Methodology In this section, classic logit, Bayesian, and asymmetric Bayesian logit models are
described in detail. Because it is well-known, logit and probit models would be the highest well known models concerning binary outcomes. A binary response model can be a Tianeptine sodium salt
Epigenetics regression model in which the dependent variable Y can be a binary random variable that requires only the values zero and one particular. In our case, the variable y = 1 if a tourist
rents a car and y = 0 otherwise. Within this post, we make use of the logit model to estimate the probability of renting vehicles given a set of qualities of your event; that may be, provided the
predictor X, we estimate Pr(1| X = x), i.e., the conditional probability that y = 1 given the value with the predictor. As is known, the logit specification can be a certain instance of a generalized
linear model (see Weisberg 2005, chp. 12, for details). On the other hand, the logistic link function is a moderately not confusing alteration in the prediction curve and yields odds ratios. Each
traits make it well-received amongst researchers in front from the probit regression. The typical logistic distribution has a closed-form expression in addition to a shape notably comparable for the
regular distribution. Logit models have been utilized extensively in a number of fields, which includes medicine, biology, psychology, economics, insurance coverage, politics, etc. Recent
applications of binary response specification in vehicle renting are Gomes de Menezes and Uzagalieva (2012), Masiero and Zoltan (2013), Dimatulac et al. (2018) or Narsaria et al. (2020), among other
folks. Gomes de Menezes and Uzagalieva (2012) analyze the demand function of car or truck rentals within the Azores, taking into account the asymmetry by estimating a family members of zero-inflated
models. 2.1. Logistic Specification To make the paper self-contained, we describe the logistic specification briefly. Let Yi be a continuous and unobserved random variable connected with all the
occasion of renting a car for a person i which can be specified as Yi = xi i , where = ( 1 , , k) is often a k 1 vector of regression coefficients, which represents the impact of each variable within
the model, and it need to be estimated and xi = ( xi1 , …, xik) is usually a vector (explanatory variables) of known constants, which can include an intercept, the vector of covariates for the
tourist i in our case. The random variable is a disturbance term. We assume that Yi = 1 Yi = 0 if Yi 0, otherwise.J. Threat Monetary Manag. 2021, 14,four ofThus, we’ve pi = Pr(Yi = 1) = Pr( xi i 0) =
1 – F (- xi), exactly where F ( will be the cumulative distribution function of your random variable . Moreover, the marginal effect on pi for any alter in xk outcomes f (- xi) k , where f ( is the
probability density function of your random variable . If we assume F ( to become the regular typical cdf, , we get the probit model, and if we assume the logistic distribution, we’ve got the
logistic regression, which will be viewed as here. Then, for observation i within a sample of size n, we assume that pi = Pr(Yi = 1) = exp( xi) 1 = , 1 exp(- xi) 1 exp( xi)and Pr(Yi = 0) = 1 – pi .
Recall that the probability density function of the regular logistic distribution is symmetric about 0. In summary, the logit specification adopts the following form: log pi 1 – pi= xi ,i = 1, 2, . .
. , n.As a result, the likelihood is given by(y| x,) = [ F ( xi)]yi [1 – F ( xi)]1-yi ,i =n(1)where the parameters are often estimated by the maximum likelihood approach. Within this way, the model
offers the probab. | {"url":"https://www.calpaininhibitor.com/2022/07/11/8432/","timestamp":"2024-11-04T11:52:37Z","content_type":"text/html","content_length":"67828","record_id":"<urn:uuid:c2006862-ca0d-4cab-b1aa-d198596f9c34>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00007.warc.gz"} |
Count possible ways to construct buildings
Hello Everyone,
Given an input number of sections and each section has 2 plots on either sides of the road. Find all possible ways to construct buildings in the plots such that there is a space between any 2
Example :
N = 1 Output = 4 Place a building on one side. Place a building on other side Do not place any building. Place a building on both sides. N = 3 Output = 25 3 sections, which means possible ways for
one side are BSS, BSB, SSS, SBS, SSB where B represents a building and S represents an empty space Total possible ways are 25, because a way to place on one side can correspond to any of 5 ways on
other side. N = 4 Output = 64
We can simplify the problem to first calculate for one side only. If we know the result for one side, we can always do square of the result and get result for two sides.
A new building can be placed on a section if section just before it has space. A space can be placed anywhere (it doesn’t matter whether the previous section has a building or not).
Let countB(i) be count of possible ways with i sections ending with a building. countS(i) be count of possible ways with i sections ending with a space. // A space can be added after a building or
after a space. countS(N) = countB(N-1) + countS(N-1) // A building can only be added after a space. countB[N] = countS(N-1) // Result for one side is sum of the above two counts. result1(N) = countS
(N) + countB(N) // Result for two sides is square of result1(N) result2(N) = result1(N) * result1(N)
Below is the implementation of above idea.
// C++ program to count all possible way to construct buildings
using namespace std;
// Returns count of possible ways for N sections
int countWays( int N)
// Base case
if (N == 1)
return 4; // 2 for one side and 4 for two sides
// countB is count of ways with a building at the end
// countS is count of ways with a space at the end
// prev_countB and prev_countS are previous values of
// countB and countS respectively.
// Initialize countB and countS for one side
int countB=1, countS=1, prev_countB, prev_countS;
// Use the above recursive formula for calculating
// countB and countS using previous values
for ( int i=2; i<=N; i++)
prev_countB = countB;
prev_countS = countS;
countS = prev_countB + prev_countS;
countB = prev_countS;
// Result for one side is sum of ways ending with building
// and ending with space
int result = countS + countB;
// Result for 2 sides is square of result for one side
return (result*result);
// Driver program
int main()
int N = 3;
cout << "Count of ways for " << N
<< " sections is " << countWays(N);
return 0;
Output :
Count of ways for 3 sections is 25
Time complexity: O(N)
Auxiliary Space: O(1)
Algorithmic Paradigm: Dynamic Programming
Optimized Solution:
Note that the above solution can be further optimized. If we take closer look at the results, for different values, we can notice that the results for two sides are squares of Fibonacci Numbers.
N = 1, result = 4 [result for one side = 2]
N = 2, result = 9 [result for one side = 3]
N = 3, result = 25 [result for one side = 5]
N = 4, result = 64 [result for one side = 8]
N = 5, result = 169 [result for one side = 13]
In general, we can say
result(N) = fib(N+2)2 fib(N) is a function that returns N’th Fibonacci Number. | {"url":"https://discuss.boardinfinity.com/t/count-possible-ways-to-construct-buildings/6818","timestamp":"2024-11-06T17:36:03Z","content_type":"text/html","content_length":"19764","record_id":"<urn:uuid:dce7a8e5-d7e5-45c1-99b2-02542567e271>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00431.warc.gz"} |
[Solved] Graph each system of linear inequalities. | SolutionInn
Graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the
Graph each system of linear inequalities. Tell whether the graph is bounded or unbounded, and label the corner points.
Transcribed Image Text:
х2 0 y 2 0 х+ у 2 2х + Зу S 12 < 12 Зх + у< 12
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Students also viewed these Mathematics questions
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linear programming
A procedure for finding the maximum or minimum of a linear function where the arguments are subject to linear constraints. The simplex method is one well known algorithm.
Last updated: 1995-04-06
Nearby terms:
linear logic ♦ linear map ♦ linear programming ♦ linear space ♦ linear topology
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A backtracking Solution to the Pattern132 Problem
Pattern132 is an interesting problem I've seen floating around on various message boards. It is a constraint satisfaction problem that I've seen all manner of solutions for ranging from dynamic
programming, to straight brute force iteration. When I first scoped the problem I immediately thought it was the kind of problem that would be fun to solve with backtracking, and so I present a
backtracking solution to the Pattern132 problem.
The Problem
As already mentioned, the Pattern 132 problem is a constraint satisfaction problem. Given a list of numbers where the values i, j, and k are the index of items in the list. Find if there is a
sequence of values in the provided list that meet the following criteria:
i < j < k AND list[i] < list[k] < list[j]
For example, the input array, {1, 3, 4, 2} matches Pattern132 with the given i, j, k of (0, 1, 3) and (0, 2, 3).
Finding a solution with backtracking
Backtracking algorithms all follow a basic pattern, which i describe below in pseudocode:
bool backtracking(input list, index i, index j, index k) {
if (the input parameters match a solution) {
display solution
return true;
if (the next input permutation is promising) {
backtracking(input list, new index i, new index j, new index k)
Starting from our framework we know we are going to need a few things to implement our backtracking algorithm:
• A test to determine if our current configuration of inputs is a solution
• A test to determine if the next input permutation looks promising.
It is this promising function that allows us to trim the search space, and differentiates this method from being a brute force enumeration, this in combination with the recursive nature of our depth
first search through the search space is what makes backtracking algorithms efficient.
The test to check for a solution is simple:
bool isSolution(input list, int i, int j, int k) {
if (i < j < k && list[i] < list[k] < list[j]) {
return true;
return false;
So now we need our promising function. To do this, we need to know how we are going to implement the permutation, to know if its worth looking at. As our backtracking function stands right now, we
have the input list, and index i, index j, and index k of the list. To implement backtracking we will use the following calls:
backtracking(input list, i + 1, j, k)
backtracking(input list, i, j + 1, k)
backtracking(input list, i, j, k + 1)
So our promising function should check: does our input permutation satisfy the first condition of our solution, that is, i < j < k? if so, we can proceed to check if it has a solution. This means the
above function calls should be wrapped like this:
bool isPromising(int i, int j, int k) {
return (i < j) && (j < k);
if (isPromising(i+1, j, k)) backtracking(input list, i + 1, j, k)
if (isPromising(i, j+1, k)) backtracking(input list, i, j + 1, k)
if (isPromising(i, j, k+1)) backtracking(input list, i, j, k + 1)
Putting it all together
Now that we have our plan of attack, its a simple matter of putting it all together into a working algorithm:
public class Pattern132 {
private static boolean isValidPattern(int[] a, int i, int j, int k) {
if (isPromising(i, j, k)) {
if (a[i] < a[k] && a[k] < a[j]) {
return true;
return false;
public static boolean isPromising(int i, int j, int k) {
return (i < j && j < k);
public static boolean backtracking132(int[] a, int i, int j, int k) {
if (isValidPattern(a, i, j, k)) {
System.out.println("[" + i + ", " + j + ", " + k + "]");
System.out.println("(" + a[i] + ", " + a[j] + ", " + a[k] + ")");
return true;
if (i+1 < a.length && isPromising(i+1, j, k)) backtracking132(a, i+1, j, k);
if (j+1 < a.length && isPromising(i, j+1, k)) backtracking132(a, i, j+1, k);
if (k+1 < a.length && isPromising(i, j, k+1)) backtracking132(a, i, j, k+1);
return false;
public static void main(String[] args) {
int[] a = new int[]{1, 3, 4, 2};
backtracking132(a, 0, 1, 2);
You'll notice the initial call to the backtracking function has the indexes 0, 1, 2 to start. We COULD use 0, 0, 0 as starting indices but we already know it would fail our promising function, so we
might as well start with the first possible combination of indices that could satisfy i < j < k: 0, 1, 2. | {"url":"http://maxgcoding.com/a-backtracking-solution-to-the-pattern132-problem","timestamp":"2024-11-03T07:42:58Z","content_type":"text/html","content_length":"13892","record_id":"<urn:uuid:9128793f-7c2b-4c68-af37-3688764609fb>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00057.warc.gz"} |
The Taub Faculty of Computer Science
The Taub Faculty of Computer Science Events and Talks
Dean Leitersdorf (CS, Technion)
Wednesday, 08.12.2021, 12:30
Given a distributed network, represented by a graph of computation nodes connected by communication edges, a fundamental problem is computing distances between nodes. Our recent line of works show
that while exactly computing all distances (All-Pairs-Shortest-Paths, APSP) in certain distributed settings currently has O(n^{1/3}) complexity, it is possible to find very good distance
approximations even with an O(1) complexity. This talk will present a unified view of our various papers developing these interesting advances in distributed computing. The central theme uniting
these developments is designing sparsity-aware algorithms, and then applying them to problems on general, non-sparse, graphs. Primarily, the main tools used are a series of distributed sparse matrix
multiplication algorithms we develop. We then use these tools in novel manners to develop a toolkit for basic distance computations, such as computing for each node the distances to the O(n^{2/3})
nodes closest to it. Next, we use these tools to solve end-goal distance problems, in O(log^2 n) rounds. Subsequently, we adapt the load-balancing techniques used in the matrix multiplication
algorithms to show combinatorial algorithms which directly compute APSP approximations, without passing through other intermediate distance tools. This allows us to show O(1) round algorithms.
Finally, the talk concludes with observing the connection of the above addressed works to other developments in the realm of distributed computing, specifically MST computation and subgraph existence | {"url":"https://cs.technion.ac.il/events/view-event.php?evid=10149","timestamp":"2024-11-12T16:01:18Z","content_type":"text/html","content_length":"15770","record_id":"<urn:uuid:5794f723-1d7d-42ba-9c41-eae041c65dc4>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00227.warc.gz"} |
Applied Mechanics of Solids (A.F. Bower) Problems 5: Solutions for
Elastic Solids - 5.4 3D Static Problems
Problems for Chapter 5
Analytical Techniques and Solutions for Linear Elastic Solids
5.4. Solutions to 3D Static Problems
5.4.1. Consider the Papkovich-Neuber potentials
${\Psi }_{i}=\frac{\left(1-u \right){\sigma }_{0}}{\left(1+u \right)}{x}_{3}{\delta }_{i3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace
{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{u \left(1-u \right){\sigma }_{0}}{\left(1+u \right)}\left(3{x}_
5.4.1.1. Verify that the potentials satisfy the equilibrium equations
5.4.1.2. Show that the fields generated from the potentials correspond to a state of uniaxial stress, with magnitude ${\sigma }_{0}$ acting parallel to the ${e}_{3}$ direction of an infinite solid
5.4.2. Consider the fields derived from the Papkovich-Neuber potentials
${\Psi }_{i}=\frac{\left(1-u \right)p}{\left(1+u \right)}{x}_{i}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\
text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{2u \left(1-u \right)p}{\left(1+u \right)}{R}^{2}$
5.4.2.1. Verify that the potentials satisfy the equilibrium equations
5.4.2.2. Show that the fields generated from the potentials correspond to a state of hydrostatic tension ${\sigma }_{ij}=p{\delta }_{ij}$
5.4.3. Consider the Papkovich-Neuber potentials
${\Psi }_{i}=\alpha {x}_{i}+\beta \frac{{x}_{i}}{{R}^{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\
hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\alpha {R}^{2}-\frac{3\beta }{R}$
5.4.3.1. Verify that the potentials satisfy the governing equations
5.4.3.2. Show that the potentials generate a spherically symmetric displacement field
5.4.3.3. Calculate values of $\alpha$ and $\beta$ that generate the solution to an internally pressurized spherical shell, with pressure p acting at R=a and with surface at R=b traction free.
5.4.4. Verify that the Papkovich-Neuber potential
${\Psi }_{i}=\frac{{P}_{i}}{4\pi R}⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄⥄\varphi =0$
generates the fields for a point force $P={P}_{1}{e}_{1}+{P}_{2}{e}_{2}+{P}_{3}{e}_{3}$ acting at the origin of a large (infinite) elastic solid with Young’s modulus E and Poisson’s ratio $u$. To
this end:
5.4.4.1. Verify that the potentials satisfy the governing equation
5.4.4.2. Calculate the stresses
5.4.4.3. Consider a spherical region with radius R surrounding the origin. Calculate the resultant force exerted by the stress on the outer surface of this sphere, and show that they are in
equilibrium with a force P.
5.4.5. Consider an infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio $u$. Suppose that the solid contains a rigid spherical particle (an inclusion) with radius a
and center at the origin. The particle is perfectly bonded to the elastic matrix, so that ${u}_{i}=0$ at the particle/matrix interface. The solid is subjected to a uniaxial tensile stress ${\sigma
}_{33}={\sigma }_{0}$ at infinity. Calculate the stress field in the elastic solid. To proceed, note that the potentials
${\Psi }_{i}=\frac{\left(1-u \right){\sigma }_{0}}{\left(1+u \right)}{x}_{3}{\delta }_{i3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace
{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{u \left(1-u \right){\sigma }_{0}}{\left(1+u \right)}\left(3{x}_
generate a uniform, uniaxial stress ${\sigma }_{33}={\sigma }_{0}$ (see problem 1). The potentials
${\Psi }_{i}=\frac{{a}^{3}{p}_{ik}^{T}{x}_{k}}{3{R}^{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\
hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{{a}^{3}{p}_{ij}^{T}}{15{R}^{3}}\left(\left(5{R}^{2}-{a}^{2}
\right){\delta }_{ij}+3{a}^{2}\frac{{x}_{i}{x}_{j}}{{R}^{2}}\right)$
are a special case of the Eshelby problem described in Section 5.4.6, and generate the stresses outside a spherical inclusion, which is subjected to a uniform transformation strain. Let ${p}_{ij}^
{T}=A{\delta }_{ij}+B{\delta }_{i3}{\delta }_{j3}$, where A and B are constants to be determined. The two pairs of potentials can be superposed to generate the required solution.
5.4.6. Consider an infinite, isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio $u$. Suppose that the solid contains a spherical particle (an inclusion) with radius a and
center at the origin. The particle has Young’s modulus ${E}_{p}$ and Poisson’s ratio ${u }_{p}$, and is perfectly bonded to the matrix, so that the displacement and radial stress are equal in both
particle and matrix at the particle/matrix interface. The solid is subjected to a uniaxial tensile stress ${\sigma }_{33}={\sigma }_{0}$ at infinity. The objective of this problem is to calculate
the stress field in the elastic inclusion.
5.4.6.1. Assume that the stress field inside the inclusion is given by ${\sigma }_{ij}=A{\sigma }_{0}{\delta }_{ij}+B{\sigma }_{0}{\delta }_{i3}{\delta }_{j3}$. Calculate the displacement field in
the inclusion (assume that the displacement and rotation of the solid vanish at the origin).
5.4.6.2. The stress field outside the inclusion can be generated from Papkovich-Neuber potentials
${\Psi }_{i}=\frac{\left(1-u \right){\sigma }_{0}}{\left(1+u \right)}{x}_{3}{\delta }_{i3}+\frac{{a}^{3}{p}_{ik}^{T}{x}_{k}}{3{R}^{3}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace
{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\varphi =\frac{u \left(1-u \right){\
sigma }_{0}}{\left(1+u \right)}\left(3{x}_{3}^{2}-{R}^{2}\right)+\frac{{a}^{3}{p}_{ij}^{T}}{15{R}^{3}}\left(\left(5{R}^{2}-{a}^{2}\right){\delta }_{ij}+3{a}^{2}\frac{{x}_{i}{x}_{j}}{{R}^{2}}\right)$
where ${p}_{ij}^{T}=C{\sigma }_{0}{\delta }_{ij}+D{\sigma }_{0}{\delta }_{i3}{\delta }_{j3}$, and C and D are constants to be determined.
5.4.6.3. Use the conditions at r=a to find expressions for A,B,C,D in terms of geometric and material properties.
5.4.6.4. Hence, find the stress field inside the inclusion.
5.4.7. Consider the Eshelby inclusion problem described in Section 5.4.6. An infinite homogeneous, stress free, linear elastic solid has Young’s modulus E and Poisson’s ratio $u$. The solid is
initially stress free. An inelastic strain distribution ${\epsilon }_{ij}^{T}$ is introduced into an ellipsoidal region of the solid B (e.g. due to thermal expansion, or a phase transformation).
Let ${u}_{i}$ denote the displacement field, ${\epsilon }_{ij}^{}={\epsilon }_{ij}^{e}+{\epsilon }_{ij}^{T}$ denote the total strain distribution, and let ${\sigma }_{ij}$ denote the stress field in
the solid.
5.4.7.1. Write down an expression for the total strain energy ${\Phi }_{I}$ within the ellipsoidal region, in terms of ${\sigma }_{ij}$, ${\epsilon }_{ij}$ and ${\epsilon }_{ij}^{T}$.
5.4.7.2. Write down an expression for the total strain energy outside the ellipsoidal region, expressing your answer as a volume integral in terms of ${\epsilon }_{ij}$ and ${\sigma }_{ij}$. Using
the divergence theorem, show that the result can also be expressed as
${\Phi }_{O}=-\frac{1}{2}\underset{S}{\int }{\sigma }_{ij}{n}_{j}{u}_{i}dA$
where S denotes the surface of the ellipsoid, and ${n}_{j}$ are the components of an outward unit vector normal to B. Note that, when applying the divergence theorem, you need to show that the
integral taken over the (arbitrary) boundary of the solid at infinity does not contribute to the energy $–$ you can do this by using the asymptotic formula given in Section 5.4.6 for the
displacements far from an Eshelby inclusion.
5.4.7.3. The Eshelby solution shows that the strain ${\epsilon }_{ij}^{}={\epsilon }_{ij}^{e}+{\epsilon }_{ij}^{T}$ inside B is uniform. Write down the displacement field inside the ellipsoidal
region, in terms of ${\epsilon }_{ij}$ (take the displacement and rotation of the solid at the origin to be zero). Hence, show that the result of 7.2 can be re-written as
${\Phi }_{O}=-\frac{1}{2}\underset{S}{\int }{\sigma }_{ij}{\epsilon }_{ik}{x}_{k}{n}_{j}dA$
5.4.7.4. Finally, use the results of 7.1 and 7.3, together with the divergence theorem, to show that the total strain energy of the solid can be calculated as
$\Phi ={\Phi }_{O}+{\Phi }_{I}=-\frac{1}{2}\underset{B}{\int }{\sigma }_{ij}{\epsilon }_{ij}^{T}⥄dV$
5.4.8. Using the solution to Problem 7, calculate the total strain energy of an initially stress-free isotropic, linear elastic solid with Young’s modulus E and Poisson’s ratio $u$, after an
inelastic strain ${\epsilon }_{ij}^{T}$ is introduced into a spherical region with radius a in the solid.
5.4.9. A steel ball-bearing with radius 1cm is pushed into a flat steel surface by a force P. Neglect friction between the contacting surfaces. Typical ball-bearing steels have uniaxial tensile
yield stress of order 2.8 GPa. Calculate the maximum load that the ball-bearing can withstand without causing yield, and calculate the radius of contact and maximum contact pressure at this load.
5.4.10. The contact between the wheel of a locomotive and the head of a rail may be approximated as the (frictionless) contact between two cylinders, with identical radius R as illustrated in the
figure. The rail and wheel can be idealized as elastic-perfectly plastic solids with identical Young’s modulus E, Poisson’s ratio $u$ and yield stress Y. Find expressions for the radius of the
contact patch, the contact area, and the contact pressure as a function of the load acting on the wheel and relevant geometric and material properties. By estimating values for relevant quantities,
calculate the maximum load that can be applied to the wheel without causing the rail to yield.
5.4.11. The figure shows a rolling element bearing. The inner raceway has radius R, and the balls have radius r, and both inner and outer raceways are designed so that the area of contact between
the ball and the raceway is circular. The balls are equally spaced circumferentially around the ring. The bearing is free of stress when unloaded. The bearing is then subjected to a force P as
shown. This load is transmitted through the bearings at the contacts between the raceways and the balls marked A, B, C in the figure (the remaining balls lose contact with the raceways but are held
in place by a cage, which is not shown). Assume that the entire assembly is made from an elastic material with Young’s modulus $E$ and Poisson’s ratio $u$
5.4.11.1. Assume that the load causes the center of the inner raceway to move vertically upwards by a distance $\Delta$, while the outer raceway remains fixed. Write down the change in the gap
between inner and outer raceway at A, B, C, in terms of $\Delta$
5.4.11.2. Hence, calculate the resultant contact forces between the balls at A, B, C and the raceways, in terms of $\Delta$ and relevant geometrical and material properties.
5.4.11.3. Finally, calculate the contact forces in terms of P
5.4.11.4. If the materials have uniaxial tensile yield stress Y, find an expression for the maximum force P that the bearing can withstand before yielding.
5.4.12. A rigid, conical indenter with apex angle $2\beta$ is pressed into the surface of an isotropic, linear elastic solid with Young’s modulus $E$ and Poisson’s ratio $u$.
5.4.12.1. Write down the initial gap between the two surfaces $g\left(r\right)$
5.4.12.2. Find the relationship between the depth of penetration h of the indenter and the radius of contact a
5.4.12.3. Find the relationship between the force applied to the contact and the radius of contact, and hence deduce the relationship between penetration depth and force. Verify that the contact
stiffness is given by $\frac{dP}{dh}=2{E}^{*}a$
5.4.12.4. Calculate the distribution of contact pressure that acts between the contacting surfaces.
5.4.13. A sphere, which has radius R, is dropped from height h onto the flat surface of a large solid. The sphere has mass density $\rho$, and both the sphere and the surface can be idealized as
linear elastic solids, with Young’s modulus $E$ and Poisson’s ratio $u$. As a rough approximation, the impact can be idealized as a quasi-static elastic indentation.
