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Project Example: Age prediction using bone microstructure for archaeometric analyses
Research Objective
In contrast to the determination of the age of an excavated skeleton, e.g. using radio-carbon method, it is rather difficult to predict which age a person reached at their death. This is a key
question for further research in prehistorical anthropology. Therefore, the goal of this project was the construction of a prediction model for age at the time of death based on bone micro-structure
from human remains that have been buried. Furthermore, we were interested in whether the person's profession and reason of death influenced their bone micro-structure. For the analysis, we had data
from 103 skeletons from a cemetery in Basel, Switzerland. The bone structure was described by twelve continuous variables, that were measured using similar criteria and hence were highly correlated.
We used a flexible elastic net model as an alternative to the simple linear model, which resulted in surprising precise age prediction.
Statistical Methodology
• linear model with step-wise forward variable selection
• penalized regression using elastic net to consider variable collinearity and perform variable selection
• Cross-validation based on Mean Squared Error of Prediction
• MANOVA
Open-source software R
• Statistical consulting project with Andreas Mayr and Paul Schmidt in context of the lecture 'Statistical Praktikum' at LMU Munich. | {"url":"http://manitz.org/bone.html","timestamp":"2024-11-07T20:26:46Z","content_type":"text/html","content_length":"3876","record_id":"<urn:uuid:9827cc66-157d-4ceb-b76c-55c9a97b4064>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00358.warc.gz"} |
What should I do if I'm carrying an abstract algebra book around and someone thinks I'm doing remedial algebra (Quora Answer)
With all due respect, you haven't cared about such people since you first started rocking Serre's 'A Course in Arithmetic' or Jacobson's 'Basic Algebra II'. The whole point of your lifestyle
choice is to scare away lesser spirits so you can get some serious math done.
And this way, when you meet a pair of identical twin swimsuit models who say 'Wow. Want to come over to our place and pick our way through Grothendieck's later correspondence?' you'll know
they're serious about what really matters, and not a couple of airheads trying to distract you with meaningless physical intimacy.
- Eric Weinstein on Quora | {"url":"https://theportal.wiki/wiki/What_should_I_do_if_I%27m_carrying_an_abstract_algebra_book_around_and_someone_thinks_I%27m_doing_remedial_algebra_(Quora_Answer)","timestamp":"2024-11-13T15:02:18Z","content_type":"text/html","content_length":"30338","record_id":"<urn:uuid:88ec9c9b-5ad6-438e-aa30-048eeb87da4d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00658.warc.gz"} |
Machine Learning- Job Interview Questions and Answers in Karachi Pakistan Dubai
Machine Learning is the heart of Artificial Intelligence. It consists of techniques that lay out the basic structure for constructing algorithms. These algorithms are used to give functionalities to
make automated machines carry out tasks without being explicitly programmed. This basic structure of Machine Learning and various ML algorithms are the key areas where interviewers would check a
candidate’s compatibility. So, to leverage your skillset while facing the interview, we have come up with a comprehensive blog on ‘Top 30 Machine Learning Interview Questions and Answers for 2020.’
Machine Learning Interview Questions
1. What are the types of Machine Learning?
In all the ML Interview Questions that we would be going to discuss, this is one of the most basic question.
So, basically, there are three types of Machine Learning techniques:
Supervised Learning: In this type of the Machine Learning technique, machines learn under the supervision of labeled data. There is a training dataset on which the machine is trained, and it gives
the output according to its training.
Unsupervised Learning: Unlike supervised learning, it has unlabeled data. So, there is no supervision under which it works on the data. Basically, unsupervised learning tries to identify patterns in
data and make clusters of similar entities. After that, when a new input data is fed into the model, it does not identify the entity; rather, it puts the entity in a cluster of similar objects.
Reinforcement Learning: Reinforcement learning includes models that learn and traverse to find the best possible move. The algorithms for reinforcement learning are constructed in a way that they try
to find the best possible suite of action on the basis of the reward and punishment theory.
2. Differentiate between classification and regression in Machine Learning.
In Machine Learning, there are various types of prediction problems based on supervised and unsupervised learning. These are classification, regression, clustering, and association. Here, we will
discuss about classification and regression.
Classification: In classification, we try to create a Machine Learning model that assists us in differentiating data into separate categories. The data is labeled and categorized based on the input
For example, imagine that we want to make predictions on the churning out customers for a particular product based on some data recorded. Either the customers will churn out or they will not. So, the
labels for this would be ‘Yes’ and ‘No.’
Regression: It is the process of creating a model for distinguishing data into continuous real values, instead of using classes or discrete values. It can also identify the distribution movement
depending on the historical data. It is used for predicting the occurrence of an event depending on the degree of association of variables.
For example, the prediction of weather condition depends on factors such as temperature, air pressure, solar radiation, elevation of the area, and distance from sea. The relation between these
factors assists us in predicting the weather condition.
3. What is Linear Regression?
Linear Regression is a supervised Machine Learning algorithm. It is used to find the linear relationship between the dependent and the independent variables for predictive analysis.
The equation for Linear Regression:
• X is the input or the independent variable
• Y is the output or the dependent variable
• a is the intercept and b is the coefficient of X
Below is the best fit line that shows the data of weight (Y or the dependent variable) and height (X or the independent variable) of 21-years-old candidates scattered over the plot. This straight
line shows the best linear relationship that would help in predicting the weight of candidates according to their height.
To get this best fit line, we will try to find the best values of a and b. By adjusting the values of a and b, we will try to reduce errors in the prediction of Y.
This is how linear regression helps in finding the linear relationship and predicting the output.
4. How will you determine the Machine Learning algorithm that is suitable for your problem?
To identify the Machine Learning algorithm for our problem, we should follow the below steps:
Step 1: Problem Classification: Classification of the problem depends on the classification of input and output:
• Classifying the input: Classification of the input depends on whether we have data labeled (supervised learning) or unlabeled (unsupervised learning), or whether we have to create a model that
interacts with the environment and improves itself (reinforcement learning).
• Classifying the output: If we want the output of our model as a class, then we need to use some classification techniques.
If it is giving the output as a number, then we must use regression techniques and, if the output is a different cluster of inputs, then we should use clustering techniques.
Step 2: Checking the algorithms in hand: After classifying the problem, we have to look for the available algorithms that can be deployed for solving the classified problem.
Step 3: Implementing the algorithms: If there are multiple algorithms available, then we will implement each one of them, one by one. Finally, we would select the algorithm that gives the best
5. What are Bias and Variance?
• Bias is the difference between the average prediction of our model and the correct value. If the bias value is high, then the prediction of the model is not accurate. Hence, the bias value should
be as low as possible to make the desired predictions.
• Variance is the number that gives the difference of prediction over a training set and the anticipated value of other training sets. High variance may lead to large fluctuation in the output.
Therefore, the model’s output should have low variance.
The below diagram shows the bias–variance trade off:
Here, the desired result is the blue circle at the center. If we get off from the blue section, then the prediction goes wrong.
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6. What is Variance Inflation Factor?
Variance Inflation Factor (VIF) is the estimate of the volume of multicollinearity in a collection of many regression variables.
VIF = Variance of the model / Variance of the model with a single independent variable
We have to calculate this ratio for every independent variable. If VIF is high, then it shows the high collinearity of the independent variables.
7. Explain false negative, false positive, true negative, and true positive with a simple example.
True Positive (TP): When the Machine Learning model correctly predicts the condition, it is said to have a True Positive value.
True Negative (TN): When the Machine Learning model correctly predicts the negative condition or class, then it is said to have a True Negative value.
False Positive (FP): When the Machine Learning model incorrectly predicts a negative class or condition, then it is said to have a False Positive value.
False Negative (FN): When the Machine Learning model incorrectly predicts a positive class or condition, then it is said to have a False Negative value.
8. What is a Confusion Matrix?
Confusion matrix is used to explain a model’s performance and gives the summary of predictions on the classification problems. It assists in identifying the uncertainty between classes.
A confusion matrix gives the count of correct and incorrect values and also the error types.Accuracy of the model:
For example, consider this confusion matrix. It consists of values as True Positive, True Negative, False Positive, and False Negative for a classification model. Now, the accuracy of the model can
be calculated as follows:
Thus, in our example:
Accuracy = (200 + 50) / (200 + 50 + 10 + 60) = 0.78
This means that the model’s accuracy is 0.78, corresponding to its True Positive, True Negative, False Positive, and False Negative values.
9. What do you understand by Type I and Type II errors?
Type I Error: Type I error (False Positive) is an error where the outcome of a test shows the non-acceptance of a true condition.
For example, a cricket match is going on and, when a batsman is not out, the umpire declares that he is out. This is a false positive condition. Here, the test does not accept the true condition that
the batsman is not out.
Type II Error: Type II error (False Negative) is an error where the outcome of a test shows the acceptance of a false condition.
For example, the CT scan of a person shows that he is not having a disease but, in reality, he is having it. Here, the test accepts the false condition that the person is not having the disease.
10. When should you use classification over regression?
Both classification and regression are associated with prediction. Classification involves the identification of values or entities that lie in a specific group. The regression method, on the other
hand, entails predicting a response value from a consecutive set of outcomes.
The classification method is chosen over regression when the output of the model needs to yield the belongingness of data points in a dataset to a particular category.
For example, we have some names of bikes and cars. We would not be interested in finding how these names are correlated to bikes and cars. Rather, we would check whether each name belongs to the bike
category or to the car category.
11. Explain Logistic Regression.
Logistic regression is the proper regression analysis used when the dependent variable is categorical or binary. Like all regression analyses, logistic regression is a technique for predictive
analysis. Logistic regression is used to explain data and the relationship between one dependent binary variable and one or more independent variables. Also, it is employed to predict the probability
of a categorical dependent variable.
We can use logistic regression in the following scenarios:
• To predict whether a citizen is a Senior Citizen (1) or not (0)
• To check whether a person is having a disease (Yes) or not (No)
There are three types of logistic regression:
• Binary Logistic Regression: In this, there are only two outcomes possible.
Example: To predict whether it will rain (1) or not (0)
• Multinomial Logistic Regression: In this, the output consists of three or more unordered categories.
Example: Prediction on the regional languages (Kannada, Telugu, Marathi, etc.)
• Ordinal Logistic Regression: In ordinal logistic regression, the output consists of three or more ordered categories.
Example: Rating an Android application from 1 to 5 stars.
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12. Imagine, you are given a dataset consisting of variables having more than 30% missing values. Let’s say, out of 50 variables, 8 variables have missing values, which is higher than 30%. How will
you deal with them?
To deal with the missing values, we will do the following:
• We will specify a different class for the missing values.
• Now, we will check the distribution of values, and we would hold those missing values that are defining a pattern.
• Then, we will charge these into a yet another class, while eliminating others.
13. How do you handle the missing or corrupted data in a dataset?
In Python Pandas, there are two methods that are very useful. We can use these two methods to locate the lost or corrupted data and discard those values:
• isNull(): For detecting the missing values, we can use the isNull() method.
• dropna(): For removing the columns/rows with null values, we can use the dropna() method.
Also, we can use fillna() to fill the void values with a placeholder value.
14. Explain Principal Component Analysis (PCA).
Firstly, this is one of the most important Machine Learning Interview Questions.
In the real world, we deal with multi-dimensional data. Thus, data visualization and computation become more challenging with the increase in dimensions. In such a scenario, we might have to reduce
the dimensions to analyze and visualize the data easily. We do this by:
• Removing irrelevant dimensions
• Keeping only the most relevant dimensions
This is where we use Principal Component Analysis (PCA).
Finding a fresh collection of uncorrelated dimensions (orthogonal) and ranking them on the basis of variance are the goals of Principal Component Analysis.
The Mechanism of PCA:
• Compute the covariance matrix for data objects
• Compute the Eigen vectors and the Eigen values in a descending order
• To get the new dimensions, select the initial N Eigen vectors
• Finally, change the initial n-dimensional data objects into N-dimensions
Example: Below are the two graphs showing data points (objects) and two directions: one is ‘green’ and the other is ‘yellow.’ We got the Graph 2 by rotating the Graph 1 so that the x-axis and y-axis
represent the ‘green’ and ‘yellow’ directions, respectively.
After the rotation of the data points, we can infer that the green direction (x-axis) gives us the line that best fits the data points.
Here, we are representing 2-dimensional data. But in real-life, the data would be multi-dimensional and complex. So, after recognizing the importance of each direction, we can reduce the area of
dimensional analysis by cutting off the less-significant ‘directions.’
Now, we will look into another important Machine Learning Interview Question on PCA.
15. Why rotation is required in PCA? What will happen if you don’t rotate the components?
Rotation is a significant step in PCA as it maximizes the separation within the variance obtained by components. Due to this, the interpretation of components becomes easier.
The motive behind doing PCA is to choose fewer components that can explain the greatest variance in a dataset. When rotation is performed, the original coordinates of the points get changed. However,
there is no change in the relative position of the components.
If the components are not rotated, then we need more extended components to describe the variance.
16. We know that one hot encoding increases the dimensionality of a dataset, but label encoding doesn’t. How?
When we use one hot encoding, there is an increase in the dimensionality of a dataset. The reason for the increase in dimensionality is that, for every class in the categorical variables, it forms a
different variable.
Example: Suppose, there is a variable ‘Color.’ It has three sub-levels as Yellow, Purple, and Orange. So, one hot encoding ‘Color’ will create three different variables as Color.Yellow, Color.Porple,
and Color.Orange.
In label encoding, the sub-classes of a certain variable get the value as 0 and 1. So, we use label encoding only for binary variables.
This is the reason that one hot encoding increases the dimensionality of data and label encoding does not.
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17. How can you avoid overfitting?
Overfitting happens when a machine has an inadequate dataset and it tries to learn from it. So, overfitting is inversely proportional to the amount of data.
For small databases, we can bypass overfitting by the cross-validation method. In this approach, we will divide the dataset into two sections. These two sections will comprise testing and training
sets. To train the model, we will use the training dataset and, for testing the model for new inputs, we will use the testing dataset.
This is how we can avoid overfitting.
18. Why do we need a validation set and a test set?
We split the data into three different categories while creating a model:
1. Training set: We use the training set for building the model and adjusting the model’s variables. But, we cannot rely on the correctness of the model build on top of the training set. The model
might give incorrect outputs on feeding new inputs.
2. Validation set: We use a validation set to look into the model’s response on top of the samples that don’t exist in the training dataset. Then, we will tune hyperparameters on the basis of the
estimated benchmark of the validation data.
When we are evaluating the model’s response using the validation set, we are indirectly training the model with the validation set. This may lead to the overfitting of the model to specific data. So,
this model won’t be strong enough to give the desired response to the real-world data.
1. Test set: The test dataset is the subset of the actual dataset, which is not yet used to train the model. The model is unaware of this dataset. So, by using the test dataset, we can compute the
response of the created model on hidden data. We evaluate the model’s performance on the basis of the test dataset.
Note: We always expose the model to the test dataset after tuning the hyperparameters on top of the validation set.
As we know, the evaluation of the model on the basis of the validation set would not be enough. Thus, we use a test set for computing the efficiency of the model.
19. What is a Decision Tree?
A decision tree is used to explain the sequence of actions that must be performed to get the desired output. It is a hierarchical diagram that shows the actions.
We can create an algorithm for a decision tree on the basis of the hierarchy of actions that we have set.
In the above decision tree diagram, we have made a sequence of actions for driving a vehicle with/without a license.
20. Explain the difference between KNN and K-means Clustering.
K-nearest neighbors: It is a supervised Machine Learning algorithm. In KNN, we give the identified (labeled) data to the model. Then, the model matches the points based on the distance from the
closest points.
K-means clustering: It is an unsupervised Machine Learning algorithm. In this, we give the unidentified (unlabeled) data to the model. Then, the algorithm creates batches of points based on the
average of the distances between distinct points.
21. What is Dimensionality Reduction?
In the real world, we build Machine Learning models on top of features and parameters. These features can be multi-dimensional and large in number. Sometimes, the features may be irrelevant and it
becomes a difficult task to visualize them.
Here, we use dimensionality reduction to cut down the irrelevant and redundant features with the help of principal variables. These principal variables are the subgroup of the parent variables that
conserve the feature of the parent variables.
22. Both being tree-based algorithms, how is Random Forest different from Gradient Boosting Algorithm (GBM)?
The main difference between a random forest and GBM is the use of techniques. Random forest advances predictions using a technique called ‘bagging.’ On the other hand, GBM advances predictions with
the help of a technique called ‘boosting.’
• Bagging: In bagging, we apply arbitrary sampling and we divide the dataset into N After that, we build a model by employing a single training algorithm. Following, we combine the final
predictions by polling. Bagging helps increase the efficiency of the model by decreasing the variance to eschew overfitting.
• Boosting: In boosting, the algorithm tries to review and correct the inadmissible predictions at the initial iteration. After that, the algorithm’s sequence of iterations for correction continues
until we get the desired prediction. Boosting assists in reducing bias and variance, both, for making the weak learners strong.
23. Suppose, you found that your model is suffering from high variance. Which algorithm do you think could handle this situation and why?
Handling High Variance
• For handling issues of high variance, we should use the bagging algorithm.
• Bagging algorithm would split data into sub-groups with replicated sampling of random data.
• Once the algorithm splits the data, we use random data to create rules using a particular training algorithm.
• After that, we use polling for combining the predictions of the model.
24. What is ROC curve and what does it represent?
ROC stands for ‘Receiver Operating Characteristic.’ We use ROC curves to represent the trade-off between True and False positive rates, graphically.
In ROC, AUC (Area Under the Curve) gives us an idea about the accuracy of the model.
The above graph shows an ROC curve. Greater the Area Under the Curve better the performance of the model.
Next, we would be looking at Machine Learning Interview Questions on Rescaling, Binarizing, and Standardizing.
25. What is Rescaling of data and how is it done?
In real-world scenarios, the attributes present in data will be in a varying pattern. So, rescaling of the characteristics to a common scale gives benefit to algorithms to process the data
We can rescale the data using Scikit-learn. The code for rescaling the data using MinMaxScaler is as follows:
#Rescaling data
import pandas
import scipy
import numpy
from sklearn.preprocessing import MinMaxScaler
names = ['Abhi', 'Piyush', 'Pranay', 'Sourav', 'Sid', 'Mike', 'pedi', 'Jack', 'Tim']
Dataframe = pandas.read_csv(url, names=names)
Array = dataframe.values
# Splitting the array into input and output
X = array[:,0:8]
Y = array[:,8]
Scaler = MinMaxScaler(feature_range=(0, 1))
rescaledX = scaler.fit_transform(X)
# Summarizing the modified data
26. What is Binarizing of data? How to Binarize?
In most of the Machine Learning Interviews, apart from theoretical questions, interviewers focus on the implementation part. So, this ML Interview Questions in focused on the implementation of the
theoretical concepts.
Converting data into binary values on the basis of threshold values is known as the binarizing of data. The values that are less than the threshold are set to 0 and the values that are greater than
the threshold are set to 1. This process is useful when we have to perform feature engineering, and we can also use it for adding unique features.
We can binarize data using Scikit-learn. The code for binarizing the data using Binarizer is as follows:
from sklearn.preprocessing import Binarizer
import pandas
import numpy
names = ['Abhi', 'Piyush', 'Pranay', 'Sourav', 'Sid', 'Mike', 'pedi', 'Jack', 'Tim']
dataframe = pandas.read_csv(url, names=names)
array = dataframe.values
# Splitting the array into input and output
X = array[:,0:8]
Y = array[:,8]
binarizer = Binarizer(threshold=0.0).fit(X)
binaryX = binarizer.transform(X)
# Summarizing the modified data
27. How to Standardize data?
Standardization is the method that is used for rescaling data attributes. The attributes would likely have a value of mean as 0 and the value of standard deviation as 1. The main objective of
standardization is to prompt the mean and standard deviation for the attributes.
We can standardize the data using Scikit-learn. The code for standardizing the data using StandardScaler is as follows:
# Python code to Standardize data (0 mean, 1 stdev)
from sklearn.preprocessing import StandardScaler
import pandas
import numpy
names = ['Abhi', 'Piyush', 'Pranay', 'Sourav', 'Sid', 'Mike', 'pedi', 'Jack', 'Tim']
dataframe = pandas.read_csv(url, names=names)
array = dataframe.values
# Separate the array into input and output components
X = array[:,0:8]
Y = array[:,8]
scaler = StandardScaler().fit(X)
rescaledX = scaler.transform(X)
# Summarize the transformed data
28. Executing a binary classification tree algorithm is a simple task. But, how does a tree splitting take place? How does the tree determine which variable to break at the root node and which at its
child nodes?
Gini index and Node Entropy assist the binary classification tree to take decisions. Basically, the tree algorithm determines the feasible feature that is used to distribute data into the most
genuine child nodes.
According to Gini index, if we arbitrarily pick a pair of objects from a group, then they should be of identical class and the possibility for this event should be 1.
To compute the Gini index, we should do the following:
1. Compute Gini for sub-nodes with the formula: The sum of the square of probability for success and failure (p^2 + q^2)
2. Compute Gini for split by weighted Gini rate of every node of the split
Now, Entropy is the degree of indecency that is given by the following:
where a and b are the probabilities of success and failure of the node
When Entropy = 0, the node is homogenous
When Entropy is high, both groups are present at 50–50 percent in the node.
Finally, to determine the suitability of the node as a root node, the entropy should be very low.
29. What is SVM (Support Vector Machines)?
SVM is a Machine Learning algorithm that is majorly used for classification. It is used on top of the high dimensionality of the characteristic vector.
Below is the code for the SVM classifier:
# Introducing required libraries
from sklearn import datasets
from sklearn.metrics import confusion_matrix
from sklearn.model_selection import train_test_split
# Stacking the Iris dataset
iris = datasets.load_iris()
# A -> features and B -> label
A = iris.data
B = iris.target
# Breaking A and B into train and test data
A_train, A_test, B_train, B_test = train_test_split(A, B, random_state = 0)
# Training a linear SVM classifier
from sklearn.svm import SVC
svm_model_linear = SVC(kernel = 'linear', C = 1).fit(A_train, B_train)
svm_predictions = svm_model_linear.predict(A_test)
# Model accuracy for A_test
accuracy = svm_model_linear.score(A_test, B_test)
# Creating a confusion matrix
cm = confusion_matrix(B_test, svm_predictions)
30. Implement the KNN classification algorithm.
We will use the Iris dataset for implementing the KNN classification algorithm.
# KNN classification algorithm
from sklearn.datasets import load_iris
from sklearn.neighbors import KNeighborsClassifier
import numpy as np
from sklearn.model_selection import train_test_split
A_train, A_test, B_train, B_test = train_test_split(iris_dataset["data"], iris_dataset["target"], random_state=0)
kn = KNeighborsClassifier(n_neighbors=1)
kn.fit(A_train, B_train)
A_new = np.array([[8, 2.5, 1, 1.2]])
prediction = kn.predict(A_new)
print("Predicted target value: {}\n".format(prediction))
print("Predicted feature name: {}\n".format
print("Test score: {:.2f}".format(kn.score(A_test, B_test)))
Predicted Target Name: [0]
Predicted Feature Name: [‘ Setosa’]
Test Score: 0.92
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Introduction to Aerospace Flight Vehicles
22 Energy Equation & Bernoulli’s Equation
Energy is defined as the ability to do work, which is a very useful property in engineering. There are many types of energy, including potential, kinetic, chemical, electrical, nuclear, etc. For
thermodynamic analyses, energy can be classified into two categories: macroscopic energy and microscopic energy. Macroscopic energy is energy that a whole system possesses with respect to a fixed
external reference, and in thermodynamics, these are potential and kinetic energies. Microscopic energy is the energy contained in the system at the molecular level. A “system” in this context can be
defined as a collection of matter of fixed identity.
When developing the energy equation for a fluid flow, the applicable physical principle is a thermodynamic one in that energy cannot be created or destroyed but only converted from one form to
another. This principle is formally embodied in the first law of thermodynamics applied to a system of a given (fixed) mass, i.e.,
where heat added to the system (such as from a thermal source), work done on the system (such as mechanical work), and energy of the mass inside that system, the principle being illustrated in the
schematic below. Notice that positive when work is done on the system and negative when the system does work on the fluid.
The first law of thermodynamics states that the change in internal energy of a fixed system is equal to the net heat added less the net work done on that system.
Internal energy can be considered the sum of the macroscopic kinetic and potential energy of the molecules comprising the fluid and their microscopic internal energy, e.g., temperature effects.
Indeed, when dealing with thermodynamic relations, the internal energy is a function of temperature alone for a perfect gas. Therefore, another way of writing the first law of thermodynamics for a
fixed system is
If the system does no external work, all the heat added will manifest as an increase in internal energy, e.g., an increase in temperature. Of course, heat and work can be added to or subtracted from
an engineering system, i.e., transferred across the system’s boundaries. If
By convention, heat added to the system and the rate of work done on the system are defined as positive contributions. The system’s energy can also be viewed as being associated with that system’s
specific amount of mass. Therefore, for a fluid flow, the energy terms in the resulting equations are usually written as an energy per unit mass, i.e., in the spirit of the differential calculus,
This latter approach is also consistent with the previous derivations of the equations that applied to the conservation of mass and momentum.
• Set up the most general form of the energy equation in integral form.
• Know how to simplify the energy equation into various, more practical forms.
• Understand how to derive a surrogate for the energy equation for steady, incompressible, inviscid flow, called the Bernoulli equation.
• Learn how to solve fundamental engineering problems using the energy equation.
Setting Up the Energy Equation
As before, consider a fluid flow through a fixed finite control volume (C.V.) of volume
A finite control volume fixed in space, as used to establish the energy equation in its most general form.
The objective is to develop an analogous equation to the continuity and momentum equations for energy conservation as it applies to a fluid flow. As will be shown, however, the resulting energy
equation describes a power balance equation for the fluid system, i.e., a balance of the rate of work done on the flow versus the rate of energy added to the flow and/or conversion from one form of
energy in the flow to another.
Heat Addition
An increment in energy,
If the flow is viscous, then frictional effects can also add heat. This latter effect can be important on a high-speed (supersonic) aircraft or a spacecraft during re-entry into the Earth’s
atmosphere. While viscous heating effects must be recognized as a complicated process involving the consideration of shear stresses and thermal effects from shock wave formation in a high-speed fluid
flow, for now, the effects can be cumulatively denoted by
Alert: Symbol conflict!
Symbol conflict is common between branches of engineering. In the situation just examined,
If some work is done on the system, i.e.,
In the case of pressure effects, a pressure force arises from the pressure acting over an area, i.e., for a point on the C.S. then
Similarly, the effects of body forces can be written as
In the case of viscous work, this latter contribution can be written as
Finally, some mechanical work could be added to the system, say
Internal Energy
The next step is to examine the internal energy inside the C.V. Energy can do work, so this is an important part of the energy equation. This form of energy will comprise some “temperature” energy or
the microscopic kinetic energy per unit mass of the fluid molecules,
where the resultant velocity
Notice that the microscopic energy of the fluid molecules,
The microscopic energy of the fluid molecules,
The temperature of the fluid, Boltzmann’s constant. In this latter form, however, it strictly holds for monoatomic gases because they have three degrees of translational freedom, i.e., kinetic energy
only. A thermal gradient, for example, will cause the molecules to move faster toward increasing temperature. Thermal energy will be transferred from the hotter part of the gas to the cooler part,
basically a statement of the second law of thermodynamics, i.e., energy flows “downhill.”
Diatomic gases, such as nitrogen and oxygen, which comprise about 98% of air, also have two degrees of freedom of rotational motion and two degrees of vibrational motion, the latter being important
only at higher temperatures. Therefore, their total internal energy will be related using
The macroscopic kinetic energy per unit mass of the fluid per unit mass,
The macroscopic kinetic and potential energy of the fluid can be interpreted within the framework of a continuum model.
Energy In & Out of Control Volume
The total energy will then be obtained by integration over the entire mass of fluid contained within the C.V., remembering that mass can also flow across the C.S. from the C.V. Across the element
In addition, unsteady effects may be present so that the energy in the system can change because of temporal variations of the flow field properties inside
and so the time rate of change of energy inside the C.V. will be
Finally, adding these two latter terms together (Eqs. 12 and 14) gives
Final Form of the Energy Equation
Now, all of the parts that make up the final form of the energy equation are available in its integral form. To this end, the principle of conservation of energy has been applied to a fluid flowing
through a finite C.V. that may have heat added or removed, and/or mechanical work is done on the fluid or extracted from it., i.e.,
which is mathematically the most general form of the energy equation.
Because energy is conserved, then in words, it can be stated that: “The rate of heat added plus the rate of doing work on the flow will be equal to the total rate of change of energy of the flow,”
which is just the first law of thermodynamics, where
Rewriting the energy equation again without the components being identified gives
This latter equation (Eq. 18) now completes the set of three conservation equations, i.e., mass (continuity), momentum, and energy.
Recall for completeness that the continuity equation is
and the momentum equation is
In such collective equations, the unknown parameters may include the velocities, pressures, densities, and temperatures, although not all quantities may be unknown or required in any given problem. A
fourth equation is also available to round out the set, namely the equation of state
Simplifications of the Energy Equation
The general form of the energy equation in Eq. 18 may look somewhat forbidding, and it is if written out in its entirety. Nevertheless, as with the other conservation equations, the general form can
be written in various simplified forms depending on the assumptions that might be made (and justified!), e.g., steady flow, absence of body forces, no heat added to the system, no mechanical work, no
viscous forces, one-dimensional flow, etc. However, because thermodynamic principles are used here, caution must be applied when using the energy equation to ensure that the needed terms are retained
in all potential simplification processes. Remember that in engineering problem solving, any over-simplification of the governing equations without careful justification will likely result in
disastrous predictive outcomes.
Single Stream System
By way of illustrating the steps that could be used in the simplifications of the energy equation, it should be recognized that many practical engineering problems involve fluid systems with just one
inlet and one outlet, the mass flow rate through such a system being constant, which are sometimes referred to as single-stream systems. A general depiction of such a single-stream system is shown in
the figure below, where some flow comes in at one side of the system and exits on the other. Notice that, in general, there could be a difference in the heights of the inlet and the outlet, so the
gravitational potential energy or “head” terms must be retained.
For such a single-stream system, the general form of the energy equation can be reduced to the form
where the subscripts 1 and 2 refer to the inlet and outlet conditions, respectively.
Look carefully at Eq. 18 and consider how Eq. 21 is obtained. As you do that, remember that a mass flow rate or the “21 becomes equivalent to a net pressure contribution times the mass flow rate
divided by density.
The left-hand side of the preceding equation represents the energy input, and the right-hand side represents the energy output. The energy input, in this case, comes from the rate of heat transfer to
the fluid
In such a system, the mass flow through the system is conserved (mass inside the system is constant if the flow is steady), so
Based on per unit mass, which means dividing the terms in the equations by 21 becomes
which is a common form of the steady flow energy balance in thermodynamics. Rearranging this equation gives
The latter term in Eq. 24 can be viewed as the sum of the frictional losses, i.e., the value of
Another way to write this previous equation is just
which is often called the mechanical energy equation in terms of work per unit mass.
What are the units of Eq. 26? For this equation to be dimensionally homogeneous, then the units of each of the terms must be the same, i.e.,
Therefore, the equation is confirmed to be dimensionally homogeneous with units of L
Particular Case: No Losses
Now, suppose the flow is ideal with no irreversible processes such as turbulence, friction, or viscosity, then the total energy must be conserved. Then the energy loss term
or in terms of power, so bringing the
Mechanical Work
Remember that adding or subtracting mechanical work,
In the case of a basic pump, which adds energy and does work on the fluid system, then
Finally, if the fluid is assumed to be incompressible, i.e.,
or in terms of per unit mass (again, divide by
which is called the extended Bernoulli equation.
Energy Equation in Terms of “Head”
Civil and hydraulic engineers, who deal primarily with liquids rather than gases, often express the energy equation in terms of pressure head and specific weight. The pressure head or the static
pressure head,
Specific Weight
The specific weight is the weight of a unit volume of a fluid. The symbol
or by rearrangement, then
Using the specific weight, which can be obtained by dividing through Eq. 38 by
where the units of all terms are now length (m or ft). If frictional losses are included, which can also be expressed in terms of head, i.e.,
Pressure Head of a Pump or Turbine
The power,
The mass flow rate,
Of course, no pump or turbine can be 100% efficient, so if the efficiency is
Therefore, in the case of a pump (energy in) of power
In the case of a turbine (energy out) of power
Caution: Do not confuse the pressure head for a pump,
Check Your Understanding #1 – Using the energy equation to calculate pumping power
Use conservation of energy principles to calculate the power of a hydraulic pump that must deliver the fluid (oil) from one location to another. The pump is 75% efficient in converting mechanical
input work to pressure. The inlet pressure, ^3/hr. The height difference, ^3. Assume the internal fluid losses are equivalent to 1.4 m of static pressure head.
Show solution/hide solution.
First, it is necessary to find the inlet and outlet velocities,
and for
The relevant form of the energy equation is
So solving for
Inserting the known values gives
The power required from the pump is
Inserting the numerical values gives
Bernoulli’s Equation
By assuming incompressible flow and that no mechanical work is introduced into or taken out of the fluid system, then 38 becomes
This latter equation can then be rearranged into the form
or simply that
which is known as the Bernoulli equation or Bernoulli’s principle after Daniel Bernoulli.
Notice that the Bernoulli equation has units of pressure and not energy. For this reason, the Bernoulli equation is often referred to as a surrogate for the energy under the conditions of steady,
incompressible flow without energy addition. It will be apparent that the Bernoulli equation is still a statement of energy conservation in that a fluid exchanges its specific kinetic energy for
pressure, either static or potential, the specific kinetic energy being the kinetic energy per unit volume, as shown in the figure below. Notice that the sum of the static and dynamic pressure is
called the total pressure.
The terms in the Bernoulli equation are static, dynamic, and hydrostatic pressure, which are conserved.
A more general form of the Bernoulli equation, which often appears in many sources, is to leave it in an unintegrated form, i.e.
After integration then
Assuming incompressible flow gives
as has been written down previously.
The Bernoulli equation is one of the most famous fluid mechanics equations, and it can be used to solve many practical problems. It has been derived here as a particular degenerate case of the
general energy equation for a steady, inviscid, incompressible flow. Still, it can also be derived in several other ways. Remember that it is not an energy equation per se because it has units of
pressure, so it is often referred to as a surrogate of the energy equation.
Another Derivation of Bernoulli’s Equation
In terms of another physically intuitive derivation of the Bernoulli equation, consider the figure below, which represents a streamtube flow of an ideal fluid that is steady, incompressible, and
inviscid. At the inlet to the streamtube, the cross-sectional area is
Streamtube flow model used for the derivation of the Bernoulli equation.
remembering that work is equal to force times distance. Let
Notice that the minus sign here indicates that the pressure force here acts upstream in the opposite direction to the fluid flow. In addition, there will be work done under gravity, which will be
where the mass into the fluid system to increase its potential energy. Therefore, the total external work is
The kinetic energy of the flow must now be considered. The change in kinetic energy of the fluid as it moves through the streamtube is
The application of the principle of conservation of energy requires that
so that
Notice also that conservation of mass (the continuity equation) requires that
or just
Therefore, inserting the various terms gives
Cancelling out the volume
which, again, is the Bernoulli equation.
Caution: Remember that in the derivation of the Bernoulli equation, the flow has been assumed to be steady and incompressible with negligible frictional or viscous losses (i.e., an ideal fluid) and
where no mechanical work is added or subtracted. While the Bernoulli equation is found to have many practical applications, it should be remembered that it has been derived based on these prior
assumptions, so its practical use requires considerable justification and caution in actual application.
Check Your Understanding #2 – Pressure change in a contraction
Air flows at low speed through a pipe with a volume flow rate of 0.135 m
Show solution/hide solution.
From the information, assuming a one-dimensional, steady, incompressible, inviscid flow seems reasonable. The flow rates, velocities, and Mach numbers are low enough that compressibility effects can
be neglected. Let the inlet be condition 1 and the outlet condition 2. The continuity equation relates the inlet and outlet conditions, i.e.,
Or because the flow is assumed incompressible, then just
It follows that
Calculating the areas gives
Now the velocities can be calculated, i.e.,
which are much lower than a Mach number of 0.3, so the assumption of incompressible flow is justified.
The pressure difference between points 1 and 2 is needed, which requires some form of the energy equation. Using the Bernoulli equation is justified because the information given is for an
incompressible, frictionless flow. The Bernoulli equation is
Inserting the values for
In terms of
so solving for
Other Forms of the Bernoulli Equation
Other forms of the Bernoulli equation are used. For example, the unintegrated form of the Bernoulli equation is
Recall that for an incompressible flow, the first term is
so the classic Bernoulli equation is recovered, i.e.,
Isothermal Process
For an isothermal process, then
and so the Bernoulli equation for an isothermal flow is
or between two points 1 and 2 (which are on the same streamline), then
Adiabatic Process
Another case of a compressible flow is an adiabatic process characterized by the relationship
As previously derived, the unintegrated form of the Bernoulli equation can be written as
Substituting for
Therefore, another form of the Bernoulli equation for the steady, adiabatic, compressible flow of a gas becomes
Quasi-Steady Effects
The Bernoulli equation can also be used for certain classes of quasi-steady flows. A quasi-steady flow means the flow is considered to change slowly over time, so the unsteady effects are accounted
for, but the assumption is that the unsteady changes occur smoothly and gradually, without large, rapid fluctuations, i.e., in this case,
where the first term is equivalent to the additional energy involved in accelerating the fluid along the streamline. Such effects usually appear in quasi-steady aerodynamics problems, appearing as
“apparent” or added “mass” in the forces and moments acting on a body. In true unsteady flow situations, where rapid changes occur (such as in turbulent flows, flows with strong vortices, or flows
with large-scale unsteadiness, the Bernoulli equation cannot be applied. In such cases, a more general form of the energy equation would be needed.
This quasi-steady form of the Bernoulli equation can be integrated between two points, say point 1 (located at a distance
Notice that the “unsteady” pressure term, accounting for acceleration effects in the flow in the direction of
In practice, this unsteady term is not easy to calculate except in some simple unsteady flows.
For example, consider an accelerating flow where the velocity at a point changes linearly with time, i.e., constant acceleration, where the velocity is given by
To find the unsteady contribution, it must be integrated along a streamline from point
In this case,
Notice that
so the unsteady term for this flow becomes
This result is simply part of the flow’s energy balance, accounting for its acceleration.
Energy Equation from the RTE
The Reynolds Transport Theorem (RTE) for the total energy
For an incompressible flow,
Therefore, the energy equation in differential form is
This equation represents energy conservation for a fluid, accounting for changes in internal energy
From the energy equation, then
Assuming steady, inviscid flow where
This result implies that the specific internal energy
Assuming constant density, i.e.,
Once again, this is the Bernoulli equation for steady, inviscid, and incompressible flow.
Summary & Closure
Applying the principle of energy conservation to fluid flow results in a rather formidable-looking equation in its more general form. The energy equation is developed from thermodynamic principles,
i.e., the first law of thermodynamics, and uses the concepts of heat, work, and power. The types of energy are internal, potential, and kinetic energy, all of which are involved in most fluid
problems. The energy equation is a power equation based on its units, i.e., the rate of doing work. When thermodynamic principles are involved in fluid flow problems, the equation of state is also
helpful in establishing the relationships between the known and unknown quantities.
The most common application of the energy equation is to so-called single-stream systems, in which a certain amount of fluid energy comes into a system where work is added or extracted. Then, the
energy can come out of the system in another form. Further simplifying the energy equation to incompressible, inviscid flows without energy addition leads to the Bernoulli equation, which can be used
successfully in many practical problems to relate pressures and flow velocities. The Bernoulli equation is also a statement of energy conservation, i.e., fluid exchanges its specific kinetic energy
for static or potential pressure. Nevertheless, the Bernoulli equation must be used carefully and consistently applied within the assumptions and limitations of its derivation.
5-Question Self-Assessment Quickquiz
For Further Thought or Discussion
• The energy equation is often called a redundant equation for analyzing incompressible, inviscid flows. Why?
• What are the three main assumptions used in deriving the Bernoulli equation?
• The flow through the turbine blades of a jet engine can be modeled using the incompressible form of the Bernoulli equation. True or false? Explain.
• In conjunction with the conservation of mass and momentum, the Bernoulli equation can be used to analyze the flow through a propeller operating at low flow speeds. Could you explain how to do
Additional Online Resources
To learn more about the energy equation, as well as the Bernoulli equation and its uses, check out some of these online resources: | {"url":"https://eaglepubs.erau.edu/introductiontoaerospaceflightvehicles/chapter/conservation-of-energy-energy-equation-bernoullis-equation/","timestamp":"2024-11-10T17:28:49Z","content_type":"text/html","content_length":"413547","record_id":"<urn:uuid:1d284fa5-1843-45fd-ad52-b9fc47063813>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.61/warc/CC-MAIN-20241110170046-20241110200046-00436.warc.gz"} |
Mathematics and Statistics
M. Cristina Pereyra, Chair
Department of Mathematics and Statistics
Science and Math Learning Center
MSC01 1115
1 University of New Mexico
Albuquerque, NM 87131-0001
(505) 277-4613
Matthew Blair, Ph.D., University of Washington
Alexandru Buium, Ph.D., University of Bucharest (Romania)
Ronald Christensen, Ph.D., University of Minnesota
Gabriel Huerta, Ph.D., Duke University
Jens Lorenz, Ph.D., University of Münster (Germany)
Terry A. Loring, Ph.D., University of California, Berkeley
Pavel M. Lushnikov, Ph.D., L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences
Michael J. Nakamaye, Ph.D., Yale University
Monika Nitsche, Ph.D., University of Michigan
Maria C. Pereyra, Ph.D., Yale University
Deborah L. Sulsky, Ph.D., New York University
Dimiter Vassilev, Ph.D., Purdue University
Helen Wearing, Ph.D., Heriot-Watt University (Scotland)
Associate Professors
James Degnan, Ph.D., University of New Mexico
Erik B. Erhardt, Ph.D., University of New Mexico
Hongnian Huang, Ph.D., University of Wisconsin, Madison
Alexander O. Korotkevich, Ph.D., L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences
Stephen Lau, Ph.D., University of North Carolina, Chapel Hill
Li Li, Ph.D., University of South Carolina
Yan Lu, Ph.D., Arizona State University
Mohammad Motamed, Ph.D., Royal Institute of Technology (Sweden)
Anna Skripka, Ph.D., University of Missouri
Janet Vassilev, Ph.D., University of California, Los Angeles
Guoyi Zhang, Ph.D., University of Arizona
Maxim Zinchenko, Ph.D., University of Missouri
Assistant Professors
Jehanzeb H. Chaudhry, Ph.D., University of Illinois Urbana, Champaign
Fletcher Christensen, Ph.D., University of California, Irvine
Jacob Schroeder, Ph.D., University of Illinois Urbana, Champaign
Timothy Berkopec, M.S., University of Illinois, Urbana
Jurg Bolli, M.S., University of Zurich (Switzerland)
Karen Champine, M.A., University of New Mexico
Nina Greenberg, M.S., University of New Mexico
Derek Martinez, Ph.D., University of New Mexico
Patricia Oakley, Ph.D., Northwestern University
Kathleen Sorensen, M.S., University of Alaska, Fairbanks
Karen Sorensen-Unruh, M.S., University of New Mexico
Professors Emeriti/Retired
Charles P. Boyer, Ph.D., Pennsylvania State University
Evangelos A. Coutsias, Ph.D., California Institute of Technology
James A. Ellison, Ph.D., California Institute of Technology
Pedro F. Embid, Ph.D., University of California, Berkeley
Roger C. Entringer, Ph.D., University of New Mexico
Archie G. Gibson, Ph.D., University of Colorado
Frank L. Gilfeather, Ph.D., University of California, Irvine
Nancy A. Gonzales, Ed.D., Harvard University
Richard J. Griego, Ph.D., University of Illinois
Liang-Shin Hahn, Ph.D., Stanford University
Reuben Hersh, Ph.D., New York University
Wojciech Kucharz, Ph.D., Jagiellonian University (Poland)
Cornelis W. Onneweer, Ph.D., Wayne State University
Pramod K. Pathak, Ph.D., Indian Statistical Institute
Clifford R. Qualls, Ph.D., University of California, Riverside
Ronald M. Schrader, Ph.D., Pennsylvania State University
Stanly L. Steinberg, Ph.D., Stanford University
William J. Zimmer, Ph.D., Purdue University
Mathematics is fundamental to the formulation and analysis of scientific theories, is a rich and independent field of inquiry, and its study is excellent preparation for life in our highly
specialized society. Active research throughout the mathematical sub-disciplines, spurred on in part by advances in computing technology, leads to new perspectives and applications. The major in
mathematics combines broad study of fundamental theories with in-depth investigation of particular subjects chosen from pure, applied and computational mathematics. A degree in mathematics, either
alone or in combination with study in another field, is excellent preparation for careers in industry, academia, and research institutes.
Statistics is the science of collecting and analyzing data. Statisticians interact with researchers in all the various disciplines of science, engineering, medicine, social science and business to
develop scientifically sound methods in those areas. Most course work in the department is devoted to understanding current methods and the reasoning behind them. A degree in statistics prepares
students for careers in industry, government, academia, and research institutes, as well as being excellent preparation for professional programs in medicine, law, business administration and public
policy and administration.
High School Students: To prepare for college-level Mathematics or Statistics, high school students must take two years of algebra and one year of geometry prior to admission. Students should take
mathematics during their senior year of high school and also take the ACT or SAT examination during that year, for the best preparation and placement into mathematics courses at the University of New
Mexico. Students planning to major in any scientific or technological field should take advanced mathematics courses (Trigonometry, Pre-Calculus, Calculus, etc.) in high school. Placement in
Mathematics or Statistics courses at UNM is based on the highest ACT/SAT Math scores or UNM Placement Exam Math scores.
A beginning student who wishes to take MATH 1522 or a more advanced course must have College Board Advanced Placement scores as described in the Admissions section of this Catalog.
A student who wishes to enroll in a course requiring a prerequisite must earn a grade of "C" (not "C-") or better in the prerequisite course.
Credit Conflicts and Restrictions
1. Content on specific courses overlaps enough to necessitate restricting credit of both courses toward a student’s degree. These courses are not considered equivalent and the completion of the
second course in a pair will not affect a student’s earned hours on the transcript. Students should consult their advisor if they feel the incorrect course is applied for credit on their degree
Students will be allowed to apply only one of the following courses in each pair for credit towards a degree:
• MATH 1430 and MATH 1512.
• MATH 1440 and MATH 1522.
• MATH **314 and MATH **321.
2. Students who have credit for MATH 1220 College Algebra or higher may not then take MATH 1215X, 1215Y, and 1215Z Intermediate Algebra for credit.
3. Mathematics or Statistics coursework dating back more than five years cannot automatically be counted as fulfillment of a prerequisite. Students with older coursework take the placement exam
offered through the University of New Mexico Testing Center to determine what Mathematics or Statistics courses to register for based on their skill level.
Undergraduate courses in Mathematics (MATH) may be categorized as Introductory Course, or as Courses for Teachers and Education Students. Courses in these categories are identified in parenthesis at
the end of the course description according to the following legend:
Introductory Courses (I), Courses for Teachers and Education Students (T). | {"url":"https://catalog.unm.edu/catalogs/2021-2022/colleges/arts-sciences/mathematics-statistics/index.html","timestamp":"2024-11-12T09:45:41Z","content_type":"application/xhtml+xml","content_length":"105281","record_id":"<urn:uuid:edfc58d7-8088-496b-838b-1fbe3d606e8e>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00008.warc.gz"} |
ICHEP 2022
Stefano Anselmi (Istituto Nazionale di Fisica Nucleare)
Baryon Acoustic Oscillations (BAO) are one of the most useful and used cosmological probes to measure cosmological distances independently of the underlying background cosmology. However, in the
current measurements, the inference is done using a theoretical clustering correlation function template where the cosmological and the non-linear damping parameters are kept fixed to fiducial LCDM
values. How can we then claim that the measured distances are model-independent and so useful to select cosmological models?
Motivated by this compelling question we introduce a rigorous tool to measure cosmological distances without assuming a specific background cosmology: the “Purely-Geometric-BAO”. I will explain how
to practically implement this tool with clustering data. This allows us to quantify the effects of some of the standard measurements’ assumptions.
However, the inference is still plagued by the ambiguity of choosing a specific correlation function template to measure cosmological distances. We address this issue by introducing a new approach to
the problem that leverages a novel BAO cosmological standard ruler: the “Linear Point”. Its standard ruler properties allow us to estimate cosmological distances without the need of modeling the
poorly-known late-time nonlinear corrections to the linear correlation function. Last but not least, it also provides smaller statistical uncertainties with respect to the correlation function
template fit. All these features make the Linear Point a promising candidate to properly measure cosmic distances with the upcoming Euclid galaxy survey.
In-person participation Yes | {"url":"https://agenda.infn.it/event/28874/contributions/169370/","timestamp":"2024-11-05T05:49:33Z","content_type":"text/html","content_length":"103083","record_id":"<urn:uuid:b9352774-2ee9-4e13-b615-4775f9021ac7>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00637.warc.gz"} |
What is the SSAT?
The Secondary School Admission Test (SSAT) consists of two parts: a brief essay and a multiple-choice aptitude test, which measures a student's ability to solve mathematics problems, use language,
and comprehend what is read.
The test is administered on two levels:
Lower (for students currently in grades 5-7)
Upper (for students currently in grades 8-11)
SSAT Test Format
The test is divided into five sections. There is a 25-minute writing sample, a 40-minute reading section (40 questions based on about 7 reading passages), and 30 minutes each for the remaining
multiple-choice sections. There are two sections of Math comprising 25 questions each, while the verbal section consists of 30 synonym and 30 analogy questions. The writing sample is not included in
the report sent to students. All questions on the SSAT are equal in value, and scores are based on the number of questions answered correctly, minus one-quarter point for each question answered
incorrectly. Although no points are awarded or deducted for questions left unanswered, students are penalized for questions answered incorrectly or with more than one response.
What is the ISEE?
The Independent School Entrance Examination (ISEE) is a three-hour admission test for entrance into grades five through twelve.
The test has three levels:
• A Lower Level for students in grades four and five who are candidates for admission to grades five and six,
• A Middle Level for students in grades six and seven who are candidates for admission to grades seven and eight
• An Upper Level for students in grades eight to eleven who are candidates for admission to grades nine through twelve.
ISEE Test Format
The ISEE consists of verbal and quantitative reasoning tests that measure a student's capability for learning and reading comprehension and mathematics achievement tests that provide specific
information about an individual's strengths and weaknesses in those areas. All levels include a timed essay written in response to an assigned topic. The essay is not scored, but a copy is forwarded
to the recipient schools along with the Individual Student Report, which shows scaled scores, percentiles, and stanines. The ISEE may only be taken once within a six-month period, and it must be
taken for admission to a school, not as a practice test.
What is the HSPT?
The HSPT, or High School Placement Test, is used by many Catholic schools as a tool to compare applicants from diverse middle schools. The HSPT, unlike the ISEE and SSAT, is given by individual
schools and is generally taken at the school to which a student is applying.
HSPT Test Format
The HSPT has five multiple-choice sections: a 60-question, 16-minute Verbal Skills section; a 52-question, 30-minute Quantitative Skills section; a 62-question, 25-minute Reading section; a
64-question, 45-minute Mathematics section; and a 60-question, 25-minute Language section.
The verbal skills section incorporates synonyms, antonyms, analogies, logic, and verbal classification questions. The quantitative skills section includes series, geometric comparisons, non-geometric
comparisons, and number manipulations. The reading section comprises questions on short passages on a variety of topics. The mathematics section includes mathematical concepts and problem-solving,
covering arithmetic, elementary algebra, and basic geometry. The language section tests capitalization, punctuation, usage, spelling, and composition.
Our Program
Sessions to prepare students for the SSAT/ISEE.HSPT starts in October and meets once a week for 3 hours – one-and-a-half-hour Verbal and one-and-a-half-hour math session – and is designed for
students to take the standardized tests in December.
Classes are small-group sessions, and students are arranged into sessions according to their strengths and weaknesses. For the Verbal prep, students learn 50-70 vocabulary words each week in order to
prepare them better for answering the Synonym and Analogies portion of the test and are also taught specific reading strategies in order to help them ace the reading portions of the test.
In math, students cover all the units that will be tested under these tests, and students also learn specific skills of how to understand the questions, break them down, and then attempt to solve
Teachers teaching the classes are experts at teaching test prep and have over fifteen years of experience teaching such classes. | {"url":"https://ncaslearningcentre.com/ssat-isee-hspt/","timestamp":"2024-11-06T08:59:42Z","content_type":"text/html","content_length":"259263","record_id":"<urn:uuid:8c2fe54f-d08e-4d83-96d0-eaccaa6c89ee>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00247.warc.gz"} |
Math Maze Template
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Constraint Programming
Constraint Programming
Constraint Programming is a relatively new paradigm used to model and solve combinatorial optimization problems. It is most effective on highly combinatorial problem domains such as timetabling,
sequencing, and resource-constrained scheduling. Successful industrial applications utilizing constraint programming technology include the gate allocation system at Hong Kong airport, the yard
planning system at the port of Singapore, and the train timetable generation of Dutch Railways.
Constraint Programming in AIMMS
This chapter discusses the special identifier types and language constructs that AIMMS offers for formulating and solving constraint programming problems. We will see that constraint programming
offers a much wider range of modeling constructs than, for example, integer linear programming or nonlinear programming. Different variable types can be used, while restrictions can be formed using
arbitrary algebraic and logical expressions or by the use of special constraint types, such as alldifferent. In addition, AIMMS offers a specific syntax to express scheduling problems in an intuitive
way, taking advantage of the algorithmic power that underlies constraint-based scheduling.
This chapter
In this chapter, the basic constraint programming concepts are first presented, including different variable types and restrictions in Constraint Programming Essentials. Scheduling Problems discusses
the AIMMS syntax for modeling constraint-based scheduling problems. The final section of this chapter discusses issues related to modeling and solving constraint programs in AIMMS.
An in-depth discussion on constraint programming is given in [RBTW06] and more details on constraint-based scheduling can be found in [BPN01].
Online resources
First, the Association for Constraint Programming organizes an annual summer school, the material for which is posted online. This material can be accessed at http://4c.ucc.ie/a4cp/. The CPAIOR
conference series organizes tutorials alongside each event, the materials of which are posted online. The CPAIOR 2009 tutorial provides an introduction to constraint programming and hybrid methods,
and available online at http://www.tepper.cmu.edu/cpaior09. | {"url":"https://documentation.aimms.com/language-reference/optimization-modeling-components/constraint-programming/index.html","timestamp":"2024-11-04T20:51:32Z","content_type":"text/html","content_length":"18543","record_id":"<urn:uuid:662b49a6-9d90-4d54-8e49-3f738bbacc3a>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00320.warc.gz"} |
Reciprocity laws and K-theory
We associate to a full flag F in an n-dimensional variety X over a field k, a “symbol map” μ[F]: K(F[X])→∑^nK(k). Here, F[X] is the field of rational functions on X, and K (∙) is the K-theory
spectrum. We prove a “reciprocity law” for these symbols: Given a partial flag, the sum of all symbols of full flags refining it is 0. Examining this result on the level of K-groups, we derive the
following known reciprocity laws: The degree of a principal divisor is zero, the Weil reciprocity law, the residue theorem, the Contou-Carrère reciprocity law (when X is a smooth complete curve), as
well as the Parshin reciprocity law and the higher residue reciprocity law (when X is higher-dimensional).
• Contou-Carrère symbol
• K-theory
• Parshin reciprocity
• Parshin symbol
• Reciprocity laws
• Symbols in arithmetic
• Tate vector spaces
All Science Journal Classification (ASJC) codes
• Analysis
• Geometry and Topology
Dive into the research topics of 'Reciprocity laws and K-theory'. Together they form a unique fingerprint. | {"url":"https://cris.iucc.ac.il/en/publications/reciprocity-laws-and-k-theory-2","timestamp":"2024-11-05T16:06:02Z","content_type":"text/html","content_length":"47699","record_id":"<urn:uuid:260e4e9a-239d-485e-90d1-bfe1422ed94a>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00513.warc.gz"} |
Graphical representation of speed in uniform motion
When we apply the time expressed in seconds to the horizontal axis of the coordinate system, and traveled path in meters to the vertical axis, we get a graph of the distance as a function the time,
(es-te) s-t diagram. Specific values of time have specific values of distance. Both are inserted into the table (example of the table is shown below the graph). We transfer the points from the table
to the graph and connect them with line.
For each value of time we construct an appropriate point on the graph. When we connect them, we get a line that represents a graph of the path of uniform movement. The angle that covers the graph
with the positive part of the horizontal axis is greater if the cars moves at a higher speed. The speed is calculated by the given relation for the velocity
v = s / t
In the picture, the blue on the graph shows the velocity as a function of the time, v – t diagram. At any moment the movement in the uniform motion of the velocity is the same, constant. The speed in
the first second is 66.66 m / s, in the second 66.66 m / s, the third and fourth 66.66 m / s.
When we apply these points to a graph and connect them, we get the line which is the parallel to the axis of time. That means that the speed in uniform motion is the same at any time.
The smallest white arrow in the image indicates the initial position, and the longest end indicates the final position of the body. On the other graph, the body returns to the starting position, the
graph of the distance decreases.
The figure shows the velocity graph for the same movement during t, 2t, 3t, the velocity does not change, the graph of the velocity is parallel with the time axis. | {"url":"https://aziza-physics.com/en/graphical-representation-of-speed-in-uniform-motion/","timestamp":"2024-11-10T16:02:46Z","content_type":"text/html","content_length":"90125","record_id":"<urn:uuid:d9403720-753c-4e9a-a14d-6268fd3e72ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00353.warc.gz"} |
How to Practice with the I Ching
Ted Hall, L.Ac.
The I Ching is an ancient book that basically describes how things happen, a guide to navigating the process of being a human. It has been read and consulted by millions of people for over two
thousand years, and remains a profound and useful source of insight and understanding.
This post will provide a basic practical introduction and how-to guide for anyone who’s interested in practicing with the I Ching, and doesn’t have a lot of prior experience with it.
Many people will ask a question about something that is on their minds or happening in their lives when consulting the I Ching, which is part of how it can be used for divination. And of course
that’s an excellent way to use the book, however you don’t need to ask a question, you can simply read your hexagram, and you will feel no less connected to the process than if you asked it something
There are more than a few versions of the I Ching floating around out there in book-land, and it can seem daunting at first glance. Some of them are excellent, and some are not. I’m writing one, but
until that unfathomably ideal version is completed, the best the best thing to do is to check several of them out and see what you like. You could even go to an actual book store that has physical
books in it and look at a few of them. I have a few practical favorites listed at the end of this post.
There are of course website and apps that can do this for you – generate a hexagram, and provide some material for you to read, but I strongly recommend against using any of those “e-convenience”
methods. Here’s why: doing so will separate you from this process and I believe something of value will be lost. Without getting into a whole philosophical discussion here, just please trust me, and
go out and get an actual book and a few pennies, it makes all the difference in the world.
How the book is organized
The I Ching is primarily composed of 64 hexagrams. These are images made up of six lines. The I Ching uses two lines, a solid line to represent Yang _______ or a broken line to represent Yin ___ ___
If we combine these two lines into a trigram (an image with three lines), there are 8 possible combinations. These are the basic 8 trigrams of the I Ching.
And if we combine the trigrams, then there are 64 possible combinations, and these make up the 64 hexagrams of the I Ching. Each of the 64 hexagrams is composed of two of the 8 trigrams, and each of
the trigrams carries some basic meaning, as follows:
________ Heaven– creative, strength, power, advancing
___ ___ Earth– receptive, yielding, reticent, steadfast
___ ___ Thunder– arousing, movement, shock, growth
________ Water – abyss, danger, difficult, profound
___ ___ Mountain– stillness, rest, meditation, immobility
________ Wind (or wood) – gentle, penetrating, small effects
___ ___ Fire– clinging, illuminating, clarity, dependence
________ Lake– joy, openness, satisfaction, excess
Each of the 64 hexagrams is a combination of two of the 8 trigrams, and describes a general circumstance and provides opportunity to understand some basic principles.
Most versions of the I Ching will have several paragraphs or a couple of sections for each of the 64 hexagrams, as well as some information for each of the six lines that make up the hexagram. The
individual lines can offer some insight into the meaning of the hexagram, and some of the lines are sometimes indicated when choosing the hexagram(s) to read. Let’s go over that process.
Throwing the I Ching – The Coin Oracle
The I Ching is customarily read by randomly consulting one or more of the book’s 64 hexagrams. There are a number of different ways to do this, and here we’ll go over the coin-toss method.
The method of tossing coins to select the hexagram is a simple traditional way to let your hexagram be chosen for you. You don’t need special coins, you can use any coins you have, pennies work
The coin-toss method involves throwing 3 coins in the air six times to build a hexagram, one toss of the 3 coins to determine each of the six lines of the hexagram.
If you toss 3 coins into the air, there can only be four possible outcomes –
We will assign a numerical value to both heads and tails to determine the outcome. Let heads represent the Yang side of the coin and it will carry a numerical value of 3. And tails will then
represent the Yin side of the coin and it will carry a numerical value of 2.
An even number is considered complete and therefore Yin in traditional Chinese cosmology, and an odd number is considered incomplete and more active, and therefore Yang.
So the four possible outcomes are as follows:
2 tails & 1 head: 2+2+3 = 7 – fixed yang line _________
2 heads & 1 tail: 3+3+2 = 8 – fixed yin line ____ ____
3 heads: 3+3+3 = 9 – moving yang line _____0_____
3 tails: 2+2+2 = 6 – moving yin line ____ X ____
As you can see, each toss of the coins will produce either a Yin line or a Yang line, and since there are four possible ways the coins can land, there are two types of Yin lines we can get, and two
types of Yang lines.
A line that is all tails is all Yin – and since one of the properties of Yin & Yang is that each one must at its extreme transform into its opposite (just as night must become day, and an inhale must
give way to an exhale, and vice verse), then an all-Yin line will become a Yang line. So a Yin line from 3 tails is called a Yin Moving line – it is unstable and in motion, and it will cause the
hexagram to transform into a second hexagram, in which any moving lines becoming their opposites.
Similarly, a line that is all heads is all Yang, and is likewise unstable, and so in the same way, an all-Yang line will transform into a Yin line. So a Yang line from 3 heads is called a Yang Moving
line – it represents a more extreme & unstable version of Yang, and will cause the hexagram to transform into a second hexagram, in which any moving lines becoming their opposites.
A line that is a mix of heads and tails will produce either a Yin line (2 heads & 1 tail) or a Yang line (2tails & 1 head), and these lines are called Fixed lines. When both Yin and Yang are
represented in the coin toss, then the pattern is considered more balanced and therefore more stable, so these lines are fixed, and unlike the moving lines, they do not transform into their
So by tossing 3 coins six times, we generate a hexagram, and that hexagram can be composed of either moving or fixed lines, or both – moving lines will change to their opposites & form a second
hexagram, while fixed lines will not change.
Moving lines represent a more acute or extreme version of either Yin or Yang than do their fixed counterparts. So a hexagram with one or more moving lines will create a second hexagram in which any
moving lines become their opposite in the second hexagram. A hexagram with no moving lines, only fixed lines, will not form a second hexagram.
The hexagram is constructed from the ground up, so the 1sttoss of the coins determines the first or bottom position line of the hexagram, and so on until the 6thtoss determines the sixth or top
position line of the hexagram.
If one or more moving lines are thrown, then a second hexagram will be formed by rebuilding the first one, except with any moving lines becoming their opposite, so a Yin moving line in the first
hexagram would become a Yang line in the second hexagram. And a Yang Moving line in the original hexagram would become a Yin line in the second hexagram. There will not be any moving lines in the
second hexagram.
Only a hexagram with one or more moving lines will create a second hexagram.
If you have one or move moving lines in your hexagram, you will read the first hexagram along with any of the lines in the positions that were moving lines – do not read any of the fixed lines. And
then you will read the resulting second hexagram, and no lines are read with the second hexagram. If no moving lines are thrown, only fixed lines, the hexagram it forms is read, and no lines are
consulted (as none were indicated).
Let’s go over an example of tossing coins to find a hexagram. In this example, let’s say the first toss is heads + tails + tails. This would have a numerical value of 7 (see above for numerical
values & their outcomes), so this is a Fixed Yang line, which looks like a solid line, like so: _________ The first toss forms the bottom line of the hexagram.
Let’s say the second toss renders a similar outcome, heads + tails + tails, so this also caries the numerical value of 7. So this is also a Fixed Yang line, which looks the same, like so: _________
This is the second line, the one above the bottom line we just threw.
The third toss of our example hexagram will be three tails (no heads) so this carries a numerical value of 6 and produces a Moving Yin line, which looks like a broken Yin line with an “x” in the
middle of it, like so: ___ X ___
Our forth toss will be 2 heads and a tail, which carries a numeric value of 8 and is a Fixed Yin line, depicted as a broken line, like so: ____ ____
And let’s say the fifth toss of the coins is all heads, which carries a numerical value of 9, and forms a Moving Yang line, which looks like a solid line with a circle in the middle of it, like so:
And the sixth toss of the coins will be 2 heads and a tail, a numeric value of 8, which is a Fixed Yin line, depicted like so: ____ ____ This is the sixth and final toss of the coins, forming the top
line of the hexagram.
Here’s what the hexagram we just threw looks like:
You can then look up the hexagram in the chart in your I Ching – most versions of the book will have a chart that you can use to find which number hexagram you threw. The bottom trigram is Lake
(Yang-Yang-Yin), and the top trigram is Water (Yin-Yang-Yin). This forms hexagram 60, often translated as Opposition.
And since we had two moving lines in the hexagram we threw here (lines 3 and 5 were moving lines), then a second hexagram will be formed. The second hexagram happens because the moving lines are
considered unstable and are in the process of changing into their opposites. So the second hexagram generated looks exactly the same as the first one, except for the lines in the 3rdand 5thpositions
are their opposites, like so:
In this second hexagram, the moving lines form the original hexagram have changed into their opposites and the fixed lines from the original hexagram have remained unchanged. The bottom trigram is
now Heaven and the top trigram is Earth, forming hexagram 11, often translated as Peace.
If there had been no moving lines in the first hexagram we threw, only fixed lines, then a second hexagram would not be generated.
So the idea here is that two hexagrams we got are depicting a situation where one circumstance, represented by the initial hexagram, is moving into or could potentially transform towards another
circumstance, represented by the second hexagram. There is some aspect of relationship, connection or even transformation between the two circumstances depicted. In this example, that’d be Limitation
moving towards Peace.
So we read the information for the first hexagram, number 60, Limitation, to understand the ideas presented by that circumstance. Then we had two moving lines in the original hexagram, the 3rdand
5thlines, so after we read about hexagram 60, we read about Hexagram 60 line 3 and hexagram 60 line 5, since those two lines were in motion, and form the circumstance around which Limitation(60)
moves to Peace (11).
Then we go to hexagram 11 and read the information about that hexagram. We do not read any lines in the second hexagram, only the initial hexagram we threw had moving lines, the lines are no longer
moving in the second hexagram.
The initial foray into this process can seem a tiny bit ornate at first, but after a few go-arounds, you’ll find it simple enough and rather easy to do. The I Ching is a fun and insightful oracle to
work with, I’ve enjoyed spending many years with it, and I hope you will find it an engaging tool for your personal development.
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Insertion to Linked List - Explanation, Types and Implementation | Codingeek
Insertion to Linked List – Explanation, Types and Implementation
Insertion to linked list can be done in three ways:
1. Inserting a node at the front of linked list.
2. Inserting a node at the end of linked list.
3. Inserting a node at a specified location of linked list.
Read more – Introduction to Linked List – Explanation and Implementation
Say we have a linked list containing the elements 10, 20 and 30. Need to insert an element 50.
Linked list
1. Inserting a node at the Front of Linked List
As the name implied, inserting the new element 50 before the element 10. A new node will be created containing element 50 pointing to the first node above list. Head will now point to this new node.
Node insertion to Front
Implementation in C
struct node
int data;
struct node *next;
// This function insert a new node to the front of linked list
struct node* insertToFront(struct node *head, int data)
struct node *temp = (struct node*)malloc(sizeof(struct node));
temp->data = data;
temp->next = head;
head = temp;
return head;
void printLinkedList(struct node *list)
while (list)
printf(" %d ", list->data);
list = list->next; // accessing and assigning the next element of list
// Main method to initialize the program
int main()
//Declare and allocate 3 nodes in the heap
struct node* head;
struct node* first = (struct node*)malloc(sizeof(struct node));
struct node* second = (struct node*)malloc(sizeof(struct node));
struct node* third = (struct node*)malloc(sizeof(struct node));
first->data = 10; //assign data in first node
first->next = second; // Link first node with second
second->data = 20; //assign data to second node
second->next = third;
third->data = 30; //assign data to third node
third->next = NULL;
head = first; //head pointing to the linked list
head = insertToFront(head, 50);
return 0;
2. Insertion a node at the End of Linked List
As the name implied, inserting the new element 50 after the 30. A new node will be created containing element 50 which points null and last node of the list point to the new node.
Node insertion to end
Implementation in C
struct node
int data;
struct node *next;
// This function insert a new node to the end of linked list
void insertToEnd(struct node *list, int data)
struct node *temp = (struct node*)malloc(sizeof(struct node));
temp->data = data;
temp->next = NULL;
list = list->next;
list->next = temp;
void printLinkedList(struct node *list)
while (list)
printf(" %d ", list->data);
list = list->next; // accessing and assigning the next element of list
// Main method to initialize the program
int main()
//Declare and allocate 3 nodes in the heap
struct node* head;
struct node* first = (struct node*)malloc(sizeof(struct node));
struct node* second = (struct node*)malloc(sizeof(struct node));
struct node* third = (struct node*)malloc(sizeof(struct node));
first->data = 10; //assign data in first node
first->next = second; // Link first node with second
second->data = 20; //assign data to second node
second->next = third;
third->data = 30; //assign data to third node
third->next = NULL;
head = first; //head pointing to the linked list
insertToEnd(head, 50);
return 0;
3. Insertion a node at the Specified location in linked list
Say want to insert element 50 after element 20. A new node containing 50 will be created and Node containing 20 will point New node and that new node will point the node containing 30.
Node insertion after element 20
Implementation in C
struct node
int data;
struct node *next;
// This function insert a new node after 20
void insertToPosition(struct node *list, int knownData, int data)
struct node *temp = (struct node*)malloc(sizeof(struct node));
temp->data = data;
while(list->data != 20){
list = list->next;
temp->next = list->next;
list->next = temp;
void printLinkedList(struct node *list)
while (list)
printf(" %d ", list->data);
list = list->next; // accessing and assigning the next element of list
// Main method to initialize the program
int main()
//Declare and allocate 3 nodes in the heap
struct node* head;
struct node* first = (struct node*)malloc(sizeof(struct node));
struct node* second = (struct node*)malloc(sizeof(struct node));
struct node* third = (struct node*)malloc(sizeof(struct node));
first->data = 10; //assign data in first node
first->next = second; // Link first node with second
second->data = 20; //assign data to second node
second->next = third;
third->data = 30; //assign data to third node
third->next = NULL;
head = first; //head pointing to the linked list
//Insert after 20
insertToPosition(head, 20, 50);
return 0;
There is one more variant of insertion a node a specified location in which instead of value, a location is given where we have to insert the new node. For ex – insert a node at 3rd position from the
beginning, insert element as the second last element of the list etc. Here element is not mentioned rather position is provided. In such scenarios instead of checking condition on the values, we keep
a track on the number of nodes processed and then insert the node at the specified location.
Knowledge is most useful when liberated and shared. Share this to motivate us to keep writing such online tutorials for free and do comment if anything is missing or wrong or you need any kind of
Keep Learning… Happy Learning.. 🙂
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Covariance matrix preparation for quantum principal component analysis
It was recently shown that quantum PCA can provide an exponential speedup over classical methods for PCA. However, no method to prepare the covariance matrix on a quantum device was known. We fill
this gap with a simple approach, unlocking the possibility of near-term quantum PCA.
We find a simple, near-term method for preparing an approximation of the covariance matrix on a quantum computer, allowing for quantum principal component analysis (quantum PCA). For quantum data,
our method is exact (no approximation). For classical data, our method is PCA without centering, and we provide rigorous bounds on the accuracy.
Our main results are theorems proving that the ensemble average density matrix for a given dataset is either equal to the covariance matrix (for quantum data) or a close approximation of it (for
classical data). We also illustrate this with numerics (see right panel).
For details see
“Covariance matrix preparation for quantum principal component analysis”, M. Hunter Gordon, M. Cerezo, L. Cincio, P.J. Coles. arXiv:2204.03495. | {"url":"https://overqc.sandia.gov/2022/07/22/covariance-matrix-preparation-for-quantum-principal-component-analysis/","timestamp":"2024-11-13T08:46:31Z","content_type":"text/html","content_length":"32011","record_id":"<urn:uuid:49c98ca7-13b8-44a1-bb00-ba80ed961cc3>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00327.warc.gz"} |
llround, llroundf, llroundl - round to nearest integer value
c99 [ flag... ] file... -lm [ library... ]
#include <math.h>
long long llround(double x);
long long llroundf(float x);
long long llroundl(long double x);
These functions rounds their argument to the nearest integer value, rounding halfway cases away from 0 regardless of the current rounding direction.
Upon successful completion, these functions return the rounded integer value.
If x is NaN, a domain error occurs and an unspecified value is returned.
If x is +Inf, a domain error occurs and an unspecified value is returned.
If x is -Inf, a domain error occurs and an unspecified value is returned.
If the correct value is positive and too large to represent as a long long, a domain error occurs and an unspecified value is returned.
If the correct value is negative and too large to represent as a long long, a domain error occurs and an unspecified value is returned.
These functions will fail if:
Domain Error
argument is NaN or ±Inf, or the correct value is not representable as an integer.
If the integer expression (math_errhandling & MATH_ERREXCEPT) is non-zero, then the invalid floating-point exception will be raised.
An application wanting to check for exceptions should call feclearexcept(FE_ALL_EXCEPT) before calling these functions. On return, if fetestexcept(FE_INVALID | FE_DIVBYZERO | FE_OVERFLOW |
FE_UNDERFLOW) is non-zero, an exception has been raised. An application should either examine the return value or check the floating point exception flags to detect exceptions.
These functions differ from the llrint(3M) functions in that the default rounding direction for the llround() functions round halfway cases away from 0 and need not raise the inexact floating-point
exception for non-integer arguments that round to within the range of the return type.
See attributes(7) for descriptions of the following attributes:
ATTRIBUTE TYPE ATTRIBUTE VALUE
Interface Stability Standard
MT-Level MT-Safe | {"url":"https://man.omnios.org/man3m/llround.3m","timestamp":"2024-11-08T12:25:32Z","content_type":"text/html","content_length":"8119","record_id":"<urn:uuid:ffc88349-5723-4efd-a7a9-2fbc775dea33>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00857.warc.gz"} |
Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space
We present a novel method for fast retrieval of exact Minkowski sums of pairs of convex polytopes in ℝ^3, where one of the polytopes frequently rotates. The algorithm is based on pre-computing a
so-called criticality map, which records the changes in the underlying graph-structure of the Minkowski sum, while one of the polytopes rotates. We give tight combinatorial bounds on the complexity
of the criticality map when the rotating polytope rotates about one, two, or three axes. The criticality map can be rather large already for rotations about one axis, even for summand polytopes with
a moderate number of vertices each. We therefore focus on the restricted case of rotations about a single, though arbitrary, axis. Our work targets applications that require exact collisiondetection
such as motion planning with narrow corridors and assembly maintenance where high accuracy is required. Our implementation handles all degeneracies and produces exact results. It efficiently handles
the algebra of exact rotations about an arbitrary axis in ℝ^3, and it well balances between preprocessing time and space on the one hand, and query time on the other. We use Cgal arrangements and in
particular the support for spherical Gaussian-maps to efficiently compute the exact Minkowski sum of two polytopes. We conducted several experiments to verify the correctness of the algorithm and its
implementation, and to compare its efficiency with an alternative (static) exact method. The results are reported.
Original language English
Title of host publication Proceedings - 14th ACM Symposium on Solid and Physical Modeling, SPM'10
Pages 1-10
Number of pages 10
State Published - 2010
Event 14th ACM Symposium on Solid and Physical Modeling, SPM'10 - Haifa, Israel
Duration: 1 Sep 2010 → 3 Sep 2010
Publication series
Name Proceedings - 14th ACM Symposium on Solid and Physical Modeling, SPM'10
Conference 14th ACM Symposium on Solid and Physical Modeling, SPM'10
Country/Territory Israel
City Haifa
Period 1/09/10 → 3/09/10
Dive into the research topics of 'Fast and robust retrieval of Minkowski sums of rotating convex polyhedra in 3-space'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/fast-and-robust-retrieval-of-minkowski-sums-of-rotating-convex-po-2","timestamp":"2024-11-02T08:24:50Z","content_type":"text/html","content_length":"52851","record_id":"<urn:uuid:2bbed235-c713-482f-ac11-98a73006d929>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00658.warc.gz"} |
328 views
Answer to a math question (-5/6)-(-5/4)
96 Answers
-\frac{5}{6}-\left(-\frac{5}{4}\right) = -\frac{5}{6}+\frac{5}{4} Here, LCM is 24. \frac{-5\left(4\right)+5(6))}{24}=\frac{(-20+30)}{24}=\frac{10}{24}=\frac{5}{12}
Frequently asked questions (FAQs)
Find the value of sin(π/4) + cos(π/3) - tan(π/6) + cot(π/2)
Math question: What is the x-coordinate of the point where the graph of the logarithmic function y = log(x) intersects the x-axis?
Math Question: In a slope-intercept equation, the slope is 3 and the y-intercept is -2. Find the equation and graph it on a coordinate plane. | {"url":"https://math-master.org/general/5-6-5-4","timestamp":"2024-11-07T03:39:03Z","content_type":"text/html","content_length":"240137","record_id":"<urn:uuid:c31e727f-d65f-4b08-ab86-4dd52319b797>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00380.warc.gz"} |
Drawing (Complete) Binary Tanglegrams: Hardness, Approximation, Fixed-Parameter Tractability
Algorithmica , Volume onlin
A \emph{binary tanglegram} is a pair $<S,T>$ of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in
phylogenetics, it is essential that both trees are drawn without edge crossing and that the inter-tree edges have as few crossings as possible. It is known that finding a drawing with the minimum
number of crossings is NP-hard and that the problem is fixed-parameter tractable with respect to that number. We prove that under the Unique Games Conjecture there is no constant-factor approximation
for general binary trees. We show that the problem is hard even if both trees are complete binary trees. For this case we give an $O(n^3)$-time 2-approximation and a new and simple fixed-parameter
algorithm. We show that the maximization version of the dual problem for general binary trees can be reduced to a version of \textsc{MaxCut} for which the algorithm of Goemans and Williamson yields a
Additional Metadata
, ,
Byrka, J., Buchin, K., Buchin, M., Nollenburg, M., Okamoto, Y., Silveira, R. I., & Wolff, A. (2010). Drawing (Complete) Binary Tanglegrams: Hardness, Approximation, Fixed-Parameter Tractability.
Algorithmica, onlin. | {"url":"https://ir.cwi.nl/pub/17148","timestamp":"2024-11-15T00:46:36Z","content_type":"text/html","content_length":"26180","record_id":"<urn:uuid:9509661b-4556-4a4b-9412-4e71a73b3491>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00778.warc.gz"} |
Differential Equation and Probabilistic Models of Transport Phenomena in Fluid Flows
Printable PDF
Department of Mathematics,
University of California San Diego
Math 295 - Mathematics Colloquium
Jack Xin
Department of Mathematics, UC Irvine
Differential Equation and Probabilistic Models of Transport Phenomena in Fluid Flows
Transport phenomena in fluid flows are observed ubiquitously in nature such as smoke rings in the air, pollutants in the aquifers, plankton blooms in the ocean, flames in combustion engines, and
stirring a few drops of cream in a cup of coffee. We begin with examples of two dimensional Hamiltonian systems modeling incompressible planar flows, and illustrate the transition from ordered to
chaotic flows as the Hamiltonian becomes more time dependent. We discuss diffusive, sub-diffusive, and residual diffusive behaviors, and their analysis via stochastic differential equation and a so
called elephant random walk model. We then turn to level-set Hamilton-Jacobi models of the flames, and properties of the effective flame speeds in fluid flows under smoothing (such as regular
diffusion and curvature) as well as stretching.
Host: Bo Li
June 8, 2017
4:00 PM
AP&M 6402 | {"url":"https://math.ucsd.edu/seminar/differential-equation-and-probabilistic-models-transport-phenomena-fluid-flows","timestamp":"2024-11-04T13:50:45Z","content_type":"text/html","content_length":"33769","record_id":"<urn:uuid:7c31591b-bbd7-4dc3-b614-710f40d5f574>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00426.warc.gz"} |
Functions and Relations: Proving R is a Function from A to B
• MHB
• Thread starter Sharon
• Start date
In summary, the conversation discussed the conditions for a binary relation R to be a function, namely that R^-1(not) R \subseteq idB and Rnot aR^-1 \supseteq both hold. The identity relation/idA was
defined as a function {(a,a)|a A} over A (respectively, B). The conversation also mentioned the composition of relations and how it relates to functions, with an example of how proving R is a
function implies the two given inclusions. The speaker also requested more information and adherence to rules for future questions.
Let R\subseteq A*B be a binary relation from A to B , show that R is a function if and only if R^-1(not) R \subseteq idB and Rnot aR^-1 \supseteq both hold. Remember that Ida(idB) denotes the
identity relation/ Function {(a.a)|a A} over A ( respectively ,B)
Please see the attachment ,I couldn't write the question properly, and this is only one question but I need help with another one too.
$\text{id}_A\subseteq R\circ R^{-1}$ means that for every $a\in A$ we have $(a,a)\in R\circ R^{-1}$. By the definition of composition of relation, there exists a $b\in B$ such that $(a,b)\in R$ and $
(b,a)\in R^{-1}$. In fact, $(a,b)\in R$ implies $(b,a)\in R^{-1}$, so $(b,a)\in R^{-1}$ does not add useful information, but we have shown that for every $a\in A$ there exists a $b\in B$ such that $
(a,b)\in R$.
Suppose now that $(a,b)\in R$ and $(a,b')\in R$ for some $a\in A$ and $b,b'\in B$. Then $(b,a)\in R^{-1}$, so $(b,b')\in R^{-1}\circ R$. But since $R^{-1}\circ R\subseteq\text{id}_B$, it follows that
It is left to prove the other direction, where the fact that $R$ is a function implies the two inclusions.
Concerning problem 7, could you write what you have done and what is not clear to you? Also, please read the https://mathhelpboards.com/rules/, especially rule #11 for the future.
FAQ: Functions and Relations: Proving R is a Function from A to B
What is the difference between a function and a relation?
A function is a mathematical concept that maps each element of one set (the domain) to a unique element in another set (the range). This means that each input has only one output. A relation, on the
other hand, is a set of ordered pairs where the first element of each pair is related to the second element. This means that one input can have multiple outputs.
What is a domain and range of a function?
The domain of a function is the set of all possible input values for the function. The range is the set of all possible output values. In other words, the domain is the set of x-values and the range
is the set of y-values.
How do you determine if a relation is a function?
To determine if a relation is a function, you can use the vertical line test. If a vertical line intersects the graph of the relation at more than one point, then the relation is not a function. If a
vertical line only intersects the graph at one point, then the relation is a function.
What is a one-to-one function?
A one-to-one function is a function where every output has a unique input. This means that each input has only one output, and each output has only one input. In other words, there are no repeated
x-values or y-values in the function's domain or range.
What is the difference between a linear and a nonlinear function?
A linear function is a function where the graph is a straight line. This means that the rate of change (slope) of the function is constant. A nonlinear function, on the other hand, is a function
where the graph is not a straight line. This means that the rate of change (slope) of the function is not constant. | {"url":"https://www.physicsforums.com/threads/functions-and-relations-proving-r-is-a-function-from-a-to-b.1040525/","timestamp":"2024-11-10T15:03:49Z","content_type":"text/html","content_length":"82600","record_id":"<urn:uuid:56ba6ab7-d340-40c3-be7a-1be9e9a5d747>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00840.warc.gz"} |
Network-Based Statistic (NBS)
ANCOVA rank deficient
Showing 1-15 of 15 posts
ANCOVA rank deficient
Dear NBS' users,
I am running an analysis with two groups (HC and patients) where I want to control for age and gender.
This is a short example of the matrix I am using to do so:
1 1 0 -1 0.5
1 1 0 1 -05
1 1 0 -1 0.5
1 0 1 1 -05
1 0 1 1 -05
1 0 1 -1 0.5
Where the first column is the intercept, the second column the first group, 3rd the second group, 4th column is gender and 5th age (demeaned).
My contrast would be [0 1 -1 0 0] running an F-test.
Now, when I run the analysis with these settings MATLAB gives me the following warning:
Warning: Rank deficient, rank = 4, tol = 1.154632e-14.
Is it possible this is caused by the sample size being too small (HC=26, patients=26)?
What else could the problem be? How is it possible to solve this?
Thank you so much!
RE: ANCOVA rank deficient
Hi Giulia,
Use the following design matrix
The 1st column is intercept, 2nd is patient or control; 3rd is gender and 4th is age.
To test for a between-group difference, use a "t-test" and the contrast:
[0 1 0 0]
[0 -1 0 0]
The first one is patients > controls and the second is patients < controls.
Originally posted by Giulia Forcellini:
Dear NBS' users,
I am running an analysis with two groups (HC and patients) where I want to control for age and gender.
This is a short example of the matrix I am using to do so:
1 1 0 -1 0.5
1 1 0 1 -05
1 1 0 -1 0.5
1 0 1 1 -05
1 0 1 1 -05
1 0 1 -1 0.5
Where the first column is the intercept, the second column the first group, 3rd the second group, 4th column is gender and 5th age (demeaned).
My contrast would be [0 1 -1 0 0] running an F-test.
Now, when I run the analysis with these settings MATLAB gives me the following warning:
Warning: Rank deficient, rank = 4, tol = 1.154632e-14.
Is it possible this is caused by the sample size being too small (HC=26, patients=26)?
What else could the problem be? How is it possible to solve this?
Thank you so much!
RE: ANCOVA rank deficient
Dear Andrew and Giulia
Thanks for the example , it is very useful.
Still have two questions maybe not related to NBS but to statistics in general that you could please kindly acknowledge...
1. under this ancova example , do we need to perform a multiple comparison correction if we perform any additional ttest on the same matrices but with a different group definition ? in other words i
found a significant network with this design but also want to look for differences re-arranging a sub group distribution according to clinical properties.. lets say i split the patients group in two
and compare them with individual t test using the same design agaisnt all the controls.
2. is there a way to define a contrast that only shows the effect of covariates alone? (not accounting for the groups differences) or shall i modify the design matrix ?
Thanks for your patience and grateful help!
sincerely, Juan P
RE: ANCOVA rank deficient
Hi Juan, this is just like a typical F-test: you may want to perform subsequent testing to identify the groups that are driving the result. This can be done using independent t-tests, as you suggest,
although there are more fancy ways of doing this. To test the significance of a single covariate, use a t-test and set the contrast vector equal to 1 in the position corresponding to the covariate.
All other elements in the contrast vector should be zero. This will give you the significance of the covariate.
Originally posted by Juan Pablo Princich:
Dear Andrew and Giulia
Thanks for the example , it is very useful.
Still have two questions maybe not related to NBS but to statistics in general that you could please kindly acknowledge...
1. under this ancova example , do we need to perform a multiple comparison correction if we perform any additional ttest on the same matrices but with a different group definition ? in other words i
found a significant network with this design but also want to look for differences re-arranging a sub group distribution according to clinical properties.. lets say i split the patients group in two
and compare them with individual t test using the same design agaisnt all the controls.
2. is there a way to define a contrast that only shows the effect of covariates alone? (not accounting for the groups differences) or shall i modify the design matrix ?
Thanks for your patience and grateful help!
sincerely, Juan P
RE: ANCOVA rank deficient
Dear Andrew,
I used NBS to compare 2 groups as stated in this post accounting for age and sex differences using a t-test, with a [0 1 0 0] contrast type as you suggest.
I found one component with 6 connection pairs that shall have higher connectivity in patients than in controls as the matrices were arranged in the same way as mentioned in this post (patients first
followed by controls).
But i get confused interpreting results because when i extract the identified connections for all participants and perform individual tests (mann withney, non parametric) on each connection pair
between patients and controls , the values on patients are indeed significantly lower than in controls... contrary to the NBS results.
Will really appreciate your comments and possible explanations for this ambiguity. Please be aware that age and sex effects are not regressed out for the individual connections test, but significance
levels ares still very low (p.001).
Juan P
RE: ANCOVA rank deficient
You might want to check
that you are interpreting the contrast correctly. If you code patients as 1 and controls as 0 (or -1), then the contrast of [0 1 0 0] will give patients greater than controls. Otherwise, if patients
are coded as 0 (or -1) and controls as 1, then contrast [ 0 1 0 0] will give controls > patients.
You might also want to check that your connectivity matrices are ordered correctly.
Originally posted by Juan Pablo Princich:
Dear Andrew,
I used NBS to compare 2 groups as stated in this post accounting for age and sex differences using a t-test, with a [0 1 0 0] contrast type as you suggest.
I found one component with 6 connection pairs that shall have higher connectivity in patients than in controls as the matrices were arranged in the same way as mentioned in this post (patients first
followed by controls).
But i get confused interpreting results because when i extract the identified connections for all participants and perform individual tests (mann withney, non parametric) on each connection pair
between patients and controls , the values on patients are indeed significantly lower than in controls... contrary to the NBS results.
Will really appreciate your comments and possible explanations for this ambiguity. Please be aware that age and sex effects are not regressed out for the individual connections test, but significance
levels ares still very low (p.001).
Juan P
RE: ANCOVA rank deficient
Thanks for the quick response as usual.
Unfortunately the the coding looks correct.. first column intercept term, second patients (1) and controls (0), followed by one column for sex and the last accounting for age. ie..
If i followed you correctly the contras [0 1 0 0] shows higher connectivity in patients than controls.
I used the function (combine.m, kindly provided by you) to combine all subjects in a 4D matrix that were alphabetically ordered.. will recheck visually.
Is it possible to have some connection pairs showing higher connectivity values and at the same time others connections within the same component with lower values? or NBS should show separate
components in that case?
Thanks again for your support!
RE: ANCOVA rank deficient
Thanks, i just realised that matrices lost alphabetic order when compressed in a 3D stack, because controls filenames all started with a capital letter and all patients were low case names.
It resulted in an inverted order for the contrasts i 've used and consequently inverted results.
Just my 2 cents.
thanks for the advice.
Sep 1, 2024 05:09 AM |
Ru-Kai Chen
fujian medicial university
RE: ANCOVA rank deficient
Dear NBS' users
I am running an analysis were I want to control for age and sex.
My study design has two groups (patients and controls) and I would like to use age and gender as covariates to analyse the brain network connections that differ between the two groups
What about setting up that part of the design matrix which models the covariate in NBS like this (in the example I put 3 subjects per group)
The first column represents the constant, the second column is the group (1 for patient, 0 for control), the third column is the sex (1 for male, 0 for female), and the fourth column is the gender,
which seems to be possible to design in this way as I looked through the forum discussions, but I don't know if it is correct or not
Finally how should I look at the main effect of the group and the choice of the test（two-sample t-test？？？）。
Thank you in advance for any kind of advice.
Kind regards
RE: ANCOVA rank deficient
Hi Ru-Kai,
I assume the 4th column is age (not gender). The design matrix looks fine.
The main effect of group would be tested with the contrast of [0 1 0 0] or [0 -1 0 0]. Select two-sample t-test.
If you really want to model both sex and gender separately, this may be difficult because sex and gender would probably be highly correlated, leading to rank issues in the design matrix.
Originally posted by Ru-Kai Chen:
Dear NBS' users
I am running an analysis were I want to control for age and sex.
My study design has two groups (patients and controls) and I would like to use age and gender as covariates to analyse the brain network connections that differ between the two groups
What about setting up that part of the design matrix which models the covariate in NBS like this (in the example I put 3 subjects per group)
The first column represents the constant, the second column is the group (1 for patient, 0 for control), the third column is the sex (1 for male, 0 for female), and the fourth column is the
gender, which seems to be possible to design in this way as I looked through the forum discussions, but I don't know if it is correct or not
Finally how should I look at the main effect of the group and the choice of the test（two-sample t-test？？？）。
Thank you in advance for any kind of advice.
Kind regards
Sep 1, 2024 09:09 AM |
Ru-Kai Chen
fujian medicial university
RE: ANCOVA rank deficient
Dear Andrew
Thank you very much for your answer!
My intention is to use gender and age together as covariates, does this seem feasible?
RE: ANCOVA rank deficient
Yes - that sounds reasonable.
Originally posted by Ru-Kai Chen:
Dear Andrew
Thank you very much for your answer!
My intention is to use gender and age together as covariates, does this seem feasible?
Sep 4, 2024 11:09 AM |
Ru-Kai Chen
fujian medicial university
RE: ANCOVA rank deficient
Dear Andrew
Thank you very much for your answer!
I seem to have a new problem.
When I use NBS to calculate significant connections, I find that at different thresholds, the NBS results file shows different numbers, either 3 or 1. Are they all connections that represent
significance? Is there something wrong with this? (Figure below)
Very much looking forward to your answer?
RE: ANCOVA rank deficient
Each of the three matrices represent a distinct network that was found to be signifcant.
It is possible for multiple networks to be signifciant - not just one.
Originally posted by Ru-Kai Chen:
Dear Andrew
Thank you very much for your answer!
I seem to have a new problem.
When I use NBS to calculate significant connections, I find that at different thresholds, the NBS results file shows different numbers, either 3 or 1. Are they all connections that represent
significance? Is there something wrong with this? (Figure below)
Very much looking forward to your answer?
Sep 4, 2024 11:09 AM |
Ru-Kai Chen
fujian medicial university
RE: ANCOVA rank deficient
Dear Andrew
Thank you very much for your answer!
It seems that different cells represent different significant differences in connectivity（0.0002,0.0134,0.0048,0.0418） | {"url":"https://www.nitrc.org/forum/forum.php?thread_id=6785&forum_id=3444","timestamp":"2024-11-04T22:21:57Z","content_type":"text/html","content_length":"629428","record_id":"<urn:uuid:d7dcc70f-2a45-45ee-bfb0-257d65c047a6>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00002.warc.gz"} |
Input Datasets
Input Datasets for SURF¶
To execute the SURF-NEMO package, several input datasets are required. These include bathymetry datasets, which provide seafloor elevation; coastline datasets, which define the borders between land
and sea; the initial condition dataset, containing initial values for model prognostic variables; and the boundary condition datasets, which contain variables necessary to impose boundary conditions
for mass, momentum, and energy flows at the surface and lateral open boundaries of the domain.
Figure 6.1 illustrates the interfaces and external forcings acting on a typical computational domain.
Figure 6.1. Schematized representation of the interface and external forcing acting on a typical computational domain.
Input Datasets¶
The input datasets for the model are provided in the classic NetCDF format for bathymetry, initial conditions, and lateral boundary conditions. NetCDF is a widely used file format in atmospheric and
oceanic research which allows storage of different types of array based data, along with a short data description. The coastline datasets are instead provided in Shapefile format, a digital vector
data format for geographic information system (GIS) software.
SURF allows also to use, if needed, two different model type of input data during the execution: analysis data for the spin-up period and forecast data afterward. Users can configure these parameters
in the setParFree.json file, specifying path, filename, dimensions, variables, and other data characteristics.
Bathymetry Dataset¶
The bathymetry dataset contains the sea floor elevation and is required for generating the child meshmask file. Users need to configure the necessary parameters in the set_dataDownlBat section of the
configuration file. The data are distributed on a curvilinear spherical grid (which may be regular or irregular) over a region containing the nested domain.
The bathymetry file contains an elevation variable (in meters) with a specific horizontal resolution. Elevation values are relative to a reference level and can either increase (positive values) or
decrease (negative values) with increasing water depth. The coordinate variables (latitude and longitude) can be structured as one- or two-dimensional arrays.
An example CDL (Common Data Language) representation of a bathymetry file is shown in Listing 6.1.1.
netcdf bathymetry_filename {
x = 300;
y = 200;
variables: \\
float lon(y,x);
lon: units = "degrees_east";
float lat(y,x);
lat: units = "degrees_north";
float elevation(y,x);
elevation: units = "m";
Users must specify the following logical parameters in the set_dataDownlBat_fileName section of the user configuration file:
• fileBat_lcompression: Indicates whether the file is compressed (.gzip) or not.
• fileBat_llonFlip: Specifies if the longitude coordinate is defined in the range [0:360] or [-180:+180].
• fileBat_llatInv: Indicates whether the dataset contains latitude values that decrease towards the poles.
• fileBat_ldepthIncr: Defines if the dataset contains seafloor elevation (positive values) that increases with increasing water depth.
• fileBat_lkeepSrcFull: Specifies whether the original downloaded file should be deleted after cropping it for the nested domain.
The available input bathymetry dataset included in the SURF package is the General Bathymetric Chart of the Oceans (GEBCO) 2014, a publicly accessible dataset that provides global bathymetric
coverage with a resolution of 30 arc-seconds. More information can be found on the official GEBCO website here.
Coastline Dataset¶
The coastline dataset defines the borders between land and sea areas and is stored in shapefile format. It is required for generating the child meshmask. Users must configure the necessary parameters
in the set_dataDownlCoast section of the configuration file.
The available coastline dataset within the SURF package is the Global Self-consistent Hierarchical High-resolution Geography (GSHHG) dataset, produced by the National Oceanic and Atmospheric
Administration (NOAA). This dataset includes 20 shapefiles, each containing hierarchically arranged polygons that define shorelines. More information about GSHHG can be found on their official
website here.
The GSHHG data are provided at five different resolutions, each stored as a separate shapefile:
• f (full): highest resolution with a resolution of xx m
• h (high): high resolution with a resolution of xx m
• i (intermediate): intermediate resolution with a resolution of xx m
• l (low): low resolution with a resolution of xx m
• c (coarse): coarsest resolution with a resolution of xx m
Each resolution level contains shorelines organized into four hierarchical categories:
• L1: Boundary between land and ocean
• L2: Boundary between lake and land
• L3: Boundary between islands within lakes and the lakes
• L4: Boundary between ponds within islands and the islands
Initial Condition Datasets¶
To initiate a model run, the initial values for the model's prognostic variables must be provided. These include temperature, salinity, sea surface height, and the zonal and meridional velocity
components. Initial condition datasets are typically derived from coarse-grid model outputs. Users must configure the necessary parameters in the set_dataDownlOceIC section of the configuration file.
The data can be provided on a curvilinear spherical grid (which may be regular or irregular) using an unstaggered or staggered Arakawa-C grid arrangement within a region that contains the nested
domain. The model assumes that all input ocean variables are defined on the same grid.
The coarse-resolution ocean files contain the following variables at a specified horizontal resolution:
• Potential Temperature [\(C\)]
• Salinity [\(PSU\)]
• Sea Surface Height [\(m\)]
• Zonal Velocity [\(ms^{-1}\)]
• Meridional Velocity [\(ms^{-1}\)]
An example CDL (Common Data Language) representation of an initial condition file is shown in Listing 6.1.2.
netcdf fields_filename {
x = 40 ;
y = 35 ;
z = 72 ;
time = UNLIMITED ; // (1 currently)
float lont(y, x) ;
lont:units = "degrees_east" ;
float latt(y, x) ;
latt:units = "degrees_north" ;
float deptht(z) ;
deptht:units = "m" ;
double time(time) ;
time_counter:units = "seconds since
1970-01-01 00:00:00" ;
float temperature(time, z, y, x) ;
temperature:units = "degC" ;
To perform the extrapolation (SOL) of ocean fields (see Section 4.3 for more details), the parent land-sea mask file must be provided as an input dataset. The user needs to configure the required
parameters in the set_dataDownlOceICMesh section of the configuration file.
This file contains information about the coarse-resolution ocean model grids, including the following variables:
• Longitude on TUVF grid points [\(degree\)]
• Latitude on TUVF grid points [\(degree\)]
• Depth on TUVF grid points [\(m\)]
• Land-sea mask on TUVF grid points [0-1]
• Scale factors on TUVF grid points [\(m\)]
An example CDL text representation of this file is shown in Listing 6.1.3.
netcdf meshmask_filename {
dimensions :
x = 677;
y = 253;
z = 72;
t = UNLIMITED; // (7 currently)
variables : \\
float lon(y,x);
float lat(y,x);
float lev(z);
double time(t);
byte tmask(t,z,y,x);
byte umask(t,z,y,x);
byte vmask(t,z,y,x);
byte fmask(t,z,y,x);
float glamt(t,y,x);
float glamu(t,y,x);
float glamv(t,y,x);
float glamf(t,y,x);
float gphit(t,y,x);
float gphiu(t,y,x);
float gphiv(t,y,x);
float gphif(t,y,x);
double e1t(t,y,x);
double e1u(t,y,x);
double e1v(t,y,x);
double e1f(t,y,x);
double e2t(t,y,x);
double e2u(t,y,x);
double e2v(t,y,x);
double e2f(t,y,x);
double e3t(t,z,y,x);
double e3u(t,z,y,x);
double e3v(t,z,y,x);
double e3w(t,z,y,x);
Lateral Open Boundary Condition Datasets¶
In order to integrate the primitive equations, the NEMO ocean model needs to impose appropriate boundary conditions at the ocean-ocean interface (i.e., the sides of the domain not bounded by land).
Lateral open boundary values for the model's prognostic variables must be specified for the entire simulation period. These include fields such as temperature, salinity, sea surface height, and
Users must configure the necessary parameters in the set_dataDownlOceBC_preSpinup and set_dataDownlOceBC_postSpinup sections of the configuration file. The data can be distributed on a curvilinear
spherical grid (regular or irregular) with an unstaggered or staggered Arakawa-C grid arrangement covering the nested domain. The model assumes that all input ocean variables during the pre- and
post-spinup periods are defined on the same grid.
The coarse-resolution ocean files contain the following variables at a given horizontal resolution and temporal frequency:
• Potential Temperature [\(C\)]
• Salinity [\(PSU\)]
• Sea Surface Height [\(m\)]
• Zonal Velocity [\(ms^{-1}\)]
• Meridional Velocity [\(ms^{-1}\)]
An example CDL (Common Data Language) representation of this file is shown in Listing 6.1.4.
netcdf fields_filename {
dimensions :
x = 677;
y = 253;
z = 72;
t = UNLIMITED; // (7 currently)
variables : \\
float lont(x);
lont: units = "degrees_east";
float latt(y);
latt: units = "degrees_north";
float deptht(z);
deptht: units = "m";
double time(t);
time: units = "seconds since
1970-01-01 00:00:00";
float temperature(t,z,y,x);
temperature: units = "degC";
To perform the extrapolation (SOL) of ocean fields (see Section 4.3 for more details), the parent land-sea mask file must be provided as an input dataset. Users must configure the necessary
parameters in the set_dataDownlOceBCMesh section of the configuration file.
This file contains the necessary information about the coarse-resolution ocean model grids and includes the following variables:
• Longitude on TUVF grid points [\(degree\)]
• Latitude on TUVF grid points [\(degree\)]
• Depth on TUVF grid points [\(m\)]
• Land-sea mask on TUVF grid points [0-1]
• Scale factor on TUVF grid points [\(m\)]
An example CDL text representation of this file is shown in Listing 6.1.5.
netcdf meshmask_filename {
dimensions :
x = 677;
y = 253;
z = 72;
t = UNLIMITED; // (7 currently)
variables : \\
float lon(y,x);
float lat(y,x);
float lev(z);
double time(t);
byte tmask(t,z,y,x);
byte umask(t,z,y,x);
byte vmask(t,z,y,x);
byte fmask(t,z,y,x);
float glamt(t,y,x);
float glamu(t,y,x);
float glamv(t,y,x);
float glamf(t,y,x);
float gphit(t,y,x);
float gphiu(t,y,x);
float gphiv(t,y,x);
float gphif(t,y,x);
double e1t(t,y,x);
double e1u(t,y,x);
double e1v(t,y,x);
double e1f(t,y,x);
double e2t(t,y,x);
double e2u(t,y,x);
double e2v(t,y,x);
double e2f(t,y,x);
double e3t(t,z,y,x);
double e3u(t,z,y,x);
double e3v(t,z,y,x);
double e3w(t,z,y,x);
Tidal Datasets for the open boundaries¶
For barotropic solutions, there is an option to include tidal harmonic forcing at the open boundaries in addition to other external data. These tidal datasets include the harmonic constituents for
amplitude and phase of surface height and velocity. Users must configure the required parameters in the set_dataDownlTide section of the configuration file.
The tidal data are distributed on a regular curvilinear spherical grid with either unstaggered or staggered Arakawa-C grid arrangements covering the nested domain. The model assumes that all input
tidal harmonic variables are defined on the same grid.
The barotropic tide files contain the following variables for each harmonic constituent at a given horizontal resolution:
• Tidal elevation complex amplitude (Real and Imaginary parts) [\(mm\)]
• Tidal WE transport complex amplitude (Real and Imaginary parts) [\(cm^2/s\)]
• Tidal SN transport complex amplitude (Real and Imaginary parts) [\(cm^2/s\)]
An example CDL text representation of this file is shown in Listing 6.1.6.
netcdf uv.k1_tpxo8_atlas_30c_v1 {
nx = 10800 ;
ny = 5401 ;
double lon_u(nx) ;
lon_u:units = "degree_east" ;
double lat_u(ny) ;
lat_u:units = "degree_north" ;
double lon_v(nx) ;
lon_v:units = "degree_east" ;
double lat_v(ny) ;
lat_v:units = "degree_north" ;
int uRe(nx, ny) ;
uRe:units = "centimeter^2/sec" ;
int uIm(nx, ny) ;
uIm:units = "centimeter^2/sec" ;
int vRe(nx, ny) ;
vRe:units = "centimeter^2/sec" ;
int vIm(nx, ny) ;
vIm:units = "centimeter^2/sec" ;
A tidal model bathymetry file also needs to be provided as part of the input dataset. Users must configure the required parameters in the set_dataDownlTideMesh section of the configuration file.
This file contains information about the tidal model grids and depth, including the following variables:
• Longitude on TUV grid points [\(degree\)]
• Latitude on TUV grid points [\(degree\)]
• Bathymetry at TUV grid points [\(m\)]
An example CDL text representation of this file is shown in Listing 6.1.7.
netcdf grid_tpxo8atlas_30_v1 {
nx = 10800 ;
ny = 5401 ;
double lon_z(nx) ;
lon_z:units = "degree_east" ;
double lat_z(ny) ;
lat_z:units = "degree_north" ;
double lon_u(nx) ;
lon_u:units = "degree_east" ;
double lat_u(ny) ;
lat_u:units = "degree_north" ;
double lon_v(nx) ;
lon_v:units = "degree_east" ;
double lat_v(ny) ;
lat_v:units = "degree_north" ;
float hz(nx, ny) ;
hz:units = "meter" ;
float hu(nx, ny) ;
hu:units = "meter" ;
float hv(nx, ny) ;
hv:units = "meter" ;
The available input barotropic tide datasets included in the SURF package are derived from the Topex Poseidon Cross-Over (TPXO8-ATLAS) global inverse tide model, obtained using the OTIS (OSU Tidal
Inversion Software) package implementing methods described in Egbert and Erofeeva,2002. For more information, visit the TPXO8-ATLAS website.