5.4.13.1. Write down the relationship between the force P acting on the sphere and the displacement of the center of the sphere below ${x}_{2}=R$
5.4.13.2. Calculate the maximum vertical displacement of the sphere below the point of initial contact.
5.4.13.3. Deduce the maximum force and contact pressure acting on the sphere
5.4.13.4. Suppose that the two solids have yield stress in uniaxial tension Y. Find an expression for the critical value of h which will cause the solids to yield
5.4.13.5. Calculate a value of h if the materials are steel, and the sphere has a 1 cm radius. | {"url":"http://solidmechanics.org/problems/Chapter5_4/Chapter5_4.php","timestamp":"2024-11-15T01:32:15Z","content_type":"text/html","content_length":"103952","record_id":"<urn:uuid:2f78bd52-ab06-435c-b6c6-287a14df336e>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00058.warc.gz"} |
Rearranging Numbers Worksheets - 15 Worksheets.com
Rearranging Numbers Worksheets
About These 15 Worksheets
These worksheets aim to strengthen students’ understanding of number patterns, place value, and numerical relationships. Most of the worksheets specifically target place value concepts by asking
students to rearrange the digits within a number to create the largest or smallest possible value. This activity enhances students’ understanding of the significance of each digit and how they
contribute to the overall value of a number.
The purpose of rearranging numbers worksheets is to strengthen students’ numeracy skills, enhance their ability to recognize patterns, and improve their logical thinking. These worksheets promote
critical thinking, problem-solving, and numerical fluency.
How to Rearrange Numbers to Make the Largest or Smallest Value Possible
Rearranging numbers to make the largest possible value involves organizing the given digits or numbers in a specific order. For the largest value, you want the largest integers in the highest
possible place value. In the end you would arrange the integers from highest to lowest from the greatest place value to the least.
Here’s a step-by-step guide to rearranging numbers to create the largest value:
List the Numbers – Write down the given numbers or digits that you need to rearrange. For example, let’s assume you have the digits 5, 2, 9, 1, and 4.
Compare and Rearrange – Compare the digits based on their values, starting from the left-most position. Rearrange the digits in descending order, placing the largest digit in the leftmost position.
In our example, you would place the digit 9 first, as it is the largest among the given digits. Continue the process of comparing and rearranging the remaining digits in descending order. In our
example, the next largest digit is 5, followed by 4, 2, and finally 1.
Combine the Digits – Combine the rearranged digits to form the largest possible value. In our example, the largest value created by rearranging the given digits is 95,421.
Here is a quick example for you. Let’s say you were given these digits: 5, 7, 1, 9, 3
If you were asked to form the largest value possible, you would take these steps:
• The largest digit is 9, so it will be placed in the leftmost position.
• The next largest digit is 7, so it will be placed to the right of 9.
• Arrange the remaining digits in descending order: 5, 3, 1.
• Combine the digits in the determined order: 97531 or 97,531.
What If We Wanted the Smallest Value Possible?
You would take similar steps from what we just did, but in reverse. Here are the steps required to make the smallest value possible:
• Look at the given numbers and identify the digits or numbers involved.
• Determine the total number of digits or numbers to be used.
• Place the smallest digit or number in the leftmost position (most significant digit) and continue in ascending order for the remaining positions.
• If there are duplicate digits or numbers, arrange them in ascending order as well.
• Combine the digits or numbers in the determined order to form the smallest possible value.
Example: Given the digits: 5, 7, 1, 9, 3
If you were asked to form the smallest value possible, you would take these steps:
• To make the smallest value:
• The smallest digit is 1, so it will be placed in the leftmost position.
• The next smallest digit is 3, so it will be placed to the right of 1.
• Arrange the remaining digits in ascending order: 5, 7, 9.
• Combine the digits in the determined order: 13579 or 13,579. | {"url":"https://15worksheets.com/worksheet-category/rearranging-numbers/","timestamp":"2024-11-08T01:52:59Z","content_type":"text/html","content_length":"128020","record_id":"<urn:uuid:101d53a7-c9a4-4275-81e2-124bc001cc15>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00859.warc.gz"} |
Find the product and give the domain of (y+3)/(y+2)xx(4y+4+y^-Turito
Are you sure you want to logout?
Find the product and give the domain of
The expansions of certain identities are:
We are asked to find the product of the expression and hence find the domain.
The correct answer is: y 3 = 0 y = 3
Step 1 of 2:
Simplify the expression and hence find he product:
Thus, the simplified expression,
Step 2 of 2:
To find the domain of the expression, exclude the values for which the denominator gets zero value. That is,
Thus, the domain is:
When you find the domain of a rational expression exclude the values that bring zero to the denominator.
Get an Expert Advice From Turito. | {"url":"https://www.turito.com/ask-a-doubt/find-the-product-and-give-the-domain-of-y-3-y-2-xx-4y-4-y-2-y-2-9-q9c23b3b7","timestamp":"2024-11-11T17:44:02Z","content_type":"application/xhtml+xml","content_length":"364685","record_id":"<urn:uuid:86b78405-6dcd-4175-84b3-6a1525c913c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00009.warc.gz"} |
Chaos Game Fractal Generator
Create fractals using the chaos game method.
Charlie Harrison
From Wikipedia: In mathematics, the term chaos game, as coined by Michael Barnsley, originally referred to a method of creating a fractal, using a polygon and an initial point selected at random
inside it. The fractal is created by iteratively creating a sequence of points, starting with the initial random point, in which each point in the sequence is a given fraction of the distance between
the previous point and one of the vertices of the polygon; the vertex is chosen at random in each iteration. Repeating this iterative process a large number of times, selecting the vertex at random
on each iteration, and throwing out the first few points in the sequence, will often (but not always) produce a fractal shape. Using a regular triangle and the factor 1/2 will result in the
Sierpinski triangle.
Q and A control the x-component factor for all vertices
W and S control the y-component factor for all vertices
R and F control the x factor for a highlighted vertex
T and G control the y factor for a highlighted vertex
(EDIT: sorry I totally forgot these essential controls!)
SPACE adds a vertex
D removes a vertex
If you're looking for inspiration, try making a square with 3 boxes per side and an x and y factor of .7 for Sierpinski's Gasket.
Chaos Game Fractal Generator 1.0 — 21 Jan, 2011
Pygame.org account Comments | {"url":"https://www.pygame.org/project/1754","timestamp":"2024-11-04T16:48:13Z","content_type":"text/html","content_length":"29687","record_id":"<urn:uuid:8b0339f8-c691-4831-ba4b-488301471bb5>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00114.warc.gz"} |
System of equations calculator
A system of linear equations consists of multiple linear equations. Linear equations with two variables correspond to lines in the coordinate plane, so this linear equation system is nothing more
than asking if, and if yes, where the two lines intersect. This implies it can have no solution (if the lines are parallel), one solution (if they intersect) or infinitely many solutions (if the
lines are equal).
There are three important ways to solve such systems: by insertion, by equalization and by adding.
Equalization means you solve both equations for the same variable and then equalize them. This means, one variable remains and the calculation is then easy.
Equation systems
This is the system of equations calculator of Mathepower. Enter two or more equations containing many variables. Mathepower tries to solve them step-by-step. | {"url":"https://www.mathepower.com/en/system_of_equations.php","timestamp":"2024-11-01T23:26:36Z","content_type":"text/html","content_length":"48124","record_id":"<urn:uuid:e155318f-199c-496f-8240-56aa38b4fd46>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00284.warc.gz"} |
Numerical Modeling of Ejector and Development of Improved Methods for the Design of Ejector-Assisted Refrigeration System
Thermal Energy Systems Laboratory, Korea Institute of Energy Research, Daejeon 305-343, Korea
Wah Engineering College, University of Wah, Wah Cantt, Punjab 47040, Pakistan
Department of Mechanical Engineering, Air University Islamabad, Aerospace and Aviation Campus Kamra, Kamra 43570, Pakistan
Mechanical Engineering and Design, School of Engineering and Applied Sciences, Aston University, Birmingham B4 7ET, UK
Department of Mechanical Engineering, Capital University of Science and Technology, Islamabad 44000, Pakistan
School of Chemical and Materials Engineering (SCME), National University of Sciences and Technology (NUST), Sector H-12, Islamabad 44000, Pakistan
Authors to whom correspondence should be addressed.
Both the authors contributed equally to this work.
Submission received: 30 September 2020 / Revised: 26 October 2020 / Accepted: 3 November 2020 / Published: 9 November 2020
An ejector is a simple mechanical device that can be integrated with power generation or the refrigeration cycle to enhance their performance. Owing to the complex flow behavior in the ejector, the
performance prediction of the ejector is done by numerical simulations. However, to evaluate the performance of an ejector integrated power cycle or refrigeration cycle, the need for simpler and more
reliable thermodynamic models to estimate the performance of the ejector persists. This research, therefore, aims at developing a single mathematical correlation that can predict the ejector
performance with reasonable accuracy. The proposed correlation relates the entrainment ratio and the pressure rise across the ejector to the area ratio and the mass flow rate of the primary flow.
R141b is selected as the ejector refrigerant, and the results obtained through the proposed correlation are validated through numerical solutions. The comparison between the analytical and numerical
with experimental results provided an error of less than 8.4% and 4.29%, respectively.
1. Introduction
An ejector is a simple mechanical device that uses the low pressure created by the expansion of high-pressure motive fluid (primary) and generates a vacuum, which is then used to entrain and
subsequently compress a secondary fluid [
]. The widespread application of an ejector includes: the removal of ash from flue gases [
]; in desalination plants [
]; solid oxide fuel cells [
]; refrigeration machines [
]; fueling of hydrogen vehicles [
]; and power generation cycle [
]. Ejectors could be integrated into various energy conversion processes to improve the energy efficiency, and have been actively researched since the 1950s [
]. However, the ejector-assisted refrigeration systems, or the ejector expansion refrigeration cycle (EERC), have gained the most interest, due to their superior performance, operational flexibility,
and ease of control [
]. Recently, the interest in ejector-assisted refrigeration machines has been greatly renewed, since they can be operated using low-grade heat [
In an EERC, the ejector is either applied to replace the mechanical compressor, as in a steam jet cooling system, or reduce the irreversibility during the throttling process by harnessing the kinetic
energy released during the process [
]. In the latter case, the kinetic energy is utilized to raise the suction pressure higher than the evaporation pressure, thereby reducing the compression power. A schematic of a typical ejector
expansion refrigeration cycle (EERC) and the corresponding pressure enthalpy (P-h) diagram is shown in
Figure 1
]. A 17% increase in coefficient of performance (COP) has been reported using EERCs [
]. In contrast to the conventional refrigeration machine, in an EERC, the expansion valve is replaced by an ejector. Instead of throttling the high-pressure refrigerant (motive fluid) from the
condenser, it is expanded in the convergent divergent (CD) primary nozzle of the ejector. The expansion of the refrigerant in the CD generates a vacuum and entrains the refrigerant from the
evaporator (secondary fluid). The two streams subsequently mix in the mixing chamber, undergo a shock wave, and are fed into a diffuser to increase the pressure of the mixed stream. The ejector
pre-compresses the working fluid to back pressure in the EERC, and, consequently, reduces the compression load [
The ejector is at the core of the EERC, and an integral step in the design and performance estimation of an EERC is the modeling of the ejector. The performance of the ejector is quantified in terms
of two main global parameters; (i) entrainment ratio, and (ii) compression pressure ratio. The entrainment ratio is the ratio of secondary fluid flow rate to the primary (motive) fluid flow rate,
while the compression pressure ratio reflects the pressure gain of the secondary fluid [
]. Many thermodynamic models were developed to predict the ejector performance. Huang et al. [
] pioneered the development of a 1D model for the prediction of ejector performance. The model was based on ideal gas assumption, isentropic flow relations with constant isentropic efficiency, and
the constant value for specific heat, and showed 15% deviation from the experimental results. Over time, many improved models to estimate the ejector performance have been developed and proposed [
]. In the Huang model, the choice of isentropic efficiencies is the major source of error, therefore, Haghparast et al. [
] replaced the isentropic efficiencies with polytropic efficiency to improve the model accuracy. Selvaraju et al. [
] proposed another analysis code that induces the effect of friction in the ejector, which is normally neglected by its predecessors. The accuracy of the 1D analytical model has been improved;
nonetheless, they are based on assumptions of ideal gas flow and normal shock, which does not reflect the actual flow behavior in the ejector [
The fluid flow through the ejector is compressible and supersonic, involving a series of complex flow interactions, such as shock wave–boundary layer interaction and flow mixing [
]. The 1D analytical models do not incorporate the complex flow interactions occurring in the ejector; however, the complex flow structure plays a significant role in estimating the performance of
the ejector [
]. Therefore, various researchers [
] employed computational fluid dynamics (CFD) models to better visualize the fluid flow and provide a detailed description of complex fluid flow inside the ejector. These models essentially solve the
Reynolds-averaged Navier-Stokes (RANS) equation by using turbulence models and numerical methods [
]. Desevaux et al. [
] used a standard k-ε turbulence model in FLUENT to study the ejector flow. However, their study was based on ideal gas assumption, and underpredicted the back pressure by about 20%. Del Valle et al.
] presented a CFD model that considered real gas properties, and the results showed better agreement with the experimental results. Han et al. [
] utilized the CFD model to comprehensively investigate the mechanism of boundary layer separation in the ejector and its effect on ejector performance. Their study concluded that either a too small
or too big mixing chamber diameter can induce boundary layer separation, which is the main cause of back flow in the ejector. Elbarghthi et al. [
], as well, employed CFD to explore the performance of an EERC with alternative refrigerants, such as R-1234ze(E) and R-1234yf. An insightful review of system and component level numerical modeling
of ejector was done by Little et al. [
], in which they highlighted that the pathway to improving the ejector analysis is the advance modeling of turbulence effects and the phase changes.
The 1D, as well as CFD, analysis of ejectors have been continuously evolving. Although the 1D models pose certain limitations, however, they are readily adaptable and compatible for the system level
analysis of EERCs. The constant isentropic efficiencies are a major source of discrepancies in the 1D analytical models. Therefore, to improve the choice of isentropic efficiencies for the sizing and
performance prediction of the ejector, Haghparast et al. [
] used CFD tools to approximate the isentropic efficiencies. In their study, the appropriate efficiencies are extracted from the CFD model, and subsequently used in the 1D model to estimate the
ejector performance. Riaz et al. [
] extended the same integration of CFD tools and 1D models to design and optimize the low-grade waste heat driven ejector refrigeration system. Rogié et al. [
] utilized a similar technique in the investigation of the ejector as a potential replacement of expansion valve in the hydrogen fueling station to minimize the energy expenditure and the fueling
time. The CFD analysis provides more credibility in the analysis results; however, they lack the ability to be readily integrated for a system level analysis. The varying operating conditions leading
to high computation and time requirements make CFD models unfit for real time implementation in the practical systems [
The CFD models considerably improve the quality of ejector analysis; however, they are focused on the component level study and, hence, are not compatible for the performance analysis of a complete
system. On the other hand, the accuracy of 1D models for ejectors is still questionable. Therefore, in this study, empirical correlations, based on experimental data of Huang at al. [
], have been developed to predict ejector performance parameters. The study describes how the performance of an ejector is mainly governed by two parameters: (i) area ratio of the initial CD nozzle
and the mixing chamber, and (ii) the ratio of motive fluid inlet pressure to the secondary inlet pressure. Therefore, a multivariate polynomial regression technique is applied to predict the
entrainment ratio and compression pressure ratio of the ejector as a function of major performance influencing parameters. Furthermore, a CFD model is developed to investigate the ejector
performance. Zhu et al. [
] explored the ejector flow by using experimental and numerical methods. Their study investigated four turbulence models and found out the k-ε model agrees best with measurements for predictions of
mass flow rates, which is adopted in this study. The CFD model is validated against the experimental results. The validated CFD model is subsequently used to evaluate the accuracy of the results
calculated using the empirical correlations. The empirical correlations have the advantage of easy integration with the overall system model, and facilitate the real time implementation of ejectors
into practical refrigeration systems.
2. Ejector Description and Modeling
In the following section, the 1D model of ejector analysis, with the necessary assumptions, is given. Subsequently, the development and implementation of the numerical (CFD) model is also discussed
in detail, with the focus on geometry construction, meshing, selection of turbulence model, and the related FLUENT settings.
2.1. Mathematical Solution of Ejectors
The thermodynamic models are the early models developed to understand and estimate the ejector performance [
]. Basically, these 1D and 2D mathematical models analytically solve the gas dynamic relations for compressible flow to model the flow field inside the ejector [
]. Although these mathematical models provide a certain level of understanding of fluid flow through the ejector, these models are still far behind in fully explaining the ejector flow field, because
of involvement of certain flow discrepancies, such as normal shock waves, primary and secondary flow mixing, and boundary layer separation, as well as use of isentropic coefficients [
]. These flow features obstruct the development of detailed models that could fully analyze the ejector flow, as then the complexity of governing equations go beyond the capability of analytical
procedures. The existing analytical models simulate the flow by making various assumptions such that the governing system of equations is simplified [
]. All these models are used to evaluate the ejector performance in terms of global flow parameters, such as entrainment ratio (
) and compression ratio (
) [
]. The geometry of an ejector is shown in
Figure 2
. The important boundary conditions include throat area of primary nozzle (
), area of mixing chamber (
), and the inlet pressures of motive fluid (
) and secondary fluid (
In this study, the 1D thermodynamic model of Huang et al. [
] is utilized for describing the performance of the ejector at various values of
corresponding to saturated vapor temperature from 351 K to 368 K, which lies within the range of ejector refrigeration machinery. The various geometrical specifications of ejector that were used in
this study are given in
Table 1
The system of equations used to solve the analytic ejector model shown in
Figure 2
are summarized in
Table 2
, with the flowing assumptions.
• The working fluid acts as an ideal gas, having constant specific heat (C[p]) and specific heat ratio (γ).
• Steady, adiabatic, and 1D flow.
• Negligible kinetic energy at secondary flow inlet, primary nozzle inlet, and diffuser exit.
• Use of isentropic relations and constant mixing chamber efficiency.
• The primary and secondary fluid flow mixes at hypothetical throat located within constant area section (Section 2–3).
• Constant pressure mixing (P[pH] = P[sH]).
• Choking of entrained flow at hypothetical throat and Mach number of M[sH] = 1 is assumed.
• Adiabatic ejector walls.
The limitations of abovementioned analytical model include (a) constant values of isentropic coefficients and isentropic efficiencies of various components, and (b) the procedure for analytically
solving the governing equations, mentioned above, is an iterative process.
2.2. Mathematical Solution of Ejectors
In the present study, a systematic approach was proposed to develop a simple method to calculate the optimal values of
without the need of iterations. A regression analysis was conducted to identify the likely relation between ejector performance parameters and ejector operating conditions and geometrical
specifications. To correlate the inputs (
) with outputs (
), regression equations were developed using the least square method. As
are two dependent variables, two regression equations were proposed, which required both area ratio (
) and pressure ratio (
) as an input (independent variables). The relationship between dependent variable ω with
is shown
Figure 3
The figure clearly shows a nonlinear relation between the ω with
, and there exists a negative nonlinear association between ω and
, that is, decreasing the value of
results in increased entrainment ratio. Similarly, the association between the other dependent variable (
) and independent variables is also shown by
Figure 4
. Each graphical representation shows that the ejector performance parameters are highly influenced under the variations in inlet flow pressures (
) and ejector geometry (
The correlations were developed using the experimental data available in the study conducted by Huang et al. [
], and is provided in
Appendix A
. This data includes experimentally determined values of
, compression ratio (
, and
. An empirical correlation of second-degree polynomial (that predicts ejector performance in terms of ω as a function of
), was developed using polynomial regression techniques because of the existence of the nonlinear behavior among the predictor and response variables, as shown in
Figure 3
Figure 4
. Using a forward selection method, a second-degree polynomial, in terms of area ratio and pressure ratio, was selected. The general equation with second-degree polynomial [
] in the present case becomes:
$y = β 0 + β 1 x 1 + β 2 x 2 + β 11 x 1 2 + β 22 x 2 2 + β 12 x 1 x 2 + ε$
: dependent variable;
: independent variable;
: intercept;
: linear effect parameters;
: quadratic effect parameters;
: interaction parameter; and ε: error. The above equation can be represented in terms of specific independent variables, as given below:
$E ( y ) = ω = β 0 + β 1 ( A 3 / A t ) + β 2 ( P g / P e ) + β 11 ( A 3 / A t ) 2 + β 22 ( P g / P e ) 2 + β 12 ( A 3 / A t ) ( P g / P e )$
The above expression is also termed as response surface [
] where
, and
are coefficient parameters of the equation. These equation parameters were found by applying the least square approach on the matrix form of the above equations, which generated the following matrix
$[ β ] = ( [ X ] * [ X t ] ) − 1 × [ X t ] [ Y ]$
where the matrix [
] contained n values of all the independent variables (
), and matrix [
] contained all n values of dependent variable (
) associated with specific values of independent variables. The above matrix expression was then solved in MATLAB, and the following empirical correlation was obtained:
$ω = 0.1705 + 0.1479 ( A 3 A t ) − 0.07002 ( P g P e ) + 0.0014 ( A 3 A t ) 2 + 0.0035 ( P g P e ) 2 − 0.0074 ( A 3 A t × P g P e )$
Similarly, the second empirical correlation for predicting the compression ratio was obtained following the same procedure, except that, due to the linear relation between the response and predictor
variables, the general regression equation is shown below, and the values of coefficients were calculated as explained above and the developed empirical equation is also shown below [
]. In the later section, both empirical equations are validated against the CFD model developed, to predict ejector performance at varying ejector specifications.
$y = β 0 + β 1 x 1 + β 2 x 2 + ε P c n P e = 2.3204 − 0.12314 ( A 3 A t ) + 0.1713 ( P g P e )$
Figure 5
shows a comparison between performance evaluation of ejector done through the conventional 1D thermodynamics model and the empirical correlations. The figure clearly demonstrates how the complex
solution procedure of the 1D model can be replaced by a single-step empirical correlation.
CFD is a robust tool that helps in better visualization of ejector flow phenomena, such as supersonic flow, flow mixing, shock trains, and boundary layers, as discussed above. In this study, a CFD
model has been developed, utilizing the commercially available simulation package ANSYS FLUENT 2019. The software package uses finite volume techniques, that are based on discretization of governing
equations, by dividing the physical geometry into smaller elements, forming a control volume mesh. Originally, the fluid flow through the ejector is essentially unsteady in a 3D space, mainly due to
the turbulent flow nature. However, by utilizing the RANS equations, which are used for determining averaged values of flow quantities by time averaging over long intervals [
], the problem is assumed to be steady and axisymmetric, providing acceptable level of accuracy in both global and local flow phenomena [
]. Therefore, in this study, the ejector is modeled utilizing axisymmetric geometry, as literature studies have shown that a 2D axisymmetric model will produce identical results to a 3D flow model,
but with less computation effort [
]. By utilizing axis symmetric condition, 2D governing equations defined in terms of radial components are solved using a finite volume discretization method, with second-order upwind scheme [
]. These axisymmetric equations for simulating ejector flow include standard conservation equations of energy, mass, and momentum, as used in studies by Zhang et al. [
] and Bartosiewicz et al. [
Discretization is an integral part of a numerical study. The accuracy and quality of results of any CFD model is highly influenced by the mesh density. A well-refined mesh is a necessity to have
negligible discretization error. A mesh independence study was performed by refining the mesh size, and results showing variation of entrainment ratio with number of cells is given in
Figure 6
. It can be seen that when the number of cells are increased further from 500,000 to 600,000 for a structured mesh, the entrainment ratio rises only by 0.1%, and, therefore, mesh density of 500,000
cells is shortlisted in this study as a tradeoff between accuracy of results and corresponding computation effort (time). Furthermore, use of structured mesh allows much better control over mesh
quality, and quadrilateral elements are used for meshing the fluid domain. The visualization of flow domain is shown in
Figure 7
. The mesh quality at the onset of mixing of primary and secondary fluid is refined to capture the complex flow interaction and oblique shocks. After the mixing, the mixed flow undergoes normal
shock, which reduces the flow velocity from supersonic to subsonic [
]. The magnified view of mixing region and normal shock is also given in
Figure 7
Generally, a density-based solver is recommended widely in the literature for simulating compressible fluid flows. However, the recent studies show that a pressure-based solver is also equally
capable of solving compressible flows in less time than a density-based solver [
]. Hence, a pressure-based solver with coupled algorithm for pressure–velocity coupling is selected. The stagnation pressure boundary condition was applied at both the primary and secondary inlet.
The CFD study was performed using experimental data of Huang et al. [
], and for R141b working fluid, the thermodynamic properties were obtained from the REFPROP v9.1.
The solution of RANS equations involves various turbulent fluid models, and proper selection of these models highly influences the flow behavior inside the ejector. Two turbulence models are
shortlisted through the literature: k-ε (k epsilon) and k-ω (k omega) [
]. Among the two models, the k-ω model is widely recommended in the literature, as compared to the k-ε model, because the former provides a more accurate global and local flow feature estimate than
the latter [
]. In addition, the k-ε model requires higher number of iterations, and provides poor performance, as compared to the k-ω-sst (k omega sheer stress transport) model in terms of convergence, thermal,
and flow field prediction [
]. Also, according to various other CFD studies [
], the k-ω-sst model provides better prediction for shock wave location and intensity. Hence, in this paper, a CFD study was conducted using a k-ω-sst turbulence model for all simulations, and
summary of numerical setup adopted, boundary conditions detail, and mesh employed is provided in
Table 3
. The convergence criteria were satisfied typically around 1500 iterations when all the residuals fell below 1 × 10
2.3. CFD Model Validation
For validating the CFD model, the geometry (AA, AB, AC, AD) from already published experimental work [
] is utilized, where
Figure 8
shows fixed geometrical configuration [
] and values of variable parameters, such as
, and
, are taken from
Table 1
Under these geometrical specifications and experimental operating conditions (P[g]: 0.4–0.604 MPa, P[e]: 0.04–0.047 MPa and P[C]), the developed CFD model is employed to validate the results obtained
through empirical correlation of entrainment ratio and compression ratio. In addition, these comparisons provide the information about the reliability of the developed correlations.