The TPXO8 tidal model consists of a multi-resolution bathymetric grid solution, with 1/6° resolution in the global open ocean and 1/30° resolution in shallow-water regions for improved modeling. The
dataset includes complex amplitudes of tidal sea-surface elevations and transports for eight primary (M2, S2, N2, K2, K1, O1, P1, Q1), two long-period (Mf, Mm), and three non-linear (M4, MS4, MN4)
harmonic constituents.
Atmospheric Forcing Datasets¶
To integrate the primitive equations, the NEMO ocean model needs to impose appropriate boundary conditions for the exchange of mass, momentum, and energy at the atmosphere-ocean interface. The
following six fields must be provided for the integration domain:
1. Zonal components of surface ocean stress,
2. Meridional components of surface ocean stress,
3. Solar heat fluxes (Qsr),
4. Non-solar heat fluxes (Qns),
5. Water flux exchanged with the atmosphere (E-P), representing evaporation minus precipitation.
Additionally, an optional field can be provided:
6. Atmospheric pressure at the ocean surface (pa).
The NEMO model offers different methods for providing these fields, controlled by namelist variables (refer to the NEMO Manual). In the SURF platform, the choice of atmospheric forcing is determined
by setting the sbc_iformulat parameter in the user configuration file:
• sbc_iformulat=0 for the MFS bulk formulae,
• sbc_iformulat=1 for the Flux formulation,
• sbc_iformulat=2 for the CORE bulk formulae.
The atmospheric data are distributed on a regular unstaggered grid covering the nested domain. The model assumes that the input atmospheric variables for both the pre- and post-spinup periods are
defined on the same mesh, though different meshes are allowed for different variables. The user must configure the required parameters in the set_dataDownlAtm_preSpinup and
set_dataDownlAtm_postSpinup sections of the configuration file.
MFS bulk formulae¶
The MFS Bulk Formulae are selected by setting sbc_iformulat=0 in the user configuration file.
The atmospheric forcing files contain the following variables at a specific horizontal resolution and temporal frequency:
• 10 m Zonal Wind Component [\(ms^{-1}\)],
• 10 m Meridional Wind Component [\(ms^{-1}\)],
• 2 m Air Temperature [\(K\)],
• 2 m Dew Point Temperature [\(K\)],
• Mean Sea Level Pressure [\(Pa\)],
• Total Cloud Cover [%],
• Total Precipitation [\(m\)].
An example CDL text representation for an atmospheric forcing file with a 3-hour temporal frequency is shown in Listing 6.1.8.
netcdf atmFields_filename {
dimensions :
lon = 245;
lat = 73;
time = UNLIMITED; // (8 currently)
variables : \\
float lon(lon);
lon: units = "degrees_east";
float lat(lat);
lat: units = "degrees_north";
float time(time);
time: units = "seconds since
1970-01-01 00:00:00";
float T2M(time,lat,lon);
T2M: units = "K";
Core bulk formulae¶
The CORE Bulk Formulae are selected by setting sbc_iformulat=2 in the user configuration file.
The atmospheric forcing files contain the following variables at a specified horizontal resolution and temporal frequency:
• 10 m Zonal Wind Component [\(ms^{-1}\)],
• 10 m Meridional Wind Component [\(ms^{-1}\)],
• 2 m Air Temperature [\(K\)],
• 2 m Specific Humidity [%],
• Incoming Long-Wave Radiation [\(W m^{-2}\)],
• Incoming Short-Wave Radiation [\(W m^{-2}\)],
• Total Precipitation (Liquid + Solid) [\(Kg m^{-2} s^{-1}\)],
• Solid Precipitation [\(Kg m^{-2} s^{-1}\)].
An example CDL text representation for the atmospheric forcing file with a 3-hour temporal frequency is shown in Listing 6.1.9.
netcdf atmFields_filename {
dimensions :
lon = 245;
lat = 73;
time = UNLIMITED; // (8 currently)
variables : \\
float lon(lon);
lon: units = "degrees_east";
float lat(lat);
lat: units = "degrees_north";
float time(time);
time: units = "seconds since
1970-01-01 00:00:00";
float T2M(time,lat,lon);
T2M: units = "K";
Flux formulation¶
The Flux Formulation is selected by setting sbc_iformulat=1 in the user configuration file.
The atmospheric forcing files contain the following variables at a specific horizontal resolution and temporal frequency:
• Zonal Wind Stress [0 - 1],
• Meridional Wind Stress [0 - 1],
• Total Heat Flux [0 - 1],
• Solar Radiation Penetration [0 - 1],
• Mass Flux Exchanged [0 - 1],
• Surface Temperature [0 - 1],
• Surface Salinity [0 - 1].
An example CDL text representation for the atmospheric forcing file with a 3-hour temporal frequency is shown in Listing 6.1.10.
netcdf atmFields_filename {
dimensions :
lon = 245;
lat = 73;
time = UNLIMITED; // (8 currently)
variables : \\
float lon(lon);
lon: units = "degrees_east";
float lat(lat);
lat: units = "degrees_north";
float time(time);
time: units = "seconds since
1970-01-01 00:00:00";
float T2M(time,lat,lon);
T2M: units = "K";
To perform the extrapolation (SOL) of atmospheric fields (see Section 4.3 for more details), the atmospheric meshmask file must be provided as an input dataset. Users need to configure the required
parameters in the set_dataDownlAtmMesh section of the configuration file.
The atmospheric meshmask file contains the land-sea mask variable [0-1].
An example of CDL text representation of the atmospheric land-sea mask file is shown in Listing 6.1.11. The time dimension and coordinate variables can also be omitted if necessary.
netcdf atmFields_filename {
dimensions :
lon = 245;
lat = 73;
time = UNLIMITED; // (1 currently)
variables : \\
float lon(lon);
lon: units = "degrees_east";
float lat(lat);
lat: units = "degrees_north";
float time(time);
time: units = "seconds since
1970-01-01 00:00:00";
float LSM(time,lat,lon);
LSM: units = "0-1"; | {"url":"https://www.surf-platform.org/docs/surf_nemo/latest/ch6_1_input-model-datasets/","timestamp":"2024-11-09T19:50:55Z","content_type":"text/html","content_length":"61889","record_id":"<urn:uuid:473bed50-fe05-490e-b081-b785123c6e0b>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00744.warc.gz"} |
Gradient descent is an algorithm which minimizes functions. A set of parameters defines a function, and the gradient descent algorithm starts with the initial set of param values and iteratively
moves toward a set of param values that minimizes the function.
This iterative minimization is achieved using calculus, taking steps in the negative direction of the function gradient, as can be seen in the following diagram:
Gradient descent is the most successful optimization algorithm. As mentioned earlier, it is used to do weights updates in a neural network so that we minimize the loss function. Let's now talk about
an important neural network method called backpropagation, in which we firstly propagate forward and calculate the dot product of inputs with their corresponding weights, and then apply an activation
function to the sum of products which transforms... | {"url":"https://subscription.packtpub.com/book/data/9781788390392/3/ch03lvl1sec16/gradient-descent","timestamp":"2024-11-04T17:53:15Z","content_type":"text/html","content_length":"103610","record_id":"<urn:uuid:60c13d77-a611-4497-bd8d-e3db33daaa2d>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00814.warc.gz"} |
Identifying How Current Changes Depending on the Number of Parallel Paths in a Circuit
Question Video: Identifying How Current Changes Depending on the Number of Parallel Paths in a Circuit Physics • Third Year of Secondary School
A student sets up the circuit shown in the diagram. Initially, the switch is open. When the student closes the switch, will the current flowing through the circuit increase or decrease?
Video Transcript
A student sets up the circuit shown in the diagram. Initially, the switch is open. When the student closes the switch, will the current flowing through the circuit increase or decrease?
Taking a look at this diagram, we see in it a simple circuit. There is a cell providing voltage. And there are two resistors arranged in parallel with one another. On one of the resistor branches, we
see there is a switch which currently is open. We want to figure out when the student closes the switch, so that this bottom section of the circuit is now a complete loop and current can follow
through it, will that overall current in the circuit increase or decrease? It’s an interesting question. And before we figure out what happens to the current when the switch is closed, let’s first
map out where the current can move when the switch is open.
Looking at our cell, we see that conventional current will move in a counterclockwise direction. So, it will come around here, and then it will reach this junction point in the circuit. If the switch
at the bottom part of the circuit were closed, that would mean that the current has the option of travelling through this lower branch. But of course the switch is open. And the current has no such
option. That means 100 percent of the current will flow through the upper branch, through that resistor, and then back around to the negative terminal of the cell. That’s the current in the circuit
with the switch open.
But then we’re told the student closes the switch. When that happens, now it’s possible when current reaches this part of the circuit to continue on downward. There is a closed path for it to follow.
So it’s able to traverse the other resistor and then come up and join the other branch. And as we said, the question is, what happens to the current when the switch is closed? Does it go up overall
or does it go down overall?
Let’s start answering this question by recalling a law of electrical circuits, Ohm’s law. This law says that for a resistor of constant resistance, if we multiply that resistance value by the current
running through it, then that’s equal to the potential difference across the resistor. Looking at our circuit, we see that it has virtually no labels. But that doesn’t mean we can’t apply some
ourselves. What if we go to our cell and say that this cell produces a potential difference of 𝑉? And not only that, but we can also give labels to our unnamed resistors. Let’s call the top resistor
𝑅 sub one. And we’ll call the bottom resistor 𝑅 sub two. Notice that we haven’t specified what these values are or even how they relate to one another. We’re just giving them names.
We’re interested overall in two states of this circuit. One we could call the before state. That’s when the switch is still open and current only flows through the one resistor. If we apply Ohm’s law
in this scenario before the switch is closed, then we can say that 𝑉, the potential difference created by the cell, is equal to 𝐼 — we’ll call it 𝐼 sub 𝑏, the current in this circuit before the
switch is closed — multiplied by the only resistor in the circuit, which is 𝑅 sub one. Recall that when the switch was open, the current had no choice but to travel through this upper branch in the
circuit through the resistor 𝑅 sub one. It didn’t go through 𝑅 sub two at all.
So then, before the switch was closed, this is our Ohm’s law equation. And we can rearrange it to solve for that current, 𝐼 sub 𝑏. 𝐼 sub 𝑏, the current in the circuit before the switch was closed, is
equal to the potential provided by the cell divided by 𝑅 sub one. Of course we don’t know what 𝑉 and 𝑅 sub one are, but we don’t need to. We only need to solve for 𝐼 sub 𝑏 in relation to the current
after the switch is closed, what we’ll call 𝐼 sub 𝑐. So, let’s look at that now. Let’s look at the current in the circuit after the switch is closed.
Before we apply Ohm’s law to the state after the switch is closed, there is one important point to make. When the switch is closed, we noted that the current divides up across the two parallel
branches of the circuit. In other words, if we picked a point on either one of these two parallel branches, we wouldn’t get the total circuit current. But since it is the total circuit current that
we want to solve for, we’re gonna analyze this circuit at a point where the current is undivided, before it splits between the branches. Let’s analyze the circuit right at this point there.
Applying Ohm’s law to this scenario, we can say once again that the potential difference supplied by the cell is equal to the total current in the circuit at that point, what we’ll call 𝐼 sub 𝑐,
multiplied by the total resistance, right now we’ll just call it 𝑅 sub 𝑡, of our circuit. Briefly comparing this equation with our previous equation that we used to solve for 𝐼 sub 𝑏, we can see that
the crux of the matter is 𝑅 sub 𝑡, the total resistance of the circuit, compared to 𝑅 sub one. It’s the comparison between those two values which will determine whether 𝐼 sub 𝑐 is greater than or
less than 𝐼 sub 𝑏. That is, whether the current has increased or decreased.
So then, what is 𝑅 sub 𝑡? What is the total equivalent resistance of our circuit when the switch is closed? Under this condition, we have two resistors arranged in parallel. We’ve called them 𝑅 sub
one and 𝑅 sub two. When resistors are arranged this way and there is exactly two of them, there is a particular mathematical relationship for their overall or equivalent resistance. We can call that
equivalent resistance 𝑅 sub two 𝑝, when two resistors are arranged in parallel. In that case, their overall resistance is equal to the product of their individual resistances divided by their sum.
Let’s now apply this relationship to our scenario with 𝑅 sub one and 𝑅 sub two. When we do this, we see that the total resistance of our parallel circuit, when the switch is closed, is equal to 𝑅 one
times 𝑅 two divided by 𝑅 one plus 𝑅 two. At first glance, substituting that in for 𝑅 sub 𝑡 may not look like it helps us very much. After all, when we rearrange to solve for 𝐼 sub 𝑐, the total
circuit current after the switch is closed, all we get on the right-hand side is this expression. And remember, we don’t know what 𝑉, 𝑅 sub one, or 𝑅 sub two are. But thankfully, we don’t have to.
All we need to know is which is bigger, 𝐼 sub 𝑐 or 𝐼 sub 𝑏.
The way we can figure that out is by rewriting this denominator expression in our 𝐼 sub 𝑐 equation. Notice that it has 𝑅 sub one multiplied by 𝑅 sub two divided by the sum of 𝑅 sub one plus 𝑅 sub
two. That means we could write this a different way. We could express that as 𝑅 sub one multiplied by the quantity, 𝑅 sub two divided by 𝑅 sub one plus 𝑅 sub two. The significance of doing this is
that now we have 𝑉 divided by 𝑅 sub one, which is what we have over for 𝐼 sub 𝑏. That’s equal, simply, to 𝑉 divided by 𝑅 sub one.
So then, our whole question now hinges on this. Is this expression here, 𝑅 sub two divided by 𝑅 sub one plus 𝑅 sub two, is that greater than or less than one? If it’s greater than one, then that
means when we multiply it by 𝑅 sub one, overall, compared to 𝐼 sub 𝑏, 𝐼 sub 𝑐 will be lower. On the other hand, if 𝑅 sub two divided by 𝑅 sub one plus 𝑅 sub two is less than one, then that means when
we multiply it by 𝑅 sub one, our overall current, 𝐼 sub 𝑐, will go up compared to 𝐼 sub 𝑏. So, which is it? Is 𝑅 sub two divided by 𝑅 sub one plus 𝑅 sub two greater than one? Or is it less than one?
We can think about it like this. 𝑅 sub two appears in both the numerator and the denominator. If 𝑅 sub one was equal to zero, then this whole fraction would simply be one. It would be 𝑅 sub two
divided by 𝑅 sub two. But we know that 𝑅 sub one is there in the denominator. And we can assume that 𝑅 sub one is greater than zero. In other words, that resistor really is there in the circuit. And
so, when we add that nonzero number for 𝑅 sub one to the value for 𝑅 sub two and then divide that sum into 𝑅 sub two, what we’ll get is a result which is less than one. And that’s because the
denominator, 𝑅 sub two plus 𝑅 sub one, is greater than the numerator, 𝑅 sub two, by itself. So, if this fraction here is less than one, and we just found that it is, then when we multiply that by 𝑅
sub one and then divide it into 𝑉, what we’re doing is dividing into one numerator by a relatively smaller denominator compared to the denominator for 𝐼 sub 𝑏.
A way of clarifying this mathematically would be to take 𝐼 sub 𝑏, which is equal to 𝑉 divided by 𝑅 sub one, and substitute that in for the expression in 𝐼 sub 𝑐. When we do that, as well as rearrange
the fraction in the denominator of 𝐼 sub 𝑐 as expression, then we find that 𝐼 sub 𝑐 is equal to 𝐼 sub 𝑏 multiplied by this value, which is greater than one. This tells us that the current after the
switch is closed goes up compared to where it was before the switch was closed. That means we can answer this question by saying that the current will increase when the switch is closed. | {"url":"https://www.nagwa.com/en/videos/729183815436/","timestamp":"2024-11-12T07:36:33Z","content_type":"text/html","content_length":"266124","record_id":"<urn:uuid:6cc84f50-e2e0-43a6-aee5-4c64797efd42>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00532.warc.gz"} |
Uniform Polyhedra
People have always been fascinated by symmetry and symmetric objects. Uniform polyhedra are a series of well-defined, highly symmetric solids. Some of them were discovered in ancient times, but the
last addition to the collection dates from 1975.
Uniform Polyhedra are polyhedra with the following properties:
• all faces are regular polygons (which may include star polygons like pentagrams)
• all vertices are equivalent
A few special cases:
• the five regular or Platonic solids (all faces identical convex polygons)
• the thirteen semi-regular or Archimedean solids (all faces convex polygons, but not all identical)
• the four Kepler-Poinsot solids (non-convex, but all faces identical polygons)
• an infinite number of prisms and antiprisms
Excluding the prisms, there are 76 uniform polyhedra. Three of them have tetrahedral symmetry (T[d]), eighteen have octahedral symmetry (17 O[h], 1 O), the remaining fifty-five have icosahedral
symmetry (47 I[h], 8 I). I have made paper models of all of them.
Low-resolution photos of the individual solids can be found by following the links below (loading may be slow, though!).
Descriptions of how to make these models can be found in the book by Wenninger (see below). Unfortunately, the drawings in that book are not completely accurate and will not work for the most complex
There are a number of generalizations of the concept of uniform polyhedra, the most important ones being space-filling tesselations and extension to different numbers of dimensions.
A web-site about uniform polyhedra: http://www.georgehart.com/virtual-polyhedra/uniform-info.html.
Some literature relevant to uniform polyhedra:
Alicia Boole Stott, "On Certain Series of Sections of the Regular Four-dimensional Hypersolids", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam (1e sectie) 1900, 7, 1-24
Alicia Boole Stott, "Geometrical Deduction of Semiregular from Regular Polytopes and Space Fillings", Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam (1e sectie) 1910, 11,
H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, "Uniform Polyhedra", Phil. Trans. Royal. Soc. London A 1953, 246, 401
Magnus J. Wenninger, "Polyhedron Models", Cambridge University Press, 1971
H.S.M. Coxeter, "Regular Polytopes", 3rd edition, Dover Publications, New York, 1973
J. Skilling, "The Complete Set of Uniform Polyhedra", Phil. Trans. Royal. Soc. London A 1975, 278, 111-135
Roman E. Maeder, "Uniform Polyhedra", The Mathematica Journal 1983, 3, 48-57
Zvi Har'El, "Uniform Solution for Uniform Polyhedra", Geometriae Dedicata 1993, 47, 57-110
Peter R. Cromwell, "Kepler's Work on Polyhedra", The Mathematical Intelligencer 1995, 17, 23-33 | {"url":"https://www.phmbudzelaar.com/Polyhedra/Uniform.html","timestamp":"2024-11-13T04:31:46Z","content_type":"application/xhtml+xml","content_length":"4792","record_id":"<urn:uuid:5e97d2c1-ae4c-4141-9558-b11e1e426043>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00374.warc.gz"} |
Analytic theory of GL(3) automorphic forms and applications
November 17 to November 21, 2008
at the
American Institute of Mathematics, Palo Alto, California
organized by
Henryk Iwaniec, Philippe Michel, and K. Soundararajan
This workshop, sponsored by AIM and the NSF, has the goal of providing a description of GL[3] automorphic forms and their L-functions amenable to analytic number theorists and to explain the various
approaches available to perform harmonic analysis on these spaces. A second objective will be to discuss the extension of some of the important tools existing in the GL[2] theory to the GL[3]
context: a typical example is Kuznetzov's formula. A third objective will be to list some important problems known for GL[2] and to identify the main obstructions to the extension of these to GL[3]:
typical problems are non-vanishing problems for central values of L-functions and subconvexity problem. To achieve these goals we plan to bring together analytic number theorists and specialists from
the theory of automorphic forms and related fields who are interested in analytic questions.
In addition to introductory lectures, the workshop will be centered around various "practical activities" conducted by different, possibly non-disjoint teams of people: the goal will be to study some
specific problem of interest, possibly including
• Develop usable tools to perform averages in families of GL[3] automorphic forms and in families GL[3] L-functions in the various possible aspects: the goal will be to make as explicit as possible
the spectral decomposition of the space of automorphic forms. A possibility would be to develop a workable form of the analog of the Kuznetzov formula in the GL[3] context.
• Investigate the behaviour of the automorphic forms as one of the parameters attached to them vary.
• Prove non-vanishing results for GL[3]-L-functions in various aspects hopefully by using the mollification method; that should not be intrinsically difficult, the goal would be to get people used
to the combinatorics underlying the Hecke algebra for GL[3].
• Try to understand the difficulty underlying more serious problems like the subconvexity problem in its various aspects.
The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the
workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the
field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel
working sessions.
The deadline to apply for support to participate in this workshop has passed.
For more information email workshops@aimath.org
Plain text announcement or brief announcement.
Go to the American Institute of Mathematics.
Go to the list of upcoming workshops. | {"url":"https://aimath.org/ARCC/workshops/gl3.html","timestamp":"2024-11-09T19:02:06Z","content_type":"text/html","content_length":"4625","record_id":"<urn:uuid:68f24a0f-88e9-43ed-9275-c4f701b603af>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00623.warc.gz"} |
The absorption transition setween the first and the fourth ener... | Filo
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The absorption transition setween the first and the fourth energy states of hydrogen atom are 3. The emission transitions between these states will be
a. 3
b. 5
c. 4
d. 6
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Question Text The absorption transition setween the first and the fourth energy states of hydrogen atom are 3. The emission transitions between these states will be
Updated On Jan 5, 2023
Topic Calculus
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 81
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Ch. 9 Problems - University Physics Volume 3 | OpenStax
9.1 Types of Molecular Bonds
The electron configuration of carbon is $1s22s22p2.1s22s22p2.$ Given this electron configuration, what other element might exhibit the same type of hybridization as carbon?
Potassium chloride (KCl) is a molecule formed by an ionic bond. At equilibrium separation the atoms are $r0=0.279nmr0=0.279nm$ apart. Determine the electrostatic potential energy of the atoms.
The electron affinity of Cl is 3.89 eV and the ionization energy of K is 4.34 eV. Use the preceding problem to find the dissociation energy. (Neglect the energy of repulsion.)
The measured energy dissociated energy of KCl is 4.43 eV. Use the results of the preceding problem to determine the energy of repulsion of the ions due to the exclusion principle.
9.2 Molecular Spectra
In a physics lab, you measure the vibrational-rotational spectrum of HCl. The estimated separation between absorption peaks is $Δf≈5.5×1011HzΔf≈5.5×1011Hz$. The central frequency of the band is $f0=
9.0×1013Hzf0=9.0×1013Hz$. (a) What is the moment of inertia (I)? (b) What is the energy of vibration for the molecule?
For the preceding problem, find the equilibrium separation of the H and Cl atoms. Compare this with the actual value.
The separation between oxygen atoms in an $O2O2$ molecule is about 0.121 nm. Determine the characteristic energy of rotation in eV.
The characteristic energy of the $N2N2$ molecule is $2.48×10−4eV2.48×10−4eV$. Determine the separation distance between the nitrogen atoms
The characteristic energy for KCl is $1.4×10−5eV.1.4×10−5eV.$ (a) Determine $μμ$ for the KCl molecule. (b) Find the separation distance between the K and Cl atoms.
A diatomic $F2F2$ molecule is in the $l=1l=1$ state. (a) What is the energy of the molecule? (b) How much energy is radiated in a transition from a $l=2l=2$ to a $l=1l=1$ state?
In a physics lab, you measure the vibrational-rotational spectrum of potassium bromide (KBr). The estimated separation between absorption peaks is $Δf≈5.35×1010HzΔf≈5.35×1010Hz$. The central
frequency of the band is $f0=8.75×1012Hzf0=8.75×1012Hz$. (a) What is the moment of inertia (I)? (b) What is the energy of vibration for the molecule?
9.3 Bonding in Crystalline Solids
The CsI crystal structure is BCC. The equilibrium spacing is approximately $r0=0.46nmr0=0.46nm$. If $Cs+Cs+$ ion occupies a cubic volume of $r03r03$, what is the distance of this ion to its “nearest
neighbor” $I+I+$ ion?
The potential energy of a crystal is $−8.10eV−8.10eV$/ion pair. Find the dissociation energy for four moles of the crystal.
The measured density of a NaF crystal is $2.558g/cm32.558g/cm3$. What is the equilibrium separate distance of $Na+Na+$ and $Fl−Fl−$ ions?
What value of the repulsion constant, n, gives the measured dissociation energy of 221 kcal/mole for NaF?
Determine the dissociation energy of 12 moles of sodium chloride (NaCl). (Hint: the repulsion constant n is approximately 8.)
The measured density of a KCl crystal is $1.984g/cm3.1.984g/cm3.$ What is the equilibrium separation distance of $K+K+$ and $Cl−Cl−$ ions?
What value of the repulsion constant, n, gives the measured dissociation energy of 171 kcal/mol for KCl?
The measured density of a CsCl crystal is $3.988g/cm33.988g/cm3$. What is the equilibrium separate distance of $Cs+Cs+$ and $Cl−Cl−$ ions?
9.4 Free Electron Model of Metals
What is the difference in energy between the $nx=ny=nz=4nx=ny=nz=4$ state and the state with the next higher energy? What is the percentage change in the energy between the $nx=ny=nz=4nx=ny=nz=4$
state and the state with the next higher energy? (b) Compare these with the difference in energy and the percentage change in the energy between the $nx=ny=nz=400nx=ny=nz=400$ state and the state
with the next higher energy.
An electron is confined to a metal cube of $l=0.8cml=0.8cm$ on each side. Determine the density of states at (a) $E=0.80eVE=0.80eV$; (b) $E=2.2eVE=2.2eV$; and (c) $E=5.0eVE=5.0eV$.
What value of energy corresponds to a density of states of $1.10×1024eV−11.10×1024eV−1$ ?
Compare the density of states at 2.5 eV and 0.25 eV.
Consider a cube of copper with edges 1.50 mm long. Estimate the number of electron quantum states in this cube whose energies are in the range 3.75 to 3.77 eV.
If there is one free electron per atom of copper, what is the electron number density of this metal?
Determine the Fermi energy and temperature for copper at $T=0KT=0K$.
9.5 Band Theory of Solids
For a one-dimensional crystal, write the lattice spacing (a) in terms of the electron wavelength.
What is the main difference between an insulator and a semiconductor?
What is the longest wavelength for a photon that can excite a valence electron into the conduction band across an energy gap of 0.80 eV?
A valence electron in a crystal absorbs a photon of wavelength, $λ=0.300nmλ=0.300nm$. This is just enough energy to allow the electron to jump from the valence band to the conduction band. What is
the size of the energy gap?
9.6 Semiconductors and Doping
An experiment is performed to demonstrate the Hall effect. A thin rectangular strip of semiconductor with width 10 cm and length 30 cm is attached to a battery and immersed in a 1.50-T field
perpendicular to its surface. This produced a Hall voltage of 12 V. What is the drift velocity of the charge carriers?
Suppose that the cross-sectional area of the strip (the area of the face perpendicular to the electric current) presented to the in the preceding problem is $1mm21mm2$ and the current is
independently measured to be 2 mA. What is the number density of the charge carriers?
A current-carrying copper wire with cross-section $σ=2mm2σ=2mm2$ has a drift velocity of 0.02 cm/s. Find the total current running through the wire.
The Hall effect is demonstrated in the laboratory. A thin rectangular strip of semiconductor with width 5 cm and cross-sectional area $2mm22mm2$ is attached to a battery and immersed in a field
perpendicular to its surface. The Hall voltage reads 12.5 V and the measured drift velocity is 50 m/s. What is the magnetic field?
9.7 Semiconductor Devices
Show that for V less than zero, $Inet≈−I0.Inet≈−I0.$
A p-n diode has a reverse saturation current $1.44×10−8A1.44×10−8A$. It is forward biased so that it has a current of $6.78×10−1A6.78×10−1A$ moving through it. What bias voltage is being applied if
the temperature is 300 K?
The collector current of a transistor is 3.4 A for a base current of 4.2 mA. What is the current gain?
Applying the positive end of a battery to the p-side and the negative end to the n-side of a p-n junction, the measured current is $8.76×10−1A8.76×10−1A$. Reversing this polarity give a reverse
saturation current of $4.41×10−8A4.41×10−8A$. What is the temperature if the bias voltage is 1.2 V?
The base current of a transistor is 4.4 A, and its current gain 1126. What is the collector current?
9.8 Superconductivity
At what temperature, in terms of $TCTC$, is the critical field of a superconductor one-half its value at $T=0KT=0K$ ?
What is the critical magnetic field for lead at $T=2.8KT=2.8K$ ?
A Pb wire wound in a tight solenoid of diameter of 4.0 mm is cooled to a temperature of 5.0 K. The wire is connected in series with a $50-Ω50-Ω$ resistor and a variable source of emf. As the emf is
increased, what value does it have when the superconductivity of the wire is destroyed?
A tightly wound solenoid at 4.0 K is 50 cm long and is constructed from Nb wire of radius 1.5 mm. What maximum current can the solenoid carry if the wire is to remain superconducting? | {"url":"https://openstax.org/books/university-physics-volume-3/pages/9-problems","timestamp":"2024-11-11T13:09:56Z","content_type":"text/html","content_length":"415581","record_id":"<urn:uuid:b57a1fff-8c2c-48e3-ae35-cf7bbddbf35f>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00005.warc.gz"} |
Perron-Frobenius theory of seminorms: A topological approach
For nonnegative matrices A, the well known Perron-Frobenius theory studies the spectral radius ρ(A). Rump has offered a way to generalize the theory to arbitrary complex matrices. He replaced the
usual eigenvalue problem with the equation |Ax| = λ|x| and he replaced ρ(A) by the signed spectral radius, which is the maximum λ that admits a nontrivial solution to that equation. We generalize
this notion by replacing the linear transformation A by a map f:Cn→R whose coordinates are seminorms, and we use the same definition of Rump for the signed spectral radius. Many of the features of
the Perron-Frobenius theory remain true in this setting. At the center of our discussion there is an alternative theorem relating the inequalities f(x) ≥ λ|x| and f(x) < λ|x|, which follows from
topological principals. This enables us to free the theory from matrix theoretic considerations and discuss it in the generality of seminorms. Some consequences for P-matrices and D-stable matrices
are discussed.
Funders Funder number
National Science Foundation
Directorate for Mathematical and Physical Sciences 0201333
• Nonnegative matrices
• P-matrices
• Seminorms
• Theorems on alternative
Dive into the research topics of 'Perron-Frobenius theory of seminorms: A topological approach'. Together they form a unique fingerprint. | {"url":"https://cris.tau.ac.il/en/publications/perron-frobenius-theory-of-seminorms-a-topological-approach","timestamp":"2024-11-08T15:47:29Z","content_type":"text/html","content_length":"51657","record_id":"<urn:uuid:10b02077-5068-4e2c-9847-22e819b1cd42>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00174.warc.gz"} |
AVERAGEIFS प्रकार्य
Returns the arithmetic mean of all cells in a range that satisfy given multiple criteria. The AVERAGEIFS function sums up all the results that match the logical tests and divides this sum by the
quantity of selected values.
AVERAGEIFS(Func_Range; Range1; Criterion[; Range2; Criterion2][; … ; [Range127; Criterion127]])
Func_range – required argument. It is a range of cells, a name of a named range or a label of a column or a row containing values for calculating the mean.
Simple usage
Calculates the average for values of the range B2:B6 that are greater than or equal to 20. Returns 25, because the fifth row does not meet the criterion.
Calculates the average for values of the range C2:C6 that are greater than 70 and correspond to cells of B2:B6 with values greater than or equal to 20. Returns 137.5, because the second and fifth
rows do not meet at least one criterion.
Using regular expressions and nested functions
Calculates the average for values of the range C2:C6 that correspond to all values of the range B2:B6 except its minimum and maximum. Returns 127.5, because the third and fifth rows do not meet at
least one criterion.
Calculates the average for values of the range C2:C6 that correspond to all cells of the A2:A6 range starting with "pen" and to all cells of the B2:B6 range except its maximum. Returns 65, because
only second row meets all criteria.
Reference to a cell as a criterion
If you need to change a criterion easily, you may want to specify it in a separate cell and use a reference to this cell in the condition of AVERAGEIFS function. For example, the above function can
be rewritten as follows:
If E2 = pen, the function returns 65, because the link to the cell is substituted with its content. | {"url":"https://help.libreoffice.org/latest/ne/text/scalc/01/func_averageifs.html","timestamp":"2024-11-05T19:38:27Z","content_type":"text/html","content_length":"27128","record_id":"<urn:uuid:85eb0596-1860-4a3b-99d1-778e2608ca1a>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00264.warc.gz"} |
Boolean Algebra: Definition, Operation, Rules, And Application – Mod Education
Boolean algebra: Definition, Operation, Rules, and Application
Boolean algebra is a fundamental branch of mathematics and a critical concept in digital electronics and computer science. Understanding Boolean algebra is essential for anyone working with digital
systems and logic-based decision-making processes.
Boolean algebra, which is often used in computer science, digital electronics, and numerous areas for decision-making and logical analysis, offers a framework for describing and manipulating binary
data and logical connections using operations like AND, OR, and NOT.
In this article, we will explore the of definition of Boolean algebra, Rules, and application of Boolean algebra. Moreover, for a better understanding of Boolean algebra, we will discuss a detailed
Definition of Boolean algebra
Boolean algebra is a branch of mathematics that deals with variables that have only two distinct values, typically denoted as true (1) or false (0). It is named after George Boole, an English
mathematician from the 19th century who introduced the concept.
The significance of Boolean algebra lies in its applicability to computer science and digital logic design. In computer systems, where values can be either “on” (1) or “off” (0), Boolean algebra
provides the foundation for designing and understanding how logical circuits, such as those found in processors and memory systems, operate.
Boolean Algebra: Operations
Here, we discuss the operation of Boolean algebra.
• AND Operation (· or ∩): If both elements are true or one in AND operation the result is true.
• OR Operation (+ or ∪): The OR operation results in true (1) when at least one of its operands is true. Using the same example, A + B is true if it is raining (A = 1), you have an umbrella (B =
1), or both.
• NOT Operation (¬ or ~): The NOT operation negates the value of a binary variable. In NOT operation if the value of A is true then the result in not operation is True.
Boolean Algebra: Rules
In this section, we will discuss the rules of Boolean algebra and their formula.
AND Identity: A · 1 = A
OR Identity: A + 0 = A
AND Null: A · 0 = 0
OR Null: A + 1 = 1
AND Domination: A · A’ = 0
OR Domination: A + A’ = 1
AND Idempotent: A · A = A
OR Idempotent: A + A = A
AND Complement: A · A’ = 0
OR Complement: A + A’ = 1
¬(¬A) = A
AND De Morgan: ¬ (A · B) = ¬A + ¬B
OR De Morgan: ¬ (A + B) = ¬A · ¬B
AND Absorption: A · (A + B) = A
OR Absorption: A + (A · B) = A
Boolean algebra: Application
Boolean algebra has many practical applications in different fields. Here, we discuss some applications of Boolean algebra.
Boolean algebra is foundational in designing digital circuits, such as computers, microcontrollers, and integrated circuits. The AND, OR, NOT, etc. are the Logic gates used to perform logical
operations, and Boolean expressions help in designing and optimizing these circuits.
Boolean variables and expressions are fundamental in computer programming and software development. They are used for decision-making, control flow, and conditional statements. For example, in
programming, “if” statements evaluate Boolean expressions to determine program behaviour.
In search engines and information retrieval systems, Boolean operators enable users to refine search queries by specifying conditions for the desired results. For instance, you can use “AND” to find
results that meet multiple criteria.
Engineers and designers use Boolean algebra to simulate and test digital circuits before physically building them. Simulation tools help ensure that the circuits will function correctly.
Boolean logic is essential for programming robots to make decisions based on sensor inputs, enabling them to navigate, interact with the environment, and perform tasks autonomously.
How to solve the problems of Boolean algebra?
Solving problems in Boolean algebra involves manipulating Boolean expressions according to a set of postulates and theorems to simplify or prove equivalence between expressions.
Example 1:
Determine the truth table of the following expression P∨P∧Q
With the help of Boolean theorems, we solve the given expression
Factoring (1∨Q) P
Identity Law A ∨1 =1
Identity Law
A∧ 1 = 1
Table of (P∨P∧Q)
P Q P∨P (P∨P) ∧Q
The problems of Boolean algebra can also be solved by using an online Boolean calculator to get the result quickly and accurately.
Example 2:
Determine the truth table of the following expression P(~P) ∧Q
Given data
P(~P) ∧Q
We solve the given expression with the help of Boolean theorem
Idempotent law
Identity Law
A∧ 0 = 0
Boolean Table for P(~P) ∧Q
P Q ~P P∧(~p) (P∧(~P) ∧Q
Final Words
In this article, we have discussed the Boolean algebra article with the help of the definition, Rules, and application of Boolean algebra. Also, with the help of detailed examples Boolean algebra
more explain. Anyone can defend Boolean algebra easily after studying this article.
What fundamental rules govern Boolean algebra?
Some basic laws of Boolean algebra include identity laws, null laws, domination laws, idempotent laws, complement laws, and more. These laws govern how you can simplify and manipulate Boolean
How is Boolean algebra used in digital logic design?
Boolean algebra is essential in designing digital circuits and logic gates. It helps engineers create efficient and functional digital systems.
Can Boolean algebra be applied in everyday life?
Yes, Boolean algebra concepts are used in everyday life in decision-making processes, such as determining whether to take an umbrella if it’s raining or deciding if you have all the ingredients to
cook a meal.
NCERT Solutions for Class 10 Social Science History: India and the Contemporary World-II
NCERT Solutions for Class 10 Social Science Geography: Contemporary India-II
NCERT Solutions for Class 10 Social Science Civics (Political Science): Democratic Politics-II
NCERT Solutions for Class 10 Social Science Economics: Understanding Economic Development – II | {"url":"https://modeducation.com/boolean-algebra-calculator/","timestamp":"2024-11-10T09:40:18Z","content_type":"text/html","content_length":"144841","record_id":"<urn:uuid:646706ad-65d2-4e64-bb35-2927676c048d>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00621.warc.gz"} |
36 University
ACT Math Tip
Vectors: Direction and Magnitude
Brush up on your right triangle skills to handle the most difficult of vector items! #newACTcontent #math #ACT #36U
Step 1: Use the Pythagorean theorem to find the magnitude of the vector.
The vector’s magnitude is approximately 3.5 meters per second.
Step 2: Use a trig ratio to find the direction of the vector.
In this case, we set up the tangent ratio to find the missing angle. The vector’s direction is approximately 31 degrees south of east. | {"url":"https://36university.com/tag/vectors/","timestamp":"2024-11-14T07:24:56Z","content_type":"text/html","content_length":"54282","record_id":"<urn:uuid:2b973954-6b49-4e45-a82a-55ca810f62bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00095.warc.gz"} |
Is crypto-agility the key to quantum-safe security?
Are our current cryptographic methods ready for the quantum revolution? Let’s explore the challenges and solutions shaping the future of digital security in the quantum age.
Tobias Fehenberger
Strong cryptography safeguards our data
With ever-increasing digitization in all areas of our business and social lives, more and more sensitive information is being transferred between IT systems and stored for further processing. It is
obvious that the confidentiality of this data is of paramount importance in communication networks – no one should be able to read sensitive data transferred from our laptops to a data center.
Strong cryptography ensures that data remains private. A central task of cryptography is to exchange a secure key over an insecure channel. This so-called public-key (also known as asymmetric)
cryptography is based on mathematical problems that are simple to calculate in one direction but extremely complex in the other. A well-known example of such trapdoor functions is the prime
factorization that underlies the RSA cryptosystem. This method has found widespread use and is utilized for key exchange and message signing. Prime factorization is suitable for cryptographic
applications because it is extremely computationally intensive to decompose a large number into its prime factors. In contrast, the inverse operation, i.e., the multiplication of large prime numbers,
can be easily performed on ordinary classical computers.
In the past, there have also been successful attacks on symmetric encryption methods that are used for encrypting data rather than exchanging the keys. The Data Encryption Standard (DES), which was
widely used at the time, has fallen victim to sophisticated cryptanalysis and increasing computing power. This led to DES being replaced in 2000 by the Advanced Encryption Standard (AES), which has
since then become the standard for symmetric encryption.
Attacks on cryptosystems
Cryptographic procedures for key exchange such as RSA are, in the first place, abstract mathematical formulations that must be implemented in software for use in practice. Attacks on cryptosystems
are therefore possible from two different directions. On the one hand, flaws can occur in implementing the crypto algorithms, so the complexity of the underlying mathematical problem no longer offers
any protection. An example of this would be if the encryption operation’s computation time allows inferences about the key or the plaintext.
The second type of attack on cryptosystems directly targets the theoretical foundations. If new algorithms can be found that can quickly solve “hard nuts” like the prime factorization mentioned
above, the security of the encryption is no longer guaranteed. Such an algorithm has been known for almost 30 years. Named after Peter Shor, Shor’s algorithm makes it possible to break most classical
cryptographic algorithms for key exchange. To execute Shor’s algorithm, however, a powerful quantum computer is required. Such quantum computers are the subject of intense research in academia and
industry, but the most powerful currently available are still many orders of magnitude from the computational power required to break currently used public-key cryptography. Unfortunately, however,
it is possible today to store highly sensitive data encrypted using classical methods on a large scale. This can then be decrypted in the future using a powerful quantum computer. This attack
scenario is called “store now – decrypt later” and represents a practical threat that needs to be addressed today.
Consequently, the advent of quantum computers makes the key exchange procedures in today’s cryptosystems vulnerable to attacks. The AES algorithm used to encrypt the payload data is, however,
considered quantum secure if a key with a length of at least 256 bits is used.
Modern cryptosystems must embody crypto-agility.
Post-quantum cryptography
Post-quantum cryptography (PQC), an emerging quantum-safe encryption method, is necessary if highly sensitive data is to remain secure for a long time. Quantum security here means that there is no
known efficient algorithm for cracking the process, even on a quantum computer, at this point. In 2016, the US National Institute of Standards and Technology (NIST) launched a project to invite
submissions and peer reviews of new quantum-safe methods. After several rounds of evaluation, NIST in July 2022 selected “CRYSTALS-Kyber” for standardization and sent four other algorithms to another
round of evaluation. (Interesting side note: The SIKE procedure chosen for further assessment was broken just weeks after moving into the fourth round.)
Standardization of Kyber by NIST is generally expected in 2024. However, even before this standardization, and certainly afterward, there are several hurdles to overcome. In addition to unclear
licensing surrounding the use of Kyber, initial implementations of Kyber and all other post-quantum techniques may have a different maturity compared to what classical crypto techniques have achieved
in decades of deployment. It must be assumed that hackers may be able to exploit inevitable flaws in the programming. In addition, no market standard has yet been established, as the recommendations
of the various national institutes and offices diverge. While NIST has selected the high-performance Kyber cryptosystem, the German Federal Office for Information Security (BSI) recommends the more
conservative “Classic McEliece” and “FrodoKEM” methods. NIST only considers these as standardization candidates in the fourth round or, in the case of FrodoKEM, not at all.
What conclusions must be drawn for the implementation of today’s cryptosystems? At this point, there is no definitive guidance on which quantum-safe methods are best suited for specific use cases. As
a result, it is crucial to have the flexibility to update production systems as new information becomes available.
Modern cryptosystems must therefore embody crypto-agility. This means that cryptographic methods should be developed and deployed in IT systems in such a way that they can be adapted to evolving
threat landscapes and cryptographic standards. Even in the event of a weakness in a procedure used, the overall system’s security must not be compromised, and the vulnerability must be patched
promptly utilizing trustworthy update mechanisms.
Agile cryptosystems are thus highly complex and require years of expertise in theoretical and practical cryptography to be implemented securely. However, this security technology can protect
sensitive data in the long term, even if novel side-channel attacks or even quantum computers are used in the attack. | {"url":"https://www.advasecurity.com/en/newsroom/blog/20230906-is-crypto-agility-the-key-to-quantum-safe-security","timestamp":"2024-11-15T01:06:53Z","content_type":"text/html","content_length":"38818","record_id":"<urn:uuid:154b2929-9065-425e-a72d-e83603ac9a52>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00087.warc.gz"} |
An algebra of orders
Did you know that you can add and multiply orders? For any two order structures $A$ and $B$, we can form the ordered sum $A+B$ and ordered product $A\otimes B$, and other natural operations, such as
the disjoint sum $A\sqcup B$, which make altogether an arithmetic of orders. We combine orders with these operations to make new orders, often with interesting properties. Let us explore the
resulting algebra of orders!
One of the most basic operations that we can use to combine two orders is the disjoint sum operation $A\sqcup B$. This is the order resulting from placing a copy of $A$ adjacent to a copy of $B$,
side-by-side, forming a combined order with no instances of the order relation between the two parts. If $A$ is the orange $\vee$-shaped order here and $B$ is the yellow linear order, for example,
then $A\sqcup B$ is the combined order with all five nodes.
Another kind of addition is the ordered sum of two orders $A+B$, which is obtained by placing a copy of $B$ above a copy of $A$, as indicated here by adding the orange copy of $A$ and the yellow
copy of $B$. Also shown is the sum $B+A$, with the summands reversed, so that we take $B$ below and $A$ on top. It is easy to check that the ordered sum of two orders is an order. One notices
immediately, of course, that the resulting ordered sums $A+B$ and $B+A$ are not the same! The order $A+B$ has a greatest element, whereas $B+A$ has two maximal elements. So the ordered sum operation
on orders is not commutative. Nevertheless, we shall still call it addition. The operation, which has many useful and interesting features, goes back at least to the 19th century with Cantor, who
defined the addition of well orders this way.
In order to illustrate further examples, I have assembled here an addition table for several simple finite orders. The choices for $A$ appear down the left side and those for $B$ at the top, with the
corresponding sum $A+B$ displayed in each cell accordingly.
We can combine the two order addition operations, forming a variety of other orders this way.
The reader is encouraged to explore further how to add various finite orders using these two forms of addition. What is the smallest order that you cannot generate from $1$ using $+$ and $\sqcup$?
Please answer in the comments.
We can also add infinite orders. Displayed here, for example, is the order $\N+(1\sqcup 1)$, the natural numbers wearing two yellow caps. The two yellow nodes at the top form a copy of $1\sqcup 1$,
while the natural numbers are the orange nodes below. Every natural number (yes, all infinitely many of them) is below each of the two nodes at the top, which are incomparable to each other. Notice
that even though we have Hasse diagrams for each summand order here, there can be no minimal Hasse diagram for the sum, because any particular line from a natural number to the top would be implied
via transitivity from higher such lines, and we would need such lines, since they are not implied by the lower lines. So there is no minimal Hasse diagram.
This order happens to illustrate what is called an exact pair, which occurs in an order when a pair of incomparable nodes bounds a chain below, with the property that any node below both members of
the pair is below something in the chain. This phenomenon occurs in sometimes unexpected contexts—any countable chain in the hierarchy of Turing degrees in computability theory, for example, admits
an exact pair.
Let us turn now to multiplication. The ordered product $A\otimes B$ is the order resulting from having $B$ many copies of $A$. That is, we replace each node of $B$ with an entire copy of the $A$
order. Within each of these copies of $A$, the order relation is just as in $A$, but the order relation between nodes in different copies of $A$, we follow the $B$ relation. It is not difficult to
check that indeed this is an order relation. We can illustrate here with the same two orders we had earlier.
In forming the ordered product $A\otimes B$, in the center here, we take the two yellow nodes of $B$, shown greatly enlarged in the background, and replace them with copies of $A$. So we have
ultimately two copies of $A$, one atop the other, just as $B$ has two nodes, one atop the other. We have added the order relations between the lower copy of $A$ and the upper copy, because in $B$ the
lower node is related to the upper node. The order $A\otimes B$ consists only of the six orange nodes—the large highlighted yellow nodes of $B$ here serve merely as a helpful indication of how the
product is formed and are not in any way part of the product order $A\otimes B$.
Similarly, with $B\otimes A$, at the right, we have the three enlarged orange nodes of $B$ in the background, which have each been replaced with copies of $A$. The nodes of each of the lower copies
of $A$ are related to the nodes in the top copy, because in $B$ the two lower nodes are related to the upper node.
I have assembled a small multiplication table here for some simple finite orders.
So far we have given an informal account of how to add and multiply ordered ordered structures. So let us briefly be a little more precise and formal with these matters.
In fact, when it comes to addition, there is a slightly irritating matter in defining what the sums $A\sqcup B$ and $A+B$ are exactly. Specifically, what are the domains? We would like to conceive of
the domains of $A\sqcup B$ and $A+B$ simply as the union the domains of $A$ and $B$—we’d like just to throw the two domains together and form the sums order using that combined domain, placing $A$ on
the $A$ part and $B$ on the $B$ part (and adding relations from the $A$ to the $B$ part for $A+B$). Indeed, this works fine when the domains of $A$ and $B$ are disjoint, that is, if they have no
points in common. But what if the domains of $A$ and $B$ overlap? In this case, we can’t seem to use the union in this straightforward manner. In general, we must disjointify the domains—we take
copies of $A$ and $B$, if necessary, on domains that are disjoint, so that we can form the sums $A\sqcup B$ and $A+B$ on the union of those nonoverlapping domains.
What do we mean precisely by “taking a copy” of an ordered structure $A$? This way of talking in mathematics partakes in the philosophy of structuralism. We only care about our mathematical
structures up to isomorphism, after all, and so it doesn’t matter which isomorphic copies of $A$ and $B$ we use; the resulting order structures $A\sqcup B$ will be isomorphic, and similarly for
$A+B$. In this sense, we are defining the sum orders only up to isomorphism.
Nevertheless, we can be definite about it, if only to verify that indeed there are copies of $A$ and $B$ available with disjoint domains. So let us construct a set-theoretically specific copy of $A$,
replacing each individual $a$ in the domain of $A$ with $(a,\text{orange})$, for example, and replacing the elements $b$ in the domain of $B$ with $(b,\text{yellow})$. If “orange” is a specific
object distinct from “yellow,” then these new domains will have no points in common, and we can form the disjoint sum $A\sqcup B$ by using the union of these new domains, placing the $A$ order on the
orange objects and the $B$ order on the yellow objects.
Although one can use this specific disjointifying construction to define what $A\sqcup B$ and $A+B$ mean as specific structures, I would find it to be a misunderstanding of the construction to take
it as a suggestion that set theory is anti-structuralist. Set theorists are generally as structuralist as they come in mathematics, and in light of Dedekind’s categorical account of the natural
numbers, one might even find the origin of the philosophy of structuralism in set theory. Rather, the disjointifying construction is part of the general proof that set theory abounds with isomorphic
copies of whatever mathematical structure we might have, and this is part of the reason why it serves well as a foundation of mathematics for the structuralist. To be a structuralist means not to
care which particular copy one has, to treat one’s mathematical structures as invariant under isomorphism.
But let me mention a certain regrettable consequence of defining the operations by means of a specific such disjointifying construction in the algebra of orders. Namely, it will turn out that neither
the disjoint sum operation nor the ordered sum operation, as operations on order structures, are associative. For example, if we use $1$ to represent the one-point order, then $1\sqcup 1$ means the
two-point side-by-side order, one orange and one yellow, but really what we mean is that the points of the domain are $\set{(a,\text{orange}),(a,\text{yellow})}$, where the original order is on
domain $\set{a}$. The order $(1\sqcup 1)\sqcup 1$ then means that we take an orange copy of that order plus a single yellow point. This will have domain
The order $1\sqcup(1\sqcup 1)$, in contrast, means that we take a single orange point plus a yellow copy of $1\sqcup 1$, leading to the domain
These domains are not the same! So as order structures, the order $(1\sqcup 1)\sqcup 1$ is not identical with $1\sqcup(1\sqcup 1)$, and therefore the disjoint sum operation is not associative. A
similar problem arises with $1+(1+1)$ and $(1+1)+1$.
But not to worry—we are structuralists and care about our orders here only up to isomorphism. Indeed, the two resulting orders are isomorphic as orders, and more generally, $(A\sqcup B)\sqcup C$ is
isomorphic to $A\sqcup(B\sqcup C)$ for any orders $A$, $B$, and $C$, and similarly with $A+(B+C)\cong(A+B)+C$, as discussed with the theorem below. Furthermore, the order isomorphism relation is a
congruence with respect to the arithmetic we have defined, which means that $A\sqcup B$ is isomorphic to $A’\sqcup B’$ whenever $A$ and $B$ are respectively isomorphic to $A’$ and $B’$, and similarly
with $A+B$ and $A\otimes B$. Consequently, we can view these operations as associative, if we simply view them not as operations on the order structures themselves, but on their order-types, that is,
on their isomorphism classes. This simple abstract switch in perspective restores the desired associativity. In light of this, we are free to omit the parentheses and write $A\sqcup B\sqcup C$ and
$A+B+C$, if care about our orders only up to isomorphism. Let us therefore adopt this structuralist perspective for the rest of our treatment of the algebra of orders.
Let us give a more precise formal definition of $A\otimes B$, which requires no disjointification. Specifically, the domain is the set of pairs $\set{(a,b)\mid a\in A, b\in B}$, and the order is
defined by $(a,b)\leq_{A\otimes B}(a’,b’)$ if and only if $b\leq_B b’$, or $b=b’$ and $a\leq_A a’$. This order is known as the reverse lexical order, since we are ordering the nodes in the dictionary
manner, except starting from the right letter first rather than the left as in an ordinary dictionary. One could of course have defined the product using the lexical order instead of the reverse
lexical order, and this would give $A\otimes B$ the meaning of “$A$ copies of $B$.” This would be a fine alternative and in my experience mathematicians who rediscover the ordered product on their
own often tend to use the lexical order, which is natural in some respects. Nevertheless, there is a huge literature with more than a century of established usage with the reverse lexical order, from
the time of Cantor, who defined ordinal multiplication $\alpha\beta$ as $\beta$ copies of $\alpha$. For this reason, it seems best to stick with the reverse lexical order and the accompanying idea
that $A\otimes B$ means “$B$ copies of $A$.” Note also that with the reverse lexical order, we shall be able to prove left distributivity $A\otimes(B+C)=A\otimes B+A\otimes C$, whereas with the
lexical order, one will instead have right distributivity $(B+C)\otimes^* A=B\otimes^* A+C\otimes^* A$.