3. Results
For validation, five different geometries were simulated from Huang et al.’s paper, with the same boundary conditions. Simulations were performed, and the Mach number contours and pressure contours
are shown in
Figure 9
Figure 10
, respectively.
From Mach number contours, it is evident that flow in ejectors is a very complex phenomenon that includes shock waves, oblique shock waves, and turbulent effects. The figure shows the varying flow
conditions along the different sections of ejector geometry. Initially, flow enters the convergent part of the primary converging diverging (CD) nozzle, where the flow decelerates until it achieves
the choking condition (Ma = 1) at the contraction (throat) of the CD nozzle. The motive flow, while passing through the divergent section of the nozzle, accelerates and continues accelerating, even
after exiting the primary nozzle, which is as per the assumption made for the analytical solution of the ejector. After the two flows (primary and secondary) mix together in the mixing section, the
velocities of these flows sum up into a single averaged velocity (of mixed flow) along the ejector length. The mixed flow undergoes a normal compression shock wave, which transforms the supersonic
flow (Ma > 1) to subsonic (Ma < 1). This mixed flow is then fed into a diffuser, where it is further compressed until it attains condenser pressure.
The secondary flow is accelerated and is then entrained by the motive flow, due to the formation of shear layer between primary and secondary flows, under the influence of high-velocity gradient in
between the motive and entrained flow at the CD nozzle exit, as also observed in
Figure 9
, where the velocity transforms to dark blue, followed by a light blue, and finally to yellow. After the motive flow exits the primary nozzle, it slows down, due to its interaction with the entrained
flow, starting from the outer boundary of flow, and, consequently, the entrained flow gets accelerated. As the flow passes through the constant area mixing section, more uniformity in color is
observed, which points to good flow mixing.
The variation in flow pressure along the ejector centerline is shown in
Figure 10
. The motive flow, as it passes through the CD nozzle, undergoes a sudden drop in pressure while it remains steady inside the constant area mixing section. Inside the mixing section, the frictional
effects slightly reduce the flow velocity, which, in turn, increases the pressure of flow. At the end, when this fluid (supersonic) is fed into a diffuser, the pressure drops due to the supersonic
flow condition, followed by an abrupt and rapid rise of pressure due to the occurrence of normal shock wave, which brings the supersonic flow to subsonic state. The deviation between values of
entrainment ratio obtained through the CFD model, empirical correlations, and experimental data are shown in
Figure 11
The comparison of empirical correlation and the experimental data is shown in
Figure 12
, where the maximum deviation from experimental results is 8.40%.
3.1. Case Study: Simple Refrigeration Machine
A simple refrigeration machine, as shown in
Figure 13
, was designed and parametrically investigated for a case study. The system is designed for a generator temperature (
) and evaporator temperature (
) of 70–100 °C and 10–20 °C, respectively. The ambient sink is considered for condenser, so the condensing temperature (
) is fixed to 40 °C. The working fluid is taken as R141b, thus the developed correlations are valid. The conditions are summarized in
Table 4
. For the case study, the refrigeration capacity (
) is fixed to 300 W. Using the known
, the msec can be calculated. Subsequently, the correlations are applied to calculate
. The calculation procedure is summarized in
Table 5
. After solving the system, the coefficient of performance (COP) can be calculated. The pumping work was too small, and is, therefore, neglected in calculating the COP.
$C O P S y s t e m = Q A d d Q C o o l$
3.2. COP Variation with Generation Temperature and the Evaporation Temperature
have a significant impact on the COP of the system, shown in
Figure 13
. Both can vary, depending on the heat source and the target temperature. For the system shown in
Figure 13
, the
can be supplied using any low-grade heat source, such as waste heat or the solar energy [
]. Furthermore,
is governed by the required target temperature and can also vary, depending on the requirement. Therefore, a parametric study with respect to
was conducted, and the results are shown in
Figure 14
. As can be seen from the figures, higher values of
both favor the performance of the proposed design. The results are rather obvious, since the increase in evaporation or generation temperature both reduce the load on a refrigeration machine.
However, the investigation demonstrates the ease of parametric investigation using empirical correlations for the ejector.
4. Conclusions
The interest in ejector-assisted refrigeration systems has been rekindled, since these compact devices can be operated using low-grade heat. The ejector is one of the crucial components in the
ejector-assisted machines, and the need for a reliable model for estimating ejector performance persists. However, the ejector flow is comprehensible, and involves complex flow features like normal
shock wave and boundary layer interaction, and, so far, all the 1D thermodynamic models are rather complex and lack the adequate accuracy. Therefore, this study proposed an empirical correlation to
predict the performance of an ejector operated with R141b in the range of 0.4–0.604 MPa primary pressure and 0.040–0.047 MPa secondary pressure. The correlation can predict both global flow
parameters of ejectors, i.e., entrainment ratio and compression ratio, with an accuracy of 8.4%, and 6.3%, respectively. Furthermore, due to the complex flow behavior in the ejector, a k-ω CFD model
was developed. The model was validated against the experimental results. Using the CFD model, Mach number and pressure contours were plotted. The results show that the maximum error in the CFD was
4.29%. Moreover, the validity of correlation was also studied, with the validated CFD results also shown. In the end, a case study was shown that successfully demonstrated the ease of using the
correlations for the design of an ejector-assisted refrigeration machine. A parametric study was conducted that concluded that both high primary fluid temperature and evaporation temperature increase
the performance of the machine.
Author Contributions
Conceptualization: H.A.M., Z.R. and B.L.; data curation: H.A.M., H.M.A., Z.R., J.C. and M.I.; formal analysis: H.A.M., H.M.A., B.L., M.M. and M.S.; funding acquisition: B.L. and M.I.; investigation:
H.A.M., H.M.A., Z.R. and M.I.; methodology: H.A.M., H.M.A. and M.M.; project administration: B.L. and Y.-J.B.; resources: B.L. and Y.-J.B.; software: H.A.M., H.M.A. and Z.R.; supervision: Z.R., B.L.
and Y.-J.B.; validation: J.C., M.S. and M.S.B.; visualization: M.M. and M.S.B.; writing—original draft: H.A.M., H.M.A. and Z.R.; writing—review and editing: B.L., Y.-J.B. and M.I. All authors have
read and agreed to the published version of the manuscript.
This work was supported by the Development Program of the Korea Institute of Energy Research (KIER) (C0-2447), (C0-2410) and by the National Research Council of Science & Technology (NST) grant by
the Ministry of Science and ICT, Republic of Korea (No. CRC-15-07-KIER).
Conflicts of Interest
The authors declare no conflict of interest.
D Diameter
A Area, m^2
C[p] Fluid specific heat at constant pressure, KJ kg^−1 K^−1
C[v] Fluid specific heat at constant volume, KJ kg^−1 K^−1
γ Ratio of specific heats (Cp/Cv)
R Specific gas constant, KJ kg^−1 K^−1
a Sonic velocity, ms^−1
V Fluid velocity, ms^−1
M Mach number
m Mass flowrate, kgs^−1
h Enthalpy, KJ kg^−1
P[g] Fluid pressure at ejector primary nozzle inlet, MPa
P[e] Fluid pressure at ejector suction inlet, MPa
P[cn] * Ejector critical back pressure, MPa
T Temperature, K
T[g] Fluid temperature at ejector primary nozzle inlet, K
T[e] Fluid temperature at ejector suction inlet, MPa
T[cn] * Saturated vapor temperature corresponding to P[cn] *, K
T[gs] Saturated-vapor temperature corresponding to P[g], K
H Hypothetical throat position
ƞ Isentropic efficiency coefficient
φ Coefficient representing flow losses
* Ejector critical operation mode
cn Condenser, Ejector exit
e Entrained flow suction port
g Primary nozzle inlet
M Mixed flow
t Primary nozzle throat
p4 Primary fluid at nozzle exit
sH Entrained flow at hypothetical throat
pH Primary flow at hypothetical throat
1 Motive nozzle throat
2 Constant area section Entrance
3 Constant area section Exit
4 Primary Nozzle Exit
Appendix A
Geometry Area Ratio (A[3]/A[t]) Expansion Ratio (P[g]/P[e]) Compression Ratio (P[c]/P[e]) Entrainment Ratio (ω)
6.44 10 2.56 0.3257
6.44 11.62 2.84 0.288
6.44 13.45 3.18 0.2246
AA 6.44 15.1 3.54 0.1859
6.44 9.89 2.45 0.3398
6.44 11.44 2.76 0.2946
6.44 12.85 3.05 0.235
EG 6.77 15.1 3.41 0.2043
6.99 10 2.3 0.3922
AB 6.99 11.62 2.66 0.3117
6.99 13.45 3.04 0.2718
EC 7.26 15.1 3.17 0.2273
7.26 12.85 2.74 0.304
7.73 10 2.26 0.4393
7.73 11.62 2.54 0.3883
7.73 13.45 2.95 0.304
AG 7.73 15.1 3.15 0.2552
7.73 8.51 1.93 0.6132
7.73 9.89 2.17 0.479
7.73 11.44 2.45 0.4034
7.73 12.85 2.69 0.3503
ED 8.25 15.1 3 0.2902
8.29 10 2.09 0.4889
AC 8.29 11.62 2.38 0.4241
8.29 13.45 2.67 0.3488
8.29 15.1 2.91 0.2814
EE 9.17 15.1 2.71 0.3505
9.17 12.85 2.31 0.4048
9.41 10 1.91 0.6227
9.41 11.62 2.18 0.5387
9.41 13.45 2.47 0.4446
9.41 15.1 2.66 0.3457
AD 9.41 8.51 1.7 0.7412
9.41 9.89 1.91 0.635
9.41 11.44 2.14 0.5422
9.41 12.85 2.33 0.4541
9.83 15.1 2.6 0.3937
9.83 12.85 2.22 0.4989
EH 10.64 15.1 2.45 0.4377
1. Li, A.; Yuen, A.C.Y.; Chen, T.B.Y.; Wang, C.; Liu, H.; Cao, R.; Yang, W.; Yeoh, G.H.; Timchenko, V. Computational study of wet steam flow to optimize steam ejector efficiency for potential fire
suppression application. Appl. Sci. 2019, 9, 1486. [Google Scholar] [CrossRef] [Green Version]
2. Nadig, R. Evacuation systems for steam surface condensers: Vacuum pumps or steam jet air ejectors? In ASME 2016 Power Conference Collocated with the ASME 2016 10th International Conference on
Energy Sustainability and the ASME 2016 14th International Conference on Fuel Cell Science, Engineering and Technology; American Society of Mechanical Engineers Digital Collection: Charlotte, NC,
USA, 2016. [Google Scholar]
3. Pourmohammadbagher, A.; Jamshidi, E.; Ale-Ebrahim, H.; Dabir, B.; Mehrabani-Zeinabad, M. Simultaneous removal of gaseous pollutants with a novel swirl wet scrubber. Chem. Eng. Process. Process
Intensif. 2011, 50, 773–779. [Google Scholar] [CrossRef]
4. Liu, J.; Wang, L.; Jia, L.; Xue, H. Thermodynamic analysis of the steam ejector for desalination applications. Appl. Therm. Eng. 2019, 159, 113883. [Google Scholar] [CrossRef]
5. Vincenzo, L.; Pagh, N.M.; Knudsen, K.S. Ejector design and performance evaluation for recirculation of anode gas in a micro combined heat and power systems based on solid oxide fuel cell. Appl.
Therm. Eng. 2013, 54, 26–34. [Google Scholar] [CrossRef]
6. Genc, O.; Timurkutluk, B.; Toros, S. Performance evaluation of ejector with different secondary flow directions and geometric properties for solid oxide fuel cell applications. J. Power Sources
2019, 421, 76–90. [Google Scholar] [CrossRef]
7. Chen, J.; Jarall, S.; Havtun, H.; Palm, B. A review on versatile ejector applications in refrigeration systems. Renew. Sustain. Energy Rev. 2015, 49, 67–90. [Google Scholar] [CrossRef]
8. Xia, J.; Wang, J.; Zhou, K.; Zhao, P.; Dai, Y. Thermodynamic and economic analysis and multi-objective optimization of a novel transcritical CO[2] Rankine cycle with an ejector driven by low
grade heat source. Energy 2018, 161, 337–351. [Google Scholar] [CrossRef]
9. Chunnanond, K.; Aphornratana, S. Ejectors: Applications in refrigeration technology. Renew. Sustain. Energy Rev. 2004, 8, 129–155. [Google Scholar] [CrossRef]
10. Elbel, S.; Lawrence, N. Review of recent developments in advanced ejector technology. Int. J. Refrig. 2016, 62, 1–18. [Google Scholar] [CrossRef]
11. Besagni, G.; Mereu, R.; Inzoli, F. Ejector refrigeration: A comprehensive review. Renew. Sustain. Energy Rev. 2016, 53, 373–407. [Google Scholar] [CrossRef] [Green Version]
12. Riaz, F.; Lee, P.S.; Chou, S.K. Thermal modelling and optimization of low-grade waste heat driven ejector refrigeration system incorporating a direct ejector model. Appl. Therm. Eng. 2020, 167,
114710. [Google Scholar] [CrossRef]
13. Lucas, C.; Koehler, J. Experimental investigation of the COP improvement of a refrigeration cycle by use of an ejector. Int. J. Refrig. 2012, 35, 1595–1603. [Google Scholar] [CrossRef]
14. Aligolzadeh, F.; Hakkaki-Fard, A. A novel methodology for designing a multi-ejector refrigeration system. Appl. Therm. Eng. 2019, 151, 26–37. [Google Scholar] [CrossRef]
15. Huang, B.J.; Chang, J.M.; Wang, C.P.; Petrenko, V.A. A 1-D analysis of ejector performance. Int. J. Refrig. 1999, 22, 354–364. [Google Scholar] [CrossRef]
16. Haghparast, P.; Sorin, M.V.; Nesreddine, H. Effects of component polytropic efficiencies on the dimensions of monophasic ejectors. Energy Convers. Manag. 2018, 162, 251–263. [Google Scholar] [
17. Selvaraju, A.; Mani, A. Analysis of an ejector with environment friendly refrigerants. Appl. Therm. Eng. 2004, 24, 827–838. [Google Scholar] [CrossRef]
18. Aidoun, Z.; Ameur, K.; Falsafioon, M.; Badache, M. Current Advances in Ejector Modeling, Experimentation and Applications for Refrigeration and Heat Pumps. Part 1: Single-Phase Ejectors.
Inventions 2019, 4, 15. [Google Scholar] [CrossRef] [Green Version]
19. Aidoun, Z.; Ameur, K.; Falsafioon, M.; Badache, M. Current Advances in Ejector Modeling, Experimentation and Applications for Refrigeration and Heat Pumps. Part 2: Two-Phase Ejectors. Inventions
2019, 4, 16. [Google Scholar] [CrossRef] [Green Version]
20. Desevaux, P.; Marynowski, T.; Khan, M. CFD prediction of supersonic ejectors performance. Int. J. Turbo Jet Engines 2006, 23, 173–182. [Google Scholar] [CrossRef]
21. Del Valle, J.G.; Sierra-Pallares, J.; Carrascal, P.G.; Ruiz, F.C. An experimental and computational study of the flow pattern in a refrigerant ejector. Validation of turbulence models and
real-gas effects. Appl. Therm. Eng. 2015, 89, 795–811. [Google Scholar] [CrossRef]
22. Han, Y.; Wang, X.; Sun, H.; Zhang, G.; Guo, L.; Tu, J. CFD simulation on the boundary layer separation in the steam ejector and its influence on the pumping performance. Energy 2019, 167,
469–483. [Google Scholar] [CrossRef]
23. Elbarghthi, A.F.; Mohamed, S.; Nguyen, V.V.; Dvorak, V. CFD Based Design for Ejector Cooling System Using HFOS (1234ze (E) and 1234yf). Energies 2020, 13, 1408. [Google Scholar] [CrossRef] [Green
24. Little, A.B.; Garimella, S. A critical review linking ejector flow phenomena with component-and system-level performance. Int. J. Refrig. 2016, 70, 243–268. [Google Scholar] [CrossRef]
25. Wen, C.; Rogie, B.; Kærn, M.R.; Rothuizen, E. A first study of the potential of integrating an ejector in hydrogen fuelling stations for fuelling high pressure hydrogen vehicles. Appl. Energy
2020, 260, 113958. [Google Scholar] [CrossRef]
26. Zhu, Y.; Jiang, P. Experimental and numerical investigation of the effect of shock wave characteristics on the ejector performance. Int. J. Refrig. 2014, 40, 31–42. [Google Scholar] [CrossRef]
27. He, S.; Li, Y.; Wang, R.Z. Progress of mathematical modeling on ejectors. Renew. Sustain. Energy Rev. 2009, 13, 1760–1780. [Google Scholar] [CrossRef]
28. Croquer, S.; Poncet, S.; Aidoun, Z. Turbulence modeling of a single-phase R134a supersonic ejector. Part 1: Numerical benchmark. Int. J. Refrig. 2016, 61, 140–152. [Google Scholar] [CrossRef]
29. Tashtoush, B.M.; Moh’d A, A.N.; Khasawneh, M.A. A comprehensive review of ejector design, performance, and applications. Appl. Energy 2019, 240, 138–172. [Google Scholar] [CrossRef]
30. Rawlings, J.O.; Pantula, S.G.; Dickey, D.A. Applied Regression Analysis: A Research Tool; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
31. Sinha, P. Multivariate polynomial regression in data mining: Methodology, problems and solutions. Int. J. Sci. Eng. Res. 2013, 4, 962–965. [Google Scholar]
32. Honra, J.; Berana, M.S.; Danao, L.A.M.; Manuel, M.C.E. CFD Analysis of Supersonic Ejector in Ejector Refrigeration System for Air Conditioning Application. In Proceedings of the World Congress on
Engineering, London, UK, 5–7 July 2017; Volume 2. [Google Scholar]
33. Carrillo, J.A.E.; de La Flor, F.J.S.; Lissén, J.M.S. Single-phase ejector geometry optimisation by means of a multi-objective evolutionary algorithm and a surrogate CFD model. Energy 2018, 164,
46–64. [Google Scholar] [CrossRef]
34. Pianthong, K.; Seehanam, W.; Behnia, M.; Sriveerakul, T.; Aphornratana, S. Investigation and improvement of ejector refrigeration system using computational fluid dynamics technique. Energy
Convers. Manag. 2007, 48, 2556–2564. [Google Scholar] [CrossRef]
35. Zhang, H.; Wang, L.; Jia, L.; Wang, X. Assessment and prediction of component efficiencies in supersonic ejector with friction losses. Appl. Therm. Eng. 2018, 129, 618–627. [Google Scholar] [
36. Bartosiewicz, Y.; Aidoun, Z.; Desevaux, P.; Mercadier, Y. Cfd-experiments integration in the evaluation of six turbulence models for supersonic ejector modeling. Integr. CFD 2003, 1, 450. [Google
37. Hanafi, A.S.; Mostafa, G.M.; Waheed, A.; Fathy, A. 1-D mathematical modeling and CFD investigation on supersonic steam ejector in MED-TVC. Energy Procedia 2015, 75, 3239–3252. [Google Scholar] [
CrossRef] [Green Version]
38. Besagni, G.; Inzoli, F. Computational fluid-dynamics modeling of supersonic ejectors: Screening of turbulence modeling approaches. Appl. Therm. Eng. 2017, 117, 122–144. [Google Scholar] [CrossRef
39. Besagni, G.; Mereu, R.; Chiesa, P.; Inzoli, F. An Integrated Lumped Parameter-CFD approach for off-design ejector performance evaluation. Energy Convers. Manag. 2015, 105, 697–715. [Google
Scholar] [CrossRef]
40. Mohamed, S.; Shatilla, Y.; Zhang, T. CFD-based design and simulation of hydrocarbon ejector for cooling. Energy 2019, 167, 346–358. [Google Scholar] [CrossRef]
41. Scott, D.; Aidoun, Z.; Bellache, O.; Ouzzane, M. CFD simulations of a supersonic ejector for use in refrigeration applications. In Proceedings of the International Refrigeration and Air
Conditioning Conference at Purdue, Paper 927. West Lafayette, IN, USA, 14–17 July 2008. [Google Scholar]
Figure 1. (A): Illustration of schematic of Ejector expansion refrigeration cycle (EERC); (B): Shows corresponding pressure enthalpy (P-h) diagram for EERC cycle.
Figure 4. Relation between compression ratio (C[p]) with area ratio (A[3]/A[t]) and pressure ratio (P[g]/P[e]).
Figure 5. Comparative flowchart of ejector performance analysis 2.3. Computational fluid dynamics (CFD) model development.
Figure 9. Mach number contours (a) AD Geometry P[e] = 0.047 MPa, P[g] = 0.538 MPa; (b) AA Geometry P[e] = 0.047 MPa, P[g] = 0.538 MPa; (c) AD Geometry P[e] = 0.04 MPa, P[g] = 0.465 MPa; (d) AC
Geometry P[e] = 0.04 MPa, P[g] = 0.465 MPa; and (e) AB Geometry P[e] = 0.04 MPa, P[g] = 0.465 MPa.
Figure 10. Total pressure contours (a) AD Geometry P[e] = 0.047 MPa, P[g] = 0.538 MPa; (b) AA Geometry P[e] = 0.047 MPa, P[g] = 0.538 MPa; (c) AD Geometry P[e] = 0.04 MPa, P[g] = 0.465 MPa; (d) AC
Geometry P[e] = 0.04 MPa, P[g] = 0.465 MPa; and (e) AB Geometry P[e] = 0.04 MPa, P[g] = 0.465 MPa.
Figure 14. The figure (a) depicts the influence of T[Eva] on system coefficient of performance (COP). The figure (b) displays the influence of T[Pri] on system coefficient of performance (COP).
Ejector Geometry
Primary Nozzle Mixing (Constant Area) Section
Serial No. Diameter (D[3])
Serial No. Throat Diameter (D[t]) Exit Diameter (D[4]) A 6.70 mm
B 6.98 mm
A 2.64 mm 4.50 mm C 7.60 mm
D 8.10 mm
E 8.54 mm
E 2.82 mm 5.10 mm G 7.34 mm
H 9.20 mm
Step Inputs Equations Output Comments
1 $P g , T g ,$$A $m ˙ p = P g A t T e γ R ( 2 γ + 1 ) γ + 1 γ − 1 η p$ $m ˙ p$ A[t] choking condition, the mass flow rate through the primary nozzle
t$ follows a gas dynamic relation.
2 $A p 4$ $A p 4 A t = 1 M p 4 2 [ 2 γ + 1 ( 1 + γ − 1 2 M p 4 2 ) ] γ + 1 γ − 1$ $M p 4 ,$ The Mach no. M[4] is calculated by using the Newton–Raphson method.
$P g P p 4 = [ 1 + γ − 1 2 M p 4 2 ] γ γ − 1$ $P p 4$
3 $P e$ $P s H = P e ( 1 + γ − 1 2 M s H 2 ) γ γ − 1$ $P s H$ Referring to the assumptions made, the Mach number of secondary flow at
hypothetical throat is M[eH] = 1.
$P s H =$$P p $P s H P p 4 = [ 1 + γ − 1 2 M p 4 2 ] γ γ − 1 [ 1 + γ − 1 2 M p H 2 ] γ γ − 1$ $M p H ,$ $φ p$: an isentropic coefficient that represents flow losses as primary
4 H$ $A p H A p 4 = ( φ p M p H ) [ 2 γ + 1 ( 1 + γ − 1 2 M p H 2 ) ] γ + 1 2 ( γ − 1 ) ( 1 M p $A p H$ fluid flow from section 4-4 to section H-H.
4 ) [ 2 γ + 1 ( 1 + γ − 1 2 M p 4 2 ) ] γ + 1 2 ( γ − 1 )$
If A[sH] < 0, calculate A[3] by using
5 $A p H$ $A s H = A 3 + A p H$ $A s H$ A[3] = A[pH] + ΔA[3], otherwise return to step 4 to recalculate A[pH], and
again the condition is checked.
$A s H , P e ,$
6 $T e , γ ,$ $m ˙ s = P e A s H T e γ R ( 2 γ + 1 ) γ + 1 γ − 1 η s$ $m ˙ s$ $η s$: Isentropic efficiency of entrained flow.
$R , η s$
7 $T g , T e ,$ $T g T p H = 1 + γ − 1 2 M p H 2$ $T p H , T Value of T[g] and T[e] can be taken from step 1 and 3, respectively.
$M p H , M s H$ $T e T s H = 1 + γ − 1 2 M s H 2$ s H$
$T p H , T s H
,$ $φ m [ m ˙ p V p H + m ˙ s V s H ] = ( m ˙ p + m ˙ s ) V m$ $V M ,$
8 $γ , R ,$ $V p H = M p H a p H , V s H = M s H a s H$ $V p H ,$ $φ M$: mixed flow friction coefficient, P[M] = P[pH] = P[sH], M[sH] = 1, and
$M p H ,$ $a p H = γ R T p H , a s H = γ R T s H$ $V s H$ M[pH] can be taken from step 4.