Let us begin to prove some basic facts about the algebra of orders.
Theorem. The following identities hold for orders $A$, $B$, and $C$.
1. Associativity of disjoint sum, ordered sum, and ordered product.\begin{eqnarray*}A\sqcup(B\sqcup C) &\iso& (A\sqcup B)\sqcup C\\ A+(B+C) &\iso& (A+B)+C\\ A\otimes(B\otimes C) &\iso& (A\otimes B)\
otimes C \end{eqnarray*}
2. Left distributivity of product over disjoint sum and ordered sum.\begin{eqnarray*} A\otimes(B\sqcup C) &\iso& (A\otimes B)\sqcup(A\otimes C)\\ A\otimes(B+C) &\iso& (A\otimes B)+(A\otimes C) \end
In each case, these identities are clear from the informal intended meaning of the orders. For example, $A+(B+C)$ is the order resulting from having a copy of $A$, and above it a copy of $B+C$, which
is a copy of $B$ and a copy of $C$ above it. So one has altogether a copy of $A$, with a copy of $B$ above that and a copy of $C$ on top. And this is the same as $(A+B)+C$, so they are isomorphic.
One can also aspire to give a detailed formal proof verifying that our color-coded disjointifying process works as desired, and the reader is encouraged to do so as an exercise. To my way of
thinking, however, such a proof offers little in the way of mathematical insight into algebra of orders. Rather, it is about checking the fine print of our disjointifying process and making sure that
things work as we expect. Several of the arguments can be described as parenthesis-rearranging arguments—one extracts the desired information from the structure of the domain order and puts that
exact same information into the correct form for the target order.
For example, if we have used the color-scheme disjointifying process described above, then the elements of $A\sqcup(B\sqcup C)$ each have one of the following forms, where $a\in A$, $b\in B$, and $c\
in C$:
We can define the color-and-parenthesis-rearranging function $\pi$ to put them into the right form for $(A\sqcup B)\sqcup C$ as follows:
\pi:(a,\text{orange})\quad&\mapsto\quad \bigl((a,\text{orange}),\text{orange}\bigl) \\
\pi:\bigl((b,\text{orange}),\text{yellow}\bigr)\quad&\mapsto\quad \bigl((b,\text{yellow}),\text{orange}\bigl) \\
\pi:\bigl((c,\text{yellow}),\text{yellow}\bigr)\quad&\mapsto\quad (c,\text{yellow})
In each case, we will preserve the order, and since the orders are side-by-side, the cases never interact in the order, and so this is an isomorphism.
Similarly, for distributivity, the elements of $A\otimes(B\sqcup C)$ have the two forms:
where $a\in A$, $b\in B$, and $c\in C$. Again we can define the desired ismorphism $\tau$ by putting these into the right form for $(A\otimes B)\sqcup(A\otimes C)$ as follows:
\tau:\bigl(a,(b,\text{orange})\bigr)\quad&\mapsto\quad \bigl((a,b),\text{orange}\bigr) \\
And again, this is an isomorphism, as desired.
Since order multiplication is not commutative, it is natural to inquire about the right-sided distributivity laws:
(B+C)\otimes A&\overset{?}{\cong}&(B\otimes A)+(C\otimes A)\\
(B\sqcup C)\otimes A&\overset{?}{\cong}&(B\otimes A)\sqcup(C\otimes A)
Unfortunately, however, these do not hold in general, and the following instances are counterexamples. Can you see what to take as $A$, $B$, and $C$? Please answer in the comments.
1. If $A$ and $B$ are linear orders, then so are $A+B$ and $A\otimes B$.
2. If $A$ and $B$ are nontrivial linear orders and both are endless, then $A+B$ is endless; if at least one of them is endless, then $A\otimes B$ is endless.
3. If $A$ is an endless dense linear order and $B$ is linear, then $A\otimes B$ is an endless dense linear order.
4. If $A$ is an endless discrete linear order and $B$ is linear, then $A\otimes B$ is an endless discrete linear order.
Proof. If both $A$ and $B$ are linear orders, then it is clear that $A+B$ is linear. Any two points within the $A$ copy are comparable, and any two points within the $B$ copy, and every point in the
$A$ copy is below any point in the $B$ copy. So any two points are comparable and thus we have a linear order. With the product $A\otimes B$, we have $B$ many copies of $A$, and this is linear since
any two points within one copy of $A$ are comparable, and otherwise they come from different copies, which are then comparable since $B$ is linear. So $A\otimes B$ is linear.
For statement (2), we know that $A+B$ and $A\otimes B$ are nontrivial linear orders. If both $A$ and $B$ are endless, then clearly $A+B$ is endless, since every node in $A$ has something below it and
every node in $B$ has something above it. For the product $A\otimes B$, if $A$ is endless, then every node in any copy of $A$ has nodes above and below it, and so this will be true in $A\otimes B$;
and if $B$ is endless, then there will always be higher and lower copies of $A$ to consider, so again $A\otimes B$ is endless, as desired.
For statement (3), assume that $A$ is an endless dense linear and that $B$ is linear. We know from (1) that $A\otimes B$ is a linear order. Suppose that $x<y$ in this order. If $x$ and $y$ live in
the same copy of $A$, then there is a node $z$ between them, because $A$ is dense. If $x$ occurs in one copy of $A$ and $B$ in another, then because $A$ is endless, there will a node $z$ above $x$ in
its same copy, leading to $x<z<y$ as desired. (Note: we don’t need $B$ to be dense.)
For statement (4), assume instead that $A$ is an endless discrete linear order and $B$ is linear. We know that $A\otimes B$ is a linear order. Every node of $A\otimes B$ lives in a copy of $A$, where
it has an immediate successor and an immediate predecessor, and these are also immediate successor and predecessor in $A\otimes B$. From this, it follows also that $A\otimes B$ is endless, and so it
is an endless discrete linear order. $\Box$
The reader is encouraged to consider as an exercise whether one can drop the “endless” hypotheses in the theorem. Please answer in the comments.
Theorem. The endless discrete linear orders are exactly those of the form $\Z\otimes L$ for some linear order $L$.
Proof. If $L$ is a linear order, then $\Z\otimes L$ is an endless discrete linear order by the theorem above, statement (4). So any order of this form has the desired feature. Conversely, suppose
that $\P$ is an endless discrete linear order. Define an equivalence relation for points in this order by which $p\sim q$ if and only $p$ and $q$ are at finite distance, in the sense that there are
only finitely many points between them. This relation is easily seen to be reflexive, transitive and symmetric, and so it is an equivalence relation. Since $\P$ is an endless discrete linear order,
every object in the order has an immediate successor and immediate predecessor, which remain $\sim$-equivalent, and from this it follows that the equivalence classes are each ordered like the
integers $\Z$, as indicated by the figure here.
The equivalence classes amount to a partition of $\P$ into disjoint segments of order type $\Z$, as in the various colored sections of the figure. Let $L$ be the induced order on the equivalence
classes. That is, the domain of $L$ consists of the equivalence classes $\P/\sim$, which are each a $\Z$ chain in the original order, and we say $[a]<_L[b]$ just in case $a<_{\P}b$. This is a linear
order on the equivalence classes. And since $\P$ is $L$ copies of its equivalence classes, each of which is ordered like $\Z$, it follows that $\P$ is isomorphic to $\Z\otimes L$, as desired. $\Box$
(Interested readers are advised that the argument above uses the axiom of choice, since in order to assemble the isomorphism of $\P$ with $\Z\otimes L$, we need in effect to choose a center point for
each equivalence class.)
If we consider the integers inside the rational order $\Z\of\Q$, it is clear that we can have a discrete suborder of a dense linear order. How about a dense suborder of a discrete linear order?
Question. Is there a discrete linear order with a suborder that is a dense linear order?
What? How could that happen? In my experience, mathematicians first coming to this topic often respond instinctively that this should be impossible. I have seen sophisticated mathematicians make such
a pronouncement when I asked the audience about it in a public lecture. The fundamental nature of a discrete order, after all, is completely at odds with density, since in a discrete order, there is
a next point up and down, and a next next point, and so on, and this is incompatible with density.
Yet, surprisingly, the answer is Yes! It is possible—there is a discrete order with a suborder that is densely ordered. Consider the extremely interesting order $\Z\otimes\Q$, which consists of $\Q$
many copies of $\Z$, laid out here increasing from left to right. Each tiny blue dot is a rational number, which has been replaced with an entire copy of the integers, as you can see in the magnified
images at $a$, $b$, and $c$.
The order is quite subtle, and so let me also provide an alternative presentation of it. We have many copies of $\Z$, and those copies are densely ordered like $\Q$, so that between any two copies of
$\Z$ is another one, like this:
Perhaps it helps to imagine that the copies of $\Z$ are getting smaller and smaller as you squeeze them in between the larger copies. But you can indeed always fit another copy of $\Z$ between, while
leaving room for the further even tinier copies of $\Z$ to come.
The order $\Z\otimes\Q$ is discrete, in light of the theorem characterizing discrete linear orders. But also, this is clear, since every point of $\Z\otimes\Q$ lives in its local copy of $\Z$, and so
has an immediate successor and predecessor there. Meanwhile, if we select exactly one point from each copy of $\Z$, the $0$ of each copy, say, then these points are ordered like $\Q$, which is dense.
Thus, we have proved:
Theorem. The order $\Z\otimes\Q$ is a discrete linear order having a dense linear order as a suborder.
One might be curious now about the order $\Q\otimes\Z$, which is $\Z$ many copies of $\Q$. This order, however, is a countable endless dense linear order, and therefore is isomorphic to $\Q$ itself.
This material is adapted from my book-in-progress, Topics in Logic, drawn from Chapter 3 on Relational Logic, which incudes an extensive section on order theory, of which this is an important
summative part.
5 thoughts on “An algebra of orders”
1. N
□ Thank you for your answer, which is correct. Can you answer my other questions?
2. Regarding the first question: it seems intuitive enough that any finite order can be generated by $1, +, \sqcup$ (bar emptysetitis, but we I guess can cure it with an empty generating procedure),
so we’re looking for an infinite order. You ask for the “smallest order” that can’t be generated that way, which suggests an ordering of orders — but which one? My only natural idea is
“isomorphic to a suborder”, but this is only a preorder on orders (I mean, even up to order-isomorphism), as $mathbb{N}$ and $\mathbb{N} \sqcup 1$ are both order-isomorphic to a suborder of each
other but are not order-isomorphic themselves. Is there a meaningful order of orders?
Another commenter suggested $\mathbb{N}$ as the smallest order, and you answered this is correct, but isn’t the countable antichain (i.e. identity, or $1 \sqcup 1 \sqcup 1 \cdots$) arguably
“smaller” than $\mathbb{N}$ (or any other infinite order, at least in my meaning above, as it will trivially be a suborder of any of those)?
Really enjoying this series of posts on order theory!
□ Actually, it is not true that every finite order can be generated. Indeed, there is an order with four elements that cannot. The earlier commentator has got it, if you can take the hint
☆ Ah, I get it now – I guess I was biased after looking at that N and forced my own intuition! Classic | {"url":"https://jdh.hamkins.org/an-algebra-of-orders/","timestamp":"2024-11-03T17:04:09Z","content_type":"text/html","content_length":"99514","record_id":"<urn:uuid:61272fbc-cfea-419c-b67d-a0dd7d18c9f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00135.warc.gz"} |
What is: Joint Mean
What is Joint Mean?
Joint Mean refers to a statistical measure that represents the average of two or more random variables considered together. This concept is particularly significant in the fields of statistics, data
analysis, and data science, where understanding the relationships between multiple variables is crucial. The Joint Mean is calculated by taking the sum of the means of the individual variables and
dividing it by the number of variables. This metric is essential for analyzing how different factors interact and influence each other, providing a more comprehensive view of the data set.
Mathematical Representation of Joint Mean
The mathematical representation of the Joint Mean can be expressed as follows: if X and Y are two random variables, the Joint Mean can be calculated using the formula: Joint Mean = (E[X] + E[Y]) / 2,
where E[X] and E[Y] denote the expected values of the variables X and Y, respectively. This formula can be extended to more than two variables, allowing for the calculation of the average across
multiple dimensions. Understanding this representation is crucial for statisticians and data analysts as it lays the groundwork for more complex analyses, such as multivariate statistics.
Applications of Joint Mean in Data Analysis
In data analysis, the Joint Mean serves various applications, particularly in the context of multivariate data sets. For instance, it can be used to assess the average performance of different
variables in a marketing campaign, such as customer engagement and conversion rates. By calculating the Joint Mean, analysts can identify trends and correlations between these variables, enabling
businesses to make informed decisions based on the collective performance rather than isolated metrics. This holistic approach is essential for optimizing strategies and improving overall outcomes.
Joint Mean vs. Marginal Mean
It is important to differentiate between Joint Mean and Marginal Mean. While the Joint Mean considers the average of multiple variables simultaneously, the Marginal Mean focuses on the average of a
single variable, irrespective of others. For example, if we have two variables, X and Y, the Marginal Mean of X would be calculated independently of Y. Understanding this distinction is vital for
data scientists, as it helps in selecting the appropriate statistical measures based on the analysis goals. The Joint Mean provides insights into the interplay between variables, while the Marginal
Mean offers a more straightforward view of individual variable behavior.
Calculating Joint Mean in Practice
To calculate the Joint Mean in practice, one must first gather the data for the variables of interest. For instance, if we are analyzing the heights and weights of a group of individuals, we would
collect the height and weight measurements. After obtaining the means of each variable, the Joint Mean can be computed using the previously mentioned formula. This process can be facilitated by
statistical software or programming languages such as R or Python, which offer built-in functions for calculating means and handling data sets efficiently.
Importance of Joint Mean in Multivariate Analysis
The significance of Joint Mean in multivariate analysis cannot be overstated. It allows researchers and analysts to explore the relationships between multiple variables simultaneously, providing a
richer understanding of the data. For example, in a study examining the impact of education level and income on job satisfaction, calculating the Joint Mean can reveal how these factors collectively
influence overall satisfaction levels. This insight can lead to more targeted interventions and policies aimed at improving job satisfaction based on a comprehensive understanding of the contributing
Limitations of Joint Mean
Despite its usefulness, the Joint Mean has limitations that analysts must consider. One major limitation is that it can obscure individual variable behaviors, particularly when the variables are
highly correlated. In such cases, the Joint Mean may not accurately represent the underlying dynamics of the data. Additionally, the Joint Mean assumes a linear relationship between the variables,
which may not always hold true in real-world scenarios. Analysts should be cautious when interpreting Joint Mean results and consider complementing this measure with other statistical analyses to
gain a more nuanced understanding of the data.
Joint Mean in Machine Learning
In the realm of machine learning, the Joint Mean can play a role in feature engineering and model evaluation. When developing predictive models, understanding the Joint Mean of input features can
help in assessing their collective impact on the target variable. For instance, in a regression model predicting house prices, the Joint Mean of features such as square footage and number of bedrooms
can provide insights into how these factors work together to influence pricing. This understanding can guide feature selection and model refinement, ultimately leading to more accurate predictions.
Conclusion on the Relevance of Joint Mean
The Joint Mean is a powerful statistical tool that enhances the analysis of multivariate data sets. By providing insights into the average behavior of multiple variables considered together, it
enables analysts to uncover relationships and trends that may not be apparent when examining variables in isolation. Whether in data analysis, machine learning, or statistical research, the Joint
Mean serves as a foundational concept that supports more complex analyses and decision-making processes. Understanding its applications, limitations, and calculations is essential for anyone working
in the fields of statistics, data analysis, and data science. | {"url":"https://statisticseasily.com/glossario/what-is-joint-mean/","timestamp":"2024-11-05T07:38:54Z","content_type":"text/html","content_length":"139352","record_id":"<urn:uuid:8b49d23c-89df-46be-81c1-8e18ff62a048>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00029.warc.gz"} |
My ancestor (maybe) shows up on Minimum Wage Historian
Because conquistadors get the longest killing streaks and unlock mad perks.
Am I related to Gaspar Correia? I don’t actually know. However, he was a historian who made up wild crap about monsters and mystical adventures because writing about actual real stuff was lame. So if
we are related at least I know where I get it from.
Vasco – Hey, Correia. What’s on the other side of those mountains? I don’t think the Portuguese have ever gone there. Correia – Oh, those mountains? That’s where there are dragon mummies that are
ridden by ninjas who fight gladiatorial battles in an arena made of solid gold. Vasco – Oh. Groovy. Hey, let’s go pillage the **** out of that village.
Because my ancestors simply did not give a crap. Outnumbered 8,000 to 1? Well, that means we might be late for dinner.
23 thoughts on “My ancestor (maybe) shows up on Minimum Wage Historian”
1. Nature over nurture
2. Sounds plausible to me. I’ve had family on two or more sides of a lot conflicts over the last 500 years. WW II was bad. So was the US Civil War. We actually had two brothers on opposite sides.
3. Here’s an interesting song from the Tories of the US Revolution: It’s called either “Long Live the King of England” or “The Ranger Song””
The Ranger Song
Murder ye bloody heathens
We’ll fight for any reason,
Long Live the King of England
And all his mercenaries;
We’ll kill the Goddamn rebels
Burn the bastards’ homes to pebbles
Long Live the King of England
Up with the flag.
chorus: Singing rape, rape the bastards’ women,
Loot, loot their rum and linen
All we shall leave
Are some bodies and some pregnant women,
Live or die it’s just as well
We’ll all meet again in hell, so
Long Live the King of England
Up with the flag.
Burn the barns and burn the stables
defecate on the dinner tables
Steal the silver and the gold
shoot the young and knife the old
We will loot when we’re done killing
We won’t leave a goddamn shilling
Long Live the King of England
Up with the flag.
We eat children when in season
though a baby is more pleasin’
We’re not crude and we’re not heathens
Just preventing future treasons
Get them while they’re still in bed
And Serve me up a boiled head
Long Live the King of England
Up with the flag.
If you seek a life of glory,
Come and join our noble comp’ny
Take the Shilling, don the redcoat
Rum is free and lasses more so
All that you can steal or carry
You can keep or even marry!
Long live the King of England
Up with the flag.
1. Not bad, how’s the tune go?
1. I don’t really know, it’s kind of a 4 point, if that makes sense. I am ignorant musically. It’s got a marching song cadence, if that makes sense.
2. I’m pretty sure this song was made up by revolutionary war reenactors during the bicentennial. It’s not an original song at all. In fact, I know some of the people who came up with the lyrics
during a drinking bout around a campfire! I could be wrong, but this is what I’ve heard from the community. Funny to see how much it’s spread.
4. >>Because my ancestors simply did not give a crap. Outnumbered 8,000 to 1? Well, that means we might be late for dinner.<<
Sounds like your ancestors were Honey Badgers!
1. Money badgers.
5. Gee, that makes mine sound like under achievers. We got tossed out of Scotland for “liftin’ th’ kai”, ejected from Ulster, and invited to leave several colonies and territories prior to 1860. Oh,
and one black sheep designed and built clipper ships, while another was Chief Justice of the U.S.
Scots-Irish vs. Spanish vs. honey-badgers. Now there would be a battle!
6. If ‘ol Gaspar had a word processor might we have seen “Heathen Hunters International” or “Pagan Hunters Crusade”?
Although number one in manuscript sales for Science Fictions and Rural Fantasy for many Sabbaths running, HHI would have of course been panned by the Papal Indulgence Times scroll inquisitors
guild as ‘pulpy ‘old school’ ‘trash’ that could only appeal to the sensibilities of the ‘lowest common serf’.
To wit:
“…’Cause anyone who’s anyone knows that there’s no such thing as heart-chucking ninja super priests atop pyramids with snarling pet Chupacabras and obsidian knives forged in the fiery depths of
hell itself by demon snakes with wings.”
“And come on? Deadly Mermaids are supposed to be harmless sea cows? Sea Cows?! Tell that to all the sailors who’ve followed their siren call off the edge of the ocean blue…Next thing you know,
Gaspar will entertain the peasants by writing tales positing how Chicken of the Sea is really just fish in a can.
Dear Mother Mary! Talk about suspension of disbelief…”
7. Conquistadors: The Honey Badgers of History….if honey badgers were evil, greedy and fanatical.
8. Hey,
I just finished reading Hard Magic. It was the first book of yours that i have come across. I wanted to thank you for such a fun experience.
I know it’s not related to the post but I figured you might actually read this.
9. Hernando Cortez off loaded his men, horses, and supplies to the beach. He then torched the ships, pointed inland and said (in medieval Spanish) “Conquer or die!”
Now there’s a man who understands “Motivation.”
He could have taught Mr. Lombardi a thing or two.
10. How long do you think a denizen of todays’ modern world would have lasted in the 15-19th Centuries? Don’t use us as your starting point. just take 10 college grads, 5 male, 5 female, from any
major university. *NONE* can be ROTC grads of any kind.
I wager they’d be toast. The Europeans might last 15 seconds longer because they might have the lingo. You know what happens to the attractive women.
Even a 3rd world peasant from South America would do better. Our societies have developed entire nations of wusses. They channel their aggressive people in to the military and police.
If you make a reference to Rome and Carthage regarding today’s GWOT, you are “too aggressive”. Uh, pardon me, but they want to cut my head off.
1. Well, yeah, people who are not used to an environment won’t do well there. How long would a 15C person last in Los Angeles? How long would either set last in New Guinea, or the Kalahari?
1. A 15thC person in LA would be running a bike gang or Crip set in a week.
11. I suggest reading a copy of Bernal Diaz del Castillo’s account of the Conquest.
It’s available for free on the gutenberg site.
12. I suggest reading a copy of Bernal Diaz del Castillo’s account of the Conquest.
It’s available for free on the gutenberg site.
13. I suggest reading a copy of Bernal Diaz del Castillo’s account of the Conquest.
It’s available for free on the gutenberg site.
( I appologize for the triple post … wordpress is being retarded again, and defaulting to too much personal info )
1. What’s that book we should read again? And where can we find it?
1. ( Now I’m getting angry … WordPress just inflicted a toolbar on my browser … I’ll need to root it out and block that as well. )
Díaz del Castillo, Bernal (1963) [1632]. The Conquest of New Spain.
That was the latest English translation of Conquistador Bernal’s original work, and is still under copyright. There are some not so good bowdlerized versions in English dating from the
1860s on the Project Gutenberg site for free. The original Spanish/Castilian version is also available on the net.
14. check Sunday’s Schlock Mercenary for Howard Tayler’s cool
caricature of Larry!!
Tetsubo included at no extra charge!
1. The characiture of Larry is great.
Has that Resigned look of “Customer has asked me one to many stupid questions and spent way to long asking them before my coffee and I might use this tetsubo to bludgeon them” | {"url":"https://monsterhunternation.com/2012/03/23/my-ancestor-maybe-shows-up-on-minimum-wage-historian/","timestamp":"2024-11-04T13:59:03Z","content_type":"text/html","content_length":"102588","record_id":"<urn:uuid:6a44f4e5-efa9-4c9f-95aa-a13b43f59adc>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00784.warc.gz"} |
class unreal.PhysicsAsset(outer: Object | None = None, name: Name | str = 'None')¶
Bases: Object
PhysicsAsset contains a set of rigid bodies and constraints that make up a single ragdoll. The asset is not limited to human ragdolls, and can be used for any physical simulation using bodies and
constraints. A SkeletalMesh has a single PhysicsAsset, which allows for easily turning ragdoll physics on or off for many SkeletalMeshComponents The asset can be configured inside the Physics
Asset Editor. see: https://docs.unrealengine.com/InteractiveExperiences/Physics/PhysicsAssetEditor see: USkeletalMesh
C++ Source:
□ Module: Engine
□ File: PhysicsAsset.h
Editor Properties: (see get_editor_property/set_editor_property)
□ constraint_profiles (Array[Name]): [Read-Write]
□ not_for_dedicated_server (bool): [Read-Write] If true, we skip instancing bodies for this PhysicsAsset on dedicated servers
□ physical_animation_profiles (Array[Name]): [Read-Write]
□ solver_iterations (SolverIterations): [Read-Only] Old solver settings shown for reference. These will be removed at some point. When you open an old asset you should see that the settings
were transferred to “SolverSettings” above. You should usually see: SolverSettings.PositionIterations = OldSettings.SolverIterations * OldSetting.JointIterations;
SolverSettings.VelocityIterations = 1; SolverSettings.ProjectionIterations = 1;
□ solver_settings (PhysicsAssetSolverSettings): [Read-Write] Solver settings when the asset is used with a RigidBody Anim Node (RBAN).
□ solver_type (PhysicsAssetSolverType): [Read-Write] Solver type used in physics asset editor. This can be used to make what you see in the asset editror more closely resembles what you see in
game (though there will be differences owing to framerate variation etc). If your asset will primarily be used as a ragdoll select “World”, but if it will be used in the AnimGraph select
□ thumbnail_info (ThumbnailInfo): [Read-Only] Information for thumbnail rendering
get_constraint_by_bone_names(bone1_name, bone2_name) ConstraintInstanceAccessor¶
Gets a constraint by its joint name
○ bone1_name (Name) – name of the first bone in the joint
○ bone2_name (Name) – name of the second bone in the joint
ConstraintInstance accessor to the constraint data
Return type:
get_constraint_by_name(constraint_name) ConstraintInstanceAccessor¶
Gets a constraint by its joint name
constraint_name (Name) – name of the constraint
ConstraintInstance accessor to the constraint data
Return type:
get_constraints(includes_terminated) Array[ConstraintInstanceAccessor]¶
Gets all constraints
includes_terminated (bool) –
out_constraints (Array[ConstraintInstanceAccessor]): returned list of constraints matching the parameters
Return type: | {"url":"https://dev.epicgames.com/documentation/en-us/unreal-engine/python-api/class/PhysicsAsset?application_version=5.3","timestamp":"2024-11-05T22:42:41Z","content_type":"text/html","content_length":"18034","record_id":"<urn:uuid:cc511150-7815-4fb5-8a56-5fafb011866e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00615.warc.gz"} |
- Middle Grades
In this Early Edge video lesson, you'll learn more about Positive/Negative/Even/Odd Numbers, so you can be successful when you take on high-school Math & Arithmetic.
6th grade math knowledge map info
This site gives a detailed explanation about solving absolute-value equations, absolute-value inequalities, and graphing them.
In this Early Edge video lesson, you'll learn more about Number Families, so you can be successful when you take on high-school Math & Arithmetic.
This video includes sample problems and step-by-step explanations of number properties and absolute value equations for the California Standards Test.
Quiz on Lines, Absolute Values, and Polynomials
This site gives a description of working with absolute value equations involving inequalities. It also shows the meaning of absolute value equations using number lines. The site also provides
examples that students can practice with. | {"url":"https://static.tutor.com/resources/math/middle-grades/absolute-values","timestamp":"2024-11-03T02:55:05Z","content_type":"application/xhtml+xml","content_length":"60865","record_id":"<urn:uuid:6becd3f8-504c-4644-a71c-68e8666f4a27>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00334.warc.gz"} |
[Solved] Let A = (3, 5) and B = (7, 11). Let R = {(a, b) : a ∈ ... | Filo
Let A = (3, 5) and B = (7, 11). Let R = {(a, b) : a ∈ A, b ∈ B, a − b is odd}. Show that R is an empty relation from A into B.
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A = (3, 5) and B = (7, 11)
R = {(a, b) : a ∈ A, b ∈ B, a − b is odd}
a are the elements of A and b are the elements of B.
So, R is an empty relation from A to B.
Hence proved.
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Question Text Let A = (3, 5) and B = (7, 11). Let R = {(a, b) : a ∈ A, b ∈ B, a − b is odd}. Show that R is an empty relation from A into B.
Topic Relations and Functions
Subject Mathematics
Class Class 11
Answer Type Text solution:1
Upvotes 78 | {"url":"https://askfilo.com/math-question-answers/let-a-3-5-and-b-7-11-let-r-a-b-a-a-b-b-a-b-is-odd-show-that-r-is-an-empty","timestamp":"2024-11-04T14:31:06Z","content_type":"text/html","content_length":"385005","record_id":"<urn:uuid:448adc64-f668-4e0c-ba87-671587365ebe>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00132.warc.gz"} |
Pizza Fraction Printable
Pizza fraction printable - You can also use the fraction number line to find which fractions are smaller or larger (smaller ones are closer to zero). 1 5 or 1 7 ? Remember to emphasise the importance
of every slice of pizza being of equal size. As of 2022, the firm managed approximately $160 billion of investor capital. 1 2 or 5 9 ? 6 7 or 4 5 ? Do you also see that. From working with a number
line to comparing fraction quantities, converting mixed numbers, and even using fractions in addition and subtraction problems, the fractions games below introduce your students to their next math
challenge as they play to rack up points and win the game. 3 4 or 5 6. Multiplication, division, fractions, and logic games that boost fourth grade math skills.
2 10 = 1 5. Fraction to pizza fractions index. 1 2 × 2 5 = 1 × 2 = 2. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the
denominators differ from each other.the value of an expression of this type is a positive rational number; Which fraction is larger in each of these pairs?
Free Pizza Fraction Printable
1 2 × 2 5. 4th grade math games for free. The firm was founded in 1984 by.
Pizza Fraction Clip It Cards Playdough To Plato
You can also use the fraction number line to find which fractions are smaller or larger (smaller ones are closer to zero). 1 2 × 2 5 = 1 × 2 = 2. Fraction games, videos, word problems, manipulatives,
and more at mathplayground.com!
Free Fraction Strips Printable Worksheets Interactive
Bain capital is an american private investment firm based in boston.it specializes in private equity, venture capital, credit, public equity, impact investing, life sciences, and real estate.bain
capital invests across a range of industry sectors and geographic regions. As of 2022, the firm managed approximately $160 billion of investor capital. 3 4 or 5 6.
Fraction Clipart
Here you can see it with pizza. Bain capital is an american private investment firm based in boston.it specializes in private equity, venture capital, credit, public equity, impact investing, life
sciences, and real estate.bain capital invests across a range of industry sectors and geographic regions. 2 10 = 1 5.
fraction equivalent Teaching fractions, Fractions, Fractions decimals
4th grade math games for free. 3 4 or 5 6. 6 7 or 4 5 ?
Pizza Fractions Project! Middle School Frolics
1 2 × 2 5 = 1 × 2 = 2. Number puzzles sum stacks number. You can also use the fraction number line to find which fractions are smaller or larger (smaller ones are closer to zero).
How to Teach Fractions Of A Set with A Free Apple Bump Game
Kids will learn to make friends with fractions in these engaging and interactive fractions games. You can also use the fraction number line to find which fractions are smaller or larger (smaller ones
are closer to zero). Multiplication, division, fractions, and logic games that boost fourth grade math skills.
1 2 × 2 5 = 1 × 2 = 2. 2 10 = 1 5. You can also use the fraction number line to find which fractions are smaller or larger (smaller ones are closer to zero). As of 2022, the firm managed
approximately $160 billion of investor capital. That is, each fraction in the expression has a numerator equal to 1 and a denominator that is a positive integer, and all the denominators differ from
each other.the value of an expression of this type is a positive rational number; Fraction to pizza fractions index. Bain capital is an american private investment firm based in boston.it specializes
in private equity, venture capital, credit, public equity, impact investing, life sciences, and real estate.bain capital invests across a range of industry sectors and geographic regions. 4th grade
math games for free. For instance the egyptian fraction above sums to.every positive rational number can be represented by an egyptian fraction. From working with a number line to comparing fraction
quantities, converting mixed numbers, and even using fractions in addition and subtraction problems, the fractions games below introduce your students to their next math challenge as they play to
rack up points and win the game.
Which fraction is larger in each of these pairs? 3 4 or 5 6. Kids will learn to make friends with fractions in these engaging and interactive fractions games. This is a simple visual representation
of a fraction, and you can adapt it to try it with ¼ too. Here you can see it with pizza. Simplify the fraction if needed. 1 2 × 2 5 = 1 × 2 2 × 5 = 2 10. Do you also see that. 1 2 or 5 9 ? 6 7 or 4
5 ?
Remember to emphasise the importance of every slice of pizza being of equal size. Multiplication, division, fractions, and logic games that boost fourth grade math skills. 1 5 or 1 7 ? 1 2 × 2 5.
Number puzzles sum stacks number. The firm was founded in 1984 by. Fraction games, videos, word problems, manipulatives, and more at mathplayground.com! However, a firm favourite in primary
classrooms is using food to represent fractions, and this is what you can do with your child at dinner time if pizza is on the menu! | {"url":"https://templates.esad.edu.br/en/pizza-fraction-printable.html","timestamp":"2024-11-04T00:59:59Z","content_type":"text/html","content_length":"113950","record_id":"<urn:uuid:b87f9245-2555-43fa-b082-571734473808>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00169.warc.gz"} |
I'm looking at self-intersection problems when dealing with integer multi-polygons.
For example, the following 2 polygons as a multipolygon:
They have a coincident edge, and therefore doing bg::intersects(this_multipolygon) will return true. Of course, bg::intersects() on each polygon individually is perfectly fine.
Is there any way I could tweak this algorithm so that coincident edges are not considered as self intersection? (can't use algorithms on this kind of multipolygons right now because of that)
Would this possibly break algorithms ? Should I simply deal with this kind of geometries as a vector of polygons and do algorithms 1 polygon at a time rather than the multipolygon concept ?
Thank you for sharing opinions.
(code below for this example)
#include <boost/geometry.hpp>
#include <boost/geometry/geometries/point_xy.hpp>
#include <boost/geometry/geometries/polygon.hpp>
#include <boost/geometry/multi/geometries/multi_polygon.hpp>
#include <iostream>
int main()
typedef boost::geometry::model::d2::point_xy<int> ipoint_xy;
typedef boost::geometry::model::polygon<ipoint_xy > ipolygon;
typedef boost::geometry::model::multi_polygon<ipolygon > imulti_polygon;
imulti_polygon adjacents;
boost::geometry::read_wkt("MULTIPOLYGON(((0 0, 0 10, 10 10, 10 0, 0 0)), ((10 5, 10 15, 20 15, 20 5, 10 5)))", adjacents);
std::cout << boost::geometry::intersects(adjacents) << std::endl;
return 0;
-------------- next part --------------
Skipped content of type multipart/related
Geometry list run by mateusz at loskot.net | {"url":"https://lists.boost.org/geometry/2011/10/1640.php","timestamp":"2024-11-11T19:38:48Z","content_type":"text/html","content_length":"11401","record_id":"<urn:uuid:2249edf0-58e7-4745-81eb-93279353978d>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00884.warc.gz"} |
Ether, a well-known anesthetic, has a density of 0.736g/cm 3 . What is the volume of 471 g of ether?
Ether, a well-known anesthetic, has a density of 0.736g/cm^3. What is the volume of 471 g of ether?
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Wey (US) to Cubic Yard Converter
⇅ Switch toCubic Yard to Wey (US) Converter
How to use this Wey (US) to Cubic Yard Converter 🤔
Follow these steps to convert given volume from the units of Wey (US) to the units of Cubic Yard.
1. Enter the input Wey (US) value in the text field.
2. The calculator converts the given Wey (US) into Cubic Yard in realtime ⌚ using the conversion formula, and displays under the Cubic Yard label. You do not need to click any button. If the input
changes, Cubic Yard value is re-calculated, just like that.
3. You may copy the resulting Cubic Yard 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 Wey (US) to Cubic Yard?
The formula to convert given volume from Wey (US) to Cubic Yard is:
Volume[(Cubic Yard)] = Volume[(Wey (US))] × 1.8436385459533606
Substitute the given value of volume in wey (us), i.e., Volume[(Wey (US))] in the above formula and simplify the right-hand side value. The resulting value is the volume in cubic yard, i.e., Volume
[(Cubic Yard)].
Calculation will be done after you enter a valid input.
Consider that a granary holds 3 wey (US) of wheat.
Convert this volume from wey (US) to Cubic Yard.
The volume in wey (us) is:
Volume[(Wey (US))] = 3
The formula to convert volume from wey (us) to cubic yard is:
Volume[(Cubic Yard)] = Volume[(Wey (US))] × 1.8436385459533606
Substitute given weight Volume[(Wey (US))] = 3 in the above formula.
Volume[(Cubic Yard)] = 3 × 1.8436385459533606
Volume[(Cubic Yard)] = 5.5309
Final Answer:
Therefore, 3 wey (US) is equal to 5.5309 yd^3.
The volume is 5.5309 yd^3, in cubic yard.
Consider that a farmer harvests 5 wey (US) of oats.
Convert this volume from wey (US) to Cubic Yard.
The volume in wey (us) is:
Volume[(Wey (US))] = 5
The formula to convert volume from wey (us) to cubic yard is:
Volume[(Cubic Yard)] = Volume[(Wey (US))] × 1.8436385459533606
Substitute given weight Volume[(Wey (US))] = 5 in the above formula.
Volume[(Cubic Yard)] = 5 × 1.8436385459533606
Volume[(Cubic Yard)] = 9.2182
Final Answer:
Therefore, 5 wey (US) is equal to 9.2182 yd^3.
The volume is 9.2182 yd^3, in cubic yard.
Wey (US) to Cubic Yard Conversion Table
The following table gives some of the most used conversions from Wey (US) to Cubic Yard.
Wey (US) (wey (US)) Cubic Yard (yd^3)
0.01 wey (US) 0.01843638546 yd^3
0.1 wey (US) 0.1844 yd^3
1 wey (US) 1.8436 yd^3
2 wey (US) 3.6873 yd^3
3 wey (US) 5.5309 yd^3
4 wey (US) 7.3746 yd^3
5 wey (US) 9.2182 yd^3
6 wey (US) 11.0618 yd^3
7 wey (US) 12.9055 yd^3
8 wey (US) 14.7491 yd^3
9 wey (US) 16.5927 yd^3
10 wey (US) 18.4364 yd^3
20 wey (US) 36.8728 yd^3
50 wey (US) 92.1819 yd^3
100 wey (US) 184.3639 yd^3
1000 wey (US) 1843.6385 yd^3
Wey (US)
The US wey is a unit of measurement used to quantify large volumes of dry goods, particularly in agriculture and trade. It is defined as 1,200 pounds or approximately 544.31 kilograms. Historically,
the wey was used to measure bulk commodities such as grain, coal, and other dry materials. Although its use has diminished in modern contexts, it remains part of historical and regional measurement
systems, providing a standard measure for large quantities of bulk goods in specific industries and historical records.
Cubic Yard
The cubic yard is a unit of measurement used to quantify three-dimensional volumes, commonly applied in construction, landscaping, and various industrial contexts. It is defined as the volume of a
cube with sides each measuring one yard in length. Originating from the Imperial system, the cubic yard provides a standardized measure for practical volume calculations. Historically, it has been
used to measure materials like soil, concrete, and gravel. Today, it is widely used in the US and other countries with Imperial systems for tasks such as calculating material quantities for
construction projects, landscaping, and waste management.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Wey (US) to Cubic Yard in Volume?
The formula to convert Wey (US) to Cubic Yard in Volume is:
Wey (US) * 1.8436385459533606
2. Is this tool free or paid?
This Volume conversion tool, which converts Wey (US) to Cubic Yard, is completely free to use.
3. How do I convert Volume from Wey (US) to Cubic Yard?
To convert Volume from Wey (US) to Cubic Yard, you can use the following formula:
Wey (US) * 1.8436385459533606
For example, if you have a value in Wey (US), you substitute that value in place of Wey (US) in the above formula, and solve the mathematical expression to get the equivalent value in Cubic Yard. | {"url":"https://convertonline.org/unit/?convert=wey_us-cubic_yard","timestamp":"2024-11-04T17:25:58Z","content_type":"text/html","content_length":"93285","record_id":"<urn:uuid:0562044c-ed42-458e-ab6d-df1584e681e2>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00321.warc.gz"} |
Valuation and Returns in Private Equity: Methods, Returns, and Performance
7.2.3 Valuation and Returns
Private equity investments are a cornerstone of modern financial markets, offering investors the potential for substantial returns. However, the valuation and return generation processes in private
equity are complex and multifaceted. This section delves into the methods used to value private equity investments, explains how these investments generate returns, discusses key performance
measures, and provides illustrative examples of calculating returns. Additionally, we summarize the factors influencing private equity performance, emphasizing the need for a long-term perspective.
Valuation Methods in Private Equity
Valuing private equity investments requires a deep understanding of various methodologies, each with its unique approach and applicability. The primary methods include Discounted Cash Flow (DCF)
Analysis, Comparable Company Analysis, and Precedent Transactions.
Discounted Cash Flow (DCF) Analysis
The DCF method is a fundamental valuation approach that involves projecting future cash flows and discounting them to their present value. This method is particularly useful for valuing companies
with predictable cash flows. The steps involved in DCF analysis are:
1. Project Future Cash Flows: Estimate the company’s future cash flows over a specific period, typically 5 to 10 years.
2. Determine the Discount Rate: The discount rate reflects the risk associated with the investment. It is often the company’s weighted average cost of capital (WACC).
3. Calculate the Present Value: Discount the projected cash flows to their present value using the discount rate.
4. Estimate Terminal Value: Calculate the company’s value beyond the projection period, often using a perpetuity growth model or exit multiple.
5. Sum of Present Values: The sum of the present values of projected cash flows and the terminal value gives the total enterprise value.
The DCF method is highly sensitive to assumptions about future cash flows and the discount rate, making it crucial to use realistic and well-researched inputs.
Comparable Company Analysis
Comparable Company Analysis involves valuing a company based on the valuation multiples of similar publicly traded companies. This method is useful for providing a market-based perspective on
valuation. The process includes:
1. Select Comparable Companies: Identify companies in the same industry with similar size, growth, and risk profiles.
2. Calculate Valuation Multiples: Common multiples include Price-to-Earnings (P/E), Enterprise Value-to-EBITDA (EV/EBITDA), and Price-to-Book (P/B).
3. Apply Multiples to Target Company: Use the median or average multiples of the comparable companies to estimate the target company’s value.
This method provides a relative valuation but may not capture unique aspects of the target company that differ from its peers.
Precedent Transactions
Precedent Transactions analysis involves using valuation multiples from previous mergers and acquisitions (M&A) deals in the same industry. This method reflects the prices paid for similar companies
in actual transactions, providing a real-world benchmark. The steps are:
1. Identify Relevant Transactions: Find past M&A deals involving companies in the same industry and with similar characteristics.
2. Determine Transaction Multiples: Calculate multiples such as EV/EBITDA, EV/Sales, or P/E based on the transaction data.
3. Apply Multiples to Target Company: Use these multiples to estimate the value of the target company.
Precedent Transactions analysis is particularly useful in assessing control premiums and synergies expected in M&A deals.
Return Generation in Private Equity
Private equity firms generate returns through various mechanisms, primarily focusing on capital appreciation, dividend payments, and management fees and carried interest.
Capital Appreciation
Capital appreciation is the increase in the value of an investment over time. Private equity firms achieve this by improving the operational performance of portfolio companies, expanding market
share, and optimizing capital structures. The ultimate goal is to sell the investment at a higher price than the purchase price, generating substantial returns.
Dividend Payments
While not the primary focus, some private equity investments provide interim returns through dividend payments. These dividends are often derived from the cash flows generated by the portfolio
companies and can provide a steady income stream to investors.
Management Fees and Carried Interest
Private equity firms earn management fees and carried interest as part of their compensation structure. Management fees are typically a percentage of the committed capital, while carried interest is
a share of the profits, usually 20% of the gains above a predetermined hurdle rate. This incentivizes private equity managers to maximize returns for their investors.
Performance Measures in Private Equity
Evaluating the performance of private equity investments involves using specific metrics that capture both the magnitude and timing of cash flows. The key performance measures include Internal Rate
of Return (IRR) and Multiple on Invested Capital (MOIC).
Internal Rate of Return (IRR)
IRR is the discount rate that makes the net present value (NPV) of cash flows equal to zero. It represents the annualized rate of return on an investment and is a critical measure for comparing the
profitability of different investments. Calculating IRR involves:
• Identifying all cash inflows and outflows associated with the investment.
• Solving for the discount rate that equates the NPV of these cash flows to zero.
IRR is particularly useful for assessing investments with varying cash flow patterns over time.
Multiple on Invested Capital (MOIC)
MOIC is a straightforward measure that calculates the total cash returns divided by the total cash invested. It provides a simple ratio indicating how many times the initial investment has been
returned. For example, if an investor puts in $1 million and receives $3 million, the MOIC is 3x.
MOIC is easy to understand and communicate, making it a popular metric among investors.
Illustrative Examples of Calculating Returns
To better understand the application of these performance measures, let’s explore examples of calculating IRR and MOIC.
IRR Calculation Example
Consider an investment of $1 million that returns $2 million in 5 years. To calculate the IRR, we set up the cash flow timeline:
• Year 0: -$1,000,000 (initial investment)
• Year 5: +$2,000,000 (return)
Using the IRR formula or a financial calculator, we find that the IRR is approximately 15%. This means the investment generates an annualized return of 15% over the 5-year period.
MOIC Calculation Example
If an investor puts in $1 million and receives $3 million, the MOIC is calculated as follows:
$$ \text{MOIC} = \frac{\text{Total Cash Returns}}{\text{Total Cash Invested}} = \frac{3,000,000}{1,000,000} = 3x $$
This indicates that the investor has tripled their initial investment.
Factors Influencing Private Equity Performance
Several factors influence the performance of private equity investments, including economic conditions, management execution, and industry trends.
Economic Conditions
Economic conditions play a significant role in private equity performance. Factors such as interest rates, inflation, and GDP growth impact valuations and exit opportunities. During economic
downturns, valuations may decline, and exit opportunities may become scarce, affecting returns.
Management Execution
The ability of private equity managers to implement value creation strategies is crucial for success. This includes improving operational efficiency, driving revenue growth, and optimizing capital
structures. Effective management execution can significantly enhance the value of portfolio companies.
Industry Trends
Industry trends, such as technological advancements, regulatory changes, and consumer preferences, can impact investment outcomes. Private equity firms must stay abreast of these trends to identify
opportunities and mitigate risks.
Diagrams and Graphs
To illustrate the relationship between investment duration and IRR, consider the following diagram:
graph LR
A[Investment Start] --> B[Year 1]
B --> C[Year 2]
C --> D[Year 3]
D --> E[Year 4]
E --> F[Year 5]
F --> G[Investment Exit]
subgraph IRR
A -->|Cash Outflow| H[Initial Investment]
F -->|Cash Inflow| I[Return]
This flowchart demonstrates how cash flows over time affect the IRR calculation, emphasizing the importance of both the magnitude and timing of cash flows.
Private equity investments offer substantial return potential, but they require a thorough understanding of valuation methods, return generation mechanisms, and performance measures. By employing
techniques such as DCF analysis, comparable company analysis, and precedent transactions, investors can accurately value private equity opportunities. Understanding how returns are generated through
capital appreciation, dividends, and management fees is essential for evaluating investment success. Performance measures like IRR and MOIC provide valuable insights into the profitability of
investments, while factors such as economic conditions, management execution, and industry trends influence outcomes. Ultimately, private equity investments demand a long-term perspective, with
careful consideration of both the magnitude and timing of cash flows.
Quiz Time!
📚✨ Quiz Time! ✨📚
### What is the primary purpose of Discounted Cash Flow (DCF) analysis in private equity valuation? - [x] To project future cash flows and discount them to present value - [ ] To compare the company
to similar publicly traded companies - [ ] To analyze past M&A deals in the industry - [ ] To calculate management fees and carried interest > **Explanation:** DCF analysis involves projecting future
cash flows and discounting them to their present value to determine the company's value. ### Which of the following is a key component of return generation in private equity? - [x] Capital
Appreciation - [ ] Comparable Company Analysis - [ ] Precedent Transactions - [ ] Terminal Value > **Explanation:** Capital appreciation refers to the increase in the value of an investment over
time, a primary source of returns in private equity. ### What does Multiple on Invested Capital (MOIC) measure? - [x] Total cash returns divided by total cash invested - [ ] The discount rate that
makes NPV of cash flows zero - [ ] The value of a company based on similar companies - [ ] The percentage of profits earned by private equity firms > **Explanation:** MOIC measures the total cash
returns divided by the total cash invested, indicating how many times the initial investment has been returned. ### How do economic conditions influence private equity performance? - [x] They affect
valuations and exit opportunities - [ ] They determine management fees - [ ] They set the hurdle rate for carried interest - [ ] They calculate the terminal value > **Explanation:** Economic
conditions impact valuations and exit opportunities, influencing the performance of private equity investments. ### What is the typical percentage of carried interest earned by private equity firms?
- [x] 20% - [ ] 10% - [ ] 30% - [ ] 5% > **Explanation:** Private equity firms typically earn 20% of the profits above a predetermined hurdle rate as carried interest. ### Which valuation method uses
multiples from previous M&A deals? - [x] Precedent Transactions - [ ] Discounted Cash Flow (DCF) Analysis - [ ] Comparable Company Analysis - [ ] Internal Rate of Return (IRR) > **Explanation:**
Precedent Transactions analysis uses multiples from previous M&A deals to value a company. ### What does Internal Rate of Return (IRR) represent? - [x] The discount rate that makes the net present
value (NPV) of cash flows equal to zero - [ ] The total cash returns divided by the total cash invested - [ ] The value of a company based on similar companies - [ ] The percentage of profits earned
by private equity firms > **Explanation:** IRR is the discount rate that makes the NPV of cash flows equal to zero, representing the annualized rate of return on an investment. ### Why is management
execution important in private equity? - [x] It enhances the value of portfolio companies through operational improvements - [ ] It determines the discount rate used in DCF analysis - [ ] It sets the
multiples for Comparable Company Analysis - [ ] It calculates the terminal value > **Explanation:** Effective management execution enhances the value of portfolio companies through operational
improvements, driving investment success. ### What is the relationship between investment duration and IRR? - [x] Longer durations can affect the IRR calculation due to the timing of cash flows - [ ]
Shorter durations always result in higher IRR - [ ] IRR is unaffected by investment duration - [ ] Investment duration determines the management fees > **Explanation:** Longer investment durations
can affect the IRR calculation due to the timing of cash flows, impacting the annualized rate of return. ### True or False: Private equity investments typically have shorter time horizons compared to
public equity investments. - [ ] True - [x] False > **Explanation:** Private equity investments often have longer time horizons, requiring a long-term perspective to measure performance effectively. | {"url":"https://csccourse.ca/7/2/3/","timestamp":"2024-11-08T22:18:27Z","content_type":"text/html","content_length":"96411","record_id":"<urn:uuid:75091595-3823-4877-81df-014c83402b79>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00038.warc.gz"} |
What is the difference between heat transfer coefficient and thermal conductivity?
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What is the difference between heat transfer coefficient and thermal conductivity?
The coefficient of heat transfer in the past that the total heat transfer coefficient. The current national standard specification unified name for the heat transfer coefficient. Heat transfer
coefficient K value, refers to the steady heat transfer conditions, the palisade structure on both sides of the air temperature of 1 degrees (K, ℃), 1 hours by 1 square meter area of heat transfer,
the unit is the watts per square metre, degrees (W / ㎡, K, K here can be substituted for ℃).
Coefficient of thermal conductivity is refers to under the condition of steady heat transfer, 1 m thick material, the surface temperature on both sides to one degree (K, ℃), within an hour, by 1
square meter area of heat transfer, the unit for W/m, c (W/m K, here for K can be substituted for ℃). Coefficient of thermal conductivity and material composition, density, moisture content,
temperature and other factors. Amorphous structure, the low density material, small thermal conductivity. The moisture content of materials, the temperature is lower, smaller thermal conductivity.
The low coefficient of thermal conductivity of materials usually referred to as thermal insulation material, and the coefficient of thermal conductivity in the 0.05 w/m, the following material is
called the efficient thermal insulation material. | {"url":"http://en.tlon.com.cn/news/4.html","timestamp":"2024-11-11T13:50:49Z","content_type":"text/html","content_length":"37020","record_id":"<urn:uuid:0ca6f5ab-d411-42b9-9ad7-98c3832ceae9>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00489.warc.gz"} |
How To Use Operators With Numbers In GO Programming Language - Code With C
How to use operators with numbers in the Go programming language? Do you know how to use operators with numbers in Golang? If not, then you don’t need to worry because, in this article, I will tell
you How to use operators with numbers in Golang.
Go is an imperative, object-oriented language. While this makes it a great language for programming in Go, it is not the easiest language to program in. In this post, I demonstrate how to use
operators with numbers.
In Go, numbers are represented in decimal format, and Go supports four different types of numbers: float, int, uint, and complex. You can think of a number as a tuple of two types, where the first
type is the base and the second type is the exponent. For example, the number 6.6 would be represented as a tuple of type (6,.6), where the first number is 6, and the second number is.6.
Go provides a set of operators that can be applied to numbers. These include addition (+), multiplication (*), subtraction (-), division (/) and modulus (%) operators.
How to use Operators with numbers in GO programming language
What is an Operator?
An operator is a symbol that is used to perform an action on a number. The list of operators is long and you can use any one of the operators as per your requirement. Some of the common operators are
as follows:
+ +
- -
* /
& | & ^
<< >>
==!= <= >= < >
&& ||
++ --
+= -= *= /= %= &= ^=
<= >=
The use of these operators is very simple, if you use two operators, then you can operate like you can use the + sign to add two numbers. If you want to know the result of the sum then you have to
use the & sign.
Types of Operators in Go Programming language
The operator is used to check the conditions or make some calculations. There are many types of operators available, here I will tell you about the most popular ones.
Arithmetic operators:
These are the most common operators that you will use in the Golang programming language. These operators are used to calculate the value of the expressions. They are as follows:
+ (Addition)
– (Subtraction)
* (Multiplication)
/ (Division)
^ (Exponentiation)
Assignment operator:
If you want to assign a variable with some value then you can use the assignment operator. This is written as:
= (Assignment Operator)
Comparison operators:
Comparison operators compare two values and return a boolean value. The comparison operators are as follows:
== (Equality)
!= (Not equal)
> (Greater than)
< (Less than)
Greater than and less than are written as > and < respectively.