$M s H ,$
$m ˙ p , m ˙ s$
$V M , γ , R ,$
$C p , T p H ,$ $m ˙ p ( C p T p H + V p H 2 2 ) + m ˙ s ( C p T s H + V s H 2 2 ) = ( m ˙ p + m ˙ s ) ( C $T M , M The first equation gives T[M], which is then used to find value of α[M] and
9 $T s H , V p H p T m + V m 2 2 )$ M$ M[M].
, V s H ,$ $a M = γ R T M , M M = V M a M$
$m ˙ p , m ˙ s$
10 $P M , M M , γ$ $P 3 = P M ( 1 + 2 γ γ + 1 ( M M 2 − 1 ) )$ $P 3 , M Flow is solved after the shock wave, and value of P[M] can be taken from
$M 3 = 1 + γ − 1 2 M M 2 γ M M 2 − γ − 1 2$ 3$ step 8.
11 $P 3 , M 3$ $P C n = P 3 ( 1 + γ − 1 2 M 3 2 ) γ γ − 1$ $P C n$ Flow pressure at diffuser exit is calculated.
12 $P C n , P C n $i f P C n ≥ P C n * t h e n A 3 = A 3 − Δ A 3$ $A 3$ $P C n *$: critical condenser pressure, and A[3] must be equal to A[3] in
*$ $i f P C n < P C n * t h e n A 3 = A 3 + Δ A 3$ step 5, otherwise procedure starts again from step 5.
13 $m ˙ p , m ˙ s$ $ω = m ˙ s m ˙ p$ $ω$ Entrainment ratio is calculated by using ṁ[s] and ṁ[p] from step 6 and step
1, respectively.
Mesh Type Structured
Number of elements 500,000
Element type Quadratic quadrilateral
Boundary Conditions
Primary flow inlet Pressure inlet
Secondary flow inlet Pressure inlet
Discharge flow outlet Pressure outlet
Numerical Model Setup
Solver Pressure based
Turbulence model k-ω-sst
Method of initialization Hybrid
Fluid density Ideal gas
Working fluid R141b
Discretization scheme Second order upwind
Convergence criteria Residuals <10^−^6
Parameters Values
Refrigeration capacity (Q[Cool]) 300 W
Required condensing saturation temperature (T[Cond][,Req]) 40 °C
Generator saturation temperature (T[Pri]) 70–100 °C
Evaporator saturation temperature (T[Eva]) 10–20 °C
Component Input Output Equations Comment
Evaporator Q[Cool], T[Eva] m[sec] $m _ sec = Q C o o l / Δ h f g$ Since the saturation temperature is known, Δh[fg] can be calculated.
Ejector m[sec], A[3]/A[T], P[Pri]/P[Eva] ω, P[Cond], m[Pri] Using the developed co-relation, the ω and P[Cond] can be calculated.
If $P C o n d ≠ P C o n d , R e q$ then iterate A[3]/A[t].
Generator T[Pri], m[Pri] Q[Add] $m P r i = Q A d d / Δ h f g$ Since the saturation temperature is known, Δh[fg] can be calculated.
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Muhammad, H.A.; Abdullah, H.M.; Rehman, Z.; Lee, B.; Baik, Y.-J.; Cho, J.; Imran, M.; Masud, M.; Saleem, M.; Butt, M.S. Numerical Modeling of Ejector and Development of Improved Methods for the
Design of Ejector-Assisted Refrigeration System. Energies 2020, 13, 5835. https://doi.org/10.3390/en13215835
AMA Style
Muhammad HA, Abdullah HM, Rehman Z, Lee B, Baik Y-J, Cho J, Imran M, Masud M, Saleem M, Butt MS. Numerical Modeling of Ejector and Development of Improved Methods for the Design of Ejector-Assisted
Refrigeration System. Energies. 2020; 13(21):5835. https://doi.org/10.3390/en13215835
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Muhammad, Hafiz Ali, Hafiz Muhammad Abdullah, Zabdur Rehman, Beomjoon Lee, Young-Jin Baik, Jongjae Cho, Muhammad Imran, Manzar Masud, Mohsin Saleem, and Muhammad Shoaib Butt. 2020. "Numerical
Modeling of Ejector and Development of Improved Methods for the Design of Ejector-Assisted Refrigeration System" Energies 13, no. 21: 5835. https://doi.org/10.3390/en13215835
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Everything You Need to Know About Interest Coverage Ratio ⭐ YouHold
In order to assess the degree of danger associated with a company's existing debt or potential new spending, the interest coverage ratio is frequently employed by lenders, investors, and debtors.
With the interest payout ratio in place, debtors and financiers can feel easy knowing they will be paid on time. Most businesses have multiple lines of credit with various lenders. Since this is the
case, financial organizations require frequent guarantees of payment, particularly for their interest.
What is the interest coverage ratio?
The interest coverage ratio (ICR) is a debt and revenue measure used to ascertain whether or not a business can comfortably make interest payments on its debt. Earnings before interest and taxes
(EBIT) are used to determine the interest coverage ratio, which is then compared to the interest cost for the same time period. As such, it is categorized as follows:
• An indicator of a company's financial framework and danger is its debt-to-equity ratio.
• A company's solvency and the existence of any imminent insolvency risks can be gauged using the solvency ratio.
Formula and Calculation of the Interest Coverage Ratio
The term "coverage" refers to the number of quarters or fiscal years that interest payments can be made out of the company's present accessible profits, hence the name "interest coverage ratio." It
is a measure of the company's ability to meet its financial commitments out of its operating profits. The algorithm is as follows:
Interest Coverage Ratio= EBIT/Interest Expense
EBIT - Earnings before interest and taxes, which is the operating profit of the company
Interest expense is the total interest payable on multiple borrowings of the company
The smaller the percentage, the more of the company's resources are going toward servicing its debt. A company's capacity to pay interest costs is suspect if the number is 1.5 or lower. The threshold
below which lenders are likely to decline to give more money to a business because the company's risk for failure may be viewed as too high is an interest coverage ratio of 1.5. If a company's
percentage is less than one, it will have to use its capital on hand or seek additional financing, both of which are challenging for the reasons given above. Otherwise, the business risks insolvency
even if monthly profits are poor for just one month.
Businesses must have reserves of cash greater than the amount needed to cover interest payments. The profitability of a business is heavily influenced by its ability to pay its interest bills, which
is a measure of stability.
Definition of the Interest Coverage Ratio
Maintaining sufficient cash flow to cover interest expenses is a constant and vital worry for any business. Any time a business starts to have trouble meeting its financial commitments, it risks
having to take on more debt or use cash that could be better put toward things like long-term asset purchases or unexpected expenses.
Analyzing interest coverage ratios over time can often provide a much better image of a company's situation and direction than looking at a single interest coverage ratio. The interest coverage ratio
is a wonderful indicator of a company's short-term financial health, and it can be analyzed by looking at the ratio on a weekly basis for a set period of time, say the past five years.
Furthermore, the amount of this percentage that is considered optimal is somewhat subjective. A less-than-ideal percentage may be acceptable to some banks or prospective bond purchasers in return for
a greater interest rate on the company's debt.
The relevance of interest coverage rate
Lenders and debtors alike can benefit from the interest payment ratio. Lenders can gauge the debtor's financial stability and punctuality with interest payments based on this percentage. Therefore, a
larger percentage indicates that the debtors have a lower likelihood of defaulting on their interest payments across their various borrowings.
There is no danger of defaulting on interest payments or going insolvent when the ICR is significant. Borrowing businesses can gain insight into a company's financial health via the ICR. If they can
get that percentage down, they can still do their job and get ahead financially.
When conducting trend analysis, in which financial records are compared, the ICR is useful. By looking at historical data in this way, they can better anticipate upcoming trends. As a consequence,
businesses have a chance to work on their success going forward.
Interest Coverage Ratio Variances
Before diving into the ratios of different businesses, it's essential to keep in mind typical variants of the interest payment ratio. Since EBIT is a key component of the aforementioned formulation,
the interest payment ratio outcomes may vary depending on the specific EBIT forms that were employed. These shifts include:
The interest coverage ratio can also be calculated in a number of other ways, one of which involves using profits before interest, taxes, depreciation, and amortization (EBITDA) rather than EBIT. Due
to the omission of depreciation and loss from EBITDA, the number of EBITDA-based computations is typically larger than that of EBIT-based calculations. Since debt costs are the same whether EBITDA or
net income is used, the percentage calculated using EBITDA will be greater.
Interest coverage ratios can also be calculated with profits before interest and taxation (EBIAT) rather than EBIT. In order to provide a true image of a company's capacity to pay its interest costs,
this has the impact of subtracting tax expenses from the total. In light of the significance of taxes to a company's bottom line, interest payment rates are more accurately calculated using EBITAT,
or earnings before interest and taxes.
• EBITDA Inverse Interest Coverage Ratio of Capital Expenditures.
The formula is:
(EBITDA - Capex) (Interest Expense - Capital Investment)
The EBITDA Less Capex coverage ratio establishes how many times operating income (EBITDA) remains after capital expenditures (Capex) have been subtracted.
• Fixed Charge Coverage Ratio (FCCR)
The formula for calculating FCCR is:
(EBITDA - Capex) (Interest Expense + Present Value of Long-Term Debt)
One way to account for leasing costs in calculating a company's capacity to meet its short-term financial responsibilities is through the fixed fee coverage ratio (FCCR).
Interest Coverage Ratio Analyzation
The interest coverage ratio, like any other financial measure, can only tell part of the story when assessing a company's financial stability. Other than the debt payment percentage, these factors
should also be considered when evaluating a business.
1. Time. Interest coverage ratio trends are more indicative of a company's capacity to meet its interest obligations than a singular ratio. A company's expansion plans may necessitate the financing
of a new building. Although the next income statement will include the higher interest expenses from the recent loan, the facility the loan financed may substantially boost operating income in
the future, which could lead to a lower interest coverage ratio for the company.
2. Consistency in Gains. A highly steady profit margin may convince lenders to extend credit to a business with a low interest payment ratio. Lenders will still have faith in a business that has an
interest payment ratio of 2, provided that it has consistently produced running revenue over a lengthy period of time. Similarly, creditors can rest assured that the firm's debt won't impede its
3. Industry. A company's interest payment ratio may be expected to be higher or lower depending on the industry it operates in. Considering the generally steady demand for energy and water,
well-established utility companies can more easily handle their obligations despite having a smaller interest payout ratio. Lenders and buyers may need to see higher interest payment rates from
businesses operating in more risky sectors. The interest coverage ratio is most useful when comparing businesses operating within the same sector due to these distinctions.
Constrictions Inherent in the Interest Coverage Ratio
The interest coverage ratio has some caveats that prospective investors should be aware of.
One thing to keep in mind is that interest coverage varies greatly between sectors and even between firms within the same industry. In certain sectors, such as utilities, a two-to-one debt payment
ratio is considered adequate for well-established businesses.
Even with a low-interest coverage ratio, a well-established utility may be able to dependably pay its debts because of the predictability of its income and output, thanks in large part to government
rules. A minimal permissible interest coverage percentage of three or higher is often seen in sectors with greater volatility, such as manufacturing.
Typically, the company of firms like these is more prone to ups and downs. For instance, the decline in vehicle purchases that occurred during the Great Recession of 2008 had a significant impact on
the automobile business. Another unforeseen occurrence that could harm interest payment rates is an employee walkout. In order to weather the inevitable lean times, businesses in these sectors must
depend on a higher interest coverage ratio.
Due to the vast differences between sectors, it is only fair to compare one company's percentage to others in the same industry, preferably with comparable business strategies and income levels.
Even though it's crucial to factor in all debt when determining the interest payment ratio, some businesses may elect to separate out or exclude specific categories of debt. Therefore, it is
essential to check if all obligations were included when contemplating a company's self-published interest payment ratio.
What Does an Interest Coverage Ratio Reveal?
The interest coverage percentage evaluates a firm's capacity to service its loan. It's a measure of debt that can help tell you how healthy a business is. Coverage refers to the typical number of
fiscal years over which interest payments can be made from the cash on hand at the business. In layman's words, it indicates the number of times annual profits can cover the company's expenses. If a
company's percentage is high, it indicates that it is in a strong financial position to meet its interest payments, while if it is low, it indicates that it may be struggling financially.
How Do You Figure Out the Interest Coverage Ratio?
EBIT (or a variant thereof) is divided by interest on loan expenditures (the cost of acquired financing) over a specified time period, typically a year, to arrive at the ratio.
What is a Good Interest Coverage Ratio?
If the number is greater than one, it shows that the business has sufficient profits and steady income to pay its interest expenses. Although experts and buyers may consider a ratio of 1.5 for
interest payment to be adequate, a ratio of 2.0 or higher is favored. The interest payment ratio may need to be significantly higher than three in order for a company with typically more fluctuating
sales to be deemed healthy.
The meaning of a Bad Interest Coverage Ratio.
Any interest coverage percentage that falls short of one indicates that a company's present profits are inadequate to pay off its debt. Even with an interest coverage ratio below 1.5, it is highly
unlikely that a company will be able to consistently cover its interest costs. This is particularly true if the business is susceptible to periodic or repetitive declines in sales.
How can we better pay our loan expenses?
Two methods exist for boosting the ICR. One way to do this is to grow income, which will, in turn, increase EBIT or profits before interest and taxes. Reducing interest and other financing expenses
is another option.
Interest coverage, also known as the Times Interest Earned (TIE) ratio, is a measure of a company's ability to meet its interest payments by dividing its operating income (net income plus
depreciation and amortization) by its interest expenditures for the same time period. If a company's interest coverage percentage is less than 1.5, it may not have sufficient cash to meet its debt
service obligations. It is ideal for comparing ratios among businesses in the same sector and with a comparable company structure because interest payment ratios differ widely across industries. | {"url":"https://youhold.com/knowledge-hub/everything-you-need-to-know-about-interest-coverage-ratio","timestamp":"2024-11-14T21:49:29Z","content_type":"text/html","content_length":"49777","record_id":"<urn:uuid:8e564e97-d889-443c-8395-56174d24f825>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00579.warc.gz"} |
[PDF] Top 20 The biological and mathematical basis of L systems - 1Library
[PDF] Top 20 The biological and mathematical basis of L systems
Has 10000 "The biological and mathematical basis of L systems" found on our website. Below are the top 20 most common "The biological and mathematical basis of L systems".
The biological and mathematical basis of L systems
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EDN News 12
Our Universe is Expanding like a ( Hyper ) Balloon Rubber Membrane Model for Dark Matter Halo Correct representation of Negative and Imaginary Numbers Einstien did not defeat time ; it was the other
way round . Neither is time an illusion as Einstein believed . For measuring curvature of
Our Universe is Expanding like a (Hyper) Balloon
Rubber Membrane Model for Dark Matter Halo
Correct representation of Negative and Imaginary Numbers
Einstien did not defeat time; it was the other way round. Neither is time an illusion as Einstein believed.
For measuring curvature of 3D (hyper) surface we need solid angles instead of plane angles.
RRCAT Physicist challenges the Standard Model of Cosmology In questions of science, the authority of a thousand is not worth the humble reasoning of a single individual.”
— Galileo Galilei INDORE, MADHYA PRADESH, INDIA, July 20, 2023/EINPresswire.com/ -- A Nature Astronomy (2020) paper has claimed with 99% confidence level that our universe is closed. Another paper
claims betting odds of 2000 ∶ 1 against an open, flat and infinite universe, thus mortally wounding the Standard Model of Cosmology, but unifying entire physics. The widely held understanding of an
infinite, three-dimensional flat cosmos is incorrect, and our universe is indeed closed and finite [#Link1]. Not only that, but the currently accepted faulty model is why the two pillars of Physics,
General Relativity and Quantum Mechanics, are incompatible. This is what Subhajit Waugh, a physicist from the Raja Ramanna Centre for Advanced Technology (RRCAT), Department of Atomic Energy,
Government of India, has proposed in a landmark paper on Quantum Mechanics and General Relativity [#Link2]. It is a view that could change a significant part of physics and cosmology as we understand
it [#Link3].
General relativity (GR), which explains events on the greatest cosmological scale, and quantum mechanics (QM), which describes phenomena on the tiniest, submicroscopic scale, are the two cornerstones
of contemporary physics. But there are bitter conflicts between these theories. The main objective of physicists around the world has been to find a means to harmonize the two for many decades.
The reality, according to Mr. Waugh, is different. The universe’s shape is a 3-dimensional hyper-surface of a (hyper) balloon. If this shape is adopted, then both GR and QM fit together like pieces
in a jigsaw. A unification of these two theories solves most of the outstanding conflicts from dark matter to dark energy, to the requirement of a fifth dimension in physics (but which experiments
and observations do not allow).
Explaining how the whole scientific world got to this position, Mr. Subhajit points out that:
“We (the scientific community) have made a series of mistakes in our mathematics, our physics, and our cosmology. And this is what I can’t stress enough; without making the corrections (which I have
made) to our historical mistakes, no scientist will ever be able to unify fundamental physics and cosmology.”
The standard model of cosmology is based on the Friedmann equations, which are derived from Einstein's field equations of gravity for the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. The
Minkowski spacetime (MST) metric, which can explain all aspects of special relativity (including time dilation, length contraction, and relative simultaneity), describes spacetime far from massive
objects. Mr. Waugh contends that the currently accepted mathematical interpretation of the MST metric (as well as the FLRW metric) is incorrect, which has prevented cosmologists from deriving the
correct shape and size of the universe. The temporal parts of both metrics are identical, and either metric can be used to determine the shape and size of the universe if the mathematical
interpretation is corrected. The universe is like the 3D hypersurface of a 4D hypersphere. Taking the radial expansion velocity of the universe to be equal to the speed of light, c (as dictated by
spacetime equations), and the age of the universe as 13.8 billion years, the value of the Hubble constant obtained using this model (71.002 km/s/Mpc) agrees well with the accepted values (69.8 and 74
km/s/Mpc calculated by two different methods). The MST metric and Hubble’s law both say the same thing, which demonstrates that Waugh’s model is correct.
Mr. Waugh proves the curvature of the 3 dimensional hyper-surface using the center of the mass equation and the Minkowski SpaceTime equation as his primary instruments. He claims that cosmologists
should have probed the curvature of 3D space by considering the sum of solid angles rather than plane angles (of the Cosmic Microwave Background spots). Mr. Waugh further demonstrates the existence
of two different frames of reference/viewpoints. Each perspective sees the universe uniquely, but it is essentially this that enables the unification of QM and GR.
The key to this new understanding, according to Mr. Waugh, and the key to unifying both GR and QM, is understanding that there are two viewpoints; the true center of the universe viewpoint (nature’s
perspective) and our viewpoint, which he equates to that of a creature trapped inside the wall of a hyper-ballon, but free to move along the wall.
“From our perspective, the radius of the universe is an impossible direction (which forces us to use imaginary numbers), and hence it is a temporal dimension. But from the center of universe
perspective, the radius is a real dimension and hence is a spatial dimension. Thus, time and space dimensions exchange roles. The radial expansion of the universe appears as the passage of time from
our viewpoint.”
Explaining how his findings and new framework reconcile these two significant theories, Waugh notes that based on his proposed new universe shape: “Both phenomena are like two sides of the same coin.
Relativity is inside the light cone phenomena (since nothing can travel faster than light), while Quantum Mechanics is outside the light cone phenomena (allowing instant communications in ‘quantum
entanglement’ experiments). Both are dictated by the scale (i.e., whether we use classical/human scale or sub-atomic scale).”
The findings have significant ramifications for several theories, including wave-particle duality, Lagrangian-Hamiltonian duality, quantum entanglement, as well as the crucial conservation principles
of Physics.
[Mr. Waugh claims that in an upcoming paper, it will be shown that the Schwarzchild metric is also a dynamic 3D hypersurface (moving with a velocity c in the fourth dimension), just like the
Minkowski SpaceTime metric. The Flamm paraboloid is an accurate mathematical representation of the Schwarzchild metric (contrary to popular belief) if the dynamic nature is considered. Hence, the
rubber membrane/sheet model (which is used to teach General Relativity in schools and colleges) should be taken literally rather than as an analogy, provided that the dynamic nature is also assumed.
The dynamic nature of the 3D field-particle hypersurafce causes the flow of time (which appears to vary with the strength of the gravity field due to varying slopes of the Flamm paraboloid at
different distances from the massive object). A hint of the (opposing) effects of this slope on spatial stretching scale and gravitational time dilation lies hidden in plain sight in the Schwarzchild
metric. The scale factors in the temporal and radial part of the metric are negative inverse of each other. This sort of negative inverse relation is seen in the slopes (m1 and m2) of two
perpendicular lines (m1.m2 = –1), which suggests the resolution of the slope into cos and sine components. Picturing gravity as stretching of 3D hypersurface rather than warping of 4D spacetime
provides a key to unlocking the still mysterious aspects of gravity. The impact of a better understanding of gravity on the subjects of dark matter, cosmic structure, and cosmic evolution shall also
be addressed.]
About Subhajit Waugh: Mr. Subhajit Waugh works as a scientific officer with the Indian government’s Department of Atomic Energy at the Raja Ramanna Centre for Advanced Technology. He graduated top of
his class with a Master’s in Physics from the National Institute of Technology in Rourkela in 2003. His research interests include solving the remaining riddles of the cosmos and fusing cosmology and
physics into a single ‘Theory of Everything’. He received the esteemed NCERT National Talent Scholarship in 1996.
subhajit waughRRCAT (Raja Ramanna Center for Advanced Technology)email us hereVisit us on social media:YouTubeOther
Mr. Subhajit Waugh appeals the audience to watch and share the United Nations Anthem video (with lyrics). The World needs the UN now more than ever before.
just read:
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File encryption using OpenSSL
Symmetic encryption
For symmetic encryption, you can use the following:
To encrypt:
openssl aes-256-cbc -salt -a -e -in foo.txt -out foo.txt.enc
To decrypt:
openssl aes-256-cbc -salt -a -d -in foo.txt.enc -out foo.txt
Asymmetric encryption
Asymmetric encryption uses private/public key. So first generate the private key and extract the public key.
openssl genrsa -aes256 -out private.key 8912
openssl rsa -in private.key -pubout -out public.key
Now wwe can use rsautl to encrypt/decrypt:
openssl rsautl -encrypt -pubin -inkey public.key -in foo.txt -out foo.txt.enc
openssl rsautl -decrypt -inkey private.key -in foo.txt.enc -out foo.txt
But: Public-key crypto is not for encrypting arbitrarily long files (from a performance point of view). That's why we can't directly encrypt a large file using rsautl. Instead we use one-time
random key.
One-Time Random Key
• We use a symmetric cipher (here: AES) to do the normal encryption.
• Each time a new random symmetric key is generated, used for the normal encryption of the large file, and then encrypted with the RSA cipher (public key).
• The ciphertext together with the encrypted symmetric key is transferred to the recipient.
• The recipient decrypts the symmetric key using his private key.
• The recipient then uses the symmetric key to decrypt the large file.
The private key is never shared, only the public key is used to encrypt the random symmetric cipher.
Generate a symmetric key because you can encrypt large files with it
openssl rand -base64 32 > key.bin
Encrypt the large file using the symmetric key
openssl enc -aes-256-cbc -salt -in foo.txt -out foo.txt.enc -pass file:./key.bin
Encrypt the symmetric key so you can safely send it to the other person and destroy the un-encrypted symmetric key so nobody finds it
openssl rsautl -encrypt -inkey public.key -pubin -in key.bin -out key.bin.enc
shred -u key.bin
At this point, you send the encrypted symmetric key (key.bin.enc) and the encrypted large file (foo.txt.enc) to the other person
The other person can then decrypt the symmetric key with their private key using
openssl rsautl -decrypt -inkey private.key -in key.bin.enc -out key.bin
Now they can use the symmetric key to decrypt the file
openssl enc -d -aes-256-cbc -in foot.txt.enc -out foo.txt -pass file:./key.bin
And you're done. The other person has the decrypted file and it was safely sent. | {"url":"https://www.echinopsys.de/en/blog/file_encryption_using_openssl.html","timestamp":"2024-11-05T03:23:26Z","content_type":"text/html","content_length":"4844","record_id":"<urn:uuid:8cd68943-b0eb-49a1-bdda-7386c598a647>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00610.warc.gz"} |
Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents
Semiclassical asymptotics for Schrödinger equations with non-smooth potentials give rise to ill-posed formal semiclassical limits. These problems have attracted a lot of attention in the last few
years, as a proxy for the treatment of eigenvalue crossings, i.e. general systems. It has recently been shown that the semiclassical limit for conical singularities is in fact well-posed, as long as
the Wigner measure (WM) stays away from singular saddle points. In this work we develop a family of refined semiclassical estimates, and use them to derive regularized transport equations for saddle
points with infinite Lyapunov exponents, extending the aforementioned recent results. In the process we answer a related question posed by P.L. Lions and T. Paul in 1993. If we consider more singular
potentials, our rigorous estimates break down. To investigate whether conical saddle points, such as -|x|, admit a regularized transport asymptotic approximation, we employ a numerical solver based
on posteriori error control. Thus rigorous upper bounds for the asymptotic error in concrete problems are generated. In particular, specific phenomena which render invalid any regularized transport
for -|x| are identified and quantified. In that sense our rigorous results are sharp. Finally, we use our findings to formulate a precise conjecture for the condition under which conical saddle
points admit a regularized transport solution for the WM. © 2016 International Press.