Logical operators:
The logical operators evaluate the truth of a Boolean expression. They are as follows:
|| (Or)
&& (And)
! (Not)
Ternary operators:
Ternary operators are the operators that have three arguments. The three arguments are evaluated and the result is returned. The ternary operators are as follows:
? : (Question mark colon)
There are some other operators as well, but the above ones are the most common.
Operator precedence
The order of the operation has been defined in the order of the precedence. You can see the precedence of the operator by adding the numbers after the name of the operator. So, if you want to know
the precedence of the multiplication operator, then you have to add 2 to the name of the operator.
Here are some of the examples of the operators:
1 + 1 = 2
1 - 1 = 0
2 / 2 = 1
4 % 4 = 0
4 & 4 = 4
8 ^ 8 = 16
4 << 2 = 8
5 == 5 = true
As you know that Golang is a programming language that originated in Google, which is known for its powerful programming tools. This language is not only a great tool but also a programming language
that uses many useful and common operators. Here we list the most important operators with numbers in the golang programming language.
Addition operator with numbers in golang Programming language:
The addition is a binary operator that means adding two values together. The addition operator is written with a plus sign (+) or a plus sign followed by a space and then the operands. For example, 4
+ 5 is the same as 4 + 5. The value of the expression is the sum of the values of the operands.
Subtraction operator with numbers in golang Programming language:
Subtraction is a binary operator that means subtracting one value from another. The subtraction operator is written with a minus sign (-) or a minus sign followed by a space and then the operands.
For example, 4 – 5 is the same as 4 – 5. The value of the expression is the difference between the values of the operands.
Multiplication operator with numbers in golang Programming language:
Multiplication is a binary operator that means multiplying two values. The multiplication operator is written with an asterisk (*) or a star sign (*) followed by a space and then the operands. For
example, 4 * 5 is the same as 4 * 5. The value of the expression is the product of the values of the operands.
Division operator with numbers in golang Programming language:
The division is a binary operator that means dividing one value by another. The division operator is written with a forward slash (/), a space, and then the operands. For example, 4 / 5 is the same
as 4 / 5. The value of the expression is the quotient of the values of the operands.
Modulus operator with numbers in golang Programming language:
Modulus is a binary operator that means finding the remainder of the division of one number by another. The modulus operator is written with a forward slash (/), a space, and then the operands. For
example, 4 % 5 is the same as 4 % 5. The value of the expression is the remainder of the division of the first operand by the second.
Floor function with numbers in golang Programming language:
The floor function is used to find the lowest whole number that is greater than or equal to the given real value. The floor function is written with a forward slash (/), a space, and then the
operands. For example, 4 / 5 is the same as 4 / 5. The value of the expression is the smallest integer that is greater than or equal to the value of the first operand and not greater than the value
of the second operand.
Ceiling function with numbers in golang Programming language:
The ceiling function is used to find the largest whole number that is less than or equal to the given real value. The ceiling function is written with a forward slash (/), a space, and then the
operands. For example, 4 / 5 is the same as 4 / 5. The value of the expression is the largest integer that is less than or equal to the value of the first operand and not less than the value of the
second operand.
Round function with numbers in golang Programming language:
The round function is used to find the nearest whole number to the given real value. The round function is written with a forward slash (/), a space, and then the operands. For example, 4 / 5 is the
same as 4 / 5. The value of the expression is the closest integer to the value of the first operand that is equal to the value of the second operand.
Floor ceiling functions with numbers in golang Programming language:
The floor ceiling function is used to find the smallest or largest integer that is greater than or equal to the given real value. The floor ceiling function is written with a forward slash (/), a
space, and then the operands. For example, 4 / 5 is the same as 4 / 5. The value of the expression is the smallest integer that is greater than or equal to the value of the first operand and not less
than the value of the second operand.
How to apply Operators with numbers in GO programming language
When you apply an operator to a number, it is always evaluated from left to right. For example, the following snippet shows the output of applying the + operator to the numbers 5 and 6.
package main
import (
func main() {
n1 := 5
n2 := 6
In the above snippet, the + operator is being applied to the numbers 5 and 6. Because the operator is being applied to the numbers from left to right, the first operand is evaluated first. The result
of evaluating the first operand is then added to the second operand. The result of the evaluation of the second operand is then added to the result of the evaluation of the first operand. This is why
it is called a left-associative operator.
Here is a more complicated example showing how multiple operands can be added together to get the result of applying the + operator to the numbers 5 and 6.
package main
import (
func main() {
n1 := 5
n2 := 6
The above snippet shows how the result of the first operand of the + operator is added to the second operand, and the result of the second operand is added to the result of the evaluation of the
first operand. This is why it is called a left-associative operator.
You can also add a number to a variable. For example, the following snippet adds the variable n to the number 1.
package main
import (
func main() {
n := 1
fmt.Println(n + n)
While the above snippet works, it is not very useful. It is better to store the value of the number and add it to the variable.
Here is another example where we are adding a variable to a number. The following snippet adds the number 10 to the variable n.
package main
import (
func main() {
n := 10
fmt.Println(n + n)
The above snippet also works, but it is more readable when written as the following snippet.
package main
import (
func main() {
n := 10
fmt.Println(n + n)
In the above snippet, the number 10 is added to the variable n. This means that the variable n is first assigned the number 10. Then, the number 10 is added to the variable n.
This can be generalized as shown below. The following snippet adds the variable x to the number y.
package main
import (
func main() {
x := 10
y := 20
fmt.Println(x + y)
You can also add a number to a variable. For example, the following snippet adds the variable x to the number y.
package main
import (
func main() {
x := 10
y := 20
fmt.Println(x + y)
Here is a more complicated example showing how you can add the variable x to the number y.
package main
import (
func main() {
x := 10
y := 20
fmt.Println(x + y)
Go is a statically typed, compiled programming language developed by Google. Go offers a simplified syntax that lets you build faster and more maintainable programs. It is designed to provide
programmers with enough power to quickly build applications while keeping the code easy to understand. This blog covers operators with numbers in the golang programming language.
In conclusion, you can use operators to manipulate the values of variables in the golang programming language. This chapter will introduce you to various operators, the different types of operands,
and how to apply these operators. It will then show you how to combine these operators and how to define your operators in your program.
Leave a comment Leave a comment | {"url":"https://www.codewithc.com/operators-with-numbers-in-go-programming-language/","timestamp":"2024-11-11T17:35:21Z","content_type":"text/html","content_length":"153213","record_id":"<urn:uuid:a8ba95d5-b9a9-47f8-a741-89aa38d4aa6d>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00093.warc.gz"} |
De Méré's Paradox/Historical Note
The Chevalier de Méré first raised this question in the $17$th century.
He believed the two events described should have the same probabilities.
Empirical investigation (in other words: he found he was losing more money than he believed he ought to have been doing) caused him to rethink this.
Hence he posed this problem to his friend mathematician Blaise Pascal, who solved it. | {"url":"https://proofwiki.org/wiki/De_M%C3%A9r%C3%A9%27s_Paradox/Historical_Note","timestamp":"2024-11-09T22:47:38Z","content_type":"text/html","content_length":"39791","record_id":"<urn:uuid:2cac4659-b4e0-4e59-b691-1bc28d19d56a>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00372.warc.gz"} |
Making a Jet Exhaust in WebGPU
I have modified the three.js example “WebGPU Particles” to create a Jet Exhaust. I would like to change the opacity so that the material is more transparent. I have created a CodePen Example. I
believe that this could be done in line 69:
let opacityNode = textureNode.a.mul(life.oneMinus());
But I don’t know enough about the opacityNode to know what change I need to make.
2 Likes
There are quite many ways, it’s more an artistic issue. For example:
let opacityNode = textureNode.a.mul(life.oneMinus(),0.05);
let opacityNode = textureNode.a.mul(life.oneMinus().pow(50),0.1);
let opacityNode = textureNode.a.mul(life.pow(20).oneMinus(),0.1);
4 Likes
I had discovered that something like:
let opacityNode = 0.05;
also worked. But I like your solutions a lot better, especially the second one.
What, exactly do those commands mean? Is the oneMinus() an indication that the value will be reduced with each frame? (It is also used with the scaleNode, which I assume means the size of the
particle is reduced with each frame.) What do the .a and the mul mean?
This particle generator is really powerful stuff and a big reason I wanted to switch to WebGPU.
2 Likes
Here is a brief explanation:
textureNode.a is the opacity component of a color value from a texture, opacity is a number from 0 (completely transparent) to 1 (completely opaque). The name “a” comes from “alpha”. It is the same
“a” as in RGBA.
The other things are just mathematical functions to build a node-friendly expressions:
• mul – multiplication: xy = mul(x,y) = x.mul(y)
• pow – power function: x^y = pow(x,y) = x.pow(y)
• oneMinus – is the function: 1-x = oneMinus(x) = x.oneMinus()
The following expression:
would look like this in pure JavaScript:
textureNode.a * Math.pow(1-life,50) * 0.1;
Some more info about these expressions can be found in Three.js Shading Language documentation.
3 Likes
For some reason, a further version of this emitter that I used to create a Ship Wake also solved a problem I was having with frustum culling - i.e. the smoke disappeared when the origin went
off-screen. This is odd because the original three.js example has the frustum culling problem. See the example in this discussion.
1 Like | {"url":"https://discourse.threejs.org/t/making-a-jet-exhaust-in-webgpu/68450","timestamp":"2024-11-09T22:07:00Z","content_type":"text/html","content_length":"32952","record_id":"<urn:uuid:2f3e68f4-40b1-4d89-9957-6407e40a8cfe>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00063.warc.gz"} |
Healthcare Application of In-Shoe Motion Sensor for Older Adults: Frailty Assessment Using Foot Motion during Gait
Biometrics Research Labs, NEC Corporation, Hinode 1131, Abiko 270-1198, Chiba, Japan
Author to whom correspondence should be addressed.
Submission received: 23 March 2023 / Revised: 25 May 2023 / Accepted: 7 June 2023 / Published: 8 June 2023
Frailty poses a threat to the daily lives of healthy older adults, highlighting the urgent need for technologies that can monitor and prevent its progression. Our objective is to demonstrate a method
for providing long-term daily frailty monitoring using an in-shoe motion sensor (IMS). We undertook two steps to achieve this goal. Firstly, we used our previously established SPM-LOSO-LASSO (SPM:
statistical parametric mapping; LOSO: leave-one-subject-out; LASSO: least absolute shrinkage and selection operator) algorithm to construct a lightweight and interpretable hand grip strength (HGS)
estimation model for an IMS. This algorithm automatically identified novel and significant gait predictors from foot motion data and selected optimal features to construct the model. We also tested
the robustness and effectiveness of the model by recruiting other groups of subjects. Secondly, we designed an analog frailty risk score that combined the performance of the HGS and gait speed with
the aid of the distribution of HGS and gait speed of the older Asian population. We then compared the effectiveness of our designed score with the clinical expert-rated score. We discovered new gait
predictors for HGS estimation via IMSs and successfully constructed a model with an “excellent” intraclass correlation coefficient and high precision. Moreover, we tested the model on separately
recruited subjects, which confirmed the robustness of our model for other older individuals. The designed frailty risk score also had a large effect size correlation with clinical expert-rated
scores. In conclusion, IMS technology shows promise for long-term daily frailty monitoring, which can help prevent or manage frailty for older adults.
1. Introduction
1.1. Background
Typically, skeletal muscle mass begins to decline gradually at around age 45, after reaching its peak in the early adult years [
]. Additionally, gait speed, which has been deemed the sixth vital sign [
], significantly decreases in older adults after age 60 [
]. The decline in skeletal muscle mass and gait speed below a critical threshold may result in physical functional impairments that limit mobility, such as walking, climbing stairs, and crossing over
obstacles [
]. These impairments may lead to sarcopenia or frailty in older adults [
] (see
Figure 1
Although the relationship between sarcopenia and frailty has yet to be fully characterized, these conditions share many commonalities. Both are linked to physical functional impairment, and
sarcopenia is an age-related, long-term process that involves the loss of muscle mass and strength, affecting mobility and nutritional status [
]. Additionally, physical frailty may result in sedentary behavior, cognitive impairment, and social isolation [
]. Frailty is closely associated with various detrimental outcomes for older adults, such as an increased risk of falls and fractures, impaired ability to perform daily activities, loss of
independence, the need for long-term care placement, and even death [
]. Fortunately, appropriate exercise and nutritional treatment can postpone, recover, and effectively manage frailty, especially if the frail condition can be assessed in daily living [
The Asian Working Group on Sarcopenia (AWGS) defines sarcopenia for Asian individuals and includes criteria for assessing muscle function in diagnosing sarcopenia [
]. These criteria consist of gait speed measurement and hand grip strength (HGS) measurement, which are the simplest well-validated protocols for assessing muscle function in clinical practice [
]. The threshold for HGS measurement is 28 kg for males and 18 kg for females, while the gait speed requirement for both sexes is 1.0 m/s. These criteria are also included in the revised Japanese
version of the Cardiovascular Health Study (J-CHS) criteria for diagnosing physical pre-frailty/frailty [
] (see
Figure 1
b). In addition, three other criteria are evaluated subjectively by the participants themselves using a questionnaire. Participants rate their conditions on a scale from 0 to 2, and those with scores
higher than 2 are categorized as “Robust”, “Pre-frail”, or “Frail”.
The assessments mentioned above generally require older adults to visit specialized facilities and undergo evaluations under the supervision of clinicians. However, in certain areas, particularly
rural regions in Japan where healthcare resources are limited, monitoring older adults’ body conditions can be challenging. Moreover, for urban senior citizens, weekly or monthly facility visits are
not always feasible as they can increase the burden on seniors and healthcare systems alike. The recent development of Internet of Things (IoT) technologies for healthcare [
] has introduced wearable technologies as a viable option to monitor pre-frailty/frailty in daily living. By monitoring physical performance in daily living over a long period, wearable technologies
can help alert users to seek further examination or appropriate treatments on demand, prevent or manage the progression of frailty, and ultimately reduce the burden on healthcare systems.
A new approach to assessing pre-frailty/frailty has been introduced, proposing that wearable sensors can enable the simple monitoring of gait parameters (GPs), including gait speed, during daily
walking. This has made gait speed monitoring “smart”, as all data processing can be conducted on an edge device [
]. However, HGS assessment remains challenging for many individuals, as it requires clinicians to perform assessments in a facility setting, following specific protocols [
]. Despite this, IoT technologies for HGS assessment have been developed by researchers such as Becerra et al. [
], who developed a wireless hand grip device for force analysis, Chen et al. [
], who proposed a hand rehabilitation system with an HGS assessment function via soft gloves, and Wang et al. [
], who developed a novel flexible sensor to assess HGS. Nevertheless, for many users, wearing sensors on their hands may be inconvenient and challenging, considering daily routines. This may pose an
obstacle to achieving reliable monitoring of frailty in daily living.
Smart shoes/insoles with motion sensors have been proposed to improve the practicality of daily gait analysis. These devices are considered promising in various healthcare applications that require
daily gait analysis, including Parkinson’s disease, gait rehabilitation, and foot deformity detection [
]. In this paper, we refer to this type of smart motion sensor as an “in-shoe motion sensor” (IMS). An IMS can easily and noninvasively gather abundant information related to gait kinematics,
including gait speed, stride length, stance phase duration, instantaneous linear and rotational foot motion, and 3-D foot angular posture [
]. Furthermore, IMSs can be placed in various types of shoes or insoles, making them an unobtrusive addition to daily life.
We considered developing a frailty risk assessment method using only an IMS as a user-friendly solution for daily frailty assessment with the following benefits: (1) helping users avoid the burden of
wearing multiple sensors and (2) simplifying the wearable sensor system for frailty assessment. To achieve this goal, we identified two necessary steps: (1) constructing an HGS estimation model using
foot motion data obtained from an IMS and (2) designing a novel index capable of continuously assessing the conditions of frailty. In the subsequent sections, we provide a detailed explanation of
these two steps.
1.2. Step 1 to Goal: Constructing HGS Assessment Model on an IMS and Related Work
1.2.1. Research Question in Step 1
Generally, IMSs can transmit detailed waveforms wirelessly to a smartphone or server for further analysis, which consumes a significant amount of power. As a result, these IMSs need to be frequently
charged, reducing their usability for practical applications. In a previous study, we developed a new type of IMS, which is small and lightweight, can be attached to insoles, and has optimally
designed power-saving operation sequences and modes for practical applications. Our study showed that this IMS achieved high usability for long-term daily measurement without the need for battery
charging for up to one year [
]. One key feature contributing to power savings is that our IMS can perform simple data processing and calculate common spatiotemporal GPs, such as gait speed, stride length, and stance phase
duration, using inertial measurement unit (IMU) signals. We have named this type of IMS A-RROWG
. These features enable A-RROWG to collect daily gait data over long periods, regardless of location and time, without the user noticing the sensor’s presence. The research question for Step 1 is how
to construct an HGS assessment model that is feasible for an A-RROWG-type IMS and that can be proven effective. However, to the best of our knowledge, no technology has been developed for assessing
HGS performance using IMSs.
1.2.2. Ideas for Solving the Research Question in Step 1
Due to the characteristics of A-RROWG, the HGS assessment model must be lightweight enough to be implemented on it. Therefore, rather than applying recent machine learning methods that require a
large computation capacity [
], we focused on developing a lightweight, high-precision estimation model via linear multivariate regression with a minimum number of predictors required. This development included two tasks: (1)
identifying predictors that highly correlate with HGS and (2) reducing redundant predictors via feature selection.
Gait speed has been suggested to correlate with HGS [
], indicating that gait features might be a useful predictor for HGS assessment. However, gait speed is not a specific predictor for HGS as it can also be influenced by other factors, such as knee
osteoarthritis [
] or depression [
], making it challenging to construct an accurate model.
To address this limitation, we proposed considering additional potential predictors for HGS assessment. Previous research has demonstrated that HGS correlates with knee extension muscles,
specifically the quadriceps [
], which play a crucial role in walking. Since gait is a periodic movement, the same motions using muscles are repeated during specific gait phases in every gait cycle (GC). Although the quadriceps
do not directly control foot motion, they should impact foot motion through their control of the knee joint and lower leg. Therefore, we considered predictors for HGS assessment that can be
determined from foot motion signals during specific gait phases, specifically those gait phases where the quadriceps are activated.
For the second task of selecting appropriate predictors, several techniques, such as LASSO [
], Bayesian methods such as Bayesian LASSO [
], deep learning methods for sparse learning [
], and multi-objective optimization methods [
], have been proposed. However, multi-objective optimization methods are suitable for optimizing multiple conflicting objectives simultaneously, which is not within the scope of linear regression
methods utilized in our study. LASSO and Bayesian LASSO are more feasible alternatives, but Bayesian LASSO may require more substantial expertise to interpret results accurately. As such, we chose to
apply LASSO for feature selection.
In conventional LASSO, cross-validation approaches [
] are commonly used to select the LASSO tuning parameter value. However, these techniques typically consider randomly selecting training and validation sets without considering variations between
individuals. To ensure model robustness and account for individual differences, we combined LASSO with a leave-one-subject-out (LOSO) process. This approach involved running multiple LASSO analyses
by looping the LOSO process for all subjects, conceptually similar to jackknife, resampling method [
], to approximate the nature of the population estimator and improve model robustness against individual differences.
In our previous studies, we developed an algorithm capable of automatically extracting novel significant gait predictors from foot motion, selecting optimal features, and constructing an assessment
model, valid for estimating adults’ foot function and older adults’ balance ability measured by the outcome of a functional reach test (FRT) [
]. In this study, we constructed an HGS estimation model using this algorithm via the following steps:
• Identifying significant gait phases with statistically significant correlation with the target variable using statistic parametric mapping (SPM) [
], which was proven effective in biomechanical studies. The significant gait phases always continuously appeared, performing as clusters on the temporal axis; thus we called them, “gait phase
clusters” (GPCs).
• Conducting predictors by averaging the foot motion signals in the GPCs to obtain IMS predictors that can be implemented on the A-RROWG-type IMS. Although there are clustering algorithms, such as
community detection algorithms [
], due to the temporal continuity of foot motion, using the integral average of the signals in GPC as a single predictor is sufficient and helpful for implementation on the A-RROWG-type IMS.
• Reducing redundant predictors and selecting appropriate predictors using our original algorithm, the leave-one-subject-out least absolute shrinkage and selection operator (LOSO-LASSO).
• Constructing a multivariate linear regression estimation model.
We refer to our approach as SPM-LOSO-LASSO, which aids in constructing a biomechanically interpretable HGS estimation model that is both lightweight enough for implementation on an edge device and
precise in its predictions. In a previous study, we demonstrated the construction and operation of the IMS predictors on an A-RROWG-type IMS [
]. In this study, we have incorporated individual physical attributes (IPAs), such as age, height, weight, and body mass index (BMI), and designed GPs, including previously proposed temporal and
spatial GPs (we list them in
Section 2.4
), as auxiliary predictors to enhance the model’s precision. Considering the gait variance between biological sexes [
], we have constructed separate estimation models for males and females. Some of our findings in this report are based on the work presented at the 44th International Conference of the IEEE
Engineering in Medicine and Biology Society (EMBC 2022) [
1.3. Step 2 to Goal: Related Work on Frailty Assessment and Designing a Frailty Risk Score
1.3.1. Research Question in Step 2
Aside from the cardiovascular health study (CHS) criteria, there are alternative methods for diagnosing frailty in clinical practice. Examples include the phenotype model [
] and accumulated deficit model [
]. To assess frailty levels in daily living, several techniques based on wearable sensor measurements have been proposed [
]. For instance, using wearable motion sensors, Schwenk et al. [
] conducted home assessments of established gait outcomes to identify pre-frailty and frailty. Razjouyan et al. [
] utilized a pendant motion sensor to develop a composite model for discriminating three frailty categories: non-frail, pre-frail, and frail. In addition, Greene et al. [
] aimed to create an automatic, non-expert quantitative assessment of the frailty state based on wearable inertial sensors.
However, previous research studies focused solely on discriminating two or three frailty levels. The transition from non-frail to pre-frail or pre-frail to frail is a gradual, long-term process.
According to a previous study [
], the pooled incidence rate of pre-frailty was 15.1%, and that of frailty was 4.3% based on multiple cohort studies. Given that body performance tends to decline with age in the absence of
intervention, it is reasonable to hypothesize that the higher the current condition’s frailty risk, the greater the likelihood of future deterioration. To assist users in delaying and managing
frailty progression adequately, merely classifying frailty levels is considered insufficient. Consequently, the research question in Step 2 is how to construct an analog frailty risk metric and
demonstrate its effectiveness.
1.3.2. Ideas for Solving the Research Question in Step 2
An analog frailty risk score could prove beneficial for various reasons, such as providing users with an intuitive representation of their body condition’s long-term changes, enabling a more
comprehensive user rating, and demonstrating the effects of exercise. Given that HGS and gait speed are critical factors in current frailty assessment, we assert that their performance must feature
significantly in frailty risk assessment. Consequently, we developed a frailty risk score in this study by merely combining the HGS and gait speed performance of the subjects. Moreover, we utilized
the HGS distribution [
] and gait speed data for the Asian population aged over 60 years [
] to design our frailty risk score.
1.4. Testing Constructed HGS Estimation Model and Frailty Risk Score
After constructing and validating the model, we conducted two separate tests on a group of older healthy adults who were recruited independently from those used for constructing the model.
The first test involved examining the precision of the HGS assessment model on the separately recruited subjects.
The second test involved testing the effectiveness of our original frailty risk score, which was used to demonstrate the possibility of evaluating frailty via IMSs in subjects who were also recruited
separately. These subjects were rated using a continuous score ranging from 0 to 100 by experts, including clinicians and physiotherapists with over 5 years of experience, who observed their gait.
The score served as a reference for their risk of frailty. We tested the correlation coefficient between the designed score and the expert-rated score.
1.5. The Development Process and Main Contributions in this Study
In summary,
Figure 1
c presents a diagram that outlines the development process of achieving frailty risk assessment via the A-RROWG-type IMS. The main contributions of this study are as follows:
We discovered novel predictors for HGS assessment obtained from foot motions.
We constructed a lightweight HGS assessment model that can be feasibly implemented in the A-RROWG-type IMS, which serves as a key module for long-term frailty assessment.
We tested the effectiveness and robustness of the constructed model using a group of separately recruited subjects.
We designed an analog frailty risk score and evaluated its effectiveness for frailty risk assessment via an IMS.
The acronyms and symbols used in this manuscript can be referenced in
Table A1
Appendix A
Figure 1. (a) Relationship between sarcopenia and frailty. (b) Revised Japanese version of Cardiovascular Health Study criteria. (c) Diagram which explains the development process of achieving
frailty risk assessment via A-RROWG-type IMS.
2. Materials and Methods
2.1. Subjects and Their Characteristics
To contribute to potential applications for frailty prevention, as well as postponing and managing its progression, we recruited healthy older adults who could participate in the experiment
independently. We recruited three separate groups of healthy older subjects with different ages, heights, and weights for model construction (Group I), Test 1 (Group II+III, combining data in Group
II and Group III together), and Test 2 (Group III). We successfully collected data from 62 subjects (27 males and 35 females) for Group I, 20 females for Group II, and 25 subjects (6 males and 19
females) for Group III. All subjects were able to walk independently without assistive devices, had no history of severe neuromuscular or orthopedic diseases, had normal or corrected-to-normal
vision, and had no communication obstacles. After explaining the experimental procedure to the subjects, we obtained their informed consent before the experiment. This study received approval from
the NEC Ethical Review Committee for Life Sciences (Approval No. LS2021-004, 2022-002) and the Ethical Review Board of Tokyo Medical and Dental University (Approval No. M2020-365). The demographic
data are summarized in
Table 1
, with HGS and gait speed serving as reference values.
The average age of both male and female subjects in all three groups was over 70 years old. Although the average BMIs indicate that most subjects had a normal body mass, we ensured that subjects with
a wide range of body mass were recruited, including those with maximum and minimum BMIs. In Group I, male and female subjects had similar age characteristics (
= 0.755), and no significant sex difference in gait speed was found (
= 0.453). The data also show that the female subjects for model construction (Group I) were similar in age to those for model testing (Group II+III) (
= 0.604), as well as in terms of HGS and gait speed (HGS:
= 0.395; gait speed:
= 0.265). However, compared with the male subjects in the two groups, the age in Group II+III was higher than that in Group I (
= 0.040). Although there was no significant difference in gait speed (
= 0.052), due to age, the HGSs in Group II+III were much lower (
= 0.021). When comparing female subjects in Groups I and III, no significant differences in age, HGS, and gait speed were found between them (
= 0.058, 0.102, 0.972). According to the J-CHS scores of the subjects in Group III, 60% of the subjects self-assessed themselves as not being frail, and none of them assessed themselves as frail.
Further details on how the J-CHS scores were calculated for the subjects are presented in
Section 2.2
2.2. Experiment
To achieve our final goal, we collected five types of data from subjects in an indoor environment, performing the following steps:
Step 1
At the start of the experiment, all subjects were asked to complete a questionnaire to provide basic information, including age, height, and weight, and for the calculation of BMI based on their
Step 2
The same questionnaire included four questions based on the J-CHS criteria:
□ Q1. Have you lost more than 2–3 kg in the past 6 months?
□ Q2. In the past two weeks, have you felt tired for no reason?
□ Q3. Do you engage in light exercise or gymnastics at least once a week?
□ Q4. Do you engage in regular exercise or sports at least once a week?
From the four questions, we calculated the J-CHS score for each subject as subjective frailty reference data.
Step 3
After answering the questionnaire, subjects were guided to measure their HGS, which served as the reference HGS value in this study.
Step 4
Subjects were asked to walk in a straight line. In this step, we collected foot motion data for calculating GPs and IMS predictors, as well as reference gait speed data for all subjects.
Additionally, for those in Group III, video recordings were made while they walked.
Step 5
We sent the walking videos to clinical experts to obtain expert-rated frailty risk scores as objective frailty reference data.
Further details on Steps 2 through 5 are explained in the following subsections.
2.2.1. Step 2 of the Experiment
For Q1 and Q2, one point was added for each “yes” answer. For Q3 and Q4, if both questions were answered with “no”, one point was added, and if either was answered with “yes”, no points were added.
The total J-CHS score was obtained by totaling the points. Finally, we checked whether the reference HGS and gait speed were below the threshold specified in the J-CHS criteria to determine the
subjects’ total J-CHS score. Subjects who scored 0, 1–2, and higher than 2 were classified as “Robust”, “Pre-frail”, and “Frail”, respectively.
2.2.2. Step 3 of the Experiment
To assess the HGS of the subjects, we used a Jamar hydraulic hand dynamometer (Lafayette Instrument Company, Lafayette, IN, USA). The measurement process followed the method suggested in a previous
study [
], as shown in
Figure 2
a. Subjects were asked to sit on an armchair with their elbow flexed at 90°, without touching the chair arms. The Jamar is a variable hand-span dynamometer with five handle positions. The dynamometer
was set to handle position “two”, and both hands were measured three times with subjects exerting their best effort. To determine the representative HGS of each subject, we calculated the mean value
of the six measurements. This mean value served as the reference value for HGS in this study.
2.2.3. Step 4 of the Experiment
To collect foot motion data, the subjects were asked to walk straight along 16 m lines for four trials at a self-determined comfortable speed. Before data collection, they were given a 2-min practice
session to familiarize themselves with the environment and procedure. While walking, their foot motions were recorded by two IMSs embedded in insoles placed under the arches of both feet near the
calcaneus side (see
Figure 2
b). This placement ensured that the subjects could walk comfortably. Please note that during the feasibility study stage of this study, foot motion data were temporarily recorded onto the onboard
memory during experiments and would later be transferred to a personal computer for data processing. The characteristics of the IMS are described in
Section 2.3
The time taken by each subject to walk 10 m along the 16 m lines was recorded using a digital stopwatch to calculate their average gait speed when walking at a uniform pace. This speed was treated as
the reference value for gait speed in this study. Subjects in Group III were also recorded while walking by two video cameras placed at the side and end of the walking path. To protect their privacy,
their faces were obscured.
2.2.4. Step 5 of the Experiment
After gait data collection was finished, the videos were sent to six clinical experts in gait evaluation. They were asked to score the subjects regarding the risk of “being diagnosed with frailty
within the next 5 years” on a 100-point scale by observing their gait. A subject considered by the rater to have the highest risk was rated as 100, and a subject with the lowest risk was rated as 0.
The relative frailty risk of the remaining subjects compared with the highest and lowest ones was scored between 0 and 100. Then, every subject had six scores. Except for observing the recorded
videos, the raters were not given any personal information about the subjects.
2.3. Characteristics of IMS
The IMSs used in this study have the same structure as A-RROWG-type IMSs. Each IMS consists of a 6-axis IMU (BMI 160, Bosch Sensortec, Reutlingen, Germany), an ARM Cortex-M4F microcontroller unit
(MCU) with Bluetooth module (nRF52832, CPU: 64 MHz, RAM: 64 KB, ROM: 512 KB, Nordic Semiconductor, Oslo, Norway), onboard memory (AT45DB641, 64 Mbit, Adesto Technologies, Santa Clara, CA, USA), a
real-time clock (RTC) (RX8130CE, EPSON, Suwa, Japan), a control circuit, and a 3V coin lithium-ion battery (CLB2032 T1, 300 mAh, Maxell, Tokyo, Japan). The device is lightweight (12 g, including the
coin battery) and compact (29 mm × 40 mm × 7 mm) enough to be placed at the arch of the foot. Please note that during the feasibility study stage, the IMSs were set to developer mode, which differed
from A-RROWG in that all calculations were performed on the device. Under this mode, raw foot motion waveform data were first recorded on the IMSs’ onboard memory and then sent to a PC via Bluetooth
after the experiment. We developed dedicated software for controlling data recording start and end in the IMSs and for downloading raw data from the onboard memory of the IMSs to a PC via Microsoft
Visual Studio (Microsoft, Redmond, WA, USA).
The IMSs can directly measure three axes of acceleration,
(medial: +, lateral: −),
(posterior: +, anterior: −), and
(superior: +, inferior: −), as well as those of angular velocity,
(sagittal plane (
): plantarflexion: +, dorsiflexion: −),
(frontal plane (
), eversion: +, inversion: −), and
(horizontal plane (
), internal rotation: +, external rotation: −). Inside the IMSs, the three axes of sole-to-ground angles (SGAs),
(roll angle, plantarflexion: +, dorsiflexion: −),
(pitch angle, eversion: +, inversion: −), and
(yaw angle, internal rotation: +, external rotation: −), were calculated using a Madgwick filter [
]. Specifically, the acceleration values were corrected to the global coordinates in each independent trial. The IMSs had a data sampling frequency of 100 Hz, and their measurement range for
acceleration was ±16 g, while that for angular velocity was ±2000 degrees/s.
2.4. Signal Processing and GPs
For all data processing, simulation, and model construction tasks, MATLAB (MathWorks, Natick, MA, USA) was used in this study.
To construct the HGS estimation model via the SPM-LOSO-LASSO algorithm, predictors from three categories were required: IPAs, temporospatial GPs, and IMS predictors. Temporospatial GPs and IMS
predictors were obtained by processing one stride of the foot motion waveform. In this section, we explain the procedures used to obtain GP predictors. The flow chart is shown in
Figure 3
During the preliminary stage, two primary tasks were completed. The first task involved processing every stride of the foot motion waveform into data matrices. The second task focused on calculating
the GPs that were extracted from each stride of the foot motion waveform. The third task was to obtain a set of average foot motion waveforms and GPs in each trial.
For the first task, to prepare the nine-dimensional foot motion signals from the IMSs for analysis, the signals were partitioned into individual strides by detecting a heel-strike (HS) event [
]. The IMS signal during the stance phase was then temporally normalized to a 1–60% gait cycle (%GC), while the swing phase was normalized to 61–100%GC to create a 9 × 100 matrix. To eliminate
potential biases, we subtracted the average signal amplitude during 21–25%GC from each stride assuming that these phases, where the foot sole fully touches the ground, can be represented as a neutral
posture. Additionally, to exclude any walking velocity bias in foot motion, we normalized the amplitude of acceleration and angular velocity waveform of each stride using the corresponding maximum
instantaneous velocity during a stride. The instantaneous walking velocity was computed by integrating
from a neutral posture to the end of the stride. It is worth noting that we excluded the first and last three strides of each trial, as they were not uniform in speed. Furthermore, we removed any
gait outliers from the remaining strides of each participant, following the exclusion criteria outlined in [
Before temporal normalization, we derived 20 temporal and spatial GPs [
] from each stride of the foot motion waveform using the algorithm depicted in [
]. These parameters are listed in
Table 2
. GP01, GP05, and GP06 were normalized by subject height. GP11-14, GP19, and GP20 were normalized by the duration of one stride. GP15, GP16, and GP18 were normalized by the maximum instantaneous
walking velocity during the swing phase.
We then calculated the average foot motion and GPs for each trial on the left and right feet for each subject. The data of the left and right feet were further averaged within each trial. This
resulted in each participant having four sets of average foot motions and GPs. Thus, a total of 108 and 140 datasets were generated for males and females in Group I, respectively, and 24 and 156
datasets were generated in Group II+III for males and females, respectively. These processed average waveforms were used to determine new predictors for HGS estimation.
2.5. SPM-LOSO-LASSO, Model Evaluation of HGS, and Precision Evaluation of Gait Speed
2.5.1. The Details of SPM-LOSO-LASSO
In this section, we explain the process of constructing and selecting predictors for HGS estimation via SPM-LOSO-LASSO, following the steps depicted in
Figure 4
a [
]. Here, IMS predictor processing is part of SPM-LOSO-LASSO.
To construct IMS predictors from foot motion signals that are significantly correlated with HGS outcome, it is necessary to determine the %GCs that have a significant correlation. For this purpose,
we used SPM, a widely used and effective method in biomechanical studies [
]. We performed SPM analysis to evaluate the correlation between HGS outcomes and foot motion signals at each %GC. SPM for correlation analysis is a stepwise process. First, a canonical correlation
analysis (CCA) with SPM (SPM-CCA) was performed [
]. The %GCs whose test statistic in the CCA exceeded a critical test statistic threshold calculated in accordance with the random field theory (RFT) [
] were determined as significant %GCs. The level of significance was set as
< 0.05. Second, as a post hoc test, only data in significant %GCs were further investigated by Pearson’s correlation (PeC) analysis with SPM (SPM-PeC) for each component of the foot motion signal.
For each component, the %GCs whose test statistic in the PeC exceeded an RFT-based critical test statistic threshold were judged as the final HGS-correlated significant %GCs for each component.
Because there were nine components in the foot motion signals, we conducted Šidák correction [
] at a level of correlation significance where
< 0.0057.
Based on biomechanical knowledge, we limited the predictors to the range of approximately 1–16%GC, 48–70%GC, and 92–100%GC, where the quadriceps are mostly activated. These defined
quadricep-activation %GCs were used as a filter, denoted as
. The intersection between the %GCs judged by SPM to be HGS-correlated and the
was taken to exclude the %GCs not related to quadricep activities. The intersections were treated as GPCs. The integral average of the signal in GPCs was then used as an IMS predictor, as expressed
by (1).
$C i = ∑ T s T e − Δ T W T + W T + Δ T 2 T e − T s Δ T , + T e − T s > 0 W T e , + T e − T s = 0$
means the
-th IMS predictor;
mean the start and end of %GCs of GPCs, respectively; and
means the waveform of the foot motion signal, where
∊ {
After collecting the subjects’ IPAs, GPs, and IMS predictors, we formed predictor candidates for model construction. We used our original algorithm, LOSO-LASSO [
], along with the “lasso” function in MATLAB to determine the best selection of predictors. We obtained multiple LASSO analysis results by looping the LOSO process for all subjects. By statistically
analyzing these results, we can approximate the nature of the population estimator and thereby make the LASSO analysis more robust against individual differences.
The details of LOSO-LASSO are shown in
Figure 4
b. In the
-th LOSO process, the data of the
-th subject are first excluded, and the remaining data are then subjected to LASSO analysis. LASSO solves the following problem:
$min β i 0 , β i 1 2 N ∑ k = 1 N y k − β i 0 − x k T β i 2 + λ i ∑ j = 1 C β i j$
Here, N is the amount of data. y[k] is the target variable. x[k] is the predictor vector of length C. λ[i] is a non-negative regularization parameter input to LASSO, which can be set freely. β[i] is
the set of fitted least-squares regression coefficients, and β[i][0] is the residual of the linear regression y[k] = x[k]^Tβ[i] + β[i][0], corresponding to λ[i], which is also the output of LASSO. β
[ij] is the j-th element of β[i]. As λ[i] increases, the number of nonzero components of β[i] decreases. For optimizing feature selection, we set 100 different λ[i]’s which formed a geometric
sequence to compose a regularization parameter 100-dimensional vector λ; thus, the index i here means the i-th element of λ. In each LOSO, 100 β[i]’s formed a coefficient matrix. Then, we substituted
nonzero elements in LASSO coefficient matrices with 1 to form label matrix B[u].
This process is repeated for each subject. After completion of the LOSO process, we can obtain U sets of B[u]’s. By summing all B[u]’s, we obtain a matrix with a total counter B[0]. The elements over
0.95 × U (25 for males and 33 for females) in this matrix are substituted with 1, while the remaining elements are substituted with 0, forming the final label matrix B. LOSO-LASSO generates 100 types
of predictor combinations (denoted as Ω[1]–Ω[100]) based on different regularization coefficient sets in LASSO. Using these features, 100 different candidate multivariate regression models can be
obtained for the dataset. We evaluated 100 candidate models (H[1]–H[100]) for estimating HGS using leave-one-subject-out cross-validation (LOSOCV) and the intraclass correlation coefficient (ICC) of
type (2, 1) as the evaluation index, denoted as ICC(2, 1). The model with the highest ICC(2, 1) value was chosen as the optimal model (M[o]).
2.5.2. Model Evaluation of HGS and Precision Evaluation of Gait Speed
After selecting
, we used LOSOCV to evaluate the degree of agreement and precision between the reference and estimated HGS, using the ICC(2, 1) and mean absolute error (MAE). Additionally, we evaluated the adjusted
coefficient of determination (
) for the multivariate regression models using all training data (not LOSOCV) and the Pearson’s coefficient of correlation (
) between predictors and the outcome of HGS. For comparison, we derived models by optimizing three other patterns of predictor combinations in the same process:
(gait speed (GP02)),
plus other GPs in one stride), and
plus IPAs) (see
Figure 4
We evaluated the average value of gait speed measured by the IMS in one trial and used ICC(2, 1) and MAE to assess the agreement and precision between the reference and measured values.
The guidelines for interpreting ICC inter-rater agreement are as follows: excellent (>0.750), good (0.600–0.749), fair (0.400–0.599), and poor (<0.400) [
]. The guidelines for interpreting
are as follows: none (<0.02), small (0.02 to 0.13), medium (0.14 to 0.26), and large (>0.26). The guidelines for interpreting r are as follows: none (<0.100), small (0.100 to 0.299), medium (0.300 to
0.499), and large (>0.499) [
2.6. Designing Frailty Risk Score
We assumed that the distribution of HGS and gait speed of our subjects would follow a normal distribution similar to that of the population of older Asian adults. According to [
], the mean values of HGS for males (
= 12,190) and females (
= 14,154) over 60 years old are 34.7 and 21.9 kg, respectively, and the standard deviations are 7.1 and 4.8 kg, respectively. In [
], the baseline demographic and health characteristics of 1686 community-dwelling Japanese were demonstrated, and no significant difference in gait speed was observed between sexes. Thus, the
calculated mean value and standard deviation of gait speed for all subjects were 1.29 and 0.24 m/s, respectively.
We utilized a probability-distribution-based method to design the frailty risk score. First, we calculated the Z-score of the HGS performance of males and females using Equations (3) and (4),
respectively, and that of gait speed using Equation (5), using the mean value and standard deviation of HGS and gait speed for older Asian adults in [
$Z H G S _ m = H G S m − 34.7 / 7.1$
$Z H G S _ f = H G S f − 21.9 / 4.8$
$Z G S = G S − 1.29 / 0.24$
Here, HGS[m], HGS[f], and GS are the HGS of the male subjects, the HGS of the female subjects, and the gait speed of all subjects (no sex difference). Z[HGS_m] and Z[HGS_f] denote the Z-scores of the
HGS performance of males and females, and Z[GS] denotes the Z-scores of the gait speed performance for the standard normal distribution.
Because Z-scores can theoretically be from −∞ to +∞, to constrain the score to 0 to 100, we used the cumulative percentage of the standard normal distribution as the frailty risk score, which was
calculated via the Z-scores mentioned before. Then, to ensure that the scores were still in the range of 0 to 100, we propose performance scores of HGS for males and females as Equations (6) and (7)
and the performance score of gait speed as (8):
$P H G S _ m = ∫ − ∞ Z H G S _ m 1 2 π exp − x 2 2 d x$
$P H G S _ f = ∫ − ∞ Z H G S _ f 1 2 π exp − x 2 2 d x$
$P G S = ∫ − ∞ Z G S 1 2 π exp − x 2 2 d x$
denote the designed score of the HGS performance of males and females, and
denotes the designed score of the gait speed performance. By following the calculation process described above, we eliminated the sex difference in the HGS distribution. Thus, the scores for males
and females had the same distribution and could be discussed together. Finally, to reflect the equal weight given to HGS and gait speed in the J-CHS criteria, we propose a frailty risk score (
) by combining the performance of the two, as expressed by Equation (9).
$P f r = P H G S _ m + P G S / 2 or P f r = P H G S _ f + P G S / 2$
2.7. Evaluation Methods in Model Tests
2.7.1. Test 1
In Test 1, we utilized Bland–Altman (BA) plots [
] to assess the limit of agreement (LoA) between IMS-assessed and reference values of gait speed and HGS. We computed both the sample-based LoA and the confidence limits of LoA in the population. To
examine the existence of a fixed and proportional bias, we applied a
-test and Pearson’s correlation test if the differences and averages between the two methods followed a normal distribution, initially tested by a Kolmogorov–Smirnov (KS) test. The LoA of the 95%
confidence interval was established from the perfect agreement (PA) line ± 1.96 × standard deviation (σ), resulting in upper and lower LoAs (ULoA and LLoA). Additionally, the 95% confidence limits of
LoA were also determined, which included the upper and lower limits of ULoA (UULoA and LULoA), as well as the upper and lower limits of LLoA (ULLoA and LLLoA). T-tests were used for comparing
differences between two groups, and ANOVA was used to compare the differences among three or more groups, with all levels of significance set at
< 0.05.
In the model testing stage, we evaluated the validity of gait speed measurement and HGS estimation based on the ratio of test data in Group II+III, whose BA plots were within the agreement range
determined by the model test data for Group I, i.e., the success rate of measurements denoted as
. We considered the measurement to be successful by the model when the difference between IMS-measured and reference values was located inside the agreement interval, determined by the data of Group
I. We used the optimistic agreement range, i.e., the range between UULoA and LLLoA. Because the test data size was limited, we utilized the probability-distribution-based method [
] to estimate
and eliminate randomness. We set the confidence level to 95%, assuming 5% of the measurements as the outliers in this study. If over 95% of data were inside the agreement interval,
was considered to be 100%.
In the probability-distribution-based method, we hypothesized that the residual of BA plots for training and test data to the PA line, denoted as
, followed a normal distribution,
) and
). Here,
s and
s mean the averages and standard deviation, respectively. Because the model was based on multivariate regression, theoretically,
≡ 0. Furthermore, because of the limited data size, we calculated the 95% confidence levels of
and obtained their upper and lower limits, (
), (
), and (
), respectively. Hence, if we use an optimistic agreement range, the agreement range of the residual should be fixed as −1.96
to 1.96
. By then,
should be in the area of
) inside the interval of −1.96
to 1.96
. Because
are independent of each other, the largest and smallest areas for
) subject to
∊ [
] and
∊ [
] would be the upper and lower limits of
, denoted as
, which can be expressed by Equations (10) and (11):
$K A U = min max ∫ − 1.96 σ A U 1.96 σ A U 1 2 π σ T i exp − x − μ T i 2 2 σ T i 2 d x / 0.95 , 1 s u b j e c t t o μ T i ∈ μ T L , μ T U , σ V i ∈ σ T L , σ T U$
$K A L = min min ∫ − 1.96 σ A U 1.96 σ A U 1 2 π σ T i exp − x − μ T i 2 2 σ T i 2 d x / 0.95 , 1 s u b j e c t t o μ T i ∈ μ T L , μ T U , σ V i ∈ σ T L , σ T U$
2.7.2. Test 2
After calculating the P[fr]s of all subjects in Group III, we compared them with the expert-rated scores and calculated the correlation (r) between them to evaluate the effectiveness of the designed
For each subject, we obtained a total of six expert-rated scores. We preliminarily tested the reliability of the six expert-rated scores based on the ICC values. The results showed that the ICC(2, 1)
was 0.490 (fair), and ICC(2,
) was 0.850 (excellent). Moreover, the KS test indicated that the mean score of all subjects corresponding to six raters followed the normal distribution (
= 0.987). These results showed that the score indicating a diagnosis of frailty within the next 5 years for the subjects in Group III could be assessed using an average of six expert-rated scores
with high reliability. Additionally, as another statistical processing method, we obtained the median values of the six expert-rated scores and the rank of subjects according to each score. For each
subject, we then calculated their average rank. Thus, for the other patterns, we used the median value and averaged rank as the reference frailty risk score of the subjects. The correlation analysis
between the reference frailty risk score for the other patterns and the designed frailty risk score is shown in the
Supplementary Materials
3. Results
3.1. SPM Analysis in HGS Estimation Model Construction
In a comparison between the males and females, their average waveforms appeared approximately similar. In contrast, the standard deviation of waveforms, particularly in the frontal and horizontal
plane (
, and
), appeared to have more different shapes (
Figure 5
According to the results of the SPM-CCA, a significant correlation was found between the foot motion signal vectors for most of the stance phase and the end of the swing phase (immediately before HS)
and the HGSs for both sexes. A post hoc SPM-PC analysis, represented by statistic SPM{
} curves, revealed the strength of the correlation between each type of foot motion signal and the HGS. Significant GC intervals, referred to as GPCs, were identified in the sections of curves that
exceeded critical thresholds and correlated with the HGSs. It is worth noting that the GPCs of the acceleration signals were more fragmented due to the smaller smoothness of the acceleration waveform
compared to the angular velocities and sole-to-ground Euler angles. The shape of the statistic SPM{
} curves and the location of the GPCs varied between males and females (see
Figure 5
). Consequently, 20 GPCs and 17 GPCs were obtained for males and females, respectively. Filtered by quadricep-activation %GCs (
), 10 GPCs and 14 GPCs ultimately remained for creating the same numbers of IMS predictors.
3.2. Feature Selection for HGS Estimation Model
To obtain the final optimal predictor combination
, consisting of IPA and GP predictors, we inputted a total of 34 and 38 candidate predictors into LOSO-LASSO for males and females, respectively. Referring to
Figure 6
, we determined
for males and females by finding the highest ICC(2, 1), which included 16 and 8 finally selected predictors, respectively. The selected predictors for constructing multivariate linear regression and
their correlation analyses with the HGS are listed in
Table 3
Table 4
Regarding the IPA predictors, age and height were selected for both males and females, with medium to large effect sizes (age: r = 0.162 and 0.271; height: r = 0.428 and 0.682). In particular, the
age for males and height for females had the highest correlation with HGS. These results indicate that the effect of age and body size on HGS was observed. Although the effect size was small (r =
0.209), weight was also selected for the estimation model for males.
Compared to females, more GP predictors were selected for males, with GP16 (r = 0.303, medium effect size; r = 0.199, small effect size) being present in the predictor list for both sexes. This
result suggests that subjects with higher HGSs have lower maximum G[x] in the dorsiflexion direction during the swing phase. Except for GP03, which had a medium effect size (r = 0.338), the remaining
GP predictors (GP05, 08, 09, 10, 18, 19) only had effect sizes classified as none or small.
For both males and females, five IMS predictors were ultimately selected (
) by LOSO-LASSO. The corresponding GPCs are shown in
Figure 7
. Besides foot motions in the sagittal (
) plane, such as
), those in the frontal (
) and horizontal (
) planes, such as
, and
C[m][12,15,16,] C[f][4–6,8]
), were suggested to be essential for HGS estimation. Temporally, major parts of GPCs for females appeared around HS (
), where both the rectus femoris (RF) and vastus muscles (VAs) in the quadriceps were mainly activated. In contrast, besides the GPCs (
) in the %GCs when both the RF and VAs activated, the male subjects also had more GPCs (
) inside the %GCs for which only the RF activated, which appeared around TO, than the female subjects (
). These results may reflect the sex differences in muscle activation patterns during gait.
By referencing the mean value and linear correlation coefficients of the selected IMS predictors with the HGS, the direction of foot motions during these phases and the changing trend as HGS
increased could be determined. Male subjects with stronger HGSs had strong acceleration in the anterior and superior direction (C[m][13,14]) immediately before and after TO. During the early
mid-stance phase, when the foot approaches the defined neutral position, male subjects with stronger HGSs had higher angular velocities in the direction of eversion and internal rotation (C[m]
[15,16]). Immediately after the heel rocker occurred, female subjects with stronger HGSs tended to have lower acceleration in the lateral direction and lower angular velocity in the internal rotation
direction (C[f][4,8]). Combining the two predictors, the results may suggest that female subjects with stronger HGSs tend to have a higher ability to land their feet stably and smoothly. After the
foot has completely hit the ground, female subjects with higher HGSs tended to have less acceleration in the medial direction (or more acceleration in the lateral direction) (C[f][5]). At the end of
the initial swing phase when the lower limb transitioned from acceleration to deceleration, the acceleration in the anterior direction (C[f][7]) of female subjects began to approach zero as HGS
Furthermore, we also list the coefficients of predictors and their
-values in a multivariate regression model in
Table 2
Table 3
. Although the linear correlation coefficient with the HGS contained predictors with effect sizes only classified as none or small, the constructed models for both males and females had large effect
sizes (
= 0.858,
< 0.001, and
= 0.773 and
< 0.001, respectively).
3.3. Precision Evaluation of Gait Speed, Model Evaluation of HGS Estimation, and Test 1
3.3.1. Gait Speed
For all subjects, we evaluated the agreement between the 10 m average gait speed calculated from stopwatch-measured time in one trial and that calculated by averaging all strides of gait speed in 10
m intervals in one trial (see
Figure 8
a). The ICC agreement reached the “excellent” level with a value of 0.978. Compared to the reference value, the IMS achieved an MAE of 0.029 m/s, which is only 2.1% of the average gait speed of all
subjects in Group I.
From the BA plots of data for Group I (see
Figure 8
b), we observed a fixed bias indicating that the IMS-measured gait speed was on average 0.014 m/s greater than the stopwatch-measured data (
< 0.001). There was also a proportional bias between the two measurements (
= −0.173,
= 0.006), indicating that IMS slightly overestimated the gait speed when the gait speed became slower (
= −0.034
+ 0.060). The agreement interval for testing data for Group II+III was determined by the BA plots generated from the data for Group I. According to
calculated using Equations (8) and (9), the IMS successfully assessed 100% of gait speed data for subjects in Group II+III with an MAE precision of 0.029 m/s.
3.3.2. HGS
The results presented in
Figure 9
a suggest that gait speed alone or combined with other common GPs is not an effective predictor for estimating HGS. From the results shown in
Figure 9
a, it can be inferred that gait speed is significantly correlated with HGS among male and female subjects, with moderate effect sizes (
= 0.384, 0.337,
= 0.048, 0.048), but estimating HGS based solely on gait speed is not feasible due to the poor ICC agreement between the estimated and reference values. However, when additional GPs were added as
predictors by using the LOSO-LASSO model (
), significant improvements were observed for ICC, MAE, and
. The ICC agreement for males and females improved from poor to fair and good, respectively, while the
improved from small to large. Specifically, for the males, the ICC agreement improved from fair to good with the aid of IPAs. Additionally, the optimal model (
) that included IMS predictors resulted in a substantial improvement in ICC agreements, MAE, and
, where the ICCs reached excellent for both males and females with MAE and
values improving to 2.88 and 2.57 kg and 0.86 and 0.77, respectively. Further details on
predictor combinations can be found in the
Supplementary Materials
The differences between the reference and estimated values of Group I data followed a normal distribution, as shown in
Figure 9
b. The Bland–Altman plots of Group I for both males and females did not reveal any proportional biases (
= 0.76, 0.09) between the measurements. In terms of the HGS model test results using Group II+III data, the HGS estimation was successful for 5/6 males and 36/39 females within the agreement
interval. According to Equations (10) and (11), 48.0–100.0% of male subjects and 89.1–100.0% of female subjects were estimated successfully. However, it appeared that HGS was overestimated for males
in Group II+III.
3.4. Test 2: Validity of Designed Frailty Risk Score with Estimated HGS
In Test 2, the scores of male and female subjects were evaluated together because the experts did not consider biological sex. For both males and females in Group III, there was no significant linear
correlation between HGS and gait speed (r = 0.025, p = 0.963, and r = 0.170, p = 0.302, respectively). Even after calculating P[HGS] and P[GS], there was still no significant linear correlation
between the performance scores (r = 0.363, p = 0.075), possibly due to the small sample size and insufficient statistical power.
The ICC agreement between the three types of performance scores based on reference and IMS-estimated values is shown in
Figure 10
had an excellent level of agreement with an ICC(2,1) of 0.959 (
Figure 10
b), while
only had a poor level with an ICC(2,1) of 0.282 (
Figure 10
a), possibly due to a few subjects who did not agree well with the reference data. However, when
were combined with
, the ICC value improved to a good level at 0.727 (
Figure 10
Figure 11
Figure 12
show the correlations between the expert-rated score and the three types of performance scores based on reference and IMS-estimated values. The expert-rated score had a significant negative
correlation with reference data-based
, with large effect sizes (
= −0.555, −0.503;
= 0.004, 0.010), but not with reference data-based
= −0.225,
= 0.280) (
Figure 11
). However, the
based on IMS-estimated data had a significant negative correlation with the expert-rated score with a large effect size (
= −0.525,
= 0.007), and the
based on IMS-estimated data had a higher effect size (
= −0.676,
< 0.001) than the reference data-based one. These results indicate that the performance scores based on IMS-estimated data are more consistent with the experts’ diagnostic reasoning.
We conducted a statistical analysis of the difference in expert-rated scores between subjects classified as pre-frail and robust based on the J-CHS score (
Figure 13
). We found no significant difference between the subject groups that scored 1 to 2 and 0, which may be due to the difficulty in precisely scoring subjects who are on the boundary of robust/pre-frail
conditions based only on gait observation. Nevertheless, the average value of the robust group was lower than that of the pre-frail group.
Furthermore, we tested the three types of performance scores on the basis of IMS-estimated data for the pre-frail and robust groups (
Figure 14
). Despite the
-test showing no significant difference between the two groups in either HGS or gait speed performance scores, the overall performance
of the robust group was significantly higher than that of the pre-frail group. This suggests that the frailty risk score was consistent with the J-CHS criteria.
4. Discussion
4.1. Some Significant GP Predictors for HGS Estimation
Although gait speed has been suggested to be correlated with HGS in previous studies [
], in this study, gait speed was not selected as a predictor for the HGS estimation model for either male or female subjects. Instead, other spatiotemporal parameters were discovered to be
significant for HGS estimation in our designed model. These parameters played a key role in the optimal model for HGS estimation based on the analysis of ICC agreement. After applying the LOSO-LASSO
method, essential GP predictors were selected.
One of the essential GP predictors is the maximum sole-to-ground angle in the dorsiflexion direction (GP03), which has a relatively high positive correlation with HGS in males. As shown in
Figure 5
, GP03 occurs immediately before heel strike. During this phase of the gait cycle, the ankle joint is in a neutral status; i.e., the foot is perpendicular to the tibia. Therefore, the value of GP03
is determined by the degree of knee extension [
]. When the knee extensor, i.e., quadriceps, becomes weaker, the knee cannot be extended sufficiently, which causes GP03 to become smaller.
Another essential GP predictor for both sexes is the maximum angular velocity in the dorsiflexion direction during the swing phase (GP16). Unlike GP03, the negative correlation coefficient between
HGS and GP16 suggests that subjects with a higher HGS have a lower absolute value of GP16, i.e., a value closer to zero. This can be explained as follows: According to
Figure 5
, GP16 is most likely to occur during the initial swing phase. During this phase, the upper leg rotates forward (blue arrow on upper leg in
Figure 15
), and the knee joint gradually increases flexion. Passively, the lower leg lifts behind the central line of the body (yellow arrow in
Figure 15
), which prevents the lower leg from rotating forward too early by overcoming the gravity force (green arrow in
Figure 15
). At the same time, the ankle joint spontaneously reduces plantarflexion, which rotates the foot forward (blue arrow on foot in
Figure 15
). The
waveform during this phase reflects the counterbalance motion of the knee and ankle joint [
]. Furthermore, Nene et al. [
] suggested that the rectus femoris muscle controls the degree of knee flexion. Therefore, when the quadriceps, especially the rectus femoris, becomes weaker, the antagonizing muscle power that
prevents the lower leg from rotating forward along with gravity also decreases. Consequently, GP16 becomes larger in the dorsiflexion direction.
4.2. Some Significant IMS Predictors for HGS Estimation
Through SPM analysis of the correlation between HGS and the foot motion waveforms, we discovered a number of effective IMS predictors, and five IMS predictors were finally selected by LOSO-LASSO. As
shown in
Figure 7
, the major parts of GPCs for the females appeared around HS (
), where both the RF and VAs in the quadriceps were mainly activated, while the male subjects also had more GPCs (
) inside the %GCs for which only the RF was activated, which appeared around TO, than the female subjects (
). Di Nardo et al. [
] suggested that female subjects have more complex activation patterns in VAs. Bailey et al. [
] indicated that in older adults’ gait, the activation level of RF for males is higher than that for females according to a study using electromyography. The results shown in
Figure 6
may reflect sex-dependent muscle activation during gait.
Kobayashi et al. [
] demonstrated the differences in GPs between sexes. Rowe et al. [
] analyzed the sex differences in kinetics and kinematics of lower limbs in detail, which indicated that more differences were found in frontal and horizontal planes. In this study, we also observed
a difference in foot motion waveforms, which also belong to the gait in the lower limbs, between male and female subjects, especially in the frontal plane and transverse plane. Our results agreed
with the findings demonstrated in these previous studies.
We analyzed the correlation between the balance ability, represented by the outcome of the FRT, and foot motion with the same subjects in Group I in our previous study [
]. We found several significant GPCs by paying attention to the gait phases related to the activation of the tibialis anterior (TA) and calf muscles (gastrocnemius (GA) and soleus (SO)). The TA has
two periods of activity: one is during the early stance phase (1–15%GC), and the other is from the late pre-swing to the end swing phase (55–100%GC). Partial quadricep-activated gait phases overlap
with the TA at the moments before and after HS. Different from the GPCs in the HGS assessment model, there were no GPCs selected at the end of the swing phase, i.e., the second period of TA activity,
in the balance ability assessment model, which may suggest that the power needed for knee extension contributes less to balance ability. In contrast, similar to the balance ability assessment model,
the HGS estimation model also has GPCs in the early stance phase (the first period of TA activity). In this period, the quadriceps control the lower limb to prevent excessive knee flexion, and at the
same time, the TA contributes to decelerating the passive plantarflexion and foot pronation to make the posture more stable [
]. A previous study discovered that HGS was significantly correlated with the outcome of FRT in the older Asian population [
]. We also found that the HGS was significantly correlated with the outcome of FRT in our study (male:
= 0.456,
= 0.017; female:
= 0.390,
= 0.020). We think that the correlation may be related to the common parts of GPCs in both models during the early stance phase right after HS.
4.3. Results Regarding Model Test and Designed Frailty Risk Score
The agreement between reference data and IMS-estimated data for
only reached a “poor” level due to the estimated HGSs of one male and two female subjects that deviated from the reference values (
Figure 10
a, marked with three dashed black circles). It appears that IMS did not estimate the HGS of these three subjects accurately compared to the hydraulic hand dynamometer which is considered as the gold
standard. However, the reference HGS only reflected the static systemic muscle strength of the upper limb.
Figure 11
a indicates that there was no significant correlation between the reference HGS and the expert-rated score, while
Figure 12
a suggests that IMS-estimated HGS was significantly correlated with the expert-rated score. Furthermore, compared to the results in
Figure 11
c and
Figure 12
c, our designed frailty risk score using IMS-estimated values agreed more with the clinical experts. These results may be due to the fact that our model focused on gait performance and reflected
dynamic muscle conditions via the lower limbs. Moreover, experienced clinicians and physiotherapists tend to rely on information extracted from gait observation for making their decisions in clinical
practice. The ICC(2,
) of the HGS between reference and IMS-estimated values reached 0.886 and 0.902 for males and females, respectively, indicating that the average value of the dynamometer and IMS-estimated HGS can be
used in clinical practice to better approach subjects’ systemic muscle strength reality.
Our designed frailty risk score was significantly correlated with the expert-rated score (r = −0.676, p < 0.001), indicating the reliability of frailty risk assessment using IMS and our designed
frailty risk score. Additionally, significant differences were found in the designed frailty risk score between subjects in the group with a J-CHS score of 0 and those with a score of 1 to 2, further
supporting the use of our proposed method for frailty assessment.
4.4. Outlook for this Technology
As a feasibility study, we temporarily recorded foot motion data in the onboard memory of IMSs during the experiments and transferred the data to a personal computer after the gait measurements were
completed. However, in daily use, a real-time algorithm for frailty assessment is necessary. In our previous studies, we proposed an online algorithm for estimating stride parameters for daily gait
analysis using an IMS [
], as well as an algorithm for integrating the process of IMS predictor construction into the online algorithm [
]. By using the same algorithms, we believe that daily frailty assessment can be performed using an IMS.
In this study, we did not diagnose whether the subjects were frail or not, as all recruited subjects were able to come to the laboratory using public transportation. Therefore, we assumed that they
were in generally good health. The characteristics of the subjects can be observed from their J-CHS scores, but the frailty risk score does not directly represent the probability of an individual
being diagnosed as frail. Instead, it reflects the relative degree of frailty in the population. To obtain more evidence supporting our frailty risk score, future longitudinal cohort studies should
be conducted to track the frailty of subjects. Additionally, an epidemiological study regarding the frailty risk score is needed to improve its interpretability in connection with the real
probability of being diagnosed as frail.
To improve the HGS estimation model’s precision, future studies should focus on increasing the sample size. To improve the agreement of the frailty risk score with experts’ diagnostic reasoning, IMS
estimation should include three additional items in the J-CHS criteria: activity level, fatigue, and weight loss. Gokalgandhi et al. [
] suggested that daily activity and calorie consumption could be monitored by smart shoes. However, an estimation method via IMSs for the other two items is still lacking. In their study, Luo et al.
] proposed a pilot method for assessing fatigue via wearable sensors that utilized vital signs such as heart rate, blood pressure, skin temperature, and steps, but did not include other GPs. Previous
kinematic studies [
] have shown that fatigue and weight loss can impact kinematic patterns. Therefore, assessing fatigue and weight loss using IMSs alone is promising but requires further investigation in the future.
5. Conclusions
In this study, we demonstrated the potential for long-term frailty assessment using IMSs, which required two key tasks. The first task was to accurately measure gait speed using IMSs and construct an
HGS estimation model via foot motion. The second task was to create a frailty risk score that can continuously assess frailty and validate its effectiveness.
For the first task, we confirmed that IMSs can measure gait speed with high accuracy, with an ICC agreement with reference data of over 0.97. By analyzing the correlation between HGS and foot motion
waveforms using SPM-LOSO-LASSO, we discovered novel GPs and IMS predictors for HGS estimation. Specifically, we found that male subjects had more GPC components inside the %GCs for which only the RF
was activated, while female subjects had more GPC components inside the %GCs for which both the VAs and RF were activated. We successfully constructed sex-dependent HGS estimation models, both of
which achieved “excellent” ICC agreement, MAEs below 2.9 kg, and large effect sizes (R^2 over 0.77). By testing the model on a separate sample of subjects, we found that 48.0–100% of males and
89.1–100% of females were within the agreement interval, indicating the robustness of our model for other older individuals.
For the second task, we successfully designed a novel analog frailty risk score by combining the HGS performance and gait speed performance of the subjects aiding by the normal distribution of HGS
and gait speed of the Asian older population. This score had a large effect size correlation with the expert-rated score, demonstrating its validity and agreement with clinical experts’ diagnostic
In the future, an epidemiological study is needed to improve the interpretability of the frailty risk score in connection with the real probability of being diagnosed with frailty. Furthermore, to
better align with clinical experts’ diagnostic reasoning, an IMS assessment of three other items related to activity, weight loss, and fatigue is needed.
Supplementary Materials
The following supporting information can be downloaded at:
. Figure S1. Results of LOSO-LASSO analysis to determine (a)
and (b)
; Table S1. Optimal predictor combination in
; Figure S2. Correlations between expert-rated median score and three types of performance scores calculated from IMS-estimated value: (a)
, (b)
, (c)
. Figure S3. Correlations between expert-rated mean rank and three types of performance scores calculated from IMS-estimated value: (a)
, (b)
, (c)
Author Contributions
Conceptualization, C.H. and F.N.; methodology, C.H.; software, C.H. and Y.N.; validation, C.H., F.N., K.F. and K.I.; formal analysis, C.H.; investigation, C.H. and F.N.; resources, C.H., H.K. and
F.N.; data curation, C.H. and F.N.; writing—original draft preparation, C.H.; writing—review and editing, C.H. and F.N.; visualization, C.H.; supervision, F.N. and K.F.; project administration, K.N.;
funding acquisition, K.N. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Institutional Review Board Statement
The study was conducted in accordance with the Declaration of Helsinki and approved by the NEC Ethical Review Committee for Life Sciences (Approval No. LS2021-004, 2022-002) and Ethical Review Board
of Tokyo Medical and Dental University (Approval No. M2020-365).
Informed Consent Statement
Informed consent was obtained from all subjects involved in the study.
Data Availability Statement
Data are unavailable due to privacy or ethical restrictions.
Partial test data in this study were provided by a joint study by the Tokyo Medical and Dental University (TMDU), NEC, and The National Institute of Advanced Industrial Science and Technology (AIST),
“Gait analysis using walking sensing insole A-RROWG” (M2020-365). We thank Koji Fujita from TMDU and Yoshiyuki Kobayashi from AIST for providing the data.
Conflicts of Interest
The authors declare no conflict of interest.
Appendix A
The acronyms and symbols used in this manuscript are listed in
Table A1
Symbol Description Symbol Description
%GC Percentage gait cycle LR Loading response
AWGS Asian Working Group on Sarcopenia LULoA Lower limit of ULoA
A[x] The acceleration signal of x-axis M[1]–M[3] Three other patterns of predictor combinations
A[y] The acceleration signal of y-axis MAE Mean absolute error
A[z] The acceleration signal of z-axis MCU Micro-control unit
BA plots Bland–Altman plots M[o] The optimal model
BMI Body mass index MSt Mid-stance
CCA Canonical correlation analysis MSw Mid-swing
CHS Cardiovascular health study PeC Pearson’s correlation
C[i] The i-th predictor variable P[fr] Frailty risk score
EMBC Conference of the IEEE Engineering in Medicine and Biology Society PGS Designed score of the gait speed performance
E[x] The SGA signal of x-axis P[HGS_f] Designed score of the HGS performance of females
E[y] The SGA signal of y-axis P[HGS_m] Designed score of the HGS performance of males
E[z] The SGA signal of z-axis PS Pre-swing
FRT Functional reach test Q[t] Quadricep-activation %GCs
GA Gastrocnemius r Pearson’s coefficient of correlation
GC Gait cycle R^2 Adjusted coefficient of determination
GP Gait parameter R[A] Residual of BA plots for training data
GPC Gait phase cluster R[T] Residual of BA plots for test data
GS Gait speed RF Rectus femoris
G[x] The angular velocity signal of x-axis RFT Random field theory
G[y] The angular velocity signal of y-axis RTC Real-time clock
G[z] The angular velocity signal of z-axis SGA Sole-to-ground angle
H[1]–H[100] 100 candidate models SO Soleus
HGS Hand grip strength SPM Statistical parametric mapping
HGS[f] HGS of the female subjects SPM{F} F statistic of vector field analysis by SPM-CCA
HGS[m] HGS of the male subjects SPM{t} Statistic of post hoc scalar trajectory linear correlation test by SPM-PC
HS Heel strike TA Tibialis anterior
ICC Intraclass correlation coefficient T[e] The end of %GCs of GPCs
IMS In-shoe motion sensor TO Toe-off
IMU Inertial measurement unit TSt Terminal stance
IPA Individual physical attribute T[s] The start of %GCs of GPCs
ISw Initial swing TSw Terminal swing
J-CHS Japanese version of the Cardiovascular Health Study ULLoA Upper limit of LLoA
K[A] The success rate of measurements ULoA Upper LoA
K[AL] Lower limit of KA UULoA Upper limit of ULoA
K[AU] Upper limit of KA VA Vastus muscle
KS test Kolmogorov–Smirnov test W The waveform of the foot motion signals
LASSO Least absolute shrinkage and selection operator Z[GS] Z-scores of the gait speed
LLLoA Lower limit of LLoA Z[HGS_m] Z-scores of the HGS performance of males
LLoA Lower LoA Z[HGS_f] Z-scores of the HGS performance of females
LoA Limit of agreement β The set of fitted least-squares regression coefficients
LOSO Leave-one-subject-out β[0] The residual of the linear regression
LOSOCV Leave-one-subject-out cross-validation λ Non-negative regularization parameter input to LASSO
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Figure 2. Schematic of (a) measurement of HGS. The subjects were asked to sit on an armchair sitting with the elbow in 90° flexion, but the elbow cannot touch the chair arms. The dynamometer was set
at handle position “two”. (b) The structure of an IMS (left side). IMS was embedded in an insole placed under the foot arch near the calcaneus side and then inserted into a sport shoe.
Figure 4. (a) Process of feature construction, feature selection, and model construction for HGS estimation. Ω[1]–Ω[100]: 100 types of features combinations in accordance with different
regularization coefficients set in LASSO for HGS estimation; H[1]–H[100]: 100 types of candidate multivariate regression models for HGS estimation; ICC[k] denotes ICC value of model H[k]; M[o]:
optimal models for HGS estimation. (b) Details of LOSO-LASSO; U: total number of participants for training data; λ[u]: u-th regularization coefficient vector for LASSO, 100 dimensions; λ[ui]: i-th
element of λ[u]; β[ui]: fitted least-squares regression coefficients corresponding to λ[ui]; B[u]: u-th label matrix obtained by substituting nonzero elements in LASSO coefficient by 1; B[0]: label
counter matrix; B: final label matrix obtained by substituting elements over and below 0.95 × U by 1 and 0 in B[0]. (c) Other three models derived by optimizing three other predictor combinations by
the same process as M[o], M[1]: gait speed (GP02), M[2]: M[1] plus other GPs in one stride, and M[3]: M[2] plus IPAs. Green dashed boxes in (c) indicate the corresponding process included in the same
box shown in (a).
Figure 5. Results of correlation analysis between foot motion and HGS using SPM for both (a) males (blue lines) and (b) females (red lines). Foot motion waveforms A[x], A[y], A[z], G[x], G[y], and G
[z] were normalized by the maximum instantaneous speed in one stride. The 95% confidence interval of a waveform is shown by double dotted lines linked to foot motion signals. Statistic curves outside
gray zones for each signal type indicate that intervals of GCs significantly correlated with HGS defined as GPCs. GC: gait cycle, SPM{F}: F statistic of vector field analysis by SPM-CCA, SPM{t}:
statistic of post hoc scalar trajectory linear correlation test by SPM-PC. Single and double dotted lines linked to SPM{F} and SPM{t} indicate critical RFT threshold of F and Šidák-corrected critical
RFT threshold of t.
Figure 6. Results of LOSO-LASSO analysis to determine optimal predictor combination, M[o]. (a) Male, (b) female. The upper panels depict the regularization coefficient input into LOSO-LASSO. The
middle panels depict the number of predictors output from LOSO-LASSO. The bottom panels depict the ICC(2, 1) values of the models constructed from each predictor combination output from LOSO-LASSO.
Figure 7. Selected IMS predictors for M[o] and their corresponding GPCs for male and female subjects. Q[t]’s are marked as black blocks surrounded by green dashed line frames. Selected GPCs of each
type of foot motion are also marked as blocks (male: blue, female: red). Q[t]: Quadricep-activation %GCs, including %GCs for which only rectus femoris (RF) activated and for both RF and vastus
muscles (VAs). LR: loading response; MSt: mid-stance; TSt: terminal stance; PS: pre-swing; IS: initial swing; MSw: mid-swing; TSw: terminal swing; HS: heel strike; TO: toe-off.
Figure 8. Precision evaluation results of gait speed. (a) Agreement plots. (b) BA plots of data in Group I (green) and Group II+III (yellow). PA line: black chained line; ULoA and LLoA: black dashed
line; UULoA, LULoA, ULLoA, and LLLoA: black dotted line; fitting proportional bias line: blue dashed line. For data in Group II+III, lower to upper limits of K[A], i.e., K[A] = K[AL] − K[AU], are
depicted in the figure.
Figure 9. (a) HGS estimation agreement plots of males and females by models constructed by predictor combinations of M[o], M[1], M[2], and M[3]. Blue and red dots mean data of males and females, and
black dashed lines in all panels of (a) mean perfect agreement. “ICC” in figures means ICC value of ICC(2, 1). (b) Bland–Altman plots of M[o] case for males and females of Group I. PA line: black
chained line; ULoA and LLoA: black dashed line; UULoA, LULoA, ULLoA, and LLLoA: black dotted line; fitting proportional bias line: blue dashed line. (c) Results of HGS estimation model test using
data from Group II+III and optimistic agreement interval determined using data from Group I shown in (b). All male subjects belonged to Group III, marked as blue triangles. Lower to upper limits of K
[A], i.e., K[A] = K[AL] − K[AU], are depicted in (c). Black dashed circle in (c) means subjects in Group III who did not agree with the reference data well.
Figure 10.
ICC agreement between three types of performance scores calculated from reference and IMS-estimated values: (
, (
, (
. Points in dashed circles mean subjects whose data are outside the agreement interval in
Figure 9
c (the same data in dashed circles in
Figure 9
c). Blue points: male subjects. Red points: female subjects.
Figure 11. Correlations between expert-rated score and three types of performance scores calculated from reference value: (a) P[HGS], (b) P[GS], (c) P[fr]. Blue points: male subjects. Red points:
female subjects.
Figure 12. Correlations between expert-rated score and three types of performance scores calculated from IMS-estimated value: (a) P[HGS], (b) P[GS], (c) P[fr]. Blue points: male subjects. Red points:
female subjects.
Figure 13. Boxplot of expert-rated score in pre-frail and robust groups. Lines in the boxes indicate the median values; crosses in the boxes indicate the mean values of each group. PF: pre-frail, R:
Figure 14. Boxplot of three types of performance scores calculated from IMS-estimated values in pre-frail and robust groups: (a) P[HGS], (b) P[GS], (c) P[fr]. The green dot in (a) means the outlier
point (values exceeding 1.5 times the interquartile range are displayed as outliers). Lines in the boxes indicate the median values; crosses in the boxes indicate the mean values of each group. PF:
pre-frail, R: robust.
Figure 15. Gait motion of early and late initial swing phase. Spring mark means rectus femoris. Red lines mean segments of lower limbs. Gray dashed line means original position of each segment. Red
circles mean approximate position of knee and ankle joints. Orange dashed line means central line of body. Black bold point means approximate position of hip joint. Blue arrow means rotational motion
direction, which increases angular velocity in dorsiflexion direction on IMS. Yellow arrow means rotational motion direction, which decreases angular velocity in dorsiflexion direction on IMS. Green
line arrow means direction of gravity, and green dashed arrow means projection of gravity vector in direction perpendicular to segment of lower leg.
Table 1. Demographic data and characteristics of subjects. Subjects for model construction (Group I), Test 1 (Group II+III), and Test 2 are summarized.
Overall Male Female
Mean ± SD Mean ± SD Mean ± SD
(Min–Max) (Min–Max) (Min–Max)
Number 62 27 35
Data size 248 108 140
Age (years) 70.6 ± 6.8 70.3 ± 7.7 70.9 ± 5.9
(60.0–84.0) (60.0–84.0) (60.0–82.0)
Height (cm) 160.0 ± 8.2 166.7 ± 4.2 154.9 ± 6.6
(140.0–176.0) (160.0–176.0) (140.0–171.0)
Group I Weight (kg) 59.9 ± 11.0 66.8 ± 8.8 54.7 ± 9.4
(37.0–89.0) (53.0–89.0) (37.0–80.0)
BMI 23.3 ± 3.1 24.0 ± 2.6 22.8 ± 3.4
(15.2–32.9) (19.2–29.4) (15.2–32.9)
HGS (kg) 27.4 ± 8.1 33.7 ± 6.1 22.6 ± 5.8
(14.0–45.2) (24.3–45.2) (14.0–38.0)
Gait speed (m/s) 1.37 ± 0.18 1.35 ± 0.20 1.39 ± 0.17
(0.91–1.83) (0.99–1.83) (0.91–1.72)
Number 45 6 39
Data size 180 24 156
Age (years) 71.1 ± 7.1 77.7 ± 5.4 70.1 ± 6.8
(50.0–86.0) (70.0–86.0) (50.0–83.0)
Height (cm) 155.3 ± 6.1 166.5 ± 4.8 153.6 ± 4.0
(146.0–172.0) (160.0–172.0) (146.0–164.5)
Group II+III Weight (kg) 53.2 ± 10.1 63.1 ± 12.2 51.7 ± 8.8
(34.0–76.0) (41.0–76.0) (34.0–73.0)
BMI 22.0 ± 3.6 22.7 ± 3.8 21.9 ± 3.6
(14.5–31.1) (15.2–26.0) (14.5–31.1)
HGS (kg) 22.3 ± 4.5 26.9 ± 5.6 21.6 ± 3.9
(13.7–35.4) (17.6–35.4) (13.7–31.6)
Gait speed (m/s) 1.33 ± 0.19 1.18 ± 0.14 1.35 ± 0.19
(0.75–1.64) (1.02–1.34) (0.75–1.64)
Number 25 6 19
Data size 100 24 76
Age (years) 75.1 ± 5.8 77.7 ± 5.4 74.2 ± 5.8
(65.0–86.0) (70.0–86.0) (65.0–83.0)
Height (cm) 156.4 ± 6.6 166.5 ± 4.8 153.4 ± 3.0
(146.0–172.0) (160.0–172.0) (146.0–160.0)
Weight (kg) 51.4 ± 10.9 63.1 ± 12.2 47.9 ± 7.5
(34.0–76.0) (41.0–76.0) (34.0–62.0)
BMI 20.9 ± 3.6 22.7 ± 3.8 20.4 ± 3.4
Group III (14.5–27.7) (15.2–26.0) (14.5–27.7)
HGS (kg) 21.9 ± 4.9 26.9 ± 5.6 20.1 ± 3.4
(13.7–35.4) (17.6–35.4) (13.7–26.5)
Gait speed (m/s) 1.33 ± 0.18 1.18 ± 0.14 1.39 ± 0.16
(1.02–1.64) (1.02–1.34) (1.09–1.64)
J-CHS score: 0 (Robust) 10 1 9
J-CHS score: 1–2 (Pre-frail) 15 5 10
J-CHS score: >2 (Frail) 0 0 0
Average expert-rated score 39.3 ± 17.1 46.6 ± 23.5 37.0 ± 14.6
(12.6–82.2) (23.9–82.2) (12.6–61.6)
SD: standard deviation. HGS and gait speed are reference values.
No. Description Unit
GP01 Stride length m
GP02 One-stride gait velocity m/s
GP03 Maximum E[x] in dorsiflexion direction deg
GP04 Maximum E[x] in plantarflexion direction deg
GP05 Maximum circumduction m
GP06 Maximum foot height m
GP07 Toe in/out angle deg
GP08 E[y] at HS deg
GP09 E[y] at TO deg
GP10 Cadence step/min
GP11 Stance phase time s
GP12 Swing phase time s
GP13 Double support time 1 (loading response) s
GP14 Double support time 2 (pre-swing) s
GP15 Maximum G[x] in plantarflexion direction during swing phase deg/s
GP16 Maximum G[x] in dorsiflexion direction during swing phase deg/s
GP17 Maximum instantaneous velocity in one stride m/s
GP18 Maximum A[z] in superior direction during swing phase 9.8 m/s^2
GP19 Duration of HS to foot flat s
GP20 Duration of foot flat s
GP: gait parameter. GP01-GP07 were calculated using the method of Fukushi et al. [29]. GP13, GP14, GP19, and GP20 were calculated using the method of Huang et al. [62]. Deg: degree.
Table 3. Predictors in constructed multivariate linear regression model and their correlation analyses with HGS for males.
No. Detail Mean (SD) r Coef. p[m]
Int. Interception 37.9 0.050
C[m1] Age 70.3 (7.7) −0.599 −0.236 0.000
C[m2] Height 166.7 (4.2) 0.428 0.185 0.055
C[m3] Weight 66.8 (8.8) 0.209 0.191 0.000
C[m4] GP03 31.64 (4.83) 0.338 −0.525 0.000
C[m5] GP05 1.88 × 10^−2 (0.75 × 10^−2) 0.204 132 0.006
C[m6] GP08 −5.21 (3.83) −0.005 0.262 0.009
C[m7] GP09 3.64 (3.88) −0.049 0.246 0.013
C[m8] GP10 112.28 (9.20) 0.052 −0.222 0.000
C[m9] GP16 −97.05 (5.46) 0.303 0.314 0.000
C[m10] GP18 6.35 × 10^−1 (0.61 × 10^−1) −0.190 15.9 0.017
C[m11] GP19 9.41 × 10^−2 (2.13 × 10^−2) 0.065 116 0.000
C[m12] A[x], 97 to 98 −2.67 × 10^−1 (1.41 × 10^−1) 0.462 11.2 0.001
C[m13] A[y], 59 −5.76 × 10^−1 (1.04 × 10^−1) −0.487 −26.0 0.000
C[m14] A[z], 61 to 63 −3.43 × 10^−1 (0.84 × 10^−1) −0.389 −8.11 0.054
C[m15] G[y], 15 to 16 −5.26 × 10^−1 (8.01 × 10^−1) −0.387 −1.58 0.000
C[m16] G[z], 12 to 16 4.35 × 10^−1 (3.75 × 10^−1) 0.582 4.98 0.000
GP03: maximum E[x] in dorsiflexion direction; GP05: maximum circumduction; GP09: E[y] at TO; GP10: cadence; GP16: maximum G[x] in dorsiflexion direction during swing phase; GP18: maximum A[z] in
superior direction during swing phase; GP19: Duration of HS to foot flat. GP05 was normalized by height of subject; GP16, GP18, and GP19 were all normalized by maximum instantaneous speed in one
stride. C[m12] to C[m16]: IMS predictors, signal type, and interval range of GPCs are depicted in “Detail” column. Interval range is in %GC. Units of IMS predictors were the same as signals. C[m12]
to C[m16] were all normalized by maximum instantaneous speed in one stride. SD: standard deviation, r: linear correlation coefficient of predictor with HGS; Coef.: coefficient of multivariate
regression model using all participants’ data; p[m]: p-value of coefficient of multivariate regression model, with significance level of p[m] < 0.05.
Table 4. Predictors in constructed multivariate linear regression model and their correlation analyses with HGS for females.
No. Detail Mean (SD) r Coef. p[m]
Int. Interception −17.3 0.178
C[f1] Age 70.9 (5.9) −0.517 −0.349 0.000
C[f2] Height 154.9 (6.6) 0.682 0.374 0.000
C[f3] GP16 −102.37 (7.67) 0.199 −0.095 0.025
C[f4] A[x], 3 −1.31 × 10^−1 (0.76 × 10^−1) 0.419 28.9 0.000
C[f5] A[x], 13 to 14 −2.43 × 10^−3 (2.43 × 10^−3) −0.529 −819 0.000
C[f6] A[x], 97 −2.25 × 10^−1 (1.30 × 10^−1) −0.362 −7.03 0.002
C[f7] A[y], 69 −2.81 × 10^−1 (0.46 × 10^−2) 0.374 22.5 0.000
C[f8] G[z], 2 14.44 (9.76) −0.325 0.244 0.000
GP16: maximum G[x] in dorsiflexion direction during swing phase, which was normalized by maximum instantaneous speed in one stride. C[f4] to C[f8]: IMS predictors, signal type, and interval range of
GPCs are depicted in “Detail” column. Interval range is in %GC. C[f][4] to C[f][8] were all normalized by maximum instantaneous speed. Units of IMS predictors were the same as signals. SD: standard
deviation; r: linear correlation coefficient of predictor with HGS, Coef.: coefficient of multivariate regression model using all participants’ data, p[m]: p-value of coefficient of multivariate
regression model, with significance level of p[m] < 0.05.
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Huang, C.; Nihey, F.; Ihara, K.; Fukushi, K.; Kajitani, H.; Nozaki, Y.; Nakahara, K. Healthcare Application of In-Shoe Motion Sensor for Older Adults: Frailty Assessment Using Foot Motion during
Gait. Sensors 2023, 23, 5446. https://doi.org/10.3390/s23125446
AMA Style
Huang C, Nihey F, Ihara K, Fukushi K, Kajitani H, Nozaki Y, Nakahara K. Healthcare Application of In-Shoe Motion Sensor for Older Adults: Frailty Assessment Using Foot Motion during Gait. Sensors.
2023; 23(12):5446. https://doi.org/10.3390/s23125446
Chicago/Turabian Style
Huang, Chenhui, Fumiyuki Nihey, Kazuki Ihara, Kenichiro Fukushi, Hiroshi Kajitani, Yoshitaka Nozaki, and Kentaro Nakahara. 2023. "Healthcare Application of In-Shoe Motion Sensor for Older Adults:
Frailty Assessment Using Foot Motion during Gait" Sensors 23, no. 12: 5446. https://doi.org/10.3390/s23125446
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Question Text old as his sister . The age of and differ by 40
Updated On Nov 21, 2022
Topic Coordinate Geometry
Subject Mathematics
Class Class 12
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Data Clustering
• Data Clustering is a type of unsupervised learning algorithm that creates a new node, [Factor_i].
• Each state of this newly-created node represents a Cluster.
• Data Clustering can be used for different purposes:
□ For finding observations that appear the same, i.e., have similar values.
□ For finding observations that behave the same, i.e., interact similarly with other nodes in the network.
□ For representing an unobserved dimension by means of an induced latent Factor.
□ For summarizing a set of nodes.
□ For compactly representing the Joint Probability Distribution.
• From a technical perspective, each cluster should be:
□ Homogeneous and pure.
□ Clearly differentiated from other clusters.
□ Stable.
• From a functional perspective, all clusters should be:
□ Easy to understand.
□ Operational.
□ A fair representation of the data.
• Data Clustering with Bayesian networks is typically based on a Naive Bayes structure, in which the newly-created latent Factor Node [Factor_i] is the parent of the so-called Manifest Nodes.
□ latent (adjective): (of a quality or state) existing but not yet developed or manifest; hidden or concealed.
□ manifest (adjective): clear or obvious to the eye or mind.
• This variable being hidden, i.e., with 100% of missing values, the marginal probability distribution of [Factor_i] and the conditional probability distributions of the Manifest variables are
initialized with random distributions.
• Thus, an Expectation-Maximization (EM) algorithm is used to fit these distributions with the data:
□ Expectation: the network is used with its current distributions for computing the posterior probabilities of [Factor_i], for the entire set of observations described in the data set; These
probabilities are used for soft imputing [Factor_i];
□ Maximization: based on these imputations, the distributions of the network are updated via Maximum-Likelihood. The algorithm goes back to Expectation until no significant changes occur to the
• You can either select a subset of nodes to be included in Data Clustering, or leave all node unselected. In the latter case, all nodes on the [Graph Panel](/Users/stefanconrady/Library/
CloudStorage/OneDrive-BayesiaUSA/ClickHelp Backup/project-backup_bayesialab_2024-06-07_08-24-26/HTML/graph-windows-graph-panel.html) will be included for Data Clustering.
Feature History
[Data Clustering] has been updated in versions [5.1]and [5.2].
New Feature: Meta-Clustering
This new feature has been added for improving the stability of the induced solution (3_rd technical quality). It consists of creating a dataset made of a subset of the Factors that have been created
while searching for the best segmentation, and using Data Clustering on these new variables. The final solution is thus a summary of the best solutions that have been found (4^th purpose).
The five Manifest variables (bottom of the graph) are used in the dataset for describing the observations.
The Factor variables [Factor_1], [Factor_2], and [Factor_3] have been induced with Data Clustering. They are** then imputed to create a new dataset.
In this example, three Factor variables are used for creating the final solution [Factor_4].
Let's use a dataset that contains house sale prices for King County (opens in a new tab), which includes the city of Seattle, Washington. It describes homes sold between May 2014 and May 2015. More
specifically, we have extracted 94 houses that are more than 100 years old, that have been renovated, and come with a basement. For simplicity, we are just describing the houses with the 5 Manifest
variables below, discretized into 2 bins.
• grade: Overall grade given to the housing unit
• sqrt_above: Square footage of house, apart from basement
• sqft_living15: Living room area in 2015
• sqft_lot: Square footage of the lot
• lat: Latitude coordinate
The wizard below shows the settings used for segmenting these houses:
After 100 steps, segmenting the houses into 4 groups is the best solution. Below, the Mapping function shows the newly created states/segments:
• the size of each segment is proportional to its marginal probability (i.e. how many houses belong to each segment),
• the intensity of the blue is proportional to the purity of the associated cluster (1^st technical quality), and
• the layout reflects the neighborhood.
This radar chart (Menu > Analysis > Report > Target > Posterior Mean Analysis > Radar Chart) allows interpreting the generated segments. As we can see, they are easily distinguishable (2^nd technical
Thus, the solution with 4 segments satisfies the first two technical qualities listed above. However, what about the 3_rd one, the stability? Below are the scores of the 10 best solutions that have
been generated while learning:
Even though the best solution is made of 4 segments, this is the only solution with 4 clusters, all the other ones have nearly the same score, but with 3 clusters. Thus, we can assume that a solution
with 3 clusters would be more stable.
Using Meta-Clustering on the 10 best solutions (10%) indeed generates a final solution made of 3 clusters.
This mapping juxtaposes the mapping of the initial solution with 4 segments (lower opacity) and of the one corresponding to the meta-clustering solution.
The relationships between the final and initial segments are as follows:
• C1 groups C4 and C2 (the main difference between C4 and C2 was Square footage of the lot),
• C2 corresponds to C3
• C3 corresponds to C1
New Feature: Multinet
As stated in the Context, Data Clustering with Bayesian networks is typically done with Expectation-Maximization (EM) on a Naive structure. Thus, it is based on the hypothesis that the Manifest
variables are conditionally independent of each other given ***[Factor_i]_**. Therefore, the Naive structure is well suited for finding observations that look the same (1^st purpose), but not so good
for finding observations that behave similarly (2^nd purpose). The behavior should be represented by direct relationships between the **Manifests**.
Our new Multinet clustering is an EM^2 algorithm based both on a Naive structure (Look) and on a set of Maximum Weight Spanning Trees (MWST) (Behavior). Once the distributions of the Naive are
randomly set, the algorithm works as follows:
1. Expectation_Naive: the Naive network is used with its current distributions for computing the posterior probabilities of [Factor_i], for the entire set of observations described in the data set;
These probabilities are used for hard-imputing [Factor_i], i.e. choosing the state with the highest posterior probability;
2. Maximization_MWST: [Factor_i] is used as a breakout variable. An MWST is learned on each subset of data.
3. Expectation_MWST: the joint probabilities of the observations are computed with each MWST and used for updating the imputation of [Factor_i].
4. Maximization_Naive: based on this updated imputation, the distributions of the Naive network are updated via **Maximum-Likelihood. Then, the algorithm goes back to Expectation_Naive, until no
significant changes occur to the distributions.
Two parameters allow changing the Look/Behavior equilibrium. They can be considered as probabilities to run the Naive and MWST steps at each iteration.
Setting a weight of 0 for Behavior defines a Data Clustering quite similar to the [usual one](/Users/stefanconrady/Library/CloudStorage/OneDrive-BayesiaUSA/ClickHelp Backup/
project-backup_bayesialab_2024-06-07_08-24-26/HTML/18351067.html#DataClustering(7.0)-EM1718232680), but based on hard imputation instead of soft imputation.
Let's use the same data set that describes houses in Seattle.
The wizard below show the settings we used for segmenting the houses:
After 100 steps, segmenting the houses into three groups is the best solution. The final network is a Naive Augmented Network, with a direct link between two Manifest variables, which are, therefore,
not independent given the segmentation, i.e. the Behavior part. Note that this dependency is valid for C3 only, which can be seen after performing inference with the network.
The radar chart allows analyzing the Look of the segments.
New Feature: Heterogeneity Weight
The assumption that the data is homogeneous, given all the Manifest Variables, can sometimes be unrealistic. There may be significant heterogeneity in the data across unobserved groups, and it can
bias the machine-learned Bayesian networks. This phenomenon is known as Unobserved Heterogeneity, i.e. an unobserved variable in the dataset.
Data Clustering represents a solution for searching for such hidden unobserved groups (3^rd purpose). However, whereas the default scoring function in Data Clustering is based on the entropy of the
data, finding heterogenous groups requires modifying the scoring function.
We thus defined a Heterogeneity Index
• Factor,
• Factor, i.e. the segments used to split the data,
• Manifest variables,
• Mutual Information between the Manifest variable Target Node
The Heterogeneity Weight allows setting a weight of the Heterogeneity Index in the score, which will, therefore, bias the selection of the solutions toward segmentations that maximize the Mutual
Information of the Manifest variables with the Target Node.
Let's use the entire data set that describes houses in Seattle, with this subset of Manifest variables:
• Renovated: indicates if the house has been renovated
• Age: Age of the house
• sqft_living15: Living room area in 2015
• long: Longitude coordinate
• lat: Latitude coordinate
• Price (K$): Price of the house**.**
After setting Price (K$) as a Target Node and selecting all the other variables, we use the following settings for Data Clustering:
This returns a solution with 2 segments, generating a Heterogeneity Index of 60%. This indicates, therefore, that using [Factor_i] as a breakout variable would increase the sum of the Mutual
Informations of the Manifest variables with the Target Node by 60 %.
The Quadrant Chart below highlights the improvement of the Mutual Information. The points correspond to the Mutual Informations on the entire data set, and the vertical scales show the variations of
the Mutual Informations by splitting the data based on the values of ***[Factor_i]._**
The Heterogeneity Index is computed on the Manifest variables that are used during the segmentation only. In order to take into account other variables in the computation of the index, these
variables have to be included in the segmentation, with a weight of 0 for preventing them to influence the segmentation.
New Feature: Random Weights
By default, the weight associated with a variable is set to 1. Whereas a weight of 0 renders the variable purely illustrative, a weight of 2 is equivalent to duplicating the variable in the dataset.
The option [Mutual Information Weight,](/Users/stefanconrady/Library/CloudStorage/OneDrive-BayesiaUSA/ClickHelp Backup/project-backup_bayesialab_2024-06-07_08-24-26/HTML/2851547.html) introduced in
version 5.1, allows weighting the variable by taking into account its Mutual Information with the Target node.
As of version 7.0, a new option, Random Weights, allows to modify the weight values randomly while trying the find the best segmentation. The amplitude of the randomness is inversely proportional to
the current number of trials, therefore starting with the maximum level of randomness and ending with almost no weight modification. This option can be useful for escaping from local minima. | {"url":"https://www.bayesia.com/bayesialab/user-guide/learning/clustering/data-clustering","timestamp":"2024-11-12T02:33:02Z","content_type":"text/html","content_length":"789531","record_id":"<urn:uuid:03860a87-4d1f-4169-bc8f-5c6dc942c852>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00623.warc.gz"} |
Millionaire Calculator
Personal Finance
Millionaire Calculator
Understanding the Millionaire Calculator
The Millionaire Calculator is an essential tool that helps users forecast their financial future by estimating the growth of their investments over time. This calculator is perfect for anyone looking
to achieve millionaire status through consistent investments and smart financial planning.
How the Millionaire Calculator Works
At the core, this tool takes into account four primary inputs: initial investment amount, annual contribution, annual return rate, and the number of years the investment will grow. By inputting these
values, the calculator projects the future value of the investment.
1. Initial Investment Amount: This is the starting capital you have invested.
2. Annual Contribution: The amount you add to the investment each year.
3. Annual Return Rate: The expected annual growth rate of your investment, expressed as a percentage.
4. Years to Grow: The period for which you’ll let your investment grow.
Benefits of Using the Millionaire Calculator
The Millionaire Calculator is invaluable for anyone seeking to understand and plan their financial trajectory. It helps you:
1. Set Realistic Goals: By understanding how different variables affect your investment, you can set achievable financial goals.
2. Make Informed Decisions: Evaluate how increasing your annual contributions or changing the duration of your investment impacts your future wealth.
3. Motivation to Save: Visualizing potential outcomes can encourage disciplined saving and investing habits.
Real-World Applications
Imagine you’re 25 years old and want to save $1,000,000 by the time you’re 65. Using the Millionaire Calculator, you can input different scenarios to see how much you need to save annually with
various expected return rates to reach your goal.
For instance, starting with an initial investment of $10,000, contributing $5,000 annually, and expecting an annual return rate of 7%, you can see how this strategy will help you reach your target
amount over 40 years.
How the Calculation is Done
The formula behind the Millionaire Calculator combines compound interest calculation and the future value of a series of investments. Essentially, it adds the future value of your initial investment
and the cumulative contributions over the years, accounting for the annual return rate.
The calculation accounts for both your initial investment growing at a compounded rate and the annual contributions accumulating over time, each contributing to your overall future wealth.
What are the primary inputs for the Millionaire Calculator?
The primary inputs are: initial investment amount, annual contribution, annual return rate, and the number of years the investment will grow.
How is the annual return rate calculated?
The annual return rate is the percentage growth expected from your investment each year. It is typically based on historical performance of similar investments or projected future returns.
Can the calculator account for inflation?
This version of the calculator does not adjust for inflation. You may need to manually adjust the final amount based on your expected inflation rate for a more accurate projection of purchasing
What happens if I skip annual contributions?
If you skip annual contributions, your investment will grow at a slower rate. Consistent contributions help amplify compound growth and significantly impact your final value.
Can I use the calculator for different currencies?
Yes, the calculator can be used for any currency. Just ensure all inputs are in the same currency for accurate results.
How accurate are the projections?
The projections are estimates based on the inputs provided. Actual investment growth can vary due to market conditions, changes in return rates, and other economic factors.
Is there a minimum initial investment amount required?
No, there is no minimum initial investment amount required. You can start your calculation with any starting capital, even zero.
What if I have multiple investments with different return rates?
For multiple investments with different return rates, you will need to calculate each investment separately and then combine the results to get your overall future value.
Can I change my annual contributions over time?
This version of the calculator assumes a constant annual contribution. If you plan to change contributions over time, you would need to make separate calculations for each period of different
How frequently does the calculator compound interest?
The calculator assumes annual compounding of interest. This means the interest is calculated and added to the investment once per year.
What is the formula used for the calculation?
The formula combines the future value of a lump sum investment and the future value of a series of annual contributions. It accounts for the annual interest rate applied to both the initial amount
and the contributions. | {"url":"https://www.onlycalculators.com/finance/personal-finance/millionaire-calculator/","timestamp":"2024-11-05T20:41:43Z","content_type":"text/html","content_length":"245163","record_id":"<urn:uuid:5674c155-9090-43f8-b66b-6f4a5d633943>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00128.warc.gz"} |
1.1.2 - Strategies for Collecting Data
1.1.2 - Strategies for Collecting Data
How can we get data? How do we select observations or measurements for a study?
There are two types of methods for collecting data, non-probability methods and probability methods.
Non-probability Methods
These might include:
Convenience sampling (haphazard): Collecting data from subjects who are conveniently obtained.
□ Example: surveying students as they pass by in the university's student union building.
Gathering volunteers: Collecting data from subjects who volunteer to provide data.
□ Example: using an advertisement in a magazine or on a website inviting people to complete a form or participate in a study.
Probability Methods
□ Simple random sample: making selections from a population where each subject in the population has an equal chance of being selected.
□ Stratified random sample: where you have first identified the population of interest, you then divide this population into strata or groups based on some characteristic (e.g. sex, geographic
region), then perform simple random sample from each strata.
□ Cluster sample: where a random cluster of subjects is taken from the population of interest. For instance, if we were to estimate the average salary for faculty members at Penn State -
University Park Campus, we could take a simple random sample of departments and find the salary of each faculty member within the sampled department. This would be our cluster sample.
There are advantages and disadvantages to both types of methods. Non-probability methods are often easier and cheaper to facilitate. When non-probability methods are used it is often the case that
the sample is not representative of the population. If it is not representative, you can make generalizations only about the sample, not the population. The primary benefit of using probability
sampling methods is the ability to make inference. We can assume that by using random sampling we attain a representative sample of the population The results can be “extended” or “generalized” to
the population from which the sample came.
Example 1-1: Survey Methods
Airline Company Survey of Passengers
Let's say that you are the owner of a large airline company and you live in Los Angeles. You want to survey your L.A. passengers on what they like and dislike about traveling on your airline. For
each of the methods, determine if a non-probability method or a probability method is used. Then determine the type of sampling.
a. Since you live in L.A. you go to the airport and just interview passengers as they approach your ticket counter.
Non-probability method; convenience sampling.
b. You have your ticket counter personnel distribute a questionnaire to each passenger requesting they complete the survey and return it at end of the flight.
Non-probability methods; Volunteer sampling
c. You randomly select a set of passengers flying on your airline and question those that you have selected.
Probability method; Simple random sampling
d. You group your passengers by the class they fly (first, business, economy), and then take a random sample from each of these groups.
Probability method: Stratified sampling
e. You group your passengers by the class they fly (first, business, economy) and randomly select such classes from various flights and survey each passenger in that class and flight selected.
Probability method; Cluster sampling
Think About it!
In predicting the 2008 Iowa Caucus results a phone survey said that Hillary Clinton would win, but instead, Obama won. Where did they go wrong?
The survey was based on landline phones, which was skewed to older people who tended to support Hillary. However, lots of younger people got involved in this election and voted for Obama. The younger
people could only be reached by cell phone.
Looking Ahead
Students interested in pursuing topics related to sampling might explore STAT 506: Sampling Theory. STAT 506 covers sampling design and analysis methods that are useful for research and management in
many fields. A well-designed sampling procedure ensures that we can summarize and analyze data with a minimum of assumptions and complications. | {"url":"https://online.stat.psu.edu/stat500/book/export/html/526","timestamp":"2024-11-12T22:48:51Z","content_type":"text/html","content_length":"14147","record_id":"<urn:uuid:37ef2721-bc1d-4c6d-aa23-e8eb20e625fb>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00540.warc.gz"} |
ACO Seminar
The ACO Seminar (2017–2018)
Feb. 1, 3:30pm, Wean 8220
Cezar Lupu, University of Pittsburgh
Multiple zeta values: analytic and combinatorial aspects
The multiple zeta values (Euler–Zagier sums) were introduced independently by Hoffman and Zagier in 1992 and they play a crucial role at the interface between analysis, number theory,
combinatorics, algebra and physics.
The central part of the talk is given by Zagier's formula for the multiple zeta values, ζ(2, 2,..., 2, 3, 2, 2,..., 2). Zagier's formula is a remarkable example of both strength and the limits of the
motivic formalism used by Brown in proving Hoffman's conjecture where the motivic argument does not give us a precise value for the special multiple zeta values ζ(2, 2,..., 2, 3, 2, 2,..., 2) as
rational linear combinations of products ζ(m)π^2n with m odd. The formula is proven indirectly by computing the generating functions of both sides in closed form and then showing that both are entire
functions of exponential growth and that they agree at sufficiently many points to force their equality.
By using the Taylor series of integer powers of arcsin function and a related result about expressing rational zeta series involving ζ(2n) as a finite sum of Q-linear combinations of odd zeta values
and powers of π, we derive a new and direct proof of Zagier's formula in the special case ζ(2, 2,..., 2, 3).
Before the talk, at 3:10pm, there will be tea and cookies in Wean 6220. | {"url":"https://aco.math.cmu.edu/abs-17-18/feb1.html","timestamp":"2024-11-05T07:21:31Z","content_type":"text/html","content_length":"3018","record_id":"<urn:uuid:9145ce5c-604e-48d8-99b6-c1edf9286b70>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00558.warc.gz"} |
How to Convert Grams Into Pounds
In order to be able to compare dimensions of something, such as length or weight, it is important that the quantity is measured or described in the same units.
There are several famous examples of unit conversion mistakes that have led to disasters, such as the metric conversion disaster that resulted in a NASA orbiter drifting off-course. Therefore,
understanding unit conversion and how to check one's work can help reduce frustrating errors or even potential disasters!
How to Convert Grams to Pounds
First, grams is a unit of measure for mass, and pounds is a unit of measure of force. Often it describes the force of gravity on an object with some mass. Not only are grams and pounds different
units, but they are also different quantities all together.
Mass is the amount of matter in an object, whereas a force is determined by the acceleration of that object. Don't forget, we are constantly rotating around the Earth's axis, and the Earth around the
sun; this results in an acceleration that gives us weight on Earth. This also means that in different parts of the solar system, an object with some mass, m[1], can weigh more or less depending on
the local acceleration of gravity.