Dive into the research topics of 'Regularized semiclassical limits: Linear flows with infinite Lyapunov exponents'. Together they form a unique fingerprint. | {"url":"https://faculty.kaust.edu.sa/en/publications/regularized-semiclassical-limits-linear-flows-with-infinite-lyapu","timestamp":"2024-11-13T09:24:47Z","content_type":"text/html","content_length":"57086","record_id":"<urn:uuid:9da16540-8122-436b-b122-a9e3c71ad8b6>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00810.warc.gz"} |
LCM of 3 Numbers Calculator | How to Calculate Least Common Multiple? - lcmgcf.com
The Least Common Multiple is the smallest integer that is divisible by all the numbers. For three integers a, b, c LCM is denoted as LCM(a, b, c). It is also known as LCD or Least Common Denominator.
For instance LCM of 12, 15, 10 is 60 the smallest number that is divisible by all three numbers. Therefore LCM(12, 15, 10) = 60.
Procedure to find the Least Common Multiple of 3 Numbers
Check out the manual procedure on how to find LCM of 3 numbers by hand using different techniques. Each of the methods is explained using step by step process along with solved examples. Understand
the concept better and choose the method that is convenient for you. If you feel any difficulty They are along the lines
• Listing Multiples
• Prime Factorization
• Cake/Ladder Method
• Division Method
• Using the Greatest Common Factor GCF
LCM of 3 Numbers using Listing Multiples
• List all the multiples of 3 numbers till you have at least one of the multiples appear on all the lists.
• Find the number that is the smallest and is present on all the lists.
• That itself is the Least Common Multiple of the numbers.
Find LCM(6, 7, 21) using Listing Multiples?
Given numbers are 6, 7, 21
Multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.
Multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56
Multiples of 21 are 21, 42, 63
As per the definition, LCM is the smallest number that is common in all 3 numbers multiples.
Thus, 42 is the LCM of 6, 7, 21.
LCM of Three Numbers using Prime Factorization
• Make a note of all the Prime Factors of the given numbers.
• List all the prime numbers found as many times they occur often for the given numbers.
• Multiply the list of prime numbers to obtain the Least Common Multiple.
Find LCM of 12, 24, 30 using Prime Factorization?
Prime Factorization of 12 = 2 × 2 × 3
Prime Factorization of 24 = 2 × 2 × 2 × 3
Prime Factorization of 30 = 2 × 3 × 5
Make a note of prime numbers that occur often for the given numbers and multiply them
= 2×2×2×3×5
= 120
How to find LCM of 3 Numbers using the Cake/ Ladder Method?
The cake or Ladder Method is the easiest method to find the Least Common Multiple of all the methods. Go through the procedure below
• Write down all the 3 numbers in a cake or row.
• Divide the numbers in the layer with a prime number that is evenly divisible in two or more numbers and bring the result into the next layer.
• If any number in the layer or row isn't divisible just bring it down.
• Continue dividing rows by prime numbers.
• When there are no more prime numbers that divide evenly two or more numbers you are done.
Find the LCM(10, 12, 15) using the Cake/ Ladder Method?
Given Numbers are 10, 12, 15
To find LCM Multiply all the prime numbers from top to bottom i.e. 2*3*5*1*2*1
= 60
Therefore, LCM(10, 12, 15) is 60
If you are looking for help on finding the LCM, GCF concepts you can always visit the portal lcmgcf.com to clear all your queries.
Steps to find LCM of 3 Numbers using the Division Method?
• Write down all the numbers in a top table row.
• Begin with the Lowest Prime Numbers and divide the numbers of row by prime numbers that are evenly divisible into at least one of the numbers and bring it to the next row.
• If any number is not divisible bring it down.
• Continue dividing with Prime Numbers that divide at least one number.
• When the last rows are 1's you are done.
Find LCM of 10, 18, 25 using the Division Method?
LCM is the Product of Prime Numbers in the first column
LCM = 2 × 3 × 3 × 5 × 5
LCM = 450
Therefore, LCM(10, 18, 25) is 450.
Finding LCD of 3 Numbers using GCF?
• Initially, find the LCM of the first two numbers.
• Later, find the LCM of the result obtained for the first two numbers and the third number.
• This way, you can find the Least Common Multiple of 3 Numbers easily.
Find the LCM(12, 24, 36) using the GCF Method?
Given Numbers are 12, 24, 36
Step 1:Find the LCM of the first 2 numbers 12, 24
LCM of 12, 24 is 24 the smallest number that is divisible by both the numbers.
Step 2: Now take the LCM of the result i.e. 24, 36
LCM(24, 36) = 72
72 is the smallest number that is divisible by both the numbers.
Therefore, the LCM(12, 24, 36) is 72. | {"url":"https://lcmgcf.com/lcm-of-3-numbers-calculator/","timestamp":"2024-11-03T04:10:38Z","content_type":"text/html","content_length":"53595","record_id":"<urn:uuid:0afed9ed-1478-4e84-876e-9657c6fc2c9f>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00085.warc.gz"} |
2D Straight Skeleton and Polygon Offsetting
Chapter 16
2D Straight Skeleton and Polygon Offsetting
Fernando Cacciola
16.1 Definitions
16.1.1 2D Contour
A 2D contour is a closed sequence (a cycle) of 3 or more connected 2D oriented straight line segments called contour edges. The endpoints of the contour edges are called vertices. Each contour edge
shares its endpoints with at least two other contour edges.
If the edges intersect only at the vertices and at most are coincident along a line but do not cross one another, the contour is classified as simple.
A contour is topologically equivalent to a disk and if it is simple, is said to be a Jordan Curve.
Contours partition the plane in two open regions: one bounded and one unbounded. If the bounded region of a contour is a singly-connected set, the contour is said to be strictly-simple.
The Orientation of a contour is given by the order of the vertices around the region they bound. It can be Clockwise (CCW) or Counter-clockwise (CW).
The bounded side of a contour edge is the side facing the bounded region of the contour. If the contour is oriented CCW, the bounded side of an edge is its left side.
A contour with a null edge (a segment of length zero given by two consecutive coincident vertices), or with edges not connected to the bounded region (an antenna: 2 consecutive edges going forth and
back along the same line), is said to be degenerate (collinear edges are not considered a degeneracy).
16.1.2 2D Polygon with Holes
A 2D polygon is a contour.
A 2D polygon with holes is a contour, called the outer contour, having zero or more contours, called inner contours, or holes, in its bounded region. The intersection of the bounded region of the
outer contour and the unbounded regions of each inner contour is the interior of the polygon with holes. The orientation of the holes must be opposite to the orientation of the outer contour and
there cannot be any intersection among any contour. A hole cannot be in the bounded region of any other hole.
A polygon with holes is strictly-simple if its interior is a singly-connected set.
The orientation of a polygon with holes is the orientation of its outer contour. The bounded side of any edge, whether of the outer contour or a hole, is the same for all edges. That is, if the outer
contour is oriented CCW and the holes CW, both contour and hole edges face the polygon interior to their left.
Throughout the rest of this chapter the term polygon will be used as a shortcut for polygon with holes.
Figure: Examples of strictly simple polygons: One with no holes and two edges coincident (left) and one with 2 holes (right).
Figure: Examples of non-simple polygons: One folding into itself, that is, non-planar (left), one with a vertex touching an edge (right), and one with a hole crossing into the outside (bottom)
16.1.3 Inward Offset of a Non-degenerate Strictly-Simple Polygon with Holes
For any 2D non-degenerate strictly-simple polygon with holes called the source, there can exist a set of 0, 1 or more inward offset polygons with holes, or just offset polygons for short, at some
euclidean distance t $>0$ (each being strictly simple and non-degenerate). Any contour edge of such offset polygon, called an offset edge corresponds to some contour edge of the source polygon,
called its source edge. An offset edge is parallel to its source edge and has the same orientation. The Euclidean distance between the lines supporting an offset edge and its source edge is exactly t
An offset edge is always located to the bounded side of its source edge (which is an oriented straight line segment).
An offset polygon can have less, equal or more sides as its source polygon.
If the source polygon has no holes, no offset polygon has holes. If the source polygon has holes, any of the offset polygons can have holes itself, but it might as well have no holes at all (if the
distance is sufficiently large).
Each offset polygon has the same orientation as the source polygon.
Figure: Offset contours of a sample polygon
16.1.4 Straight Skeleton of a 2D Non-degenerate Strictly-Simple Polygon with Holes
The 2D straight skeleton of a non-degenerate strictly-simple polygon with holes [AAAG95] is a special partitioning of the polygon interior into straight skeleton regions corresponding to the monotone
areas traced by a continuous inward offsetting of the contour edges. Each region corresponds to exactly 1 contour edge.
These regions are bounded by angular bisectors of the supporting lines of the contour edges and each such region is itself a non-convex non-degenerate strictly-simple polygon.
Figure: Straight skeleton of a complex shaggy contour
Figure: Other examples: A vertex-event (left), the case of several collinear edges (middle), and the case of a validly simple polygon with tangent edges (right).
Angular Bisecting Lines and Offset Bisectors
Given two points and a line passing through them, the perpendicular line passing through the midpoint is the bisecting line (or bisector) of those points.
Two non-parallel lines, intersecting at a point, are bisected by two other lines passing through that intersection point.
Two parallel lines are bisected by another parallel line placed halfway in between.
Given just one line, any perpendicular line can be considered the bisecting line (any bisector of any two points along the single line).
The bisecting lines of two edges are the lines bisecting the supporting lines of the edges (if the edges are parallel or collinear, there is just one bisecting line).
The halfplane to the bounded side of the line supporting a contour edge is called the offset zone of the contour edge.
Given any number of contour edges (not necessarily consecutive), the intersection of their offset zones is called their combined offset zone.
Any two contour edges define an offset bisector, as follows: If the edges are non-parallel, their bisecting lines can be decomposed as 4 rays originating at the intersection of the supporting lines.
Only one of these rays is contained in the combined offset zone of the edges (which one depends on the possible combinations of orientations). This ray is the offset bisector of the non-parallel
contour edges.
If the edges are parallel (but not collinear) and have opposite orientation, the entire and unique bisecting line is their offset bisector. If the edges are parallel but have the same orientation,
there is no offset bisector between them.
If the edges are collinear and have the same orientation, their offset bisector is given by a perpendicular ray to the left of the edges which originates at the midpoint of the combined complement of
the edges. (The complement of an edge/segment are the two rays along its supporting line which are not the segment and the combined complement of N collinear segments is the intersection of the
complements of each segment). If the edges are collinear but have opposite orientation, there is no offset bisector between them.
Faces, Edges and Vertices
Each region of the partitioning defined by a straight skeleton is called a face. Each face is bounded by straight line segments, called edges. Exactly one edge per face is a contour edge (corresponds
to a side of the polygon) and the rest of the edges, located in the interior of the polygon, are called skeleton edges, or bisectors.
The bisectors of the straight skeleton are segments of the offset bisectors as defined previously. Since an offset bisector is a ray of a bisecting line of 2 contour edges, each skeleton edge (or
bisector) is uniquely given by two contour edges. These edges are called the defining contour edges of the bisector.
The intersection of the edges are called vertices. Although in a simple polygon, only 2 edges intersect at a vertex, in a straight skeleton, 3 or more edges intersect a any given vertex. That is,
vertices in a straight skeleton have degree $>=3$.
A contour vertex is a vertex for which 2 of its incident edges are contour edges.
A skeleton vertex is a vertex who's incident edges are all skeleton edges.
A contour bisector is a bisector who's defining contour edges are consecutive. Such a bisector is incident upon 1 contour vertex and 1 skeleton vertex and touches the input polygon at exactly 1
An inner bisector is a bisector who's defining contour edges are not consecutive. Such a bisector is incident upon 2 skeleton vertices and is strictly contained in the interior of the polygon.
16.2 Representation
This CGAL package represents a straight skeleton as a specialized Halfedge Data Structure (HDS) whose vertices embeds 2D Points (see the StraightSkeleton_2 concept in the reference manual for
Its halfedges, by considering the source and target points, implicitly embeds 2D oriented straight line segments (each halfedge per see does not embed a segment explicitly).
A face of the straight skeleton is represented as a face in the HDS. Both contour and skeleton edges are represented by pairs of opposite HDS halfedges, and both contour and skeleton vertices are
represented by HDS vertices.
In a HDS, a border halfedge is a halfedge which is incident upon an unbounded face. In the case of the straight skeleton HDS, such border halfedges are oriented such that their left side faces
outwards the polygon. Therefore, the opposite halfedge of any border halfedge is oriented such that its left side faces inward the polygon.
This CGAL package requires the input polygon (with holes) to be non-degenerate, strictly-simple, and oriented counter-clockwise.
The skeleton halfedges are oriented such that their left side faces inward the region they bound. That is, the vertices (both contour and skeleton) of a face are circulated in counter-clockwise
order. There is one and only one contour halfedge incident upon any face.
The contours of the input polygon are traced by the border halfedges of the HDS (those facing outward), but in the opposite direction. That is, the vertices of the contours can only by traced from
the straight skeleton data structure by circulating the border halfedges, and the resulting vertex sequence will be reversed w.r.t the input vertex sequence.
A skeleton edge, according to the definition given in the previous section, is defined by 2 contour edges. In the representation, each one of the opposite halfedges that represent a skeleton edge is
associated with one of the opposite halfedges that correspond to one of its defining contour edges. Thus, the 2 opposite halfedges of a skeleton edge link the edge to its 2 defining contour edges.
Starting from any border contour halfedge, circulating the structure walks through border counter halfedges and traces the vertices of the polygon's contours (in opposite order).
Starting from any non-border but contour halfedge, circulating the structure walks counter-clockwise around the face corresponding to that contour halfedge. The vertices around a face always describe
a non-convex non-degenerate strictly-simple polygon.
A vertex is the intersection of contour and/or skeleton edges. Since a skeleton edge is defined by 2 contour edges, any vertex is itself defined by a unique set of contour edges. These are called the
defining contour edges of the vertex.
A vertex is identified by it's set of defining contour edges. Two vertices are distinct if they have differing sets of defining contour edges. Note that vertices can be distinct even if they are
geometrically embedded at the same point.
The degree of a vertex is the number of halfedges around the vertex incident upon (pointing to) the vertex. As with any halfedge data structure, there is one outgoing halfedge for each incoming
(incident) halfedge around a vertex. The degree of the vertex counts only incoming (incident) halfedges.
In a straight skeleton, the degree of a vertex is not only the number of incident halfedges around the vertex but also the number of defining contour halfedges. The vertex itself is the point where
all the defining contour edges simultaneously collide.
Contour vertices have exactly two defining contour halfedges, which are the contour edges incident upon the vertex; and 3 incident halfedges. One and only one of the incident halfedges is a skeleton
halfedge. The degree of a contour vertex is exactly 3.
Skeleton vertices have at least 3 defining contour halfedges and 3 incident skeleton halfedges. If more than 3 edges collide simultaneously at the same point and time (like in any regular polygon
with more than 3 sides), the corresponding skeleton vertex will have more than 3 defining contour halfedges and incident skeleton halfedges. That is, the degree of a skeleton vertex is $>=3$ (the
algorithm initially produces nodes of degree 3 but in the end all coincident nodes are merged to form higher degree nodes). All halfedges incident upon a skeleton vertex are skeleton halfedges.
The defining contour halfedges and incident halfedges around a vertex can be traced using the circulators provided by the vertex class. The degree of a vertex is not cached and cannot be directly
obtained from the vertex, but you can calculate this number by manually counting the number of incident halfedges around the vertex.
Each vertex stores a 2D point and a time, which is the euclidean distance from the vertex's point to the lines supporting each of the defining contour edges of the vertex (the distance is the same to
each line). Unless the polygon is convex, this distance is not equidistant to the edges, as in the case of a Medial Axis, therefore, the time of a skeleton vertex does not correspond to the distance
from the polygon to the vertex (so it cannot be used to obtain the deep of a region in a shape, for instance).
If the polygon is convex, the straight skeleton is exactly equivalent to the polygon's voronoi diagram and each vertex time is the equidistance to the defining edges.
Contour vertices have time zero.
Figure: Straight Skeleton Data Structure
16.3 API
The straight skeleton data structure is defined by the StraightSkeleton_2 concept and modeled in the Straight_skeleton_2<Traits,Items,Alloc> class.
The straight skeleton construction algorithm is encapsulated in the class Straight_skeleton_builder_2<Gt,Ss> which is parameterized on a geometric traits (class Straight_skeleton_builder_traits<
Kernel>) and the Straight Skeleton class (Ss).
The offset contours construction algorithm is encapsulated in the class Polygon_offset_builder_2<Ss,Gt,Container> which is parameterized on the Straight Skeleton class (Ss), a geometric traits (class
Polygon_offset_builder_traits<Kernel>) and a container type where the resulting offset polygons are generated.
To construct the straight skeleton of a polygon with holes the user must:
(1) Instantiate the straight skeleton builder.
(2) Enter one contour at a time, starting from the outer contour, via the method enter_contour. The input polygon with holes must be non-degenerate, strictly-simple and counter-clockwise oriented
(see the definitions at the beginning of this chapter). Collinear edges are allowed. The insertion order of each hole is unimportant but the outer contour must be entered first.
(3) Call construct_skeleton once all the contours have been entered. You cannot enter another contour once the skeleton has been constructed.
To construct a set of inward offset contours the user must:
(1) Construct the straight skeleton of the source polygon with holes.
(2) Instantiate the polygon offset builder passing in the straight skeleton as a parameter.
(3) Call construct_offset_contours passing the desired offset distance and an output iterator that can store a boost::shared_ptr of Container instances into a resulting sequence (typically, a back
insertion iterator)
Each element in the resulting sequence is an offset contour, given by a boost::shared_ptr holding a dynamically allocated instance of the Container type. Such a container can be any model of the
VertexContainer_2 concept, for example, a CGAL::Polygon_2, or just a std::vector of 2D points.
The resulting sequence of offset contours can contain both outer and inner contours. Each offset hole (inner offset contour) would logically belong in the interior of some of the outer offset
contours. However, this algorithm returns a sequence of contours in arbitrary order and there is no indication whatsoever of the parental relationship between inner and outer contours.
On the other hand, each outer contour is counter-clockwise oriented while each hole is clockwise-oriented. And since offset contours do form simple polygons with holes, it is guaranteed that no hole
will be inside another hole, no outer contour will be inside any other contour, and each hole will be inside exactly 1 outer contour.
Parental relationships are not automatically reconstructed by this algorithm because this relation is not directly given by the input polygon with holes and doing it robustly is a time-consuming
A user can reconstruct the parental relationships as a post processing operation by testing each inner contour (which is identified by being clockwise) against each outer contour (identified as being
counter-clockwise) for insideness.
This algorithm requires exact predicates but not exact constructions Therefore, the Exact_predicates_inexact_constructions_kernel should be used.
16.3.1 Exterior Skeletons and Exterior Offset contours
This CGAL package can only construct the straight skeleton and offset contours in the interior of a polygon with holes. However, constructing exterior skeletons and exterior offsets is possible:
Say you have some polygon made of 1 outer contour C0 and 1 hole C1, and you need to obtain some exterior offset contours.
The interior region of a polygon with holes is connected while the exterior region is not: there is an unbounded region outside the outer contour, and one bounded region inside each hole. To
construct an offset contour you need to construct an straight skeleton. Thus, to construct exterior offset contours for a polygon with holes, you need to construct, separately, the exterior skeleton
of the outer contour and the interior skeleton of each hole.
Constructing the interior skeleton of a hole is directly supported by this CGAL package; you just need to input the hole's vertices in reversed order as if it were an outer contour.
Constructing the exterior skeleton of the outer contour is possible by means of the following trick: place the contour as a hole of a big rectangle (call it frame). If the frame is sufficiently
separated from the contour, the resulting skeleton will be practically equivalent to a real exterior skeleton.
To construct exterior offset contours in the inside of each hole you just use the skeleton constructed in the interior, and, if required, revert the orientation of each resulting offset contour.
Constructing exterior offset contours in the outside of the outer contour is just a little bit more involved: Since the contour is placed as a hole of a frame, you will always obtain 2 offset
contours for any given distance; one is the offseted frame and the other is the offseted contour. Thus, from the resulting offset contour sequence, you always need to discard the offsetted frame,
easily identified as the offset contour with the largest area.
It is necessary to place the frame sufficiently away from the contour. If it is not, it could occur that the outward offset contour collides and merges with the inward offset frame, resulting in 1
instead of 2 offset contours.
However, the proper separation between the contour and the frame is not directly given by the offset distance at which you want the offset contour. That distance must be at least the desired offset
plus the largest euclidean distance between an offset vertex and its original.
This CGAL packages provides a helper function to compute the required separation: compute_outer_frame_margin
If you use this function to place the outer frame you are guaranteed to obtain an offset contour corresponding exclusively to the frame, which you can always identify as the one with the largest area
and which you can simple remove from the result (to keep just the relevant outer contours).
Figure: Exterior skeleton obtained using a frame (left) and 2 sample exterior offset contours (right)
16.3.2 Example
// This example illustrates how to use the CGAL Straight Skeleton package
// to construct an offset contour on the outside of a polygon
// This is the recommended kernel
typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
typedef Kernel::Point_2 Point_2;
typedef CGAL::Polygon_2<Kernel> Contour;
typedef boost::shared_ptr<Contour> ContourPtr;
typedef std::vector<ContourPtr> ContourSequence ;
typedef CGAL::Straight_skeleton_2<Kernel> Ss;
typedef Ss::Halfedge_iterator Halfedge_iterator;
typedef Ss::Halfedge_handle Halfedge_handle;
typedef Ss::Vertex_handle Vertex_handle;
typedef CGAL::Straight_skeleton_builder_traits_2<Kernel> SsBuilderTraits;
typedef CGAL::Straight_skeleton_builder_2<SsBuilderTraits,Ss> SsBuilder;
typedef CGAL::Polygon_offset_builder_traits_2<Kernel> OffsetBuilderTraits;
typedef CGAL::Polygon_offset_builder_2<Ss,OffsetBuilderTraits,Contour> OffsetBuilder;
int main()
// A start-shaped polygon, oriented counter-clockwise as required for outer contours.
Point_2 pts[] = { Point_2(-1,-1)
, Point_2(0,-12)
, Point_2(1,-1)
, Point_2(12,0)
, Point_2(1,1)
, Point_2(0,12)
, Point_2(-1,1)
, Point_2(-12,0)
} ;
std::vector<Point_2> star(pts,pts+8);
// We want an offset contour in the outside.
// Since the package doesn't support that operation directly, we use the following trick:
// (1) Place the polygon as a hole of a big outer frame.
// (2) Construct the skeleton on the interior of that frame (with the polygon as a hole)
// (3) Construc the offset contours
// (4) Identify the offset contour that corresponds to the frame and remove it from the result
double offset = 3 ; // The offset distance
// First we need to determine the proper separation between the polygon and the frame.
// We use this helper function provided in the package.
boost::optional<double> margin = CGAL::compute_outer_frame_margin(star.begin(),star.end(),offset);
// Proceed only if the margin was computed (an extremely sharp corner might cause overflow)
if ( margin )
// Get the bbox of the polygon
CGAL::Bbox_2 bbox = CGAL::bbox_2(star.begin(),star.end());
// Compute the boundaries of the frame
double fxmin = bbox.xmin() - *margin ;
double fxmax = bbox.xmax() + *margin ;
double fymin = bbox.ymin() - *margin ;
double fymax = bbox.ymax() + *margin ;
// Create the rectangular frame
Point_2 frame[4]= { Point_2(fxmin,fymin)
, Point_2(fxmax,fymin)
, Point_2(fxmax,fymax)
, Point_2(fxmin,fymax)
} ;
// Instantiate the skeleton builder
SsBuilder ssb ;
// Enter the frame
// Enter the polygon as a hole of the frame (NOTE: as it is a hole we insert it in the opposite orientation)
// Construct the skeleton
boost::shared_ptr<Ss> ss = ssb.construct_skeleton();
// Proceed only if the skeleton was correctly constructed.
if ( ss )
// Instantiate the container of offset contours
ContourSequence offset_contours ;
// Instantiate the offset builder with the skeleton
OffsetBuilder ob(*ss);
// Obtain the offset contours
ob.construct_offset_contours(offset, std::back_inserter(offset_contours));
// Locate the offset contour that corresponds to the frame
// That must be the outmost offset contour, which in turn must be the one
// with the largetst unsigned area.
ContourSequence::iterator f = offset_contours.end();
double lLargestArea = 0.0 ;
for (ContourSequence::iterator i = offset_contours.begin(); i != offset_contours.end(); ++ i )
double lArea = CGAL_NTS abs( (*i)->area() ) ; //Take abs() as Polygon_2::area() is signed.
if ( lArea > lLargestArea )
f = i ;
lLargestArea = lArea ;
// Remove the offset contour that corresponds to the frame.