In the Imperial system, the acceleration of gravity, a, must be defined in units of feet/sec^2, and the mass, m, in slugs, in order to use for formula
to arrive at a net force, F, in pounds. In the metric system, for a mass in grams, and acceleration in meters/sec^2, the resulting force has units of newtons.
However, due to the knowledge of an average acceleration of gravity on Earth, a simple conversion factor between grams and pounds exists: 1 pound = 453.59 grams. The nuance of units is embedded into
this conversion factor.
The General Concept of Unit Conversion
In order to convert one unit to another, we need to be able to transform the quantity into another unit, without changing the quantity represented. Therefore, the most important part of unit
conversion is knowing the conversion factor between two units. For example, there are 12 inches in 1 foot, and 100 centimeters in 1 meter; these lengths are equivalent, therefore 12 inches = 1 foot
is an accurate equation.
The reason knowing the conversion factor is the most important, is because it is a form of the number 1; and multiplying a number by 1 does not change the quantity. In the case of conversion, the
conversion factor is the multiplicative factor that equals one.
Conversion With Metric Prefixes
We have already covered the grams to pounds conversion: 1 pound = 453.59 grams. However, how can we convert kilograms to pounds?
Quite often, quantities in the metric system are described by prefixes that are used to signify the order of magnitude of the number, such as millimeters, microseconds or picograms. The standard unit
of mass in the metric system is a gram; therefore, a kilogram is 1,000 grams, where the prefix kilo- means 10^3. So we immediately know the conversion from kilograms to pounds: 0.453 kg = 1 pound.
Another unit of mass in the imperial system is an ounce, which is 1/16 of a pound. Therefore, to convert ounces to grams, we can use our previously known conversion factor and divide it by 16,
resulting in: 1 ounce = 28.35 grams.
The prefix system does not work in imperial units. Instead, small quantities are often rewritten in scientific notation.
About the Author
Lipi Gupta is currently pursuing her Ph. D. in physics at the University of Chicago. She earned her Bachelor of Arts in physics with a minor in mathematics at Cornell University in 2015, where she
was a tutor for engineering students, and was a resident advisor in a first-year dorm for three years. With this experience, when not working on her Ph. D. research, Gupta participates in STEM
outreach activities to promote young women and minorities to pursue science careers. | {"url":"https://sciencing.com/convert-grams-pounds-2239703.html","timestamp":"2024-11-03T10:47:38Z","content_type":"text/html","content_length":"401951","record_id":"<urn:uuid:5a10734a-43ee-4215-9fb8-9f89d55be867>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00289.warc.gz"} |
AppSheet & FRC: Episode 5
In AppSheet & FRC: Episode 4, I showed how to use three powerful formulas: =ARRAYFORMULA, =VLOOKUP, and =QUERY. We also reviewed some simple use cases in a sample database to grasp how each formula
Now that we have a base of knowledge regarding each formula, it’s time to put that knowledge into practice and start designing a system of analysis for the data collected by our scouters.
Since we haven’t touched Google Sheets since Episode 2, the screenshot below shows the layout of the columns we’re using based on how we designed things in AppSheet. Since then, I’ve added a few rows
of sample data to demonstrate this episode’s work better.
The first point to consider when designing our analysis tools is what we should be analyzing. We want data on each team competing at our current competition, but what data do we want on them? The
answer to this question will vary greatly year-to-year and depends on what qualities you value in other teams. For this exercise, though, I’ll demonstrate how to calculate each robot’s average
score—the total number of points a given team has scored divided by the number of matches they’ve participated in.
However, to calculate this statistic, we need a list containing each robot in our dataset. Specifically, since we don’t want to aggregate a robot’s score across multiple competitions,^1At least not
for this exercise. we need a list of each unique combination of event and robot that we have data on. To calculate this, we can use the following formula:
=SORT(UNIQUE(ARRAYFORMULA(IF(ISBLANK('Match Scouting'!A2:A),,'Match Scouting'!E2:E&"-"&'Match Scouting'!H2:H))),1,TRUE)
In the screenshot on the right, the E column is the Event column, and the H column is the Team Number column. These two columns are concatenated with a dash in between, creating a unique way of
referring to each combination of event and team number. =IF and =ISBLANK are used to replace the result of the concatenation with a blank entry if column A, the Key column, is empty. =ARRAYFORMULA is
used to ensure the formula is applied to the entirety of the columns it references and =UNIQUE removes any duplicate values. Finally, =SORT makes the list alphabetical, which will be relevant later.
Note that since I’m doing this analysis in a new sheet, referencing our dataset also requires providing the name of the sheet in single quotes followed by an exclamation mark.
The result is a column where every combination of Event and Team Number is represented, but no combination is duplicated. Our goal now is to calculate the average points for each combination in the
Of course, to calculate the average points each team has scored, we first need to calculate the total number of points each team has scored. This can be done with the following formula:
=ARRAYFORMULA(IF('Match Scouting'!J2:J,2,0)+'Match Scouting'!K2:K*4+'Match Scouting'!L2:L*2+'Match Scouting'!N2:N*2+'Match Scouting'!O2:O+SWITCH('Match Scouting'!P2:P,"Low",4,"Mid",6,"High",10,"Traversal",15,0))
Column J contains a true/false value representing whether the robot Taxied in the given match, so =IF is used to provide a score of 2 or 0 as appropriate. Columns K, L, N, and O contain how many
pieces of Cargo the robot scored in various categories. The values in these columns are multiplied by constants based on how many points each category is worth. =SWITCH is used on column P to provide
varying points depending on what climb the robot completed, defaulting to 0. Finally, all of these values are summed together and wrapped in =ARRAYFORMULA.
As seen on the left, this creates a column of values calculating the number of points scored by a robot in each match. These results are in the same order as the rows in the original table. The
series of 0s^2The 0s continue down the column indefinitely. are from empty rows.
Now that we have an array of point values, we need to aggregate them for each unique combination of robot and competition. This is a perfect job for =QUERY. The following formula will do what we
=QUERY(ARRAYFORMULA({IF(ISBLANK('Match Scouting'!A2:A),,'Match Scouting'!E2:E&"-"&'Match Scouting'!H2:H),IF('Match Scouting'!J2:J,2,0)+'Match Scouting'!K2:K*4+'Match Scouting'!L2:L*2+'Match Scouting'!N2:N*2+'Match Scouting'!O2:O+SWITCH('Match Scouting'!P2:P,"Low",4,"Mid",6,"High",10,"Traversal",15,0)}),"select avg(Col2) where Col1 is not null group by Col1 label avg(Col2) ''")
This takes the result of our earlier point-calculation =ARRAYFORMULA, along with a second column containing the concatenated results of columns E and H. It then finds the average number of points
scored for every combination of team number and event, only including those rows where the Key column has a value and leaving off the default labels. Note that =QUERY automatically sorts the results
based on the group by column.^3This cannot be relied on in all software, but Google Sheets explicitly states this is the case in its Query Language Reference. This means that it will automatically
line up with the results of =UNIQUE from earlier because both are in alphabetical order.
The final result of our formulas is shown on the left, with the =SORT(UNIQUE([...] formula in A2 and the =QUERY(ARRAYFORMULA([...] in B2.^4The =ARRAYFORMULA(IF([...] is only used as a sub-formula and
doesn’t appear in our final result. Similar formulas can be used to calculate other relevant statistics. A few examples are shown below, along with the analysis each generates using my sample data.
Alliance Carry Percentage
One statistic that 5675 found very useful during the 2022 season was something we called “Alliance Carry Percentage.” This statistic answers the question: on average, how much of their Alliance’s
score does a team account for? We noticed that some teams were ranked lower than they should have due to bad luck with their Alliance partners. Alliance Carry Percentage highlights those teams by
focusing not on the total amount of points they scored but on how well they did relative to their teammates.
Note that the screenshot on the right shows the results of the formula below formatted as a percentage, which can be done by selecting the C column and then clicking More formats > Percent, as shown
in the below-right screenshot.
=QUERY(ARRAYFORMULA({IF(ISBLANK('Match Scouting'!A2:A),,'Match Scouting'!E2:E&"-"&'Match Scouting'!H2:H),{IF('Match Scouting'!J2:J,2,0)+'Match Scouting'!K2:K*4+'Match Scouting'!L2:L*2+'Match Scouting'!N2:N*2+'Match Scouting'!O2:O+SWITCH('Match Scouting'!P2:P,"Low",4,"Mid",6,"High",10,"Traversal",15,0)}/'Match Scouting'!R2:$R}),"select avg(Col2) where Col1 is not null group by Col1 label avg(Col2) ''")
When interpreting the results of the Alliance Carry Percentage column, keep in mind that a score of 33.33% should be considered standard, with a higher or lower percentage than that indicating an
above- or below-par team, respectively.
As the name suggests, the Taxi Percentage for a team represents the percentage of matches where they managed Taxi during autonomous. Although this statistic became less valuable as the season wore on
and the quality of robots improved,^5 Nearly all teams’ Taxi Percentage approached 100% toward the end of the season. it was helpful during 5675’s first competition to see which teams had a
functioning autonomous program.
=ARRAYFORMULA(IF(ISBLANK(A2:A),,COUNTIFS(IF(ISBLANK('Match Scouting'!A2:A),,'Match Scouting'!E2:E&"-"&'Match Scouting'!H2:H),A2:A,'Match Scouting'!J2:J,TRUE)/COUNTIF(IF(ISBLANK('Match Scouting'!A2:A),,'Match Scouting'!E2:E&"-"&'Match Scouting'!H2:H),A2:A)))
Although the number of points a team scores is usually the more practical statistic, sometimes it can be helpful to know how many game pieces a team scores. The formula below calculates the average
total amount of Cargo a team scores, combining the autonomous and teleoperation periods.
=QUERY(ARRAYFORMULA({IF(ISBLANK('Match Scouting'!A2:A),,'Match Scouting'!E2:E&"-"&'Match Scouting'!H2:H),'Match Scouting'!K2:K+'Match Scouting'!L2:L+'Match Scouting'!N2:N+'Match Scouting'!O2:O}),"select avg(Col2) where Col1 is not null group by Col1 label avg(Col2) ''")
A team’s Cycle Time is how long it takes that team to go from scoring one piece of Cargo to scoring the next. This value is equal to the number of seconds in the teleoperation period of a match^6In
2022, as in most years, teleoperation lasted 135 seconds. divided by the amount of Cargo the team scores during that time.
=ARRAYFORMULA(135/QUERY(ARRAYFORMULA({IF(ISBLANK('Match Scouting'!A2:A),,'Match Scouting'!E2:E&"-"&'Match Scouting'!H2:H),'Match Scouting'!N2:N+'Match Scouting'!O2:O}),"select avg(Col2) where Col1 is not null group by Col1 label avg(Col2) ''"))
After this episode, most of the work is done. We’ve built an app that can collect data on our opponents, piped the data into Google Sheets, and developed advanced analytical tools that sift out
crucial insights.
In the next and final episode, I’ll show how to generate a professional-style report showcasing our analysis. I’ll also explain how to utilize Google Sheets’ built-in interface for designing graphs
and charts, adding visual impact to the presentation of our data. | {"url":"https://www.thequips.com/2023/02/17/appsheet-frc-episode-5/","timestamp":"2024-11-04T19:06:26Z","content_type":"text/html","content_length":"81361","record_id":"<urn:uuid:cb20f2d2-0c40-40e0-bb05-60d1bc3b4a08>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00251.warc.gz"} |
Python Math – Calculating the Average with Ease
Calculating averages is a fundamental concept in various applications, from analyzing data to forecasting trends. One of the popular programming languages used for average calculation is Python. In
this blog post, we will explore the basics of averaging, the relevance of mean average, and how to calculate it using Python.
Python is a versatile and user-friendly programming language known for its simplicity and readability. Its extensive libraries and modules make it an ideal choice for mathematical calculations,
including averaging.
Basic Concepts of Averaging
Average is a measure of central tendency that summarizes a collection of data points into a single representative value. There are several types of averages commonly used, including mean, median, and
mode. In this blog post, we will focus on mean average, also known as arithmetic mean.
The mean average is calculated by summing all the data points and dividing the sum by the total number of data points. It provides a balanced representation of the data and is often used to
understand the overall trend or central value of a dataset.
Calculating the Mean Average in Python
Python’s math module offers functions to facilitate mathematical calculations, including averaging. To calculate the mean average using Python, follow these steps:
1. Handling input data: Start by collecting or importing the dataset you wish to calculate the mean average for.
2. Summing the data points: Use the sum() function from the math module to add up all the data points in the dataset.
3. Dividing the sum by the total number of data points: Use the len() function to determine the number of data points, then divide the sum by this value to calculate the mean average.
Here’s an example code snippet that demonstrates the implementation of mean average calculation:
“`python import math
data = [5, 10, 15, 20, 25]
sum_of_data = sum(data) mean_average = sum_of_data / len(data)
print(mean_average) “`
Executing the code snippet above will output the mean average of the dataset [5, 10, 15, 20, 25] as 15. This represents the balanced representation of the dataset.
Advanced Techniques for Calculating Averages
In addition to mean average, there are other types of averages that may be relevant in specific scenarios. These include median and mode.
The median is the middle value in a sorted dataset, while the mode is the value that appears most frequently. Use the median when there is a possibility of outliers that can significantly impact the
mean average. The mode is useful when identifying the most common value in a dataset.
Python provides built-in functions to calculate the median and mode. The median can be calculated using the median() function, while the mode can be calculated using the mode() function from the
statistics module.
However, if you prefer to implement custom algorithms for calculating the median and mode, you can utilize multiple Python functions such as sorting, counting, and iterating through the dataset.
Handling Special Cases and Edge Scenarios
When working with datasets, it is important to consider special cases and edge scenarios that may affect the accuracy or validity of the averages calculated.
For instance, there may be instances where the dataset contains missing or invalid values. In such cases, it is necessary to employ proper techniques to handle these scenarios:
1. Ignoring missing or invalid values during calculation: One approach is to exclude the missing or invalid values from the calculation of the average. This can be done by selectively considering
only the valid values while excluding any missing or invalid entries.
2. Handling missing or invalid values through data manipulation techniques: Another approach is to substitute missing or invalid values with appropriate replacements. This can involve using
statistical techniques, such as imputation or interpolation, to estimate missing values based on the known data points.
By addressing these special cases and edge scenarios, you ensure that the calculated averages accurately represent the underlying dataset, providing valuable insights and analysis.
Summary and Conclusion
Calculating averages is an essential concept in various applications, providing valuable insights into datasets and facilitating trend analysis. Python, with its math module and built-in functions,
offers a powerful and user-friendly platform for average calculation.
In this blog post, we explored the basics of averaging and focused on mean average, demonstrating step-by-step how to calculate it using Python. We also discussed advanced techniques for calculating
other types of averages, such as median and mode. Additionally, we addressed handling special cases and edge scenarios to ensure accurate and valid average calculations.
In conclusion, Python’s versatile nature, combined with its extensive mathematical capabilities, makes it a valuable tool for calculating averages in various scenarios. The ease and flexibility of
Python empower data analysts and programmers to effortlessly perform averaging calculations, allowing them to gain valuable insights from their datasets. | {"url":"https://skillapp.co/blog/python-math-calculating-the-average-with-ease/","timestamp":"2024-11-11T08:23:30Z","content_type":"text/html","content_length":"110306","record_id":"<urn:uuid:2264d8ab-ac51-4bd0-b2fd-893491c6ccc5>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00676.warc.gz"} |
Here are the homework sheets for the Week of October 21-25, 2024.
If your student loses their sheet, you can find the information here.
Math & Social Living
Math Wk 11 Oct. 21-25, 2024.docx
Social Studies Vocabulary
Here are some videos that you can watch to help with different math strategies.
Open Number Line Strategy
Quick Picture Strategy
Here is a "Cheat Sheet" for adding Three-Digit Numbers. It has 6 different strategies to help students add two Three-Digit Numbers. | {"url":"https://app.oncoursesystems.com/school/webpage/10987806/2002594","timestamp":"2024-11-09T20:55:11Z","content_type":"text/html","content_length":"8234","record_id":"<urn:uuid:1b6907b9-2c6c-46d7-bf1c-8a7a88eacf1a>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00350.warc.gz"} |
Systems of Linear Equations (2024)
A Linear Equation is an equation for a line.
A linear equation is not always in the form y = 3.5 − 0.5x,
It can also be like y = 0.5(7 − x)
Or like y + 0.5x = 3.5
Or like y + 0.5x − 3.5 = 0 and more.
(Note: those are all the same linear equation!)
A System of Linear Equations is when we have two or more linear equations working together.
Example: Here are two linear equations:
Together they are a system of linear equations.
Can you discover the values of x and y yourself? (Just have a go, play with them a bit.)
Let's try to build and solve a real world example:
Example: You versus Horse
It's a race!
You can run 0.2 km every minute.
The Horse can run 0.5 km every minute. But it takes 6minutes to saddle the horse.
How far can you get before the horse catches you?
We can make two equations (d=distance in km, t=time in minutes)
• You run at 0.2km every minute, so d = 0.2t
• The horse runs at 0.5 km per minute, but we take 6 off its time: d = 0.5(t−6)
So we have a system of equations (that are linear):
We can solve it on a graph:
Do you see how the horse starts at 6 minutes, but then runs faster?
It seems you get caught after 10 minutes ... you only got 2 km away.
Run faster next time.
So now you know what a System of Linear Equations is.
Let us continue to find out more about them ....
There can be many ways to solve linear equations!
Let us see another example:
Example: Solve these two equations:
The two equations are shown on this graph:
Our task is to find where the two lines cross.
Well, we can see where they cross, so it is already solved graphically.
But now let's solve it using Algebra!
Hmmm ... how to solve this? There can be many ways! In this case both equations have "y" so let's try subtracting the whole second equation from the first:
x + y − (−3x + y) = 6 − 2
Now let us simplify it:
x + y + 3x − y = 6 − 2
4x = 4
x = 1
So now we know the lines cross at x=1.
And we can find the matching value of y using either of the two original equations (because we know they have the same value at x=1). Let's use the first one (you can try the second one yourself):
x + y = 6
1 + y = 6
y = 5
And the solution is:
x = 1 and y = 5
And the graph shows us we are right!
Linear Equations
Only simple variables are allowed in linear equations. No x^2, y^3, √x, etc:
Linear vs non-linear
A Linear Equation can be in 2 dimensions ...
(such as x and y)
... or in 3 dimensions ...
(it makes a plane)
... or 4 dimensions ...
... or more!
Common Variables
Equations that "work together" share one or more variables:
A System of Equations has two or more equations in one or more variables
Many Variables
So a System of Equations could have many equations and many variables.
Example: 3 equations in 3 variables
2x + y − 2z = 3
x − y − z = 0
x + y + 3z = 12
There can be any combination:
• 2 equations in 3 variables,
• 6 equations in 4 variables,
• 9,000 equations in 567 variables,
• etc.
When the number of equations is the same as the number of variables there is likely to be a solution. Not guaranteed, but likely.
In fact there are only three possible cases:
• No solution
• One solution
• Infinitely many solutions
When there is no solution the equations are called "inconsistent".
One or infinitely many solutions are called "consistent"
Here is a diagram for 2 equations in 2 variables:
"Independent" means that each equation gives new information.
Otherwise they are "Dependent".
Also called "Linear Independence" and "Linear Dependence"
Those equations are "Dependent", because they are really the same equation, just multiplied by 2.
So the second equation gave no new information.
Where the Equations are True
The trick is to find where all equations are true at the same time.
True? What does that mean?
Example: You versus Horse
The "you" line is true all along its length (but nowhere else).
Anywhere on that line d is equal to 0.2t
• at t=5 and d=1, the equation is true (Is d = 0.2t? Yes, as 1 = 0.2×5 is true)
• at t=5 and d=3, the equation is not true (Is d = 0.2t? No, as 3 = 0.2×5 is not true)
Likewise the "horse" line is also true all along its length (but nowhere else).
But only at the point where they cross (at t=10, d=2) are they both true.
So they have to be true simultaneously ...
... that is why some people call them "Simultaneous Linear Equations"
Solve Using Algebra
It is common to use Algebra to solve them.
Here is the "Horse" example solved using Algebra:
Example: You versus Horse
The system of equations is:
In this case it seems easiest to set them equal to each other:
d = 0.2t = 0.5(t−6)
Start with:0.2t = 0.5(t − 6)
Expand 0.5(t−6):0.2t = 0.5t − 3
Subtract 0.5t from both sides:−0.3t = −3
Divide both sides by −0.3:t = −3/−0.3 = 10 minutes
Now we know when you get caught!
Knowing t we can calculate d:d = 0.2t = 0.2×10 = 2 km
And our solution is:
t = 10 minutes and d = 2 km
Algebra vs Graphs
Why use Algebra when graphs are so easy? Because:
More than 2 variables can't be solved by a simple graph.
So Algebra comes to the rescue with two popular methods:
• Solving By Substitution
• Solving By Elimination
We will see each one, with examples in 2 variables, and in 3 variables. Here goes ...
Solving By Substitution
These are the steps:
• Write one of the equations so it is in the style "variable = ..."
• Replace (i.e. substitute) that variable in the other equation(s).
• Solve the other equation(s)
• (Repeat as necessary)
Here is an example with 2 equations in 2 variables:
We can start with any equation and any variable.
Let's use the second equation and the variable "y" (it looks the simplest equation).
Write one of the equations so it is in the style "variable = ...":
We can subtract x from both sides of x + y = 8 to get y = 8 − x. Now our equations look like this:
Now replace "y" with "8 − x" in the other equation:
• 3x + 2(8 − x) = 19
• y = 8 − x
Solve using the usual algebra methods:
Expand 2(8−x):
• 3x + 16 − 2x = 19
• y = 8 − x
Then 3x−2x = x:
And lastly 19−16=3
Now we know what x is, we can put it in the y = 8 − x equation:
And the answer is:
x = 3
y = 5
Note: because there is a solution the equations are "consistent"
Check: why don't you check to see if x = 3 and y = 5 works in both equations?
Solving By Substitution: 3 equations in 3 variables
OK! Let's move to a longer example: 3 equations in 3 variables.
This is not hard to do... it just takes a long time!
• x + z = 6
• z − 3y = 7
• 2x + y + 3z = 15
We should line up the variables neatly, or we may lose track of what we are doing:
We can start with any equation and any variable. Let's use the first equation and the variable "x".
Write one of the equations so it is in the style "variable = ...":
Now replace "x" with "6 − z" in the other equations:
(Luckily there is only one other equation with x in it)
x = 6 − z
− 3y + z = 7
2(6−z) + y + 3z = 15
Solve using the usual algebra methods:
2(6−z) + y + 3z = 15 simplifies to y + z = 3:
Good. We have made some progress, but not there yet.
Now repeat the process, but just for the last 2 equations.
Write one of the equations so it is in the style "variable = ...":
Let's choose the last equation and the variable z:
Now replace "z" with "3 − y" in the other equation:
x = 6 − z
− 3y + 3 − y = 7
z = 3 − y
Solve using the usual algebra methods:
−3y + (3−y) = 7 simplifies to −4y = 4, or in other words y = −1
Almost Done!
Knowing that y = −1 we can calculate that z = 3−y = 4:
And knowing that z = 4 we can calculate that x = 6−z = 2:
And the answer is:
x = 2
y = −1
z = 4
Check: please check this yourself.
We can use this method for 4 or more equations and variables... just do the same steps again and again until it is solved.
Conclusion: Substitution works nicely, but does take a long time to do.
Solving By Elimination
Elimination can be faster ... but needs to be kept neat.
"Eliminate" means to remove: this method works by removing variables until there is just one left.
The idea is that we can safely:
• multiply an equation by a constant (except zero),
• add (or subtract) an equation on to another equation
Like in these two examples:
CAN we safely add equations to each other?
Yes, because we are "keeping the balance".
Imagine two really simple equations:
x − 5 = 3
5 = 5
We can add the "5 = 5" to "x − 5 = 3":
x − 5 + 5 = 3 + 5
x = 8
Try that yourself but use 5 = 3+2 as the 2nd equation
It works just fine, because both sides are equal (that is what the = is for)
We can also swap equations around, so the 1st could become the 2nd, etc, if that helps.
OK, time for a full example. Let's use the 2 equations in 2 variables example from before:
Very important to keep things neat:
Now ... our aim is to eliminate a variable from an equation.
First we see there is a "2y" and a "y", so let's work on that.
Multiply the second equation by 2:
Subtract the second equation from the first equation:
Yay! Now we know what x is!
Next we see the 2nd equation has "2x", so let's halve it, and then subtract "x":
Multiply the second equation by ½ (i.e. divide by 2):
Subtract the first equation from the second equation:
And the answer is:
x = 3 and y = 5
And here is the graph:
The blue line is where 3x + 2y = 19 is true
The red line is where x + y = 8 is true
At x=3, y=5 (where the lines cross) they are both true. That is the answer.
Here is another example:
Lay it out neatly:
Multiply the first equation by 3:
Subtract the second equation from the first equation:
0 − 0 = 9 ???
What is going on here?
Quite simply, there is no solution.
They are actually parallel lines:
And lastly:
Multiply the first equation by 3:
Subtract the second equation from the first equation:
0 − 0 = 0
Well, that is actually TRUE! Zero does equal zero ...
... that is because they are really the same equation ...
... so there are an Infinite Number of Solutions
And so now we have seen an example of each of the three possible cases:
• No solution
• One solution
• Infinitely many solutions
Solving By Elimination: 3 equations in 3 variables
Before we start on the next example, let's look at an improved way to do things.
Follow this method and we are less likely to make a mistake.
First of all, eliminate the variables in order:
• Eliminate xs first (from equation 2 and 3, in order)
• then eliminate y (from equation 3)
Start with:
Eliminate in this order:
We then have this "triangle shape":
Now start at the bottom and work back up (called "Back-Substitution")
(put in z to find y, then z and y to find x):
And we are solved:
ALSO, it is easier to do some of the calculations in our head, or on scratch paper, instead of always working within the set of equations:
• x + y + z = 6
• 2y + 5z = −4
• 2x + 5y − z = 27
Written neatly:
x + y + z = 6
2y + 5z = −4
2x + 5y − z = 27
First, eliminate x from 2nd and 3rd equation.
There is no x in the 2nd equation ... move on to the 3rd equation:
Subtract 2 times the 1st equation from the 3rd equation (just do this in your head or on scratch paper):
And we get:
Next, eliminate y from 3rd equation.
We could subtract 1½ times the 2nd equation from the 3rd equation (because 1½ times 2 is 3) ...
... but we can avoid fractions if we:
• multiply the 3rd equation by 2 and
• multiply the 2nd equation by 3
and then do the subtraction ... like this:
And we end up with:
We now have that "triangle shape"!
Now go back up again "back-substituting":
We know z, so 2y+5z=−4 becomes 2y−10=−4, then 2y=6, so y=3:
Then x+y+z=6 becomes x+3−2=6, so x=6−3+2=5
And the answer is:
x = 5
y = 3
z = −2
Please check this for yourself, it is good practice.
General Advice
Once you get used to the Elimination Method it becomes easier than Substitution, because you just follow the steps and the answers appear.
But sometimes Substitution can give a quicker result.
• Substitution is often easier for small cases (like 2 equations, or sometimes 3 equations)
• Elimination is easier for larger cases
And it always pays to look over the equations first, to see if there is an easy shortcut ... so experience helps ... so get experience!
591, 592, 593, 594, 1240, 61, 1241, 2863, 8157, 8158
Linear Equations Algebra Index | {"url":"https://southsound.org/article/systems-of-linear-equations","timestamp":"2024-11-06T12:14:50Z","content_type":"text/html","content_length":"157093","record_id":"<urn:uuid:8d79c240-6632-4ae3-8018-233a87db9645>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00692.warc.gz"} |
Numbers and Counting in Turkish
These lessons will help you to learn everything about numbers in Turkish. Counting , ordinal numbers , useful sentences and numbers will be covered in Turkish.
Numbers in Turkish
Firstly, We need to learn the numbers in Turkish language from 0 to 9. It is essential to learn before you learn counting from 10 to 100.
Numbers in Turkish
Counting in Turkish
It is assumed that you have learnt well the numbers in Turkish. Follow the below rules properly to learn counting in Turkish from 10 to 100.
How to count in Turkish
It is assumed that you have learnt well the numbers in Turkish. Follow the below rules properly to learn counting in Turkish from 10 to 100.
1. Memorize the multiples of the ten. Look at the below table to learn the multiples of ten.
Multiples of ten
2. After learning numbers 1 to 10 in Turkish. You should add the appropriate multiple of ten and then add the numbers you need.
54 = 50 + 4 -> Elli + Dört -> Elli dört
34 = 30 + 4 -> Otuz + Dört -> Otuz dört
67 = 60 +7 -> Altmış + yedi -> Altmış yedi
81 = 80 + 1 -> Seksen + bir -> Seksen bir
Ordinal Numbers in Turkish
You have learnt how to count in Turkish. Not it is to easy to describe the ordinal numbers in Turkish.
Examine the below table to see how ordinal numbers in Turkish
Ordinal numbers in Turkish language
Useful phrases related with numbers in Turkish
Lots of example sentences related with numbers . It is shared with you most common questions and answers related with numbers.
They will help you to understand how numbers in Turkish can be used in the sentences.
• Kaç yaşındasın?
• How old are you?
• 32 yaşındayım.
• I am 32 years old.
• Üçüncü 3. kattayım
• I’m on the 3rd floor.
• Kaç kilo istiyorsun?
• How many kilos do you want?
• Kaç kilosun?
• What is your weight?
• Seksen üç 83 kiloyum.
• I am 83 kg.
• Bu ne kadar?
• How much does that cost?
• Altmış beş lira.
• it is 65 lira.
• Yarışı ikinci (2.) sırada bitirdim.
• I finished the race in the second (2nd) place.
Yarışı 2. sırada bitirdim.
• Sınavda birinci (1.) oldum.
• I came first in the exam.
• Üçüncü (3.) sıradayım.
• I’m in third (3rd) place.
• Dördüncü (4.) denemede soruyu çözdüm.
• I solved the problem on the fourth try.
4. denemede soruyu çözdüm.
you can share with us ın the comment section. Turkish lessons are in the page of Turkish for beginners | {"url":"https://turkishwithemre.com/turkish-for-beginners/numbers-and-counting-in-turkish/","timestamp":"2024-11-12T02:58:58Z","content_type":"text/html","content_length":"130118","record_id":"<urn:uuid:a5a36dba-542a-4125-8e04-cb118d1fd7bf>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00064.warc.gz"} |
LIKECS03 - Editorial
Problem Link
Author: Bhuvnesh Jain
Tester: Hasan Jaddouh
Editorialist: Bhuvnesh Jain
Bitwise Operations, Greedy Algorithm
Given an array and a recursive algorithm which operates on this array, find the minimum number of elements to be inserted so that the algorithm on the resultant array never goes into infinite loop.
The algorithm is as follows :
void recurse ( array a, int n )
// n = size of array
define array b currently empty
consider all 2^n subsets of a[]
x = bitwise OR of elements in the subsets
add x into "b" if it is not present yet
if (sizeof( b ) == 1 << k)
recurse ( b, sizeof( b ) );
Quick Explanation
Try to think about powers of 2. Answer is simply the number of missing powers of 2.
It suffices to consider only the unique elements in the initial array because a OR a = a.
Now, assume we stop the recursion the moment when we see there are no new addition of elements in the array. At this moment 2 things can happen, either the array will contain all numbers from 0 to (2
^K - 1) i.e. it size will be 2^K or some of the elements might we missing from the array. Let us assume for now, we have a black box which can effectively give us the final array after we stop this
recursion, i.e. generate all elements in the final array formed. If the size of array is 2^K, we do not need to insert any element and the answer is trivially 0. Otherwise, let the smallest number
that is not generated be x. We will insert this number into the array and run the same black box again. This is because, since x was not generated by the previous array, for the new array to prevent
infinite loop in the algorithm, x should be present. If we insert any larger number which was not generated, then again we can’t form x as OR operations always generates a number which is greater
than or equal to the minimum of 2 numbers.
We will now prove that the smallest number which is not found the black box is indeed a power of 2. Let y be the number of bits set in x. Then, there are exactly 3^y ordered pairs, (u, v), whose
bitwise OR is x.
One can observe that once a bit is set while doing OR operation, it will remain set in the end also. So if the algorithm given in the problem generates x, it must do so in maximum k steps of
iterations. So, if it doesn’t generate x, then there should not also not generate both of (u, v) for all u, and v, such that their bitwise OR is x. Since, x >= min(u, v). So, our claim that x is
the smallest number not generated will be wrong unless x is power of 2, as the only possible ordered pairs are (0, x), (x, 0) and (x, x).
Since, we proved x should be power of 2, so even after inserting x into the array, we will not generate y where y is power of 2 not initially present in array and is greater than x. Thus, we do this
step again and again and greedily insert only powers of 2 into the array. In the end once all powers of 2 are present in the array, then considering only subsets of these, we can generate all numbers
using OR operations. Thus, greedy strategy will be optimal here.
Hence, we simply need to find the count of powers of 2 not present in the initial array. For this, you can either mark the elements in other auxilliary space (of size 2^k) or simply insert only
powers of 2 into a set and check it’s size.
Bonus Problem
Try to above black box mentioned with complexity O(2^k \log{n}).
Time Complexity
O(N \log{n}) or O(N + 2^K)
Space Complexity
O(N) or O(2^K)
Solution Links
4 Likes
@admin . I think you have posted solutions of 4th question here…
What are the subsets of an array? Plz someone give me an example…
This problem can be solved in O(N+K). Extra space required is O(K).
Check solution here
1 Like
Should’nt we check whether the array has 0 or not?0 cannot be expressed as an OR of any other two numbers as well right?
Should’nt we check whether the array has 0 or not?0 cannot be expressed as an OR of any other two numbers as well right?
But I don’t see how 0 and the empty subset are same.For the empty subset I would say there are no elements hence do nothing at all.So I thought it essential to contain a 0, I got a WA because of
this.Can you please explain why empty subset and 0 are the same?
1 Like
This problem can also be solved with this solution
Time complexity : O(N.K)
Extra space : O(1)
The logic in both if-else condition is same(Just thought something at that time).
O(N.K) is worse than O(N+2^k), just saying.(For only this problem with its constraints)
For T=1, N=10^5, K=20
N.K = 2x10^6
N+2^k = 10^5 + 2^20 = 1148576, which is less than 2x10^6
Finally it is space vs time constraint…
(PS: How to reply to a comment? Could not find any button here)
A solution of O(n+k) time and O(2^k) space is possible.
Since we can go through just the powers of 2, and not the entire 2^k elements.
Edit: And I noticed it is already mentioned only that it is challenged for log2. I have it reversed and multiply by 2 won’t be a problem.
input :
if we follow above approach we get result as 2 since 2 and 1 are missing but if we consider all subsets of intial array we need not add any elements hence answer would be 0. pls clarify this
1 Like
While the initial eleemnts of the array can’t be \geq2^k, there is no such restriction on additional elements.
Consider the case n=2, k=2, a=\{0,3\}. The algorithm presented here would find that 2 elements are missing (since 1 and 2 are not in the initial array). But if you take 4 as additonal element, i.e.
a'=\{0,3,4\} the resulting array after one step is b=\{0,3,4,7\} which has 2^k elements. So the algorithm terminates with just one additional element.
6 Likes
i can not understand the editorial. why there are exactly 3^y ordered pairs (u, v), whose or is x.
can anyone expain it? thanks
@likecs as per the @ceilks observation, whether the test cases were weak and most of the solution would fail or test cases were intended this way?
1 Like
@likecs and @ceilks are anagrams ???
4 Likes
bro you are not considering the complexity of log2() function.
Empty subset is considered in question. So every array has 0 by default.
Sorry fr adding this as a new answer
1 Like
This sample I/o cleared it for me-
Answer is- Add only 2. Only possible if we consider empty subset as 0. It makes sense by this definition-
How will you calculate x?
int x=0;
//For(subset b)
x = bitwise OR of elements in the subsets
X needs to be 0 to give correct answer. You cant leave it uninitialised. So , empty subset added 0 to array.
1 Like | {"url":"https://discuss.codechef.com/t/likecs03-editorial/16574","timestamp":"2024-11-15T00:16:42Z","content_type":"text/html","content_length":"56630","record_id":"<urn:uuid:fb28b810-813b-4c45-9bbf-06b6ce14940c>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00260.warc.gz"} |
Pigeon holes, Markov chains and Sagemath.
On the 16/10/2013
I posted the following picture on G+
Here's what I wrote on that post:
For a while now there's been a 'game' going on with our pigeon holes where people would put random objects in other people's pigeon holes (like the water bottle you see in the picture). These
objects would then follow a random walk around the pigeon holes as each individual would find an object in their pigeon hole and absent-mindedly move it to someone else's pigeon hole.
As such each pigeon hole could be thought of as being a transient state in a Markov chain (http://en.wikipedia.org/wiki/Markov_chain). What is really awesome is that one of the PhD students here
didn't seem to care when these random objects appeared in her pigeon hole. Her pigeon hole was in fact an absorbing state. This has now resulted in more or less all random objects (including a
wedding photo that no one really knows the origin of) to be in her pigeon hole.
I thought I'd have a go at modelling this as an actual Markov chain. Here's a good video by a research student of mine (
+Jason Young
) describing the very basics of a Markov chain:
To model the movement of an object as a Markov chain we first of all need to describe the
. In our case this is pretty easy and we simply number our pigeon holes and refer to them as states. In my example there I've decided to model a situation with 12 pigeon holes.
What we now need is a set of transition probabilities which model the random behaviour of people finding an object in their pigeon hole and absent-mindedly moving it to another pigeon hole.
This will be in the form of a matrix
. Where
denotes the probability of going from state
to state
I could sit in our photocopier room (that's where our pigeon holes are) and take notes as to where the individual who owns pigeon hole
places the various objects that appear in their pigeon hole...
That would take a lot of time and sadly I don't have any time. So instead I'm going to use
+Sage Mathematical Software System
. The following code gives a random matrix:
N = 12
P = random_matrix(QQ, N, N)
This is just a random matrix over
so we need to do tiny bit of work to make it a stochastic matrix:
P = [[abs(k) for k in row] for row in P] # This ensures all our numbers are positive
P = matrix([[k / sum(row) for k in row] for row in P]) # This ensures that our rows all sum to 1
The definition of a stochastic matrix is any matrix
such that:
• $P$ is square
• $P_{ij}\geq 0$ (all probabilities are non negative)
• $\sum_{j}P_{ij}=1\;\forall\;i$ (when leaving state $i$ the probabilities of going to all other states must sum to 1)
Recall that our matrix is pretty big (12 by 12) so we the easiest way to visualise it is through a heat map:
Here's what a plot of our matrix looks like (I created a bunch of random matrix gifs
We can find the steady state probability of a given object being in any given state using a very neat result (which is not actually that hard to prove). This probability vector
denotes the probability of being in state
) will be a solution of the matrix equation:
$$\pi P = \pi$$
To solve this equation it can be shown that we simply need to find the eigenvector of
corresponding to the unit eigenvalue:
eigen = P.eigenvectors_left() # This finds the eigenvalues and eigenvectors
To normalise our eigenvector we can do this:
pi = [k[1][0] for k in eigen if k[0] == 1][0] # Find eigenvector corresponding to unit eigenvalue
pi = [k / sum(pi) for k in pi] # normalise eigenvector
Here's a bar plot of out probability vector:
We can read the probabilities from this chart and see the probability of finding any given object in a particular pigeon hole. The
function in Sage still needs a bit of work and at the moment can only print a single list of data so it automatically has the axis indexed from 0 onwards (not from 1 to 12 as we would want). We can
easily fix this using some
code (Sage is just wrapping matplotlib anyway):
import matplotlib.pyplot as plt
plt.bar(range(1, N + 1), pi)
Here's the plot:
We could of course pass a lot more options to the matplotlib plot to make it just as we want (and I'll in fact do this in a bit). The ability to use base python within Sage is really awesome.
One final thing we can do is run a little simulation of our objects going through the chain. To do this we're going to sample a sequence of states (pigeon holes
). For every
we sample a random number
$0\ r\leq 1$
and find
such that
$\sum_{j'=1}^{j}P_{ij'}. This is a random sampling technique called inverse random sampling.
import random
def nextstate(i, P):
A function that takes a transition matrix P, a current state i (assumingstarting at 0) and returns the next state j
r = random.random()
cumulativerow = [P[i][0]]
for k in P[i][1:]: # Iterate through elements of the transition matrix
cumulativerow.append(cumulativerow[-1] + k) # Obtain the cumulative distribution
for j in range(len(cumulativerow)):
if cumulativerow[j] >= r: # Find the next state using inverse sampling
return j
return j
states = [0]
numberofiterations = 1000
for k in range(numberofiterations):
We can now compare our simulation to our theoretical result:
import matplotlib.pyplot as plt
plt.bar(range(1, N + 1), pi, label='Theory') # Plots the theoretical results
plt.hist([k + 1 for k in states], color='red', bins=range(1, N + 2), alpha=0.5, normed=True, histtype='bar', label='Sim') # Plots the simulation result in a transparent red
plt.legend() # Tells matplotlib to place the legend
plt.xlim(1, N) # Changes the limit of the x axis
plt.xlabel('State') # Include a label for the x axis
plt.ylabel('Probability') # Include a label for the y axis
plt.title("After %s steps" % numberofiterations) # Write the title to the plot
We see the plot here:
A bit more
flexing of muscles
allows us to get the following animated gif in which we can see the simulation confirming the theoretical result:
This post assumes that all our states are transitive (although our random selection of
$P$ could
give us a non transitive state)
the motivation of my post is the fact that one of our students' pigeon holes was in fact absorbing. I'll write another post soon looking at that (in particular seeing which pigeon hole is most likely
to move the object to the absorbing state). | {"url":"https://drvinceknight.blogspot.com/2013/10/pigeon-holes-markov-chains-and-sagemath.html","timestamp":"2024-11-08T10:38:07Z","content_type":"application/xhtml+xml","content_length":"68588","record_id":"<urn:uuid:053976af-e3dc-4528-a2d5-32694b2fe854>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00883.warc.gz"} |
Fixed-Parameter Algorithms in Phylogenetics
We survey the use of fixed-parameter algorithms in
phylogenetics. A central computational problem in
this field is the construction of a likely phylogeny
(genealogical tree) for a set of species based on
observed differences in the phenotype, on
differences in the genotype, or on given partial
phylogenies. Ideally, one would like to construct
so-called perfect phylogenies, which arise from an
elementary evolutionary model, but in practice one
must often be content with phylogenies whose
"distance from perfection" is as small as
possible. The computation of phylogenies also has
applications in seemingly unrelated areas such as
genomic sequencing and finding and understanding
genes. The numerous computational problems arising
in phylogenetics are often NP-complete, but for many
natural parametrizations they can be solved using
fixed-parameter algorithms.
Original language English
Title of host publication Methods in Molecular Biology : Bioinformatics: Volume I: Data, Sequence Analysis and Evolution
Volume 452
Publisher Springer Berlin Heidelberg
Publication date 2008
Pages 507-535
Publication status Published - 2008
• Tantau, T., Schnoor, I., Elberfeld, M., Kuczewski, J. & Pohlmann, J.
01.01.05 → 31.12.10
Project: DFG Projects › DFG Individual Projects | {"url":"https://research.uni-luebeck.de/en/publications/fixed-parameter-algorithms-in-phylogenetics","timestamp":"2024-11-07T16:44:20Z","content_type":"text/html","content_length":"45329","record_id":"<urn:uuid:be005e5f-9c84-4806-9e56-0dcd8c4e28ad>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00681.warc.gz"} |
Key Functions
PIStream can support PCB design controlling power integrity (PI) by reducing input impedance and transfer impedance. PI problems occur when IR drop (voltage drop) relevant to IC power consumption in
the Power Distribution Network (PDN) occur. The following three key features support your PI design.
1. Input Impedance Analysis
PIStream calculates input impedance and helps you meet your target impedance by adding de-caps, changing the capacitor location, and changing the power plane thickness and shape.
This analysis shows the Z11 effect from an aggressor IC and allows you to consider optimal capacitor position by trial-and-error to keep input impedance below the target impedance.
The following figure shows the input impedance (Z11) structure.
(1) Countermeasure by Input Impedance Analysis
Case 1: Target impedance achieved by adding capacitors around target IC.
Adding capacitors around the red area near the target IC keeps the input impedance below the target impedance. You will be able to optimize capacitor location from this analysis.
Case 2: Target impedance achieved by changing thickness between power and GND plane.
This example shows that a thinner insulator thickness helps keep the input impedance below the target impedance.
(2) Auto Capacitor Placement
This function selects optimum capacitors for resonance frequency and automatically places them near IC power pins. The following figure shows the effect before and after running the auto capacitor
placement function.
As shown, the input impedance is above the target impedance before running this function. After it is run, eight capacitors are added around the IC power pins and color gradation turns to blue.
(3) Target Impedance Setting for Input Impedance Analysis
This GUI lets you to enter the allowable maximum impedance value supplied by IC venders. You can also import a CSV file into this GUI.
This setting will appear as a red line as a reference impedance value in the input impedance analysis result.
2. Transfer Impedance Analysis
PIStream calculates transfer impedance to avoid noise transition to other ICs and internal RF interference.
The following figures show the results of transfer impedance analysis.
This function allows you to consider noise distribution caused by the power source (aggressor) IC. A hot area means a high transfer impedance area, and when the aggressor IC consumes a current, it
indicates a huge voltage drop (IR drop) has occurred. To avoid such voltage drop, you can put a capacitor on the red area. By doing this, the red area turns to blue and you will be able to confirm if
the transfer impedance has been reduced and the noise distribution suppressed.
This analysis shows the Z21 (Transfer Impedance) effect using a color gradation map.
Transfer impedance is calculated by power noise emission from the power source IC.
PIStream also calculates IR Drop and displays color gradation as well as the IR Drop value for each IC. | {"url":"https://www.nec.com/en/global/prod/pistream/functions.html?","timestamp":"2024-11-08T12:26:50Z","content_type":"text/html","content_length":"30361","record_id":"<urn:uuid:f29ebf60-ed02-438d-8074-562081dff23c>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00170.warc.gz"} |
This paper is intended to equip the candidate with the knowledge and skills that will enable him/her to analyze and trade in the various types of derivative investments.
• Demonstrate an understanding of the features, structure and operations of derivatives markets
• Develop a framework for pricing various types of derivatives
• Value derivative instruments using discrete time and continuous time valuation principles.
• Price and hedge interest rate swaps
• Use financial derivative instruments for managing and hedging portfolio risk.
• Apply the framework for risk management so as to enable identification, assessment and
control of numerous sources of risk.
1. Introduction to Derivative Markets and Instruments
1.1 Introduction to Derivatives
1.2 Derivative specific definitions and terminologies
1.3 Types of Derivatives: forward commitments, contingent claims, financial futures, forward contracts, options, swaps, Exotic Derivatives, Forwards: Range forward contract, break forward contract;
Options: Asian or average-rate options, Look back options, Barrier options, Rainbow options, Compound options, Chooser options; Swaps: Interest rate swap variants, Currency swap variants, Equity swap
1.4 Overview of derivative markets; regulation, players, Trading of financial derivatives, Trading of commodities derivatives, Buying and shorting financial assets
1.5 The Structure and purpose of derivative markets
1.6 Users and uses of financial derivatives
1.7 Criticisms of derivative markets
1.8 Elementary principles of derivative pricing
1.9 Size and Scope of derivatives markets; Global and regional derivatives markets.
2. Forward Markets and Contracts
2.1 Introduction to forward markets and contracts
2.2 The structure and role of forward markets
2.3 Types of forward contracts: equity forwards contracts; bond and interest rate forward contracts; currency forward contracts; other types of forward contracts
2.4 Mechanics of Forward Markets and Contracts; Delivery and settlement of a forward contract; default risk and forward contracts; termination of a forward contract; cost of carry and transaction
2.5 Pricing and valuation of forward contracts: generic pricing and valuation of forward contracts; pricing and valuation of equity forward contracts; pricing and valuation of fixed-income and
interest-rate forward contracts; pricing and valuation of currency forward contracts
2.6 Credit risk and forward contracts
3. Futures Markets and Contracts
3.1 Introduction: Definition of Futures, Brief history of futures markets;
3.2 Types of futures contracts: short-term interest rate futures contracts; intermediate- and long-term interest rate futures contracts; Bond futures contracts; stock index futures contracts;
currency futures contracts; Commodities
futures contracts – Agricultural, Energy, Precious and Industrial metal futures
3.3 Characteristics of Futures markets: Public standardized transactions; homogenisation and liquidity; the clearinghouse; daily settlement; and performance guarantee; regulation
3.4 Futures trading: the clearinghouse, margins, and price limits; delivery and cash settlement; futures exchanges. Mechanics of trading in futures markets; Long and short positions, Profit and loss
at expiration, Closing of positions, Delivery procedures, marking to market of futures contracts, leverage effect, futures quotes
3.5 Pricing and valuation of futures contracts: generic pricing and valuation of a futures contract; pricing interest rate futures, stock index futures, and currency futures; Factors determining
contract price – CAPM, hedging pressure theory and cost of carry model; Theoretical and Reality price of futures; Comparing the calculated value of the future vs the market
3.6 Uses of financial and non-financial futures
3.7 The role of futures markets and exchanges
4. Risk Management applications of Forward and Futures strategies
4.1 Introduction to risk exposures managed by Forwards and Futures
4.2 Strategies and applications for managing interest rate risk: managing the interest rate risk of a loan using a forward contract; strategies and applications for managing bond portfolio risk
4.3 Strategies and applications for managing equity market risk: measuring and managing the risk of equities; managing the risk of an equity portfolio; creating equity out of cash; creating cash out
of equity
4.4 Asset allocation with futures: adjusting the allocation among asset classes; pre-investing in an asset class
4.5 Strategies and applications for managing foreign currency risk: managing the risk of a foreign currency receipt; managing the risk of a foreign currency payment; managing the risk of a
foreign-market asset portfolio
4.6 Hedging strategies using futures: hedge ratio, perfect hedge, basis risk and correlation risk, minimum variance hedge ratio, and hedging with several futures contracts.
5. Swap Markets and Contracts
5.1 Introduction: Definition of Swap contracts, Types of swaps: currency swaps; interest rate swaps; equity swaps; commodity and other types of swaps
5.2 Characteristics of swap contracts
5.3 The structure of global swap markets
5.4 Pricing and valuation of swaps; pricing and valuation of swaps 5.5 Swaptions: basic characteristics of swaptions; uses of swaptions; swaption payoffs; pricing and valuation of swaptions
5.6 Termination of a swap
5.7 Forward swaps
5.8 The role of swap markets
5.9 Uses of Swap Contracts: Credit risks and swaps
6. Risk management application of swap strategies
6.1 Introduction to risk exposures managed by Swaps
6.2 Strategies and applications for managing interest rate risk: using interest rate
swaps to convert a floating-rate loan to a fixed-rate loan (and vice versa); using
swaps to adjust the duration of a fixed-income portfolio; using swaps to create
and manage the risk of structured notes, reducing the cost of debt
6.3 Strategies and applications for managing exchange rate risk: converting a loan in
one currency into a loan in another currency; converting foreign cash receipts
into domestic currency; using currency swaps to create and manage the risk of a
dual-currency bond
6.4 Strategies and applications for managing equity market risk; diversifying a
concentrated portfolio; achieving international diversification; changing an asset
allocation between stocks and bonds; reducing insider exposure
6.5 Strategies and applications using swaptions; using an interest rate swaption in
anticipation of a future borrowing; using an interest rate swaption to terminate a
7. Option markets and contracts
7.1 Introduction: Basic definitions and illustrations of options contracts:
7.2 Types of options: Financial options; options on futures; commodity options; other
types of options
7.3 Characteristics of Options Contracts: some examples of options;
7.4 The concept of moneyness of an option
7.5 The structure of global options markets: over-the-counter options markets;
exchange-listed option markets
7.6 Options Valuation: Determinants of option price, Option pricing models,
sensitivity analysis options premiums
7.7 Principles of option pricing; payoff values: Boundary conditions; the effect of a
difference in exercise price; the effect of a difference in time to expiration; put-call
parity; American options, lower bounds, and early exercise; the effect of cash
flows on the underlying asset; the effect of interest rates and volatility; option
price sensitivities
7.6 Discrete-time option pricing: The binomial model; the one-period binomial model;
the two-period binomial model; binomial put option pricing; binomial interest rate
option pricing; American options: extending the binomial model
7.7 Continuous-time option pricing: The Black-Scholes-Merton model; assumptions
of the model; the black-Scholes-Merton formula; inputs to the black-ScholesMerton model; the effect of cash flows on the underlying; the critical role of
7.8 Pricing options on forward and futures contracts and an application to interest
rate option pricing: Put-call parity for options on forwards; early exercise of
American options on forward and futures contracts; the black model; application
of the black model to interest rate options
7.9 The role of options markets;
7.10 Uses of Options
8. Risk management applications of option strategies
8.1 Introduction to risk exposures managed by options
8.2 Option strategies for equity portfolios: standard long and short positions; risk management strategies with options and the underlying; money spreads; combinations of calls and puts
8.3 Interest rate option strategies using: interest rate calls with borrowing; interest rate puts with lending; an interest rate cap with a floating-rate loan; an interest rate floor with a
floating-rate loan; an interest rate collar with a floating-rate loan
8.4 Option portfolio risk management strategies: delta hedging an option over time; gamma and the risk of delta; vega and volatility risk; the Greeks.
9. Contemporary issues and emerging trends in derivatives Contracts
9.1 Numerical methods of Pricing Options: binomial model, finite difference method and Monte Carlo method.
9.2 Credit derivatives: Credit default swaps (CDS), Credit linked notes (CLN), role of credit derivatives, market participants, Valuation of credit derivatives, credit derivatives institutional
framework, spread volatility of credit default swaps
9.3 Financial Engineering; Construction, Uses and Abuses of Derivatives
9.4 Applications of Artificial intelligence and financial technology in derivatives markets
9.5 Benefits and Indispensability of derivatives
9.6 Trends and future of derivatives market globally
9.7 Effects of Crises and Pandemic on global derivatives market. | {"url":"https://college.icifa.co.ke/product/paper-no-16-derivatives-analysis/","timestamp":"2024-11-11T22:53:56Z","content_type":"text/html","content_length":"112868","record_id":"<urn:uuid:f76c5252-f735-4d28-8508-1e8522a2a559>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00235.warc.gz"} |
A small block slides without friction down an inclined plane st... | Filo
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Question Text A small block slides without friction down an inclined plane starting from rest. Let Sn be the distance travelled from time to and be the distance travelled from time to . Then is
Updated On Nov 7, 2022
Topic Mechanics
Subject Physics
Class Class 12
Answer Type Video solution: 2
Upvotes 189
Avg. Video Duration 5 min | {"url":"https://askfilo.com/user-question-answers-physics/a-small-block-slides-without-friction-down-an-inclined-plane-32323138303436","timestamp":"2024-11-07T13:19:38Z","content_type":"text/html","content_length":"297788","record_id":"<urn:uuid:4edf0405-e348-4208-8aed-2ba688170958>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00834.warc.gz"} |
twelvefold way
1—10 of 89 matching pages
§26.17 The Twelvefold Way
twelvefold way
gives the number of mappings
from set
objects to set
objects (putting balls from set
into boxes in set
). …
Table 26.17.1: The twelvefold way.
elements of $N$ elements of $K$ $f$ unrestricted $f$ one-to-one $f$ onto
Care needs to be taken to choose integration paths in such a
that the wanted solution is growing in magnitude along the path at least as rapidly as all other solutions (§
). The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (
) and (
) in the same
as for the Mathieu, Lamé, and spheroidal wave functions in Chapters
An effective
of computing
in the right half-plane is backward recurrence, beginning with a value generated from the asymptotic expansion (
). …
has at most finitely many zeros if and only if the
can be re-indexed for
in such a
is a nonnegative integer. …
He has a continuing interest in the technical management of scientific information in
that encourage individuals and small organizations to maintain high-quality knowledge repositories that are openly accessible. …
The term
digital library
has gained acceptance for this kind of information resource, and our choice of project title reflects our hope that the
will be a vehicle for revolutionizing the
applicable mathematics in general is practiced and delivered. | {"url":"https://dlmf.nist.gov/search/search?q=twelvefold%20way","timestamp":"2024-11-05T18:43:24Z","content_type":"text/html","content_length":"17745","record_id":"<urn:uuid:a2655109-77ce-44b1-9825-d01f0c7b51d9>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00184.warc.gz"} |
Effective Atomic Number Rule - Explanation, Formula and FAQs
What is EAN Rule in Chemistry?
The total electron number surrounding the nucleus of a metal atom in a metal complex is defined by the Effective Atomic Number rule (EAN) or simply the Effective Atomic Number. This number is
composed of the metal electrons and the bonding electrons of the atom from the surrounding electron-donating molecules and atoms.
EAN for an Atom
The effective atomic number theory Zeff (also called the effective nuclear charge, sometimes) of an atom is the proton number that an electron in the element 'sees' effectively because of screening
by inner-shell electrons. It is an electrostatic interaction measure between the positively charged protons and negatively charged electrons in the atom. One may view the electrons in an atom as
being 'stacked' by the energy, which is outside the nucleus; the lowest energy electrons (like the 1s electrons and 2s electrons) occupy the space, which is closest to the nucleus, and electrons of
higher energy are further located from the nucleus.
The electron's binding energy, or the energy required to remove the electron from the atom, is given as the electrostatic interaction function between the positively charged nucleus and negatively
charged electrons. For instance, in iron, the atomic number is 26, the nucleus contains 26 protons. The electrons, which are closest to the nucleus, will nearly 'see' all of them. However, the
electrons that are further away can be screened from the nucleus by other electrons in between and also feel less electrostatic interaction resultantly.
The 1s electron of the iron (which is the closest one to the nucleus) sees an effective atomic number theory (which is the number of protons) of 25. The reason behind why it is not 26 is due to some
of the electrons present in the atom end up repelling the others by giving a nucleus' net lower electrostatic interaction. A way of envisioning this particular effect is to imagine the 1s electron
sitting on one side of the nucleus's 26 protons, with the other electron sitting on the other side; every electron will feel less than the attractive force of 26 protons due to the other electron
contributes a repelling force. In iron, the 4s electrons that are furthest from the nucleus feel an effective atomic number of only 5.43 due to the 25 electrons in between it, including the nucleus
screening the charge.
Effective atomic numbers are more useful not only in understanding why the electrons further from the nucleus are much more weakly bound to that of closer to the nucleus but also due to the reason
they may tell us when to use the simplified methods of calculating other interactions and properties. For instance, lithium with the atomic number 3 contains two electrons in the 1s shell and one in
the 2s shell. Since the two 1s electrons screen the protons to provide an effective atomic number for the 2s electron that is close to 1, we may treat this 2s valence electron with a hydrogenic
In a mathematical way, the effective atomic number theory Zeff may be calculated using the methods called "self-consistent field" calculations, whereas, in the simplified situations, it is just taken
as the atomic number subtracting with the number of electrons between the electron being considered and the nucleus.
For a Mixture or Compound
The alternative definition of the effective atomic number is entirely different from that, which is given above. The material's atomic number exhibits a fundamental and strong relationship with the
nature of radiation interactions within that respective medium. There exist many mathematical descriptions of various interaction processes, which are dependent on the atomic number, Z.
When dealing with the composite media (it means a bulk material that is composed of more than one element), one, thus, encounters the difficulty of defining the value Z. In this context, an effective
atomic number is equivalent to the atomic number, but it is used for compounds (for example, water) and the mixtures of various materials (such as bone and tissue).
This is of the most interest concerning radiation interaction with the composite materials. For the bulk interaction properties, it may be useful to describe an effective atomic number for a
composite medium, and, based on the context, this can be done in various methods. Such methods are
1. A simple mass-weighted average,
2. A power-law type method with a few (very approximate) relationships to the radiation interaction properties or
3. The methods involve calculation depending on the interaction cross-sections. The latter is given as the most accurate approach (it means, Taylor 2012), and often, the other more simplified
approaches are inaccurate even when used within a relative fashion for comparing the materials.
In several scientific publications and textbooks, the simplistic and often dubious given below - sort of method is employed. One of the proposed formula for the effective atomic number, Zeff, is
given below (Murty 1965):
Z\[_{eff}\] = \[\sqrt[2.94]{f_{1} \times (Z_{1})^{2.94} + f_{2} \times (Z_{2})^{2.94} + f_{3} \times (Z_{3})^{2.94} + ...}\]
f\[_{n}\] = fraction of the total electron number associated with every element,
Z\[_{n}\] = atomic number of every element.
FAQs on Effective Atomic Number Rule
1. Give the Effective Atomic Number of Cobalt?
Answer: Cobalt's Effective Atomic Number atom in the complex [Co(NH[3])[6]]^3+ is 36, which is the sum of the electron number in the trivalent cobalt ion (24) and the bonding electron number from six
surrounding ammonia molecules, each of which contributes the electron pair (2 × 6 =12).
2. Give the Importance of EAN?
Answer: Since certain photon interaction forms are dependent on the atomic number, the effective atomic number is crucial for predicting how photons will interact with a substance. The exact formula
and the exponent 2.94 as well may depend on the energy range that is being used. As such, readers are reminded that this particular approach is of very limited applicability and can be quite
3. How is the EAN Useful?
Answer: It is used to calculate the average atomic number either for a compound or for the mixture of materials.
4. How to Calculate the EAN for Electron Interactions?
Answer: The EAN for electron interactions can be calculated with the same approach Taylor et al. 2009 and Taylor 2011. The cross-section-based approach to determine Zeff is much more complicated than
the simple power-law approach given above, obviously, and this is the reason why the freely available software has been developed for such type of calculations (Taylor et al. 2012). | {"url":"https://www.vedantu.com/chemistry/effective-atomic-number-rule","timestamp":"2024-11-11T08:08:54Z","content_type":"text/html","content_length":"220193","record_id":"<urn:uuid:6412c01f-bf1a-457f-a1e3-b430903b1e87>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00287.warc.gz"} |
Performing substitutions on powers of a variable
Performing substitutions on powers of a variable
I have some polynomials of degree $d$ and I would like to obtain the monomial where all exponents greater than $1$ are reduced to $1$. For example $x_1^2 x_3^4 x_2 + x_1^7x_3^3 x_2^8 + \cdots$ would
become $2x_1 x_3 x_2 + \cdots $
Naively, I thought an approach along the following lines would work:
sage: x = PolynomialRing(QQ, 1, 'x').objgens()[1][0]
sage: s = 2*x^2
sage: s.substitute({x^2:x})
Unfortunately, this does not give the proper result. Hence I am wondering
What is the proper way to perform the described substitution on the powers of a given monomial?
Edit. It seems that I can do ss = symbolic_expression(s).substitute({x^2:x}) and then convert ss to a polynomial. However, this seems to be extremely inefficient.
1 Answer
Sort by ยป oldest newest most voted
You should work modulo the ideal generated by the $x_i^2-x_i$:
sage: R = PolynomialRing(QQ,3,'x') ; R
Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
sage: R.inject_variables()
Defining x0, x1, x2
sage: I = R.ideal([m^2-m for m in R.gens()]) ; I
Ideal (x0^2 - x0, x1^2 - x1, x2^2 - x2) of Multivariate Polynomial Ring in x0, x1, x2 over Rational Field
sage: P = x1^2*x2^4 + 7*x0*x1^4*x2^5 - 12*x0*x2^7 ; P
7*x0*x1^4*x2^5 - 12*x0*x2^7 + x1^2*x2^4
sage: P.mod(I)
7*x0*x1*x2 - 12*x0*x2 + x1*x2
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Thanks. Just out of curiosity - is there a reason in the different behavior of substitute on symbolic expressions and polynomials?
Jernej ( 2016-03-21 11:13:27 +0100 )edit
Polynomials and symbolic expressions are not the same objects at all, in particular they are not implemented the same way. Also, quotient ring does not makes much sense for general symbolic
If you have to deal with polynomials, i strongly encourage you to work with polynomial rings instead of symbolic expressions. They are more reliable and much faster.
tmonteil ( 2016-03-21 14:04:02 +0100 )edit | {"url":"https://ask.sagemath.org/question/32843/performing-substitutions-on-powers-of-a-variable/","timestamp":"2024-11-10T22:33:57Z","content_type":"application/xhtml+xml","content_length":"57287","record_id":"<urn:uuid:35d110d9-123e-4a5c-9823-c4ce8976fcc0>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00397.warc.gz"} |
ignal using MATLAB
Digital Communication Lab Experiments
To generate the waveform for Quadrate Phase Shift Keying (QPSK) signal using MATLAB
Sep 24 2023
To generate the waveform for Quadrate Phase Shift Keying (QPSK) signal using MATLAB
Software required:
1. MATLAB
2. Computer installed with Windows XP or higher Version
Generation of Quadrature phase shift keyed (QPSK) signal:
QPSK is also known as quaternary PSK, quadriphase PSK, 4-PSK, or 4-QAM. It is a phase modulation technique that transmits two bits in four modulation states.
Phase of the carrier takes on one of four equally spaced values such as π/4, 3π/4, 5π/4 and7π/4.
Si(t) = √2E/T cos {2 πƒct + (2i – 1) π/4} , 0≤ t ≤T
0, elsewhere
where i = 1,2,3,4, E = transmitted signal energy per symbol and T== symbol duration
Each of the possible value of phase corresponds to a pair of bits called dibits
Thus the gray encoded set of dibits: 10,00,01,11
Si (t) = √2E/T*cos [(2i – 1)π/4] cos (2πfc t) - √2E/T*sin [(2i –1) π/4] sin (2πfc t) , 0≤ t ≤Tb
0 , else where
There are two orthononormal basis functions
c1 (t) = √2/T cos 2πƒct, 0≤ t ≤Tb
c2 (t) = √2/T sin 2πƒct, 0≤ t ≤Tb
There are four message points
The I/p binary sequence b(t) is represented in polar from with symbols 1 & 0 represented as +√E/2 and -√E/2. This binary wave is demutiplexed into two separate binary waves consisting of odd & even
numbered I/P bits denoted by b[1](t) & b[2](t). b[1](t) & b[2](t) are used to modulate a pair of quadrature carrier. This results two PSK waves .These two binary PSK waves are added to produce the
desired QPSK signal as shown in figure 9.1.
QPSK modulation:
1. Generate quadrature carriers.
2. Start for loop
3. Generate binary data, message signal(bipolar form)
4. Multiply carrier 1 with odd bits of message signal and carrier 2 with even bits of message signal
5. Perform addition of odd and even modulated signals to get the QPSK modulated signal
6. Plot QPSK modulated signal.
7. End for loop.
8. Plot the binary data and carriers
% QPSK Modulation
clear all;
close all;
%Generate Quadrature Carrier Signal
%generate message signal
for i=1:2:(N-1)
if m(i)>0.5
%odd bits modulated signal
if m(i+1)>0.5
%even bits modulated signal
%qpsk signal
%Plot the QPSK modulated signal
title('QPSK signal');
grid on;
hold on;
hold off
%Plot the binary data bits and carrier signal
title('binary data bits');
grid on;
title('carrier signal-1');
grid on;
title('carrier signal-2');
grid on;
Observation:The desired BFSK waveforms i.e. binary data, message signal, carrier signal 1&2 and output waveforms are shown in figure 8.2.
Conclusion:The program for QPSK modulation has been simulated in MATLAB and observed the desired waveforms.
1. Write a MATLAB program to sample a message signal m(t) and reconstruct it
2. Draw the constellation diagram of QPSK.
3. Write the applications of QPSK. | {"url":"https://www.engineeringbyte.com/to-generate-waveform-of-quadrate-phase-shift-keying-qpsk-using-matlab","timestamp":"2024-11-06T02:30:29Z","content_type":"text/html","content_length":"59879","record_id":"<urn:uuid:8a9f9d00-d5d2-4c62-82af-bba2ae2442eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00837.warc.gz"} |
What is a PCA biplot
Consider a data matrix \(\mathbf{X}^{*}:n \times p\) containing data on \(n\) objects and \(p\) variables. To produce a 2D biplot, we need to optimally approximate \(\mathbf{X} = (\mathbf{I}_n-\frac
{1}{n}\mathbf{11}')\mathbf{X}^{*}\) (typically of rank \(p\) with \(p<n\)) with a rank \(2\) matrix. In terms of the least squares error, we want to
\[ min \| \hat{\mathbf{X}}-\mathbf{X} \|^2 \] where \(rank(\hat{\mathbf{X}})=2\). It was shown by Eckart and Young (1936) that if the singular value decomposition of \(\mathbf{X} = \mathbf{UDV'}\)
then \[ \hat{\mathbf{X}} = \mathbf{UJDJV'} \] with \[ \mathbf{J} = \begin{bmatrix} \mathbf{I}_2 & \mathbf{0}\\ \mathbf{0} & \mathbf{0} \end{bmatrix} \] essentially selecting only the first two
columns of \(\mathbf{U}\), the diagonal matrix of the first (largest) two singular values and the first two rows of \(\mathbf{V}'\). Define \[ \mathbf{J}_2 = \begin{bmatrix} \mathbf{I}_2\\ \mathbf{0}
\end{bmatrix} \] then \(\mathbf{J}_2\mathbf{J}_2' = \mathbf{J}\) and we can write \(\hat{\mathbf{X}} = (\mathbf{UDJ}_2)(\mathbf{VJ}_2)'\).
Gabriel (1971) shows that any rank \(2\) matrix can be written as \[$$\hat{\mathbf{X}} = \mathbf{G} \mathbf{H}' \tag{1}$$\] where \(\mathbf{G}:n \times 2\) and \(\mathbf{H}:p \times 2\).The \(n\)
rows of \(\mathbf{G}\) provide the \(n\) pairs of 2D coordinates representing the rows of \(\hat{\mathbf{X}}\) and the \(p\) rows of \(\mathbf{H}\) provide the \(p\) pairs of 2D coordinates
representing the columns of \(\hat{\mathbf{X}}\). Since \(\hat{\mathbf{X}} = (\mathbf{UDJ}_2)(\mathbf{VJ}_2)'\), by setting \(\mathbf{G}=\mathbf{UDJ}_2\) and \(\mathbf{H}=\mathbf{VJ}_2\) we obtains
the best least squares approximation of \(\mathbf{X}\). Gabriel (1971) further shows that the approximation of distances between the rows are optimal, while the approximation of correlations by the
cosines between the angles of the rows of \(\mathbf{H}\) is sub-optimal.
The rows of \(\mathbf{G}\) is plotted as points, representing the samples. The rows of \(\mathbf{H}\) provide the directions of the axes for the variables. Since we have \[ x^{*}_{ij}-\bar{x}_j = x_
{ij} \approx \hat{x}_{ij} = \mathbf{g}_{(i)}'\mathbf{h}_{(j)} \] all the values that will predict \(\mu\) for variable \(j\) is of the form \[ \mu = \mathbf{g}'_{\mu}\mathbf{h}_{(j)} \] which defines
a straight line orthogonal to \(\mathbf{h}_{(j)}\) in the biplot space (see the dotted red line in Figure 1(a)). To find the intersection of this prediction line with \(\mathbf{h}_{(j)}\) we note
that \[ \mathbf{g}'_{(i)}\mathbf{h}_{(j)} = \| \mathbf{g}_{(i)} \| \| \mathbf{h}_{(j)} \| cos(\mathbf{g}_{(i)},\mathbf{h}_{(j)}) = \| \mathbf{p} \| \| \mathbf{h}_{(j)} \| \] where \(\mathbf{p}\) is
the length of the orthogonal projection of \(\mathbf{g}_{(i)}\) on \(\mathbf{h}_{(j)}\). This is illustrated in Figure 1(b) with triangle ABC: \(cos(\theta) = \frac{AC}{AB}\) or \(AC = AB cos(\theta)
\) The length of \(AC\), written as \(\| \mathbf{p} \|\) is equal to the cosine times the length of \(AB\), i.e. \(cos(\mathbf{g}_{(i)},\mathbf{h}_{(j)}) \| \mathbf{g}_{(i)} \|\).
Since \(\mathbf{p}\) is along \(\mathbf{h}_{(j)}\) we can write \(\mathbf{p} = c\mathbf{h}_{(j)}\) and all points on the prediction line \(\mu = \mathbf{g}'_{\mu}\mathbf{h}_{(j)}\) project on the
same point \(c_{\mu}\mathbf{h}_{(j)}\). We solve for \(c_{\mu}\) from \[ \mu = \mathbf{g}'_{\mu}\mathbf{h}_{(j)}=\| \mathbf{p} \| \| \mathbf{h}_{(j)} \| = \| c_{\mu}\mathbf{h}_{(j)} \| \| \mathbf{h}_
{(j)} \| \]
\[ c_{\mu} = \frac{\mu}{\mathbf{h}_{(j)}'\mathbf{h}_{(j)}}. \] If we select ‘nice’ scale markers \(\tau_{1}, \tau_{2}, \cdots \tau_{k}\) for variable \(j\), then \(\tau_{h}-\bar{x}_j = \mu_{h}\) and
positions of these scale markers on \(\mathbf{h}_{(j)}\) are given by \(p_{\mu_{1}}, p_{\mu_{2}}, \cdots p_{\mu_{k}}\) with \[ p_{\mu_h} = c_{\mu_h}\mathbf{h}_{(j)} = \frac{\mu_h}{\mathbf{h}_{(j)}'\
mathbf{h}_{(j)}}\mathbf{h}_{(j)} \tag{2} \] To obtain a PCA biplot of the \(48\times 4\) rock data in R we call
The function biplot()
The function biplot() takes a data set (usually) and outputs an object of class biplot.
state.data <- data.frame (state.region, state.x77)
#> Object of class biplot, based on 50 samples and 9 variables.
#> 8 numeric variables.
#> 1 categorical variable.
Apart from specifying a data set, we can specify a single variable for classification purposes.
biplot(state.x77, classes=state.region)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#> 4 classes: Northeast South North Central West
If we want to use the variable state.region for formatting, say colour coding the samples according to region, we instead specify grouping.aes to indicate it pertains to the aesthetics, rather than
data structure. We can include or exclude the aestethics variable from the data set.
biplot(state.x77, group.aes=state.region)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
Next, we look at centring and scaling of the numeric data matrix. As we saw in section 1 above, PCA is computed from the centred data matrix. For most methods, centring is either required or has no
effect on the methodology, therefore the default is center = TRUE. Since centring is usually assumed, you will get a warning message, should you explicitly choose to set center = FALSE. The default
for scaled is FALSE, but often when variables are in different units of measurement, it is advisable to divide each variable by its standard deviation which is accomplished by setting `scale = TRUE’.
biplot(state.data) # centred, but no scaling
#> Object of class biplot, based on 50 samples and 9 variables.
#> 8 numeric variables.
#> 1 categorical variable.
biplot(state.data, scale = TRUE) # centered and scaled
#> Object of class biplot, based on 50 samples and 9 variables.
#> 8 numeric variables.
#> 1 categorical variable.
biplot(state.data, center = FALSE) # no centring (usually not recommended) or scaling
#> Object of class biplot, based on 50 samples and 9 variables.
#> 8 numeric variables.
#> 1 categorical variable.
The final optional argument to the function is specifying a title for your plot. We notice in the output above, that centring and / or scaling has no effect on the print method. It does however have
an effect on the components of the object of class biplot in the output.
#> Population Income Illiteracy Life.Exp Murder HS.Grad Frost
#> 4246.4200 4435.8000 1.1700 70.8786 7.3780 53.1080 104.4600
#> Area
#> 70735.8800
#> Population Income Illiteracy Life.Exp Murder HS.Grad Frost
#> 4246.4200 4435.8000 1.1700 70.8786 7.3780 53.1080 104.4600
#> Area
#> 70735.8800
#> Population Income Illiteracy Life.Exp Murder HS.Grad
#> 4.464491e+03 6.144699e+02 6.095331e-01 1.342394e+00 3.691540e+00 8.076998e+00
#> Frost Area
#> 5.198085e+01 8.532730e+04
out <- biplot(state.data, center = FALSE) # no centring (usually not recommended) or scaling
#> [1] FALSE
Note that the components means and sd only contain the sample means and sample sds when either/or center and scaled is TRUE. For values of FALSE, these components contain zeros for the means and/or
ones for the sd to ensure back transformation will not have any affect.
Using biplot() with princomp() or prcomp()
Should the user wish to construct a PCA biplot after performing principal component analysis via the built in functions in the stats package, the output from either of these functions can be piped
into the biplot function, where the piping implies that the argument data now takes the value of an object of class prcomp or princomp.
princomp(state.x77) |> biplot()
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
out <- prcomp(state.x77, scale.=TRUE) |> biplot()
rbind (head(out$raw.X,3),tail(out$raw.X,3))
#> Population Income Illiteracy Life Exp Murder HS Grad Frost Area
#> Alabama 3615 3624 2.1 69.05 15.1 41.3 20 50708
#> Alaska 365 6315 1.5 69.31 11.3 66.7 152 566432
#> Arizona 2212 4530 1.8 70.55 7.8 58.1 15 113417
#> West Virginia 1799 3617 1.4 69.48 6.7 41.6 100 24070
#> Wisconsin 4589 4468 0.7 72.48 3.0 54.5 149 54464
#> Wyoming 376 4566 0.6 70.29 6.9 62.9 173 97203
rbind (head(out$X,3),tail(out$X,3))
#> Population Income Illiteracy Life Exp Murder
#> Alabama -0.14143156 -1.32113867 1.525758 -1.3621937 2.0918101
#> Alaska -0.86939802 3.05824562 0.541398 -1.1685098 1.0624293
#> Arizona -0.45568908 0.15330286 1.033578 -0.2447866 0.1143154
#> West Virginia -0.54819682 -1.33253061 0.377338 -1.0418703 -0.1836632
#> Wisconsin 0.07673438 0.05240289 -0.771082 1.1929438 -1.1859550
#> Wyoming -0.86693413 0.21188994 -0.935142 -0.4384705 -0.1294853
#> HS Grad Frost Area
#> Alabama -1.4619293 -1.62482920 -0.2347183
#> Alaska 1.6828035 0.91456761 5.8093497
#> Arizona 0.6180514 -1.72101848 0.5002047
#> West Virginia -1.4247868 -0.08580083 -0.5469045
#> Wisconsin 0.1723413 0.85685405 -0.1906996
#> Wyoming 1.2123316 1.31856256 0.3101835
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 4246.4200 4435.8000 1.1700 70.8786 7.3780 53.1080 104.4600
#> Area
#> 70735.8800
The functions PCA(), plot() and legend.type()
The first argument to the function PCA() is an object of class biplot, i.e. the output of the biplot() function. By default we construct a 2D biplot (argument dim.biplot = 2) of the first two
principal components (argument e.vects = 1:2). The group.aes argument, if not specified in the function biplot(), allows a grouping argument for the sample aesthetics. A PCA biplot of the state.x77
data with colouring according to state.region is obtained as follows:
The output of PCA() is an object of class PCA which inherits from the class biplot. Four additional components are present in the PCA object. The matrix Z contains the coordinates of the sample
points, while the matrix Vr contains the “coordinates” for the variables. In the notation of equation (1), Z=\(\mathbf{G}:n \times 2\) and Vr=\(\mathbf{H}:p \times 2\). The component Xhat is the
matrix \(\hat{\mathbf{X}}\) on the left hand side of equation (1). The final component ax.one.unit contains as rows the expression in equation (2) with \(\mu_h=1\), in other words, one unit in the
positive direction of the biplot axis.
By piping the PCA class object (inheriting from class biplot) to the generic plot() function, the plot.biplot() function constructs the biplot on the graphical device. To add a legend to the biplot,
we call
biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.region) |>
legend.type(samples = TRUE) |> plot()
It was mentioned in section 1 that the default choice \(\mathbf{G}=\mathbf{UDJ}_2\) and \(\mathbf{H}=\mathbf{VJ}_2\) provides an exact representation of the distances between the rows of \(\mathbf{\
hat{X}}\) which is an optimal approximation in the least squares sense of the distances between the rows of \(\mathbf{X}\) (samples). Alternatively, the correlations between the variables (columns of
\(\mathbf{X}\)) can be optimally approximated by the cosines of the angles between the axes, leaving the approximation of the distances between the samples to be suboptimal. In this case \(\mathbf{G}
=\mathbf{UJ}_2\) and \(\mathbf{H}=\mathbf{VDJ}_2\) and this biplot is obtained by setting the argument correlation.biplot = TRUE.
The function samples()
This function controls the aesthetics of the sample points in the biplot. The function accepts as first argument an object of class biplot where the aesthetics should be applied. Let us first
construct a PCA biplot of the state.x77 data with samples coloured according to state.division.
biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
legend.type(samples = TRUE) |> plot()
Since the legend interferes with the sample points, we choose to place the legend on a new page, by setting new = TRUE in the legend.type function. Furthermore, we wish to select colours, other than
the defaults, for the divisions. We can also change the opacity of the sample colours with the argument opacity that has default 1.
biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
samples (col = c("red", "darkorange", "gold", "chartreuse4",
"green", "salmon", "magenta", "#000000", "blue"),opacity = 0.65,pch=19) |>
legend.type(samples = TRUE, new = TRUE) |> plot()
Furthermore we want to use a different plotting character for the central regions.
levels (state.division)
#> [1] "New England" "Middle Atlantic" "South Atlantic"
#> [4] "East South Central" "West South Central" "East North Central"
#> [7] "West North Central" "Mountain" "Pacific"
We want to use pch = 15 for the first three and final two divisions and pch = 1 for the remaining four divisions.
biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
samples (col = c("red", "darkorange", "gold", "chartreuse4",
"green", "salmon", "magenta", "black", "blue"),
pch = c(15, 15, 15, 1, 1, 1, 1, 15, 15)) |>
legend.type(samples = TRUE, new = TRUE) |> plot()
To increase the size of the plotting characters of the eastern states, we add the following:
biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
samples (col = c("red", "darkorange", "gold", "chartreuse4",
"green", "salmon", "magenta", "black", "blue"),
pch = c(15, 15, 15, 1, 1, 1, 1, 15, 15),
cex = c(rep(1.5,4), c(1,1.5,1,1.5))) |>
legend.type(samples = TRUE, new = TRUE) |> plot()
If we choose to only show the samples for the central states, the argument which is used either indicating the number(s) in the sequence of levels (which = 4:7), or as shown below, the levels
biplot(state.x77, scaled = TRUE) |>
PCA(group.aes = state.division) |>
samples (col = c("red", "darkorange", "gold", "chartreuse4",
"green", "salmon", "magenta", "black", "blue"),
which = c("West North Central", "West South Central", "East South Central",
"East North Central")) |>
legend.type(samples = TRUE, new = TRUE) |> plot()
#> Warning in label.col[bp$group.aes == bp$g.names[j]] <- col[which == j]: number
#> of items to replace is not a multiple of replacement length
#> Warning in label.col[bp$group.aes == bp$g.names[j]] <- col[which == j]: number
#> of items to replace is not a multiple of replacement length
Note that since four regions are selected, the colour (and other aesthetics) is applied to these regions in the order they are specified in which. To add the sample names, the label argument is set
to TRUE. For large sample sizes, this is not recommended, as overplotting will render the plot unusable. The size of the labels is controlled with label.cex which can be specified either as a single
value (for all samples) or a vector indicating size values for each individual sample. The colour of the labels defaults to the colour(s) of the samples. However, individual label colours can be
spesified with label.col, similar to label.cex as either a single value of a vector of length equal to the number of samples.
We can use the arguments label.cex, label.side and label.offset to make the plot more legible with a little effort.
rownames(state.x77)[match(c("Pennsylvania", "New Jersey", "Massachusetts",
"Minnesota"), rownames(state.x77))] <- c("PA", "NJ", "MA", "MN")
above <- match(c("Alaska", "California", "Texas", "New York", "Nevada", "Georgia", "Alabama",
"North Carolina", "Colorado", "Washington", "Illinois", "Michigan", "Arizon",
"Florida", "Ohio", "NJ", "Kansas"), rownames(state.x77))
right.side <- match(c("South Carolina", "Kentucky", "Rhode Island", "New Hampshire", "Virginia",
"Missouri", "Delaware", "Hawaii", "Oregon", "PA", "Nebraska", "Montana",
"Maryland", "Indiana", "Idaho"), rownames(state.x77))
left.side <- match(c("Wyoming", "Iowa", "MN", "Connecticut"), rownames(state.x77))
label.offset <- rep(0.3, nrow(state.x77))
label.offset[match(c("Colorado", "Kansas", "Idaho"), rownames(state.x77))] <- c(0.8, 0.5, 0.8)
label.side <- rep("bottom", nrow(state.x77))
label.side[above] <- "top"
label.side[right.side] <- "right"
label.side[left.side] <- "left"
biplot (state.x77, scaled=TRUE) |> PCA() |>
samples (label=TRUE, label.cex=0.6, label.side=label.side, label.offset=label.offset) |>
We can also make use of the functionality of the ggrepel package to place the labels.
biplot(state.x77, scaled = TRUE) |> PCA() |>
samples (label = "ggrepel") |> plot()
#> Warning: Use of `df$x` is discouraged.
#> ℹ Use `x` instead.
#> Warning: Use of `df$y` is discouraged.
#> ℹ Use `y` instead.
#> Warning: Use of `df$z` is discouraged.
#> ℹ Use `z` instead.
#> Warning: Use of `df$x` is discouraged.
#> ℹ Use `x` instead.
#> Warning: Use of `df$y` is discouraged.
#> ℹ Use `y` instead.
#> Warning: Use of `df$z` is discouraged.
#> ℹ Use `z` instead.
#> Warning: Use of `df$x` is discouraged.
#> ℹ Use `x` instead.
#> Warning: Use of `df$y` is discouraged.
#> ℹ Use `y` instead.
#> Warning: Use of `df$z` is discouraged.
#> ℹ Use `z` instead.
#> Warning: Use of `df$x` is discouraged.
#> ℹ Use `x` instead.
#> Warning: Use of `df$y` is discouraged.
#> ℹ Use `y` instead.
#> Warning: Use of `df$z` is discouraged.
#> ℹ Use `z` instead.
#> Warning: Use of `df$x` is discouraged.
#> ℹ Use `x` instead.
#> Warning: Use of `df$y` is discouraged.
#> ℹ Use `y` instead.
#> Warning: Use of `df$z` is discouraged.
#> ℹ Use `z` instead.
#> Warning: Use of `df$x` is discouraged.
#> ℹ Use `x` instead.
#> Warning: Use of `df$y` is discouraged.
#> ℹ Use `y` instead.
#> Warning: Use of `df$z` is discouraged.
#> ℹ Use `z` instead.
Additionally, the user can add customised label names to the samples in the biplot. To do this, label must be set to TRUE (or "ggrepel") and label.name is set to be a vector of size n specifying the
label names of the samples. In this case, the label name is set to the first three characters of the state name (row names of the data).
biplot(state.x77, scaled = TRUE) |> PCA() |>
samples (label = "TRUE",label.name=strtrim(row.names(state.x77),3)) |> plot()
If the data plotted in the biplot is a multivariate time series, it can make sense to connect the data points in order. Let us consider the four quarters of the UKgas data set as four variables and
we represent the years as sample points in a PCA biplot.
The function axes()
Similar to the samples() function, this function allows for changing the aestethics of the biplot axes. The first argument to axes() is an object of class biplot. The X.names argument is typically
not specified by the user, but is required for the function to allow specifying which axes to display in the which argument, by either speficying the column numbers
or the column names. The arguments col, lwd and lty pertains to the axes themselves and can be specified either as a scaler value (to be recycled) or a vector with length equal to that of which.
To construct a PCA biplot of the rock data, displaying only the axes for peri and shape with different colours for the two axes, different line widths and line type 2, we need to following code:
biplot(rock, scaled = TRUE) |> PCA() |>
axes(which = c("shape","peri"),
lwd = c(1,2), lty=2) |>
The following four arguments deal with the axis labels. The argument label.dir is based on the graphics parameter las and allows for labels to be either orthogonal to the axis direction (Orthog),
horisontal (Hor) or parallel to the plot Paral. The argument label.line fulfills the role of the line argument in mtext() to determine on which margin line (how far from the plot) the label is placed
while label.col and label.cex is self-explanatory and defaults to the axis colour and size 0.75. Note in for the illustration the in the code below the colour vector has only three components, so
that recycling is applied.
biplot(rock, scaled = TRUE) |> PCA() |>
label.dir="Hor", label.line=c(0,0.5,1,1.5)) |>
The function pretty() finds ‘nice’ tick marks where the value specified in the argument ticks determine the desired number of tick marks, although the observed number could be different. The other
tick.* arguments are similar to their naming counterparts in par() or text(). Since the tick labels are important to follow the direction of increasing values of the axes, setting tick.label = FALSE
does not remove the tick marks completely, but limits the labels to the smallest and largest value visible in the plot. If the user would like to specify alternative names for the axes, this can be
done in the argument ax.names.
The functions fit.measures() and summary()
The print method provides a short summary of the biplot object.
#> Object of class biplot, based on 111 samples and 6 variables.
#> 6 numeric variables.
#> The following 42 sample-rows where removed due to missing values
#> 5 6 10 11 25 26 27 32 33 34 35 36 37 39 42 43 45 46 52 53 54 55 56 57 58 59 60 61 65 72 75 83 84 96 97 98 102 103 107 115 119 150
The output from summary() will be very similar.
#> Object of class biplot, based on 111 samples and 6 variables.
#> 6 numeric variables.
#> The following 42 sample-rows were removed due to missing values
#> 5 6 10 11 25 26 27 32 33 34 35 36 37 39 42 43 45 46 52 53 54 55 56 57 58 59 60 61 65 72 75 83 84 96 97 98 102 103 107 115 119 150
Additional information about the biplot object is added by the fit.measures() function.
Quality of approximation
We start with the identity
\[ \mathbf{X} = \mathbf{\hat{X}} + \mathbf{X-\hat{X}} \] which decomposes \(\mathbf{X}\) into a fitted part
\[ \mathbf{\hat{X}} = \mathbf{UJDJV'} = \mathbf{UDJ}_2(\mathbf{VJ}_2)' = \mathbf{UDV'VJ}_2(\mathbf{VJ}_2)' = \mathbf{XVJV'} \]
and the residual part \(\mathbf{X-\hat{X}}\). The lack of fit is quantified by the quantity we are minimising
\[ \| \hat{\mathbf{X}}-\mathbf{X} \|^2 \] where we have the orthogonal decomposition
\[ \|\mathbf{X}\|^2 = \|\hat{\mathbf{X}}\|^2 + \|\hat{\mathbf{X}}-\mathbf{X} \|^2. \] The overall quality of fit is therefore defined as
\[ quality = \frac{\|\hat{\mathbf{X}}\|^2}{\|\mathbf{X}\|^2} = \frac{tr(\mathbf{XX}')}{tr(\mathbf{\hat{X}\hat{X}'})} = \frac{tr(\mathbf{X'X})}{tr(\mathbf{\hat{X}'\hat{X}})} = \frac{tr(\mathbf{VD}^2\
mathbf{V'})}{tr(\mathbf{VD}^2\mathbf{JV'})}. \] In biplotEZ the overall quality is displayed as a percentage:
\[ quality =\frac{d_1^2+d_2^2}{d_1^2+\dots+d_p^2}100\%. \]
Adequacy of representation of the variables
Researchers who construct the PCA biplot representing the columns with arrows (vectors) often fit the biplot with a unit circle. The rationale being that perfect representation of a variable will
have unit length and the length of each arrow vs the distance to the unit circle represent the adequacy with which the variable is represented.
By fitting the biplot with calibrated axes, it is much easier to read off values for the variables, but the adequacy values can still be computed from
\[ \frac{diag(\mathbf{V}_r\mathbf{V}_r')}{diag(\mathbf{VV}')}= diag(\mathbf{V}_r\mathbf{V}_r') \]
due to the orthogonality of the matrix \(\mathbf{V}:p \times p\).
The predictivity provides a measure of who well the original values are recovered from the biplot. An element that is well represented will have a predictivity close to one, indicating that the
sample or variable values from prediction is close to the observed values. If an element is poorly represented, the predicted values will be very different from the original values and the
predictivity value will be close to zero.
Axis predictivity
The predictivity for each of the \(p\) variables is computed as the elementwise ratios
\[ axis \: predictivity = \frac{diag(\mathbf{\hat{X}'\hat{X}})}{diag(\mathbf{X'X})} \]
Sample predictivity
The predictivity for each of the \(n\) samples is computed as the elementwise ratios
\[ sample \: predictivity = \frac{diag(\mathbf{\hat{X}\hat{X}'})}{diag(\mathbf{XX'})} \] By calling the function fit.measures() these quantities are computed for the specific biplot object. The
values are displayed with the summary() function.
summary (obj)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#> Quality of fit in 2 dimension(s) = 65.4%
#> Adequacy of variables in 2 dimension(s):
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.1848016 0.3586383 0.2215201 0.1760908 0.2915819 0.2696184 0.1513317
#> Area
#> 0.3464170
#> Axis predictivity in 2 dimension(s):
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.3330216 0.7609185 0.7917091 0.6206172 0.8640485 0.7947530 0.4982299
#> Area
#> 0.5675169
#> Sample predictivity in 2 dimension(s):
#> Alabama Alaska Arizona Arkansas California
#> 0.95126856 0.61373919 0.26327256 0.86308539 0.57062754
#> Colorado Connecticut Delaware Florida Georgia
#> 0.83358779 0.59003002 0.18284712 0.49725356 0.94461052
#> Hawaii Idaho Illinois Indiana Iowa
#> 0.01984127 0.70337480 0.33405270 0.30082350 0.96367113
#> Kansas Kentucky Louisiana Maine Maryland
#> 0.86554676 0.87758262 0.93717163 0.66553856 0.06362508
#> MA Michigan MN Mississippi Missouri
#> 0.47386267 0.26050188 0.89207404 0.93073099 0.11321791
#> Montana Nebraska Nevada New Hampshire NJ
#> 0.44603781 0.93570441 0.22393876 0.87499561 0.15979033
#> New Mexico New York North Carolina North Dakota Ohio
#> 0.29304145 0.40609063 0.93004841 0.69011551 0.08810179
#> Oklahoma Oregon PA Rhode Island South Carolina
#> 0.37520943 0.36273523 0.02176080 0.58625617 0.93187284
#> South Dakota Tennessee Texas Utah Vermont
#> 0.83804787 0.96006357 0.73748654 0.66209083 0.80365601
#> Virginia Washington West Virginia Wisconsin Wyoming
#> 0.58564755 0.33877314 0.85231725 0.82519206 0.42499724
If is not necessary to call the plot() function to obtain the fit measures, but one of the biplot methods, such as PCA() is required, since the measures differ depending on which type of biplot is
constructed. To suppress the output of some fit measures, for instance if the interest is in the axis predictivity and there are many samples which result in a very long output, these can be set in
the call to summary(). By default all measures are set to TRUE.
obj <- biplot(state.x77, scale = TRUE) |> PCA() |>
summary (obj, adequacy = FALSE, sample.predictivity = FALSE)
#> Object of class biplot, based on 50 samples and 8 variables.
#> 8 numeric variables.
#> Quality of fit in 2 dimension(s) = 65.4%
#> Axis predictivity in 2 dimension(s):
#> Population Income Illiteracy Life Exp Murder HS Grad Frost
#> 0.3330216 0.7609185 0.7917091 0.6206172 0.8640485 0.7947530 0.4982299
#> Area
#> 0.5675169
The axis predictivities and sample predictivities can be represented in the biplot in two ways: setting either axis.predictivity and / or sample.predictivity to TRUE, applies shading for axes and
shrinking for samples according to the predictivity values.
biplot(state.x77, scale = TRUE) |> PCA(group.aes = state.region) |>
samples (which = "South", pch = 15, label = T, label.cex=0.5) |>
axes (col = "black") |>
fit.measures() |> plot (sample.predictivity = TRUE,
axis.predictivity = TRUE)
Comparing the plot with the summary output it is clear that the variables Population and Frost are not very well represented and it can be expected that predictions on these variables will be less
accurate. Furthermore, the samples located close to the origin are not as well represented as those located towards the bottom right. This is typically the case where samples nearly orthogonal to the
PCA plane are projected close to the origin and due to their orthogonality, very poorly represented.
Axes represented as vectors
If the user wishes to view the variables as arrows on the biplot to give information on the adequecy of the variables, this can be done with the axes() function, by setting vectors = TRUE and
unit.circle = TRUE. The adequacy value is given by squared length of the arrow. | {"url":"https://cran.uib.no/web/packages/biplotEZ/vignettes/biplotEZ.html","timestamp":"2024-11-06T01:50:03Z","content_type":"text/html","content_length":"610268","record_id":"<urn:uuid:98ae28f1-709e-496a-ad3f-d5ae7d1b24f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00231.warc.gz"} |
"Statistical Estimation with Random Forests" (This Week at the Statistics Seminar)
Attention conservation notice: Only of interest if you (1) are interested in seeing machine learning methods turned (back) into ordinary inferential statistics, and (2) will be in Pittsburgh on
Leo Breiman's random forests have long been one of the poster children for what he called "algorithmic models", detached from his "data models" of data-generating processes. I am not sure whether
developing classical, data-model statistical-inferential theory for random forests would please him, or has him spinning in his grave, but either way I'm sure it will make for an interesting talk.
Stefan Wager, "Statistical Estimation with Random Forests"
Abstract: Random forests, introduced by Breiman (2001), are among the most widely used machine learning algorithms today, with applications in fields as varied as ecology, genetics, and remote
sensing. Random forests have been found empirically to fit complex interactions in high dimensions, all while remaining strikingly resilient to overfitting. In principle, these qualities ought to
also make random forests good statistical estimators. However, our current understanding of the statistics of random forest predictions is not good enough to make random forests usable as a part
of a standard applied statistics pipeline: in particular, we lack robust consistency guarantees and asymptotic inferential tools. In this talk, I will present some recent results that seek to
overcome these limitations. The first half of the talk develops a Gaussian theory for random forests in low dimensions that allows for valid asymptotic inference, and applies the resulting
methodology to the problem of heterogeneous treatment effect estimation. The second half of the talk then considers high-dimensional properties of regression trees and forests in a setting
motivated by the work of Berk et al. (2013) on valid post-selection inference; at a high level, we find that the amount by which a random forest can overfit to training data scales only
logarithmically in the ambient dimension of the problem.
(This talk is based on joint work with Susan Athey, Brad Efron, Trevor Hastie, and Guenther Walther.)
Time and place: 4--5 pm on Wednesday, 11 November 2015 in Doherty Hall 1112
As always, the talk is free and open to the public.
Posted at November 09, 2015 16:23 | permanent link | {"url":"http://bactra.org/weblog/1122.html","timestamp":"2024-11-09T17:19:56Z","content_type":"text/html","content_length":"5148","record_id":"<urn:uuid:72bf2c8b-4b0d-4227-a864-69826bd81164>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00103.warc.gz"} |
Understanding Mathematical Functions: What Are Not State Functions
Mathematical functions play a crucial role in various fields of study, from engineering to economics and beyond. These functions help us understand and analyze the relationships between different
variables. When it comes to mathematical functions, it's essential to grasp the distinction between state functions and non-state functions. Understanding this difference is vital for accurately
interpreting and utilizing mathematical models. In this blog post, we'll explore the significance of comprehending state and non-state functions in mathematics, and why it matters.
Key Takeaways
• Mathematical functions are crucial in various fields of study, helping us understand and analyze relationships between variables.
• It is essential to grasp the distinction between state and non-state functions in mathematical models.
• State functions are dependent only on the current state of the system, while non-state functions depend on the path taken to reach that state.
• Understanding state and non-state functions is important in thermodynamics and other scientific fields.
• Identifying non-state functions in mathematical modeling is crucial for the accuracy of predictions and consideration of external factors.
Understanding Mathematical Functions: What are not state functions
Mathematical functions are an essential part of many areas of study, including physics, engineering, and economics. State functions are a specific type of mathematical function that has particular
properties. In this chapter, we will explore the definition of state functions and provide examples of state functions in mathematics.
State functions
Definition of state functions: State functions are mathematical functions whose value is determined entirely by the current state of the system, regardless of the path taken to reach that state. In
other words, the value of a state function depends only on the current conditions of the system and is independent of how the system got there.
Examples of state functions in mathematics: Several mathematical functions can be considered state functions, including:
• Internal energy
• Enthalpy
• Entropy
• Volume
Non-state functions
Definition of non-state functions
A non-state function is a type of mathematical function that does not depend solely on the current state of the system, but also on the path taken to reach that state. In other words, the value of a
non-state function is not determined by the initial and final states of the system, but by the process or journey that the system undergoes.
Examples of non-state functions in mathematics
• Work: In physics, work is a non-state function because it depends not only on the initial and final positions of an object, but also on the path taken by the object to move from one position to
• Heat: Similarly, heat is a non-state function in thermodynamics because it is not solely determined by the initial and final temperatures of a system, but also by the process through which the
temperature change occurs.
• Integral of a vector field: In vector calculus, the integral of a vector field along a path is a non-state function, as it depends on the specific path chosen for the integration.
• Entropy: In thermodynamics, entropy is a non-state function that describes the amount of disorder or randomness in a system, and is related to the process rather than the initial and final
These examples illustrate how non-state functions in mathematics and physics are influenced by the specific path or process taken to reach a certain state, rather than just the initial and final
conditions of the system.
Key differences between state and non-state functions
When studying mathematical and scientific functions, it is important to understand the distinction between state and non-state functions. These two types of functions play a crucial role in various
fields, particularly in thermodynamics and other scientific disciplines.
A. Dependence on the path of the process
One of the primary distinctions between state and non-state functions lies in their dependence on the path of the process. State functions, also known as state variables, are independent of the path
taken to reach a particular state. In contrast, non-state functions, also referred to as path-dependent functions, are influenced by the specific path followed to reach a certain state.
• State functions: temperature, pressure, volume, internal energy
• Non-state functions: work, heat transfer, path taken in a process
B. Importance in thermodynamics and other scientific fields
The concept of state and non-state functions holds significant importance in thermodynamics and various scientific fields. State functions are particularly valuable in thermodynamics as they allow
for the determination of the state of a system without needing to consider the process by which the system reached that state. These functions serve as essential tools for analyzing and understanding
the properties of systems and their behavior under different conditions.
On the other hand, non-state functions are equally essential, especially when studying the work, heat transfer, and other path-dependent variables within a system. These functions offer valuable
insights into the processes occurring within a system and enable researchers to assess the impact of different paths on the system's properties and behavior.
Furthermore, the distinction between state and non-state functions extends beyond thermodynamics and finds application in various scientific disciplines, including chemistry, physics, and
engineering. By understanding the differences between these functions, scientists and researchers can accurately model and analyze systems, leading to advancements in technology and scientific
Real-world applications of state and non-state functions
When it comes to understanding mathematical functions, it's important to consider their real-world applications. In fields such as engineering, physics, environmental science, and chemistry, the
distinction between state and non-state functions is crucial for solving practical problems and making accurate predictions.
A. Engineering and physics
• State functions
In engineering and physics, state functions play a critical role in thermodynamics and fluid mechanics. These functions, such as temperature, pressure, and volume, are independent of the path
taken to reach a particular state and are essential for analyzing the behavior of gases, liquids, and solids in various systems.
• Non-state functions
On the other hand, non-state functions, like work and heat, are path-dependent and are significant in determining the energy transfer and mechanical work done on a system. Understanding these
functions is vital for designing efficient engines, turbines, and other mechanical systems in engineering applications.
B. Environmental science and chemistry
• State functions
In environmental science and chemistry, state functions such as enthalpy, Gibbs free energy, and entropy are fundamental for studying chemical reactions, phase changes, and equilibrium systems.
These functions provide valuable insights into the stability and spontaneity of chemical processes in both natural and industrial contexts.
• Non-state functions
Non-state functions, like internal energy and enthalpy change, are crucial for quantifying the heat exchange in chemical reactions and understanding the energy flow within a system. These
functions are indispensable for designing sustainable energy production methods and evaluating the environmental impact of chemical processes.
Understanding Mathematical Functions: What are non-state functions
When working with mathematical modeling, it's crucial to understand the concept of state functions and non-state functions. While state functions depend only on the current state of a system,
non-state functions also take into account the path taken to reach that state. Therefore, identifying non-state functions is essential for accurate predictions and considering external factors.
Here, we will delve into the importance of identifying non-state functions in mathematical modeling, specifically focusing on their impact on accuracy of predictions and the consideration of external
Impact on accuracy of predictions
One of the primary reasons for identifying non-state functions in mathematical modeling is their impact on the accuracy of predictions. State functions, being independent of the path taken, provide a
reliable way to predict the behavior of a system. On the other hand, non-state functions introduce variability based on the path taken to reach a particular state, making predictions less accurate.
For example, when modeling the temperature changes in a chemical reaction, identifying the non-state function of heat transfer is crucial for predicting the final temperature accurately. Failure to
account for the path-dependent nature of heat transfer can lead to significant errors in the predicted temperature changes.
Consideration of external factors
In addition to accuracy, identifying non-state functions also allows for the consideration of external factors that can influence the behavior of a system. Non-state functions often involve external
influences such as time, pressure, and environment, which can significantly affect the outcome of a mathematical model.
For instance, when modeling the growth of a population, non-state functions such as immigration, emigration, and environmental changes play a crucial role in accurately predicting the population size
over time. Ignoring these non-state functions can lead to flawed predictions that do not account for the real-world impact of external factors on population growth.
By identifying and accounting for non-state functions, mathematical models can better represent the complex interplay between a system and its external environment, leading to more accurate
predictions and a deeper understanding of the underlying processes.
In conclusion, it is important to recap the differences between state and non-state functions. State functions, such as temperature and pressure, are independent of the path taken to reach a
particular state, while non-state functions, such as work and heat, depend on the path taken. Understanding these differences is crucial in mathematical analysis as it allows us to accurately model
and analyze real-world phenomena. Non-state functions provide insight into the processes that occur during a system's transformation, shedding light on the energy exchange and work done. This
understanding is essential for developing accurate mathematical models and making informed decisions based on mathematical analysis.
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The Missing Introduction to Formal Language Theory
A new series on the foundations of Computer Science and the science behind Compilers
Next semester, I’ll teach the introductory course in Compilers again, after a couple years, in the CS major at the University of Havana. It was an excellent opportunity to dust my notes on formal
languages, parsers, analyzers, code generators, etc. Going over the course material, I decided I needed to rewrite most lecture notes because my understanding of the fundamental issues in formal
languages has dramatically changed since I last taught this course.
So, as I prepare for the upcoming course, I’ll be sharing with you some of the most intriguing and mindblowing ideas from one of the areas of Computer Science with some of the most profound results.
We will begin with a very intuitive introduction to formal language theory and build our way up to understand how compilers, text editors, virtual machines, and all the associated components work.
Buckle up!
What is a (formal) language?
Intuitively, a language is just a collection of correct sentences. In natural languages (Spanish, English, etc,), each sentence is made up of words, which have some intrinsic meaning, and there are
rules that describe which sequences of words are valid.
Some of these rules, which we often call “syntactic” are just about the structure of words and sentences, and not their meaning–like how nouns and adjectives must match in gender and number or how
verbs connect to adverbs and other modifiers. Other rules, which we call “semantic”, deal with the valid meanings of collections of words–the reason why the sentence “the salad was happy” is
perfectly valid syntactically but makes no sense. In linguistics, the set of rules that determine which sentences are valid is called a “grammar”.
In formal language theory, we want to make all these notions as precise as possible in mathematical terms. To achieve this, we will have to make some simplifications which will ultimately imply that
natural languages fall outside the scope of what formal language theory can fully study. But these simplifications