// Print out the skeleton
Halfedge_handle null_halfedge ;
Vertex_handle null_vertex ;
// Dump the edges of the skeleton
for ( Halfedge_iterator i = ss->halfedges_begin(); i != ss->halfedges_end(); ++i )
std::string edge_type = (i->is_bisector())? "bisector" : "contour";
Vertex_handle s = i->opposite()->vertex();
Vertex_handle t = i->vertex();
std::cout << "(" << s->point() << ")->(" << t->point() << ") " << edge_type << std::endl;
// Dump the generated offset polygons
std::cout << offset_contours.size() << " offset contours obtained\n" ;
for (ContourSequence::const_iterator i = offset_contours.begin(); i != offset_contours.end(); ++ i )
// Each element in the offset_contours sequence is a shared pointer to a Polygon_2 instance.
std::cout << (*i)->size() << " vertices in offset contour\n" ;
for (Contour::Vertex_const_iterator j = (*i)->vertices_begin(); j != (*i)->vertices_end(); ++ j )
std::cout << "(" << j->x() << "," << j->y() << ")" << std::endl ;
return 0;
16.4 Straight Skeletons, Medial Axis and Voronoi Diagrams
The straight skeleton of a polygon is similar to the medial axis and the voronoi diagram of a polygon in the way it partitions it; however, unlike the medial axis and voronoi diagram, the bisectors
are not equidistant to its defining edges but to the supporting lines of such edges. As a result, Straight Skeleton bisectors might not be located in the center of the polygon and so cannot be
regarded as a proper Medial Axis in its geometrical meaning.
On the other hand, only reflex vertices (whose internal angle $>pi$) are the source of deviations of the bisectors from its center location. Therefore, for convex polygons, the straight skeleton, the
medial axis and the Voronoi diagram are exactly equivalent, and, if a non-convex polygon contains only vertices of low reflexivity, the straight skeleton bisectors will be placed nearly equidistant
to their defining edges, producing a straight skeleton pretty much alike a proper medial axis.
16.5 Usages of the Straight Skeletons
The most natural usage of straight skeletons is offsetting: growing and shrinking polygons (provided by this CGAL package).
Anther usage, perhaps its very first, is roof design: The straight skeleton of a polygonal roof directly gives the layout of each tent. If each skeleton edge is lifted from the plane a height equal
to its offset distance, the resulting roof is "correct" in that water will always fall down to the contour edges (roof border) regardless of were in the roof it falls. [LD03] gives an algorithm for
roof design based on the straight skeleton.
Just like medial axes, 2D straight skeletons can also be used for 2D shape description and matching. Essentially, all the applications of image-based skeletonization (for which there is a vast
literature) are also direct applications of the straight skeleton, specially since skeleton edges are simply straight line segments.
Consider the subgraph formed only by inner bisectors (that is, only the skeleton halfeges which are not incident upon a contour vertex). Call this subgraph a skeleton axis. Each node in the skeleton
axis whose degree is $>=3$ roots more than one skeleton tree. Each skeleton tree roughly corresponds to a region in the input topologically equivalent to a rectangle; that is, without branches. For
example, a simple letter "H" would contain 2 higher degree nodes separating the skeleton axis in 5 trees; while the letter "@" would contain just 1 higher degree node separating the skeleton axis in
2 curly trees.
Since a skeleton edge is a 2D straight line, each branch in a skeleton tree is a polyline. Thus, the path-length of the tree can be directly computed. Furthermore, the polyline for a particular tree
can be interpolated to obtain curve-related information.
Pruning each skeleton tree cutting off branches whose length is below some threshold; or smoothing a given branch, can be used to reconstruct the polygon without undesired details, or fit into a
particular canonical shape.
Each skeleton edge in a skeleton branch is associated with 2 contour edges which are facing each other. If the polygon has a bottleneck (it almost touches itself), a search in the the skeleton graph
measuring the distance between each pair of contour edges will reveal the location of the bottleneck, allowing you to cut the shape in two. Likewise, if two shapes are too close to each other along
some part of their boundaries (a near contact zone), a similar search in an exterior skeleton of the two shapes at once would reveal the parts of near contact, allowing you to stitch the shapes.
These cut and stitch operations can be directly executed in the straight skeleton itself instead of the input polygon (because the straight skeleton contains a graph of the connected contour edges).
16.6 Straight Skeleton of a General Figure in the Plane
A straight skeleton can also be defined for a general multiply-connected planar directed straight-line graph [AA95] by considering all the edges as embedded in an unbounded region. The only
difference is that in this case some faces will be only partially bounded.
The current version of this CGAL package can only construct the straight skeleton in the interior of a simple polygon with holes, that is it doesn't handle general polygonal figures in the plane. | {"url":"https://doc.cgal.org/Manual/3.2/doc_html/cgal_manual/Straight_skeleton_2/Chapter_main.html","timestamp":"2024-11-04T02:24:45Z","content_type":"text/html","content_length":"42213","record_id":"<urn:uuid:c07fa7f0-d100-4cc6-a253-a406af442a89>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00235.warc.gz"} |
Phyllotaxis Spirals - International Maths Challenge
Phyllotaxis Spirals
Phyllotaxis is a term used for the patterns that emerge in the growth of plants. Spiral phyllotaxis is observed in the heads of sunflowers, in pine-cones and pineapples, and in a variety of other
Phyllotaxis is a popular topic in mathematics recreation – it’s interesting in its own right and also related to other perennial favourites, Fibonacci numbers and the golden ratio.
The article Modeling Spiral Growth in a GSP Environment describes how to model phyllotaxis-like patterns in GSP. Although GSP works reasonably well, TinkerPlots or Fathom environments seem to be
better suited to capturing this particular model – they make the formulas more explicit and easy to manipulate, and they allow for the data generated to be viewed in a variety of ways. The images
above were created by porting this model to TinkerPlots.
As the article suggests, experimenting with the the angle between successive seeds allows you to see different resulting patterns – angles that are multiples of rational numbers create patterns of
rays while irrational numbers (actually approximate values) give spirals, or spiral/ray combinations (the rays form as the approximation gets more “rational”). A good choice for approximating actual
phyllotaxis patterns is to use tau = (1+sqrt(5))/2 in your angle. Here is a listing for the attributes required to generate the pattern in TP or Fathom. The graph/plot is simply the x attribute on
the horizontal and the y attribute on the vertical (in TP these need to be fully separated).
n = caseIndex
base_angle = pi*(1+sqrt(5))
r = sqrt(n)
theta = n*base_angle
x = r*cos(theta)
y = r*sin(theta)
The images shown in this post use a collection of 500 cases or “seeds”. The base angle is 2pi*tau, and the actual angle for a given seed is a multiple of this base angle.
The model is nice looking and easy to build, but it models only the end result of the growth process, not the process itself. It winds new seeds around the outer edge based on a pre-determined angle.
A better model would be one that mirrors what is understood to be the underlying phonomena – new seeds are added to the center and old seeds are pushed out following a set of rules. Under this
dynamic method, the angles and spirals are an emergent aspect of the system, rather than the explicit result. This website describes how such a dynamical system could be modeled.
Although the Fathom/TP model does not model the dynamical system that underlies phyllotaxis, it’s fun to play with in its own right. You can manually alter the base_angle attribute as suggested by
the GSP article. If you add a parameter (a slider) to help you vary the angle, you can obtain a whole family of spiral/ray patterns whose properties you could take a closer look at. Different
combinations of angles and sliders will give you various levels of control over the image.
For example, change the formula for base_angle to base_angle = pi*(1+sqrt(5))*base, and create a slider “base”. The image below shows the spirals obtained for base = 1…6.
Update: Here is an example of how to draw spirals like this in R.
For more such insights, log into www.international-maths-challenge.com.
*Credit for article given to dan.mackinnon* | {"url":"https://international-maths-challenge.com/phyllotaxis-spirals/","timestamp":"2024-11-08T15:30:01Z","content_type":"text/html","content_length":"146657","record_id":"<urn:uuid:82f02ad4-cba4-44db-ab19-67d64ff12f73>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00341.warc.gz"} |
Distributive Property -Meaning And 5 Examples Of Distributive Property | Example NG
Distributive property -Meaning and examples
The distributive property is useful when dealing with algebra and complex equations. Distributive property helps greatly to break the equations down into smaller parts to solve the equation. It is
used in advanced and higher multiplication, addition and algebra.
Meaning of distributive property
To “distribute” means to divide something or give a share or part of something. According to the distributive property, multiplying the sum of two or more addends by a number will give the same
result as multiplying each addend individually by the number and then adding the products together.
The distributive property helps in making difficult problems simpler. You can use the distributive property of multiplication to rewrite expression by distributing or breaking down a factor as a sum
or difference of two numbers.
Here, for instance, calculating 8 × 27 can be made easier by breaking down 27 as 20 + 7 or 30 − 3. Distributive property can be also known as the distributive law of multiplication, it’s one of the
most commonly used properties in mathematics.
Also Read: Prime Number Examples – Meaning, How To Identify, And Examples Of Prime Numbers From 1-100
Examples of Distributive Property
The following are examples of distributive property.
1. Distributive property of addition
2. Distributive property of subtraction
3. Distributive property of multiplication over addition
4. Distributive property of multiplication over subtraction
5. Distributive property of fractions
Let’s focus on the distributive property of multiplication as explained by Splash Learners.
The distributive property of multiplication states that when a number is multiplied by the sum of two numbers, the first number can be distributed to both of those numbers and multiplied by each of
them separately, then adding the two products together for the same result as multiplying the first number by the sum.
Recommended: Variable Cost – Definition, Formula, and 9 Examples
Let’s look at the distributive property with this example: distributive property of multiplication. According to the distributive property 2 × (3 + 5) will be equal to 2 × 3 + 2 x 5
2 × (3 + 5) = 2 × 8 = 16
2 × 3 + 2 × 5 = 6 + 10 = 16
In both cases we get the same result, 16, and therefore we can show that the distributive property of multiplication is correct.
Another example: Imagine one student and her two friends each have seven pens and four pencils. How many pieces of fruit do all three students have in total? In their school bags, they each have 7
pens and 4 pencils. To know the total number of pieces of pens and pencils, we need to multiply the whole thing by 3.
When you break it down, you’re multiplying 7 pens and 4 pencils by 3 students. So, you end up with 21 pens and 12 pencils for a total of 33 pieces of stationaries.
The distributive property of multiplication over addition can be used when you want to multiply a number by a sum. For example, if you want to multiply 3 by the sum of 10 + 2.
According to this property, you can add the numbers and then multiply by 3. 3(10 + 2) = 3(12) = 36. Or, you can first multiply each addend by the 3. (This is called distributing the 3.) Then, you can
add the products.
Also See: Fixed Cost – Meaning, 9 Examples, and Ultimate Guide On How To Calculate Fixed Costs
The multiplication of 3(10) and 3(2) will each be done before you add. 3(10) + 3(2) = 30 + 6 = 36. Note that the answer is the same as before.
The distributive property of multiplication over subtraction is like the distributive property of multiplication over addition. You can subtract the numbers and then multiply, or you can multiply and
then subtract as shown below. This is called “distributing the multiplier.”
5( 6 – 3) = 5(6) – 5(3)
The same number works if the is on the other side of the parentheses, as in the example below.
(6 – 3) 5 = 6(5) – 3(5)
In both cases, you can then simplify the distributed expression to arrive at your answer. The example below, in which 5 is the outside multiplier, demonstrates that this is true.
Also Read: Time Card Calculator – Top 7 Reasons To Get Your Business a Time Card Calculator
The expression on the right, which is simplified using the distributive property, is shown to be equal to 15, which is the resulting value on the left as well.
5( 6 – 3) = 5(6) – 5(3)
5(3) = 30 – 15
15= 15
www.splashlearn.com/math. Retrieved on 24th January, 2020.
www.Khan academy.org/math/pre-algebra. | {"url":"https://example.ng/distributive-property/","timestamp":"2024-11-11T12:49:29Z","content_type":"text/html","content_length":"427790","record_id":"<urn:uuid:9d087ea6-044c-49b6-830f-c454999ef3b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00651.warc.gz"} |
SOCR/DataSifterII documentation
betarate_g Sifted Data Utility Check
calWeights Calculate sampling weights for raw data imputation.
checkLink Define link function for GLMM
create.dummy Construct dummy variables for factor time-invariant vectors.
CVCF Last value carry forward and next value carry backward...
diff.imp Calculating the difference between imputed and original...
filter_lambda Optimal lambda and corresponding final GLMM LASSO model
pifv Percent of Identical Feature Values
repSifter DataSifter II Algorithm for Time-varying Data
repSifterImp Artificial Missing Introduce and Imputation Step
synthetic Create fully synthetic longitudinal dataset.
try_glmmlasso Calculating the BIC of a GLMM lasso model, given specific...
wlmmLasso Weighted GlmmLASSO
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/github/SOCR/DataSifterII/man/","timestamp":"2024-11-05T09:33:27Z","content_type":"text/html","content_length":"18308","record_id":"<urn:uuid:22f05ebe-249d-43c5-8823-69d1f638bffc>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00484.warc.gz"} |
Categorical data: Feature crosses | Machine Learning | Google for Developers
Feature crosses are created by crossing (taking the Cartesian product of) two or more categorical or bucketed features of the dataset. Like polynomial transforms, feature crosses allow linear models
to handle nonlinearities. Feature crosses also encode interactions between features.
For example, consider a leaf dataset with the categorical features:
• edges, containing values smooth, toothed, and lobed
• arrangement, containing values opposite and alternate
Assume the order above is the order of the feature columns in a one-hot representation, so that a leaf with smooth edges and opposite arrangement is represented as {(1, 0, 0), (1, 0)}.
The feature cross, or Cartesian product, of these two features would be:
{Smooth_Opposite, Smooth_Alternate, Toothed_Opposite, Toothed_Alternate, Lobed_Opposite, Lobed_Alternate}
where the value of each term is the product of the base feature values, such that:
• Smooth_Opposite = edges[0] * arrangement[0]
• Toothed_Opposite = edges[1] * arrangement[0]
• Lobed_Alternate = edges[2] * arrangement[1]
For any given example in the dataset, the feature cross will equal 1 only if both base features' original one-hot vectors were 1 for the crossed categories. That is, an oak leaf with a lobed edge and
alternate arrangement would have a value of 1 only for Lobed_Alternate, and the feature cross above would be:
{0, 0, 0, 0, 0, 1}
This dataset could be used to classify leaves by tree species, since these characteristics do not vary within a species.
Click here to compare polynomial transforms with feature crosses
Feature crosses are somewhat analogous to Polynomial transforms. Both combine multiple features into a new synthetic feature that the model can train on to learn nonlinearities. Polynomial transforms
typically combine numerical data, while feature crosses combine categorical data.
When to use feature crosses
Domain knowledge can suggest a useful combination of features to cross. Without that domain knowledge, it can be difficult to determine effective feature crosses or polynomial transforms by hand.
It's often possible, if computationally expensive, to use neural networks to automatically find and apply useful feature combinations during training.
Be careful—crossing two sparse features produces an even sparser new feature than the two original features. For example, if feature A is a 100-element sparse feature and feature B is a 200-element
sparse feature, a feature cross of A and B yields a 20,000-element sparse feature. | {"url":"https://developers.google.cn/machine-learning/crash-course/categorical-data/feature-crosses","timestamp":"2024-11-14T00:40:32Z","content_type":"text/html","content_length":"170374","record_id":"<urn:uuid:79630694-581d-478d-b264-ed5394931f4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00298.warc.gz"} |
Stability and (Obviously) Strategy-proofness in Matching Theory
Date of Submission
August 2021
Institute Name (Publisher)
Indian Statistical Institute
Document Type
Doctoral Thesis
Degree Name
Doctor of Philosophy
Subject Name
Quantitative Economics
Applied Statistics Unit (ASU-Kolkata)
Roy, Souvik (ASU-Kolkata; ISI)
Abstract (Summary of the Work)
We analyze stability and (obviously) strategy-proofness in two well-known models of matching theory, namely “assignment problem” (a one-sided matching model) and “marriage problem” (a two-sided
matching model). • In assignment problem, a set of heterogeneous indivisible objects are to be allocated over a set of agents based on the agents’ preferences over the objects. Each agent can receive
at most one object and monetary transfers are not allowed. Such problems arise when, for instance, the Government wants to assign houses to the citizens, or hospitals to doctors, or a manager wants
to allocate offices to employees, or tasks to workers, or a professor wants to assign projects to students. Agents are asked to report their preferences over the objects and the designer decides the
allocation based on these reports. • An instance of classical marriage problem involves n men and n women, each of whom specifies a preference over the members of the opposite sex. A matching μ is a
set of (man, woman) pairs such that each person belongs to exactly one pair. If (m, w) ∈ μ, we say that w is m’s partner in μ, and vice versa, and we write μ(m) = w, μ(w) = m. Agents are asked to
report their preferences and the designer decides the matching based on these reports. This thesis comprises of four chapters. A brief introduction of the chapters is provided below. In Chapter 2, we
consider assignment problems where agents are to be assigned at most one indivisible object and monetary transfers are not allowed. We provide a characterization of assignment rules that are Pareto
efficient, non-bossy, and implementable in obviously strategy-proof (OSP) mechanisms. As corollaries of our result, we obtain a characterization of OSP-implementable fixed priority top trading cycles
(FPTTC) rules, hierarchical exchange rules, and trading cycles rules. Troyan (2019) provides a characterization of OSP-implementable FPTTC rules when there are equal number of agents and objects. Our
result generalizes this for arbitrary values of those. In Chapter 3, we study the implementation of a fixed priority top trading cycles (FPTTC) rule via an obviously strategy-proof (OSP) mechanism
(Li, 2017) in the context of assignment problems with outside options, where agents are to be assigned at most one indivisible object and monetary transfers are not allowed. In a model without
outside options, Troyan (2019) gives a sufficient (but not necessary) and Mandal & Roy (2020) give a necessary and sufficient condition for an FPTTC rule to be OSPimplementable. This paper shows that
in a model with outside options, the two conditions (in Troyan (2019) and Mandal & Roy (2020)) are equivalent for an FPTTC rule, and each of them is necessary and sufficient for an FPTTC rule to be
OSP-implementable. In Chapter 4, we consider assignment problems where heterogeneous indivisible objects are to be assigned to agents so that each agent receives at most one object. Agents have
single-peaked preferences over the objects. In this setting, first we show that there is no strategy-proof, non-bossy, Pareto efficient, and strongly pairwise reallocation-proof assignment rule on a
minimally rich single-peaked domain when there are at least three agents and at least three objects in the market. Next, we characterize all strategyproof, Pareto efficient, top-envy-proof,
non-bossy, and pairwise reallocation-proof assignment rules on a minimally rich single-peaked domain as hierarchical exchange rules. We additionally show that strategyproofness and non-bossiness
together are equivalent to group strategy-proofness on a minimally rich single-peaked domain, and every hierarchical exchange rule satisfies group-wise reallocation-proofness on a minimally rich
single-peaked domain. In Chapter 5, we provide a class of algorithms, called men-women proposing deferred acceptance (MWPDA) algorithms, that can produce all stable matchings at every preference
profile for the marriage problem. Next, we provide an algorithm that produces a minimum regret stable matching at every preference profile. We also show that its outcome is always women-optimal in
the set of all minimum regret stable matchings. Finally, we provide an algorithm that produces a stable matching with given sets of forced and forbidden pairs at every preference profile, whenever
such a matching exists. As before, here too we show that the outcome of the said algorithm is women-optimal in the set of all stable matchings with given sets of forced and forbidden pairs
ProQuest Collection ID: https://www.proquest.com/pqdtlocal1010185/dissertations/fromDatabasesLayer?accountid=27563
Control Number
Recommended Citation
Mandal, Pinaki Dr., "Stability and (Obviously) Strategy-proofness in Matching Theory" (2022). Doctoral Theses. 563. | {"url":"https://digitalcommons.isical.ac.in/doctoral-theses/563/","timestamp":"2024-11-01T22:25:18Z","content_type":"text/html","content_length":"45075","record_id":"<urn:uuid:9b826a1d-fba3-4842-bbc3-7004a201e8af>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00712.warc.gz"} |
On fixation of activated random walks
We prove that for the Activated Random Walks model on transitive unimodular graphs, if there is fixation, then every particle eventually fixates, almost surely. We deduce that the critical density is
at most 1. Our methods apply for much more general processes on unimodular graphs. Roughly put, our result apply whenever the path of each particle has an automorphism invariant distribution and is
independent of other particles’ paths, and the interaction between particles is automorphism invariant and local. In particular, we do not require the particles path distribution to be Markovian.
This allows us to answer a question of Rolla and Sidoravicius [3,4], in a more general setting then had been previously known (by Shellef [5]).
• Activated Random Walks
• Interacting Particles System
Dive into the research topics of 'On fixation of activated random walks'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/on-fixation-of-activated-random-walks","timestamp":"2024-11-11T04:21:33Z","content_type":"text/html","content_length":"45783","record_id":"<urn:uuid:d4121d4f-b3f3-45d8-886a-f8c4c0840b3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00456.warc.gz"} |
Return the Blackman window.
The Blackman window is a taper formed by using the first three terms of a summation of cosines. It was designed to have close to the minimal leakage possible. It is close to optimal, only
slightly worse than a Kaiser window.
M : int
Number of points in the output window. If zero or less, an empty array is returned.
out : ndarray
The window, with the maximum value normalized to one (the value one appears only if the number of samples is odd).
The Blackman window is defined as
Most references to the Blackman window come from the signal processing literature, where it is used as one of many windowing functions for smoothing values. It is also known as an apodization
(which means “removing the foot”, i.e. smoothing discontinuities at the beginning and end of the sampled signal) or tapering function. It is known as a “near optimal” tapering function, almost as
good (by some measures) as the kaiser window.
Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York.
Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471.
>>> np.blackman(12)
array([ -1.38777878e-17, 3.26064346e-02, 1.59903635e-01,
4.14397981e-01, 7.36045180e-01, 9.67046769e-01,
9.67046769e-01, 7.36045180e-01, 4.14397981e-01,
1.59903635e-01, 3.26064346e-02, -1.38777878e-17])
Plot the window and the frequency response:
>>> from numpy.fft import fft, fftshift
>>> window = np.blackman(51)
>>> plt.plot(window)
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.title("Blackman window")
<matplotlib.text.Text object at 0x...>
>>> plt.ylabel("Amplitude")
<matplotlib.text.Text object at 0x...>
>>> plt.xlabel("Sample")
<matplotlib.text.Text object at 0x...>
>>> plt.show()
>>> plt.figure()
<matplotlib.figure.Figure object at 0x...>
>>> A = fft(window, 2048) / 25.5
>>> mag = np.abs(fftshift(A))
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(mag)
>>> response = np.clip(response, -100, 100)
>>> plt.plot(freq, response)
[<matplotlib.lines.Line2D object at 0x...>]
>>> plt.title("Frequency response of Blackman window")
<matplotlib.text.Text object at 0x...>
>>> plt.ylabel("Magnitude [dB]")
<matplotlib.text.Text object at 0x...>
>>> plt.xlabel("Normalized frequency [cycles per sample]")
<matplotlib.text.Text object at 0x...>
>>> plt.axis('tight')
(-0.5, 0.5, -100.0, ...)
>>> plt.show() | {"url":"https://docs.scipy.org/doc/numpy-1.15.1/reference/generated/numpy.blackman.html","timestamp":"2024-11-05T03:23:17Z","content_type":"text/html","content_length":"14887","record_id":"<urn:uuid:e4f6da43-98a9-4315-adae-51160f4eddc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00167.warc.gz"} |
Algebra Ch5.1 #10x
This is an error in the Algebra Solutions Manual
Mary asked,
In Algebra 1, chapter 5 lesson 1 problem 10x, the answer key says the answer is true for all positive integers. But isn’t it true for all integers? 1 to the power of x =1. If it is a negative
power, it is still one, right?
Dr. Callahan Answer:
You are correct. Should be for all real numbers – not just integers. I tend to test these in a calculator to make sure though 😉
So that is incorrect in the Solutions manual.
Just for completeness sake – my goto math tool is Wolfram – so here is there answer.
This is an error in the book by Master Books (ISBN: 978-0890519851) - the book with the blue cover showing the world. The problem is discussing hens and...
Question from Grace: We are using this test A as a practice test. We can't figure out how to solve this. Answer from Dr. Callahan: Grace, To solve this one,...
Error in Solutions Manual The Solutions Manual has the answer from the older version of the book (which used a flag of 104 x 235. Flags have grown since then....
Ch5.6 #15 As before – all are similar so I will pick the harder one. Givens AB = CD AB = 9(x-2) CD = x + 3(x+5) Therefore, 9(x-2) = x + 3(x+5) Now multiply...
Now let’s look at Ch 5.6 #14. Parts a-c are similar – so will do part c since the harder. Area = x + 20 = (x-1)4 > I am just stating all things that are...
Algebra 11.5 #7b
Algebra Chapter 16 Lesson 2 Set 1 Problem #3
Algebra Ch13 Test A Problem #13
Algebra Ch10.2 #3
Algebra Ch5.6 #15
Algebra Ch5.6 #14 | {"url":"https://askdrcallahan.com/algebra-ch5-1-10x/","timestamp":"2024-11-02T15:24:44Z","content_type":"text/html","content_length":"379912","record_id":"<urn:uuid:3f4b0f42-eb09-4aa6-be83-6cce6825a3c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00707.warc.gz"} |
Calculating the Number of Unique Items in a Delimited String
This is a guest post by David Hager. You can learn more about David on LinkedInd
There has recently been interest in handling row data that contain delimited strings. An elegant solution to split a delimited row into multiple rows using DAX queries can be found here.
Another question was asked on LinkedIn about how to count the number unique items of a delimited string. I could not figure out how to do this in PowerPivot, and I am not sure it can be done with a
single formula. However, I undertook the challenge to do this in Excel and I came up with a solution, albeit a rather lengthy one.
So, if the string a,c,d,a,b,c,d,e,f,g,h,I,j is in cell A1, the following formula will return the value 10, which is the number of unique items in the delimited string.
=SUM(N(MATCH(TRIM(MID(SUBSTITUTE(A1,",",REPT(" ",999)) ,ROW(INDIRECT("1:"&LEN(A1)-LEN(SUBSTITUTE(A1,",",""))+1))*999-998,999)),TRIM(MID(SUBSTITUTE(A1,",",REPT(" ",999)) ,ROW(INDIRECT("1:"&LEN(A1)-LEN
In order to explain how this formula works, a simplified version is shown below:
The MATCH function returns an array of n elements. Each value in the array is the MATCH function result for the nth element. For the example string, the array will look like this:
If every element in the delimited string is unique, this array would be filled by consecutive numbers from 1 to 13. It can be easily seen which elements do not fit that pattern. In order to calculate
this, the array from the MATCH function must be set equal to the unique array, or:
{1;2;3;1;5;2;3;8;9;10;11;12;13} = {1;2;3;4;5;6;7;8;9;10;11;12;13}
which affords an array of Boolean values:
Use of the N function converts this array into ones and zeros, and the SUM function returns the desired result.
Well, that was the easy part. :)
The hard part of this formula is to convert a delimited string into an array of each element in the delimited string. This was accomplished by using the following formula:
=TRIM(MID(SUBSTITUTE(A1,",",REPT(" ",999)),ROW(INDIRECT("1:"&LEN(A1)-LEN(SUBSTITUTE(A1,",",""))+1))*999-998,999))
To give credit where credit is due, the core of this formula was created by Rick Rothstein, see:
The truly amazing function of this formula is that it converts a delimited string into an array! I’m not going to go into an explanation here of how this formula works, since it is explained at the
provided link. The original formula
=TRIM(MID(SUBSTITUTE(A1,",",REPT(" ",999)),n*999-998,999))
was designed to return the nth element from a delimited string, but in this case all of the elements are returned in an array by replacing n with ROW(INDIRECT("1:"&LEN(A1)-LEN(SUBSTITUTE(A1,",",""))
+1)), which in this example returns {1;2;3;4;5;6;7;8;9;10;11;12;13}. This is also the array used for RowArray in the simplified example.
So, this formula works great in Excel, but how could it be used in PowerPivot? For those using Excel 2013, and assuming that your column of delimited strings resides within a table in your DataModel,
you can use a DAX query to bring the column into Excel, add the formula demonstrated here to the Excel table, and link the table back into the DataModel. A comprehensive example of this can be found
So, for situations where solutions may not be possible in PowerPivot (or just incredibly complex), don’t forget about the power of Excel formulas.
BTW, an offshoot of the creation of this formula is another array formula that sums (or averages, whatever) the values in a delimited string:
=SUM(N(VALUE(TRIM(MID(SUBSTITUTE(A1,",",REPT(" ",999)) ,ROW(INDIRECT("1:"&LEN(A1)-LEN(SUBSTITUTE(A1,",",""))+1))*999-998,999)))))
A solution to this problem has been pursued in the Excel community for many years, so I am happy to present this formula here.
45 thoughts on “Calculating the Number of Unique Items in a Delimited String”
1. Very slick formulas. However, if you build this out as a UDF using VBA, it’s much simpler.
Just call Chip Pearson’s Quicksort routine, compare the buckets and ignore the dupes via iterator, done.
For the Sum or Average, just do a Split, and call the Worksheet functions SUM or AVERAGE.
I did something very similar recently to create a RankUnique function since Excel’s Rank produces duplicates.
2. Another way is to split the string into a range of cells, then apply a UDF from Erlandsen Consulting which treats the range as a collection, viz, “UniqueItem()”
3. M Simms, that’s the same story I heard back in 1994 when I first started creating complex Excel formulas.
I went on to make extremely complex functions with the xlm language and later with VBA, but I continued to
appreciate the challenge of solving a problem with a native Excel formula.
4. It would be interesting to see which approach performs better.
MSFT has done a lot to optimize formula calculations in the past 2 releases.
5. @M Simms,
Here is another method that seems to work which does not involve sorting…
Function UniqueCount(ByVal S As String) As Long
Dim X As Long, Parts() As String
S = “@” & Replace(S, “,”, “@,@”) & “@”
Parts = Split(S, “,”)
For X = 0 To UBound(Parts)
If UBound(Split(S, Parts(X))) > 0 Then
UniqueCount = UniqueCount + 1
S = Replace(S, Parts(X), “,”)
End If
End Function
6. Just repeating my code, but trying something to see if I can format it better (might work or it might not work)…
Function UniqueCount(ByVal S As String) As Long
Dim X As Long, Parts() As String
S = “@” & Replace(S, “,”, “@,@”) & “@”
Parts = Split(S, “,”)
For X = 0 To UBound(Parts)
If UBound(Split(S, Parts(X))) > 0 Then
UniqueCount = UniqueCount + 1
S = Replace(S, Parts(X), “,”)
End If
End Function
7. Cool! Here’s another formula approach:
Much longer than yours: 462 characters, as opposed to your 299.
8. This is such a great challenge, that I’ve posted it in the ‘Formula Challenges’ section at Chandoo’s forum. Will be interesting to see what other approaches are tried.
This particular challenge is posted at http://chandoo.org/forums/topic/formula-challenge-016-unique-items-in-a-delimited-string and there’s a whole bunch of other very tricky challenges listed
under http://chandoo.org/forums/forum/excel-challenges if anyone is interested in testing their formula mettle.
9. This array-entered formula is slightly shorter than yours (274 characters versus your 299) and appears to work correctly in Excel 2007 and above (the nesting levels are too large for earlier
versions), which is a limitation of both your formula and Jeff’s as well…
=SUM(1*ISERROR(-SEARCH(TRIM(MID(SUBSTITUTE(A1,”,”,REPT(” “,999)),ROW(INDIRECT(“1:”&LEN(A1)-LEN(SUBSTITUTE(A1,”,”,””))+1))*999-998,999)),TRIM(LEFT(SUBSTITUTE(TRIM(SUBSTITUTE(A1,” “,”@”)),”,”,REPT
(” “,999)),(ROW(INDIRECT(“1:”&LEN(A1)-LEN(SUBSTITUTE(A1,”,”,””))+1))-1)*999)))))
10. Follow up to my last message
I note that all the quote marks in my formula will need to be replaced with proper quote marks if you copy the formula directly from my message and paste it (anywhere).
Jeff… what code tags did you use to get that scrolling display field which retained the proper quote marks?
11. ***BUG ALERT**
Do not use my previously posted formula as it will improperly count item like “aaaa,a,aaaa,a”. The following modification to my formula seems to correct the problem, but it also balloons the
character count for the formula to 329. One positive my formula has over David’s formula, though, is that it returns 0 if A1 is empty whereas David’s returns 1. Anyway, for what its worth, here
is the formula…
=SUM(1*ISERROR(-SEARCH(“,”&TRIM(MID(SUBSTITUTE(SUBSTITUTE(A1,” “,”|”),”,”,REPT(” “,999)),ROW(INDIRECT(“1:”&LEN(A1)-LEN(SUBSTITUTE(A1,”,”,””))+1))*999-998,999))&”,”,”,”&SUBSTITUTE(TRIM(LEFT
(SUBSTITUTE(TRIM(SUBSTITUTE(A1,” “,”|”)),”,”,REPT(” “,999)),(ROW(INDIRECT(“1:”&LEN(A1)-LEN(SUBSTITUTE(A1,”,”,””))+1))-1)*999)),” “,”,”)&”,”)))
12. Hi Rick. I used the word CODE surrounded by lesser than and greater than symbols, with a backslash in the 2nd tag.
13. Hi Rick. Can you elaborate further the limitation that you allude to above re David and my formula?
14. Here’s my revised method for converting a delimited string into an array:
=MID(A1,FIND("|",SUBSTITUTE(","&A1,",","|",ROW(OFFSET(A1,,,LEN(A1))))),MMULT(IFERROR(FIND("|",SUBSTITUTE(CHOOSE({1,2},","&A1,A1&","),",","|",ROW(OFFSET(A1,,,LEN(A1))))),0), {-1;1}))
At 181 characters, it’s still longer than David’s 113 character method.
This gets my formula down to 413 characters. Still much longer than David’s 299 one.
15. Whoops, forgot to post my entire formula. And now there is a wall of my ugly mug pulling a funny face staring down at y’all.
=SUM(IFERROR(N(MATCH(MID(A1,FIND("|",SUBSTITUTE(","&A1,",","|",ROW(OFFSET(A1,,,LEN(A1))))),MMULT(IFERROR(FIND("|",SUBSTITUTE(CHOOSE({1,2},","&A1,A1&","),",","|",ROW(OFFSET(A1,,,LEN(A1))))),), {-1;1})),MID(A1,FIND("|",SUBSTITUTE(","&A1,",","|",ROW(OFFSET(A1,,,LEN(A1))))),MMULT(IFERROR(FIND("|",SUBSTITUTE(CHOOSE({1,2},","&A1,A1&","),",","|",ROW(OFFSET(A1,,,LEN(A1))))),), {-1;1})),)=ROW(OFFSET(A1,,,LEN(A1)))),0))
16. Two things… First, if A1 is blank, your formula returns a #VALUE! error and David’s returns 1. Second, all of our formulas fail to work in XL2003 (and I presume earlier). I believe the problem
stems from a limitation in XL2003 of seven levels of nesting; see…
for details; XL2007 and above can handle up to 64 levels of nesting.
17. @Rick
Function UniqueCount(ByVal S As String) As Long
sn = Split(Replace("@" & S & "@", ",", "@,@"), ",")
sn = Filter(sn, sn(0), False)
UniqueCount = UniqueCount + 1
Loop Until UBound(sn) = -1
End Function
18. I used to have a little blurb about how to post code, but it appears to be gone. I wonder what happened to that. Well, it looks like this
<code>formula goes here</code>
<code lang=”vb”>vba goes here</code>
I can put <code lang=”vb” inline=”true”>code inline</code> too.
19. @Dick – Thanks for the code tags (hope I remember them).
@snb – Clever use of False for the third argument to the Filter function (I really liked it).
20. In comparison to the complexity of the Excel formula I prefer VBA:
Function F_unique_snb(c00)
sn = Split(c00, ",")
With CreateObject("scripting.dictionary")
For j = 0 To UBound(sn)
c01 = .Item(sn(j))
F_unique_snb = .Count
End With
End Function
21. From http://msdn.microsoft.com/en-us/library/84k9x471%28v=vs.84%29.aspx
If key is not found when changing an item, a new key is created with the specified newitem. If key is not found when attempting to return an existing item, a new key is created and its
corresponding item is left empty”
Why did they design it that way? So there would never be an error?
22. 195 characters long!
TRIM(MID(SUBSTITUTE(A1&REPT("",8^5),",",REPT(" ",999)),ROW($1:$999)*999-998,999)),
TRIM(MID(SUBSTITUTE(A1&REPT("",8^5),",",REPT(" ",999)),ROW($1:$999)*999-998,999)),
23. Indeed, you are avoiding any errors if you stick to work with keys only.
It performs pretty well to generate a list of unique keys.
24. Rick, your formula is pure genius.
Unless I’m missing something, you can ditch SUBSTITUTE(A1,” “,”|”) for just A1, which will make your array-generation portion identical to Davids. And you can also ditch the minus sign in front
of the SEARCH.
Also, if this were a challenge to come up wit the shortest formula length, you could replace SEARCH with FIND and also replace this:
…with this:
Not that those changes make the resulting formula any better than your existing masterpiece. Just shorter for the sake of it. Down to 283 characters in fact.
=SUM(1*ISERROR(FIND(","&TRIM(MID(SUBSTITUTE(A1,",",REPT(" ",999)),ROW(OFFSET(A1,,,LEN(A1)-LEN(SUBSTITUTE(A1,",",""))+1))*999-998,999))&",",","&SUBSTITUTE(TRIM(LEFT(SUBSTITUTE(TRIM(A1),",",REPT(" ",999)),(ROW(OFFSET(A1,,,LEN(A1)-LEN(SUBSTITUTE(A1,",",""))+1))-1)*999))," ",",")&",")))
For readers following along at home, then given data in A1 like this:
…you first split the string into two arrays:
Array 1:
Array 2:
…where array2 is just a list of incrementally concatenated elements of the string with a placeholder at the front (which offsets arary2 by one position compared to array1).
That offsetting of array2 this means that for any given element n in array1, array2(n) contains everything in the original string up to that point. Or put another way, array2(n) = CONCATENATE
(“,,” , array1(1) , array 1(2) , … , array1(n -1) )
So when we search for array1(n) within array2(n), then if there’s a match, the thing we are looking for has obviously already occurred earlier in the string.
Pure genius.
Here’s how that looks graphically (assuming WordPress doesn’t mangle things):
Result array1 array2
#VALUE! ,aa, ,,
#VALUE! ,c, ,aa,
#VALUE! ,d, ,aa,c,
1 ,aa, ,aa,c,d,
#VALUE! ,bbb, ,aa,c,d,aa,
4 ,c, ,aa,c,d,aa,bbb,
6 ,d, ,aa,c,d,aa,bbb,c,
#VALUE! ,e, ,aa,c,d,aa,bbb,c,d,
#VALUE! ,f, ,aa,c,d,aa,bbb,c,d,e,
#VALUE! ,g, ,aa,c,d,aa,bbb,c,d,e,f,
#VALUE! ,h, ,aa,c,d,aa,bbb,c,d,e,f,g,
#VALUE! ,I, ,aa,c,d,aa,bbb,c,d,e,f,g,h,
#VALUE! ,jjjjjjjj, ,aa,c,d,aa,bbb,c,d,e,f,g,h,I,
1 ,aa, ,aa,c,d,aa,bbb,c,d,e,f,g,h,I,jjjjjjjj,
25. Damn, I forgot the code tags on my 2nd array. Dick, would you kindly add code tags to Array2:
And also, is there a way to paste tablular data here, so that it shows in columns?
26. I was about to suggest a udf using the scripting dictionary, but I see snb has beaten me to it, and probably a good bit shorter than mine would have been.
Nice work.
27. You can use html in the comments box, so table tags for tabular data. I recommend joinrange with the htmltable argument.
│Result │array1 │array2 │
│#VALUE!│,aa, │,, │
│#VALUE!│,c, │,aa, │
│#VALUE!│,d, │,aa,c, │
│1 │,aa, │,aa,c,d, │
│#VALUE!│,bbb, │,aa,c,d,aa, │
│4 │,c, │,aa,c,d,aa,bbb, │
│6 │,d, │,aa,c,d,aa,bbb,c, │
│#VALUE!│,e, │,aa,c,d,aa,bbb,c,d, │
│#VALUE!│,f, │,aa,c,d,aa,bbb,c,d,e, │
│#VALUE!│,g, │,aa,c,d,aa,bbb,c,d,e,f, │
│#VALUE!│,h, │,aa,c,d,aa,bbb,c,d,e,f,g, │
│#VALUE!│,I, │,aa,c,d,aa,bbb,c,d,e,f,g,h, │
│#VALUE!│,jjjjjjjj,│,aa,c,d,aa,bbb,c,d,e,f,g,h,I, │
│1 │,aa, │,aa,c,d,aa,bbb,c,d,e,f,g,h,I,jjjjjjjj, │
28. snb – Now that is slick and elegant. Hardly any code at all.
29. You could reduce the code to:
Function F_unique_snb(c00)
With CreateObject("scripting.dictionary")
For each it in Split(c00, ",")
c01 = .Item(it)
F_unique_snb = .Count
End With
End Function
30. I think we to add a trim to the scripting dictionary version:
c01 = .Item(Trim(it))
32. Nice formula! There is a slight limitation because the maximum string length is harcoded to 999, which restricts the number of delimited values to around 66 in my tests. To work around this I
would suggest replacing 999 by LEN(A1) and 998 by (LEN(A1)-1). An alternative that also seems to work and is shorter (175 chars) – but probably no clearer :)
Agree with others that udfs are probably a better option here although maybe a bit slower than the suggested formulas for the sample string given in the post. In some basic optimisation tests, i
found that the scripting dictionary udf could be made around twice as fast if references were included and would be quicker still using a collection in place of dictionary with “on error resume
next” making it faster than the worksheet formulas as well as non-volatile. Would be nice to do a more detailed speed comparison given the time…
33. Trying above formula again:
34. to digress somewhat more:
to return the list of unique items (not only the number of unique items):
Function F_uniquelist_snb(c00)
With CreateObject("scripting.dictionary")
For Each it In Split(c00, ",")
c01 = .Item(it)
F_uniquelist_snb = Join(.keys, ",")
End With
End Function
To return a sorted list of unique items:
Function F_uniquesortedlist_snb(c00)
With CreateObject("System.Collections.arraylist")
For Each it In Split(c00, ",")
If Not .contains(it) Then .Add it
F_uniquesortedlist_snb = Join(.toarray, ",")
End With
End Function
35. Another formula approach. For readability, I have used S for the separator e.g. “,”
For example, for the string: a,b,a,c
Step 1: S&MID(A1,FIND(S,S&A1,ROW(1:999)),FIND(S,A1&S,ROW(1:999))-FIND(S,S&A1,ROW(1:999)))&S splits out the four comma-wrapped components “,a,” | “,b,” | “,a,” and “,c,”
Step 2: Find each of these components in the comma-wrapped string “,a,b,a,c,” and count 1 if they’re found in their actual position. The second “,a,” component is located at position 5, but found
at position 1, therefore must be a duplicate.
It appears to work for all well-defined strings, including spaces, e.g. it returns 4 for “a,, ,b” which I’d argue is correct. But if A1 is blank, it returns 1.
37. Stephen, that’s nice too. It’s pretty much the same as mine which uses “SUMPRODUCT(- -(” instead of “COUNT(1/SIGN(” and so gives an error for blanks. The only other differences are using a
variable length array (which does make it many times faster to recalc the test string at the expense of being volatile) and MMULT to reduce function calls.
38. That’s awesome, Lori. I’m yet to fully digest it, but noted that you can make it shorter still by replacing any instances of this:
…with this:
…leaving you this 169 character puppy:
39. Stephen, that’s a fantastic approach. Note that you can make it slightly shorter by removing the S from A1&S and instead adding 1 to the final result. Plus you don’t need the SIGN:
What’s cool about it is that it can be completely dynamic in terms of string length:
40. If we combine Stephen’s approach with Loris, then we get this masterpiece that handles any length text string up to the cell limit, in 179 characters:
43. See https://dhexcel1.wordpress.com/2017/01/03/creating-a-unique-delimited-string-from-a-delimited-string-excel-formula-method-by-david-hager/ for an extension of the concepts in this article.
Posting code? Use <pre> tags for VBA and <code> tags for inline. | {"url":"http://dailydoseofexcel.com/archives/2013/08/07/calculating-the-number-of-unique-items-in-a-delimited-string/","timestamp":"2024-11-06T08:48:25Z","content_type":"text/html","content_length":"143501","record_id":"<urn:uuid:df8b34d9-b2f2-48b8-86c6-d78b07450aa4>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00712.warc.gz"} |
Directional Derivative Of Vector | Maths For Engineers - 1 | Books | Skedbooks
Directional Derivative Of Vector
Directional Derivative of Vector:
The derivative of a point function (scalar or vector) in a particular direction is called its directional derivative along the direction. The directional derivative of a scalar point function Φ in a
given direction is the rate of change of Φ in the direction. It is given by the component of grad Φ in that direction.
The directional derivatives of a scaler point function Φ (x,y,z) in the direction of $abla \!\,$Φ. Hence the maximum directional derivatives is
Unit Normal Vector to the Surface:
If Φ (x,y,z) be the scaler function, then Φ (x,y,z) = c represents. A surface and the unit normal vector to the surface Φ is given by .
Equation of the tangent plane and normal to the surface:
The Cartesian form of the normal at (xo,yo,zo) on the surface of the Φ (x,y,z) = c is;
Hence this is the basic Directional Derivative of Vector. | {"url":"https://skedbooks.com/books/maths-for-engineers-1/directional-derivative-of-vector/","timestamp":"2024-11-06T23:36:06Z","content_type":"text/html","content_length":"94540","record_id":"<urn:uuid:f4d63e71-b94f-4408-a110-0b3e38fd54f0>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00410.warc.gz"} |
- ON-SITE INSIGHTS - WHEN STUDENTS ASK, "WHY DO WE NEED TO KNOW THIS? WHEN WILL I EVER USE THIS?" - Teachers.Net Gazette - TEACHER APPRECIATION - featuring columns and articles by top names in education and your teacher colleagues around the world! The Teachers.Net Gazette is a clearinghouse for teacher writing, from well-reasoned education essays to teacher prose, poems and humor! Bookmark the Teachers.Net Gazette and tell a friend!
On-Site Insights...
When Students ask, "Why Do We Need to Know This? When Will I Ever Use This?"
from: The Middle School Chatboard
Posted by Sci Teach:
How do you respond to questions such as "Why do we need to know this?" I always give the most stupid answers to these questions because I honestly don't know.
One thing I say is, "To broaden your experience and to know if you enjoy something or not so that you can make wise college and career decisions." Unfortunately, middle schoolers aren't looking that
far in advance and that response means nothing to them.
Currently we are studying Astronomy in 6th grade. What kind of response would you give a kid who asked "Why do we need to know about planets?" Other than state standards and the possibility that they
may someday read space.com and want to understand it, I don't know. Before I taught science I never needed to know much about the planets.
Some responses:
Posted by middle school teacher:
I admire your point of view. Many teachers do not question the standards. The standards as written in every state represent a compromise between the many, many people who had input on them and I see
many things in our state's standards and my subject's national standards that are just plain silly.
I find though that if I teach the topic in a creative enough way, I don't get those questions. The planets can be fascinating. It also helps to make it a short unit. I tend to spend more time on
those topics that are engaging of the interest of middle schoolers and less on those that clearly don't work for them. It's also true that I teach best what is fascinating to me rather than what
popped into the minds of the standards writers.
I can answer students honestly - these things are in the standards. In this state, it's thought you need to know this. Now let's get back to work and we'll master this as efficiently as we can.
Posted by Ambra:
It really helps to know why you are teaching it before you begin the lesson. In your planning stages, make it part of your routine to come up with a purpose statement. Brainstorm and think of all of
the ways the information you are presenting can come in useful in life. What sorts of skills do the students practice when learning this topic? If all else fails, you can usually respond with,
"Learning about our planets will help you to be an educated, informed member of society." Some of the most amazing discoveries have come about by people questioning what was taught to them.
Posted by DET:
Why do I need algebra?
Algebra is a critical "gateway" subject for many reasons. First of all, algebra is the gateway to all the higher maths: geometry, algebra II, trigonometry, analytic geometry, calculus and beyond.
Since all sciences (including biology, chemistry, physics, astronomy, engineering, computer science, architecture, design, many social sciences, economics, finance, even flying an airplane!) depend
on algebra and higher math, learning algebra is essential for anyone considering working in these fields.
Aside from that, learning the abstract reasoning skills that algebra teaches helps students become better abstract reasoners in general. Good abstract reasoning skills improve a student's ability to
write a coherent essay, for example, since essays require the writer to shift back and forth between abstract concepts and specific supporting facts. Many life skills, including choosing a career,
making major purchases, running a business, and managing a family also require reasoning skills that are improved by math study.
Also, California schools now require that students pass an exit exam in order to graduate with a diploma from high school. The math portion of this exam relies heavily on algebra concepts learned in
middle school and high school.
In addition, success in algebra correlates highly with success in higher education. Algebra and further math are critical to a student's chance of attending university. This was well documented in a
1990 study by the College Board. In this study, researchers found that students who take a year of algebra and follow that with a year of geometry nearly double their chances of going to college --
by doing that alone!
Beyond this direct correlation, students should be aware that the two college entrance exams, the SAT and the ACT, are loaded with algebra I questions. It is impossible to get a decent score on these
exams' math sections without a solid grasp of algebra.
This is why we study algebra.
P.S. And I think it's fun! | {"url":"https://www.teachers.net/gazette/MAY02/insight.html","timestamp":"2024-11-15T03:25:52Z","content_type":"text/html","content_length":"41380","record_id":"<urn:uuid:481acf14-e86d-4cd6-8d79-bbe6aded9b33>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00764.warc.gz"} |
Active Control of Vibration
• 1st Edition - February 8, 1996
• Authors: Christopher C. Fuller, Sharon Elliott, P. A. Nelson
• Paperback ISBN:
9 7 8 - 0 - 1 2 - 3 8 8 4 6 6 - 4
• eBook ISBN:
9 7 8 - 0 - 0 8 - 0 5 2 5 9 1 - 4
This book is a companion text to Active Control of Sound by P.A. Nelson and S.J. Elliott, also published by Academic Press. It summarizes the principles underlying active vibrat… Read more
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Immediately download your ebook while waiting for your print delivery. No promo code needed.
This book is a companion text to Active Control of Sound by P.A. Nelson and S.J. Elliott, also published by Academic Press. It summarizes the principles underlying active vibration control and its
practical applications by combining material from vibrations, mechanics, signal processing, acoustics, and control theory. The emphasis of the book is on the active control of waves in structures,
the active isolation of vibrations, the use of distributed strain actuators and sensors, and the active control of structurally radiated sound. The feedforward control of deterministic disturbances,
the active control of structural waves and the active isolation of vibrations are covered in detail, as well as the more conventional work on modal feedback. The principles of the transducers used as
actuateors and sensors for such control strategies are also given an in-depth description. The reader will find particularly interesting the two chapters on the active control of sound radiation from
structures: active structural acoustic control. The reason for controlling high frequency vibration is often to prevent sound radiation, and the principles and practical application of such
techniques are presented here for both plates and cylinders. The volume is written in textbook style and is aimed at students, practicing engineers, and researchers.
• Combines material from vibrations, signal processing, mechanics, and controls
• Summarizes new research in the field
Graduate students, researchers, and engineers in acoustics, aeronautical, aerospace, mechanical, and control engineering; applied physicists
Introduction to Mechanical Vibrations: Terminology. Single-degree-of-freedom (SDOF) Systems. Free Motion of SDOF Systems. Damped Motion of SDOF Systems. Forced Response of SDOF Systems. Transient
Response of SDOF Systems.Multi-degree-of-freedom (MDOF) Systems. Free Motion of MDOF Systems. Forced Response of MDOF Systems. Damped Motion of MDOF Systems. Finite Element Analysis of Vibrating
Mechanical Systems. Introduction to Waves in Structures: Longitudinal Waves. Flexural Waves. Flexural Response of an Infinite Beam to an Oscillating Point Force. Flexural Wave Power Flow. Flexural
Response of an Infinite Thin Beam to an Oscillating Line Moment. Free Flexural Motion of Finite Thin Beams. Response of a Finite Thin Beam to an Arbitrary Oscillating Force Distribution. Vibration of
Thin Plates. Free Vibration of Thin Plates. Response of a Thin Rectangular Simply Supported Plate to an Arbitrary Oscillating Force Distribution. Vibration of Infinite Thin Cylinders. Free Vibration
of Finite Thin Cylinders. Harmonic Forced Vibration of Infinite Thin Cylinders. Feedback Control: Single-channel Feedback Control. Stability of a Single-Channel System. Modification of the Response
of an SDOF System. The Effect of Delays inthe Feedback Loop. The State Variable Approach. Example of a Two-degree-of-freedom System. Output Feedback and State Feedback. State Estimation and
Observers. Optimal Control. Modal Control. Feedforward Control: Single Channel Feedforward Control. The Effect of Measurement Noise. Adaptive Digital Controllers. Multichannel Feedforward Control.
Adaptive Frequency Domain Controllers. Adaptive Time Domain Controllers. Equivalent Feedback Controller Interpretation. Distributed Transducers for Active Control of Vibration. Active Control of
Vibration in Structures: Feedforward Control of Finite Structures. Feedback Control of Finite Structures. Feedforward Control of Wave Transmission. Actuator Arrays for Control of Flexural Waves.
Sensor Arrays for Control of Flexural Waves. Feedforward Control of Flexural Waves. Feedback Control of Flexural Waves. Active Isolation of Vibrations: Isolation of Periodic Vibrations of an SDOF
System. Vibration Isolation From a Flexible Receiver; the Effects ofSecondary Force Location. Active Isolation of Periodic Vibrations Using Multiple Secondary Force Inputs. Finite Element Analysis of
an Active System for the Isolation of Periodic Vibrations. Practical Examples of Multi-Channel Feedforward Control for theIsolation of Periodic Vibrations. Isolation of Unpredictable Vibrations from
a Receiving Structure. Isolation of Vibrating Systems from Random External Excitation; the Possibilities for Feedforward Control. Isolation of Vibrating Systems from Random External Excitation;
Analysis of Feedback Control Strategies. Isolation of Vibrating Systems from Random External Excitation; Formulation in Terms of Modern Control Theory. Active Isolation of Vehicle Vibrations from
Road and Track Irregularities. Active Structural Acoustic Control, I. Plate Systems: Sound Radiation by Planar Vibrating Surfaces; the Rayleigh Integral. The Calculation of Radiated Sound Fields by
Using Wavenumber Fourier Transforms. Sound Power Radiation From Structures in Terms of TheirMulti-Modal Response. General Analysis of Active Structural Acoustic Control (ASAC) for Plate Systems.
Active Control of Sound Transmission Through a Rectangular Plate Using Point Force Actuators. Active Control of Structurally Radiated Sound Using Multiple Piezoelectric Actuator; Interpretation of
Behaviour in Terms of the Spatial Wavenumber Spectrum. The Use of Piezoelectric Distributed Structural Error Sensors in ASAC. An Example of the Implementation of Feedforward ASAC. Feeback Control of
Sound Radiation From a Vibrating Baffled Piston. Feedback Control of Sound Radiation From Distributed Elastic Structures. Active Structural Acoustic Control, II. Cylinder Systems: Coupled Cylinder
Acoustic Fields. Response of an Infinite Cylinder to a HarmonicForcing Function. Active Control of Cylinder Interior Acoustic Fields Using Point Forces. Active Control of Vibration and Acoustic
Transmission in Fluid-Filled Piping Systems. Active Control of Sound Radiation From Vibrating Cylinders. Active Control ofSound in Finite Cylinder Systems. Control of Interior Noise in a Full Scale
Jet Aircraft Fuselage. Appendix. References. Index.
• Published: February 8, 1996
• Paperback ISBN: 9780123884664
• eBook ISBN: 9780080525914 | {"url":"https://shop.elsevier.com/books/active-control-of-vibration/fuller/978-0-12-269440-0","timestamp":"2024-11-04T21:49:13Z","content_type":"text/html","content_length":"187625","record_id":"<urn:uuid:6db3b5ab-e7c4-42e4-b5f5-be03e90efe77>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00635.warc.gz"} |
D in cases as well as in controls. In case of | http://www.ck2inhibitor.com
D in situations also as in controls. In case of an interaction effect, the distribution in circumstances will tend toward constructive cumulative risk scores, whereas it will tend toward negative
cumulative MedChemExpress ENMD-2076 danger scores in controls. Therefore, a sample is classified as a pnas.1602641113 case if it includes a good cumulative threat score and as a handle if it has a
negative cumulative risk score. Primarily based on this classification, the coaching and PE can beli ?Additional approachesIn addition towards the GMDR, other techniques have been suggested that deal
with limitations of your original MDR to classify multifactor cells into high and low risk under specific circumstances. Robust MDR The Robust MDR extension (RMDR), proposed by Gui et al. [39],
addresses the circumstance with sparse and even empty cells and those having a case-control ratio equal or close to T. These circumstances result in a BA close to 0:5 in these cells, negatively
influencing the general fitting. The option proposed may be the introduction of a third threat group, called `unknown risk’, which is excluded from the BA calculation of the single model. Fisher’s
exact test is utilized to assign every cell to a BU-4061T web corresponding danger group: When the P-value is higher than a, it is actually labeled as `unknown risk’. Otherwise, the cell is labeled
as high danger or low danger based on the relative variety of circumstances and controls within the cell. Leaving out samples in the cells of unknown risk may well cause a biased BA, so the authors
propose to adjust the BA by the ratio of samples within the high- and low-risk groups to the total sample size. The other aspects from the original MDR method remain unchanged. Log-linear model MDR A
further approach to deal with empty or sparse cells is proposed by Lee et al. [40] and referred to as log-linear models MDR (LM-MDR). Their modification uses LM to reclassify the cells of the
greatest combination of elements, obtained as in the classical MDR. All attainable parsimonious LM are fit and compared by the goodness-of-fit test statistic. The anticipated quantity of instances
and controls per cell are provided by maximum likelihood estimates in the selected LM. The final classification of cells into high and low danger is primarily based on these expected numbers. The
original MDR is really a specific case of LM-MDR in the event the saturated LM is chosen as fallback if no parsimonious LM fits the information enough. Odds ratio MDR The naive Bayes classifier
employed by the original MDR system is ?replaced within the operate of Chung et al. [41] by the odds ratio (OR) of each multi-locus genotype to classify the corresponding cell as high or low risk.
Accordingly, their technique is called Odds Ratio MDR (OR-MDR). Their method addresses three drawbacks with the original MDR approach. 1st, the original MDR system is prone to false classifications
in the event the ratio of situations to controls is comparable to that in the complete information set or the amount of samples inside a cell is small. Second, the binary classification on the
original MDR process drops data about how effectively low or high risk is characterized. From this follows, third, that it is actually not attainable to determine genotype combinations using the
highest or lowest danger, which may possibly be of interest in practical applications. The n1 j ^ authors propose to estimate the OR of each cell by h j ?n n1 . If0j n^ j exceeds a threshold T, the
corresponding cell is labeled journal.pone.0169185 as h high threat, otherwise as low danger. If T ?1, MDR is a unique case of ^ OR-MDR. Based on h j , the multi-locus genotypes might be ordered from
highest to lowest OR. Furthermore, cell-specific self-assurance intervals for ^ j.D in cases also as in controls. In case of an interaction effect, the distribution in circumstances will tend toward
positive cumulative risk scores, whereas it’s going to tend toward adverse cumulative risk scores in controls. Therefore, a sample is classified as a pnas.1602641113 case if it includes a good
cumulative risk score and as a control if it features a adverse cumulative threat score. Primarily based on this classification, the education and PE can beli ?Additional approachesIn addition
towards the GMDR, other approaches were suggested that deal with limitations with the original MDR to classify multifactor cells into higher and low threat beneath certain circumstances. Robust MDR
The Robust MDR extension (RMDR), proposed by Gui et al. [39], addresses the scenario with sparse or even empty cells and these using a case-control ratio equal or close to T. These conditions lead to
a BA close to 0:five in these cells, negatively influencing the overall fitting. The resolution proposed is definitely the introduction of a third danger group, known as `unknown risk’, which can be
excluded from the BA calculation from the single model. Fisher’s precise test is employed to assign every single cell to a corresponding danger group: When the P-value is greater than a, it truly is
labeled as `unknown risk’. Otherwise, the cell is labeled as higher threat or low risk depending on the relative variety of instances and controls in the cell. Leaving out samples in the cells of
unknown danger may possibly cause a biased BA, so the authors propose to adjust the BA by the ratio of samples within the high- and low-risk groups to the total sample size. The other aspects with
the original MDR strategy remain unchanged. Log-linear model MDR An additional strategy to cope with empty or sparse cells is proposed by Lee et al. [40] and known as log-linear models MDR (LM-MDR).
Their modification utilizes LM to reclassify the cells from the most effective combination of factors, obtained as in the classical MDR. All feasible parsimonious LM are match and compared by the
goodness-of-fit test statistic. The anticipated variety of instances and controls per cell are offered by maximum likelihood estimates with the selected LM. The final classification of cells into
high and low risk is primarily based on these expected numbers. The original MDR is often a particular case of LM-MDR if the saturated LM is chosen as fallback if no parsimonious LM fits the data
adequate. Odds ratio MDR The naive Bayes classifier made use of by the original MDR method is ?replaced in the work of Chung et al. [41] by the odds ratio (OR) of every single multi-locus genotype to
classify the corresponding cell as high or low danger. Accordingly, their system is called Odds Ratio MDR (OR-MDR). Their method addresses 3 drawbacks of the original MDR strategy. First, the
original MDR strategy is prone to false classifications in the event the ratio of instances to controls is related to that in the entire data set or the number of samples inside a cell is little.
Second, the binary classification in the original MDR strategy drops details about how effectively low or higher risk is characterized. From this follows, third, that it really is not achievable to
determine genotype combinations with the highest or lowest threat, which may possibly be of interest in sensible applications. The n1 j ^ authors propose to estimate the OR of each cell by h j ?n n1
. If0j n^ j exceeds a threshold T, the corresponding cell is labeled journal.pone.0169185 as h high threat, otherwise as low threat. If T ?1, MDR can be a unique case of ^ OR-MDR. Based on h j , the
multi-locus genotypes may be ordered from highest to lowest OR. Additionally, cell-specific confidence intervals for ^ j. | {"url":"https://www.ck2inhibitor.com/2017/10/19/d-in-cases-as-well-as-in-controls-in-case-of/","timestamp":"2024-11-10T11:56:53Z","content_type":"text/html","content_length":"62864","record_id":"<urn:uuid:d3ec06c6-b9f4-4d17-8bab-64bca17d2cbe>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00481.warc.gz"} |
Ordinal Numbers Grammar Exercises - OrdinalNumbers.com
Ordinal Numbers Grammar – By using ordinal numbers, you can count any number of sets. They can also be used as a generalization of ordinal quantities. 1st The basic concept of mathematics is the
ordinal. It is a number indicating the place of an object in a list. In general, a number between one and … Read more | {"url":"https://www.ordinalnumbers.com/tag/ordinal-numbers-grammar-exercises/","timestamp":"2024-11-02T12:39:16Z","content_type":"text/html","content_length":"47050","record_id":"<urn:uuid:d2f9783b-504f-4f9f-9c7c-6fa22f82a1e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00706.warc.gz"} |
Area of a Running track
$A \,=\, 2lw+\pi(R^2-r^2)$
The area of running track can be expressed in mathematical form and it can be used as a formula in mathematics to find the area of any running track field. It can be derived easily in mathematical
form when you understand the geometrical science behind the construction of a running track.
Geometrically, the shape of a running track is the combination of both rectangle and annulus. So, you first have to learn how to find the area of a rectangle and the area of a circular ring.
Now, let’s denote the various dimensions of running track field algebraically as follows.
1. Denote the length of straight track lane by a letter $l$.
2. Represent the width of straight track lane by a letter $w$.
3. Denote the radius of the outer circular lane by a letter $R$.
4. Represent the radius of the inner circular lane by a letter $r$.
5. Denote the area of running track by a letter $A$.
The area of a running track is equal to the sum of two times the area of rectangular lane and the area of annulus lane.
$A \,=\, 2lw+\pi(R^2-r^2)$
Learn how to derive a formula in algebraic form to find the area of a running track in mathematics.
A worksheet of list of math questions on finding the areas of running track fields for your practice with solutions to learn how to find the area of any running track mathematically. | {"url":"https://www.mathdoubts.com/area-of-running-track/","timestamp":"2024-11-02T14:21:53Z","content_type":"text/html","content_length":"29049","record_id":"<urn:uuid:f2bdb3b5-e972-423b-ae34-1bffc950e7c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00066.warc.gz"} |
[Solved] What is the Derivative of 10^x [10 to the x]? - iMath
[Solved] What is the Derivative of 10^x [10 to the x]?
The derivative of 10^x is equal to 10^x ln 10. Here ln denotes the logarithm with base e, called the natural logarithm. In this post, we will learn to compute the derivative of 10 raised to x.
Derivative of 10^x
Question: What is the derivative of 10^x?
Answer: The derivative of 10^x is 10^x ln 10.
To find the derivative of 10 to the x, we will use the logarithmic differentiation. Let us put
We need to find $\dfrac{dy}{dx}$. Taking logarithm on both sides with base 10, we get that
$\log_{10} y =\log_{10} 10^x$
⇒ $\log_{10} y =x$ as we know that log[a] a^k=k.
Differentiating both sides with respect to x, we obtain that
$\dfrac{d}{dx}(\log_{10} y)=\dfrac{dx}{dx}$
⇒ $\dfrac{1}{\log_e 10} \dfrac{1}{y} \dfrac{dy}{dx}=1$
⇒ $\dfrac{dy}{dx}=y \cdot \log_e 10$
⇒ $\dfrac{dy}{dx}=10^x \cdot \ln 10$ as y=10^x and log[e]=ln
Thus, the derivative of 10^x is 10^x ln 10.
Question-Answer on Derivative of 10^x
Question: What is the derivative of 10^x at x=0?
From the above, we have that the derivative of 10^x is 10^x ln 10. That is,
$\dfrac{d}{dx}(10^x)=[10^x \ln 10]_{x=0}$
= 10^0 ln 10
= 1 × ln 10 as we know that a^0=1 for any non-zero number a.
= ln 10
So the derivative of 10 raised to x at x=0 is equal to ln10.
Also Read:
Q1: If y=10^x, then find dy/dx?
Answer: As the derivative of 10^x is 10^x ln 10, we have that dy/dx=10^x ln 10. | {"url":"https://www.imathist.com/derivative-of-10-x/","timestamp":"2024-11-09T17:38:10Z","content_type":"text/html","content_length":"178826","record_id":"<urn:uuid:b582e61d-e59f-41a2-9810-a9d4dc7a1ff9>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00889.warc.gz"} |
Fundamental Analysis of KERING(EPA:KER) stock
We assign a fundamental rating of 4 out of 10 to KER. KER was compared to 40 industry peers in the Textiles, Apparel & Luxury Goods industry. While KER is still in line with the averages on
profitability rating, there are concerns on its financial health. KER is valued correctly, but it does not seem to be growing. KER also has an excellent dividend rating. | {"url":"https://www.chartmill.com/stock/quote/KER.PA/fundamental-analysis","timestamp":"2024-11-11T03:32:26Z","content_type":"text/html","content_length":"720347","record_id":"<urn:uuid:55c4efd0-1eff-4a3e-8b1f-1848791d8086>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00671.warc.gz"} |
Working with Squares in Algebra
• Factoring is the opposite of multiplying or expanding. In this lesson, learn the ins and outs of factoring in algebra, and get tips on how to check your work.
Factoring is the opposite of multiplying or expanding. In this lesson, learn the ins and outs of factoring in algebra, and get tips on how to check your work.
• Continue on your algebra journey, and learn to factor the trinomial x^2 + bx + c, in order to write out the trinomial as the product of two binomials.
Continue on your algebra journey, and learn to factor the trinomial x^2 + bx + c, in order to write out the trinomial as the product of two binomials.
• Factoring is the opposite of expanding, and is covered in this algebra lesson! Learn to factor ax2 + bx + c (the general quadratic trinomial).
Factoring is the opposite of expanding, and is covered in this algebra lesson! Learn to factor ax2 + bx + c (the general quadratic trinomial).
• In this algebra lesson, review how to factor and expand. Learn how to factor the special products of Perfect Squares and the Difference of Two Squares.
In this algebra lesson, review how to factor and expand. Learn how to factor the special products of Perfect Squares and the Difference of Two Squares.
• Get ready to factor the difference and sum of two cubes! Review how to take the factored form and multiply it, and find a common factor with imperfect cubes.
Get ready to factor the difference and sum of two cubes! Review how to take the factored form and multiply it, and find a common factor with imperfect cubes.
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Digital logic gate circuits are manufactured as integrated circuits: all the constituent transistors and resistors built on a single piece of semiconductor material. The engineer, technician, or
hobbyist using small numbers of gates will likely find what he or she needs enclosed in a DIP (Dual Inline Package) housing. DIP-enclosed integrated circuits are available with even numbers of pins,
located at 0.100 inch intervals from each other for standard circuit board layout compatibility. Pin counts of 8, 14, 16, 18, and 24 are common for DIP "chips."
Part numbers given to these DIP packages specify what type of gates are enclosed, and how many.
These part numbers are industry standards, meaning that a "74LS02" manufactured by Motorola will be identical in function to a "74LS02" manufactured by Fairchild or by any other manufacturer.
Letter codes prepended to the part number are unique to the manufacturer, and are not industry standard codes. For instance, a SN74LS02 is a quad 2-input TTL NOR gate manufactured by Motorola, while
a DM74LS02 is the exact same circuit manufactured by Fairchild.
Logic circuit part numbers beginning with "74" are commercial-grade TTL. If the part number begins with the number "54", the chip is a military-grade unit: having a greater operating temperature
range, and typically more robust in regard to allowable power supply and signal voltage levels. The letters "LS" immediately following the 74/54 prefix indicate "Low-power Schottky"circuitry, using
Schottky-barrier diodes and transistors throughout, to decrease power dissipation.
Non-Schottky gate circuits consume more power, but are able to operate at higher frequencies due
to their faster switching times.
A few of the more common TTL "DIP" circuit packages are shown here for reference:
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Writer : delon
27 Nov 2006 Mon  
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Kolmogorov Complexity and the Primes
on how to derive the non-finiteness of the primes from Van der Waerden's theorem reminds me of a nice proof using Kolmogorov complexity.
A quick primer: Fixed some universal programming language. Let C(x), the Kolmogorov complexity of x, be the length of the smallest program that outputs x. One can show by a simple counting argument
for every n there is an x such that C(x) ≥ n. We call such x "random".
Suppose we had a finite list of primes p
. Then any number m can be expressed as p
. Pick n large, a random x of length n and let m be the number x expresses in binary. We can compute m from e
and a constant amount of other information, remembering that k is a constant. Each e
is at most log m and so we can describe all of them in O(log log m) bits and thus C(m) = O(log log m). But roughly C(m) = C(x) ≥ n = log m, a contradiction.
But we can do better. Again pick n large, a random x of length n and let m be the number x expresses in binary. Let p
be the largest prime that divides m where p
is the ith prime. We can describe m by p
and m/p
, or by i and m/p
. So we have C(m) ≤ C(i,m/p
) ≤ C(i) + C(m/p
) + 2 log C(p
) ≤ log i + log m/p
+ 2 log log i + c. The 2 log C(p
) term is needed to specify the separation between the program for i and the program for m/p
Since C(m) ≥ log m, we have
log m ≤ log i + log (m/p
)+ 2 log log i + c
log m ≤ log i + log m - log p
+ 2 log log i + c
log p
≤ log i + 2 log log i + c
≤ O(i (log i)
The prime number theorem has p
approximately i log i, so we get just a log factor off from optimal with simple Kolmogorov complexity.
I wrote a
short introduction
to Kolmogorov complexity with this proof. I originally got the proof from the
great text
on Kolmogorov complexity from Li and Vitányi and they give credit to Piotr Berman and John Tromp.
5 comments:
1. It should be mentioned that the bound can be improved by a better encoding, like first writing down the length of C(pi), so we get log i + log m/pi + log log i + 2 log log log i + c, and I think
pi<=O(i log i (log log i)^2) follows. Of course this can be further improved.
2. I like the first simpler proof better because Chebyshev used something almost as simple as the second proof that gets within a constant factor of the right density. The idea is to look at the
prime factorization of N=(2n choose n) = (2n)!/(n!)^2.
Consider any prime p and the exponent of p in m!. We get [m/p] terms divisible by p and [m/p^2] divisible by p^2, etc so the exponent of p in m! is just
[m/p]+[m/p^2]+[m/p^3]+... where [ ] is the floor function and we can go on forever since the terms become 0 eventually.
Therefore the exponent of p dividing (2n choose n) is precisely
[2n/p]+[2n/p^2}+[2n/[p^3] +...
- 2[n/p]-2[n/p^2}- 2[n/p^3] -...
= [2n/p]-2[n/p] + [2n/p^2]-2[n/p^2] +...+ [2n/p^k]-2[n/p^k]
where k is the largest power of p smaller than 2n.
Now for any number m, 2[n/m] > 2(n/m -1) = 2n/m - 2 so
[2n/m]-2[n/m] = 1 since it is <= 2n/m - 2[n/m] < 2.
Therefore the exponent of p in (2n choose n) is at most k and hence it contributes a factor at most p^k <= 2n to
N=(2n choose n).
Trivially N=(2n)!/(n!)^2 is at least 2^n. (I want to keep this elementary so I am willing to lose the near factor of 2 in the exponent.) Each prime dividing N must be at most 2n and its
contribution to the product N is at most 2n by the above argument. Therefore the # of primes smaller than 2n must be at least log_{2n} N > n/ log_2 (2n) which gives the right asymptotics.
3. As said, this shows only the inequality for p_i that are factors in random numbers. The previous argument shows that this is true for infinitely many i, but to get this for all large i something
else is needed (as Alexey Milovanov, if I remember correctly, pointed out)
4. There are many proofs, but how many of them are essentially the same, or use the same ideas? For example, Keith Conrad argued in his write-up that Furstenberg's proof is based on the same idea as
Euclid's proof: https://kconrad.math.uconn.edu/blurbs/ugradnumthy/primestopology.pdf
5. I think this style of reasoning can field much better results. I see how it can go further, but I have made a mistake somewhere in the proof; Is this the place to discuss this, perhaps in a new | {"url":"https://blog.computationalcomplexity.org/2017/11/kolmogorov-complexity-and-primes.html?m=0","timestamp":"2024-11-02T23:22:41Z","content_type":"application/xhtml+xml","content_length":"183921","record_id":"<urn:uuid:5a174148-50a7-48a0-983b-4cd9816f152a>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00318.warc.gz"} |
Solving Equations With Variables On Both Sides With Fractions Worksheet - Equations Worksheets
Solving Equations With Variables On Both Sides With Fractions Worksheet
Solving Equations With Variables On Both Sides With Fractions Worksheet – The goal of Expressions and Equations Worksheets is to assist your child in learning more effectively and efficiently. The
worksheets include interactive exercises and questions based on the sequence in which operations are performed. With these worksheets, kids can grasp both simple and advanced concepts in a short
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Download Free Solving Equations With Variables On Both Sides With Fractions Worksheet
A few of these worksheets are intended for students in the 5th-8th grade. The two-step word problems are created using fractions and decimals. Each worksheet contains ten problems. These worksheets
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There are many worksheets to learn about quadratic equations. Each level comes with their own worksheet. The worksheets were designed for you to help you solve problems in the fourth level. When
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Gallery of Solving Equations With Variables On Both Sides With Fractions Worksheet
Variables On Both Sides Equations Worksheet
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Solving Equations With Variables On Both Sides Worksheet Made By Teachers
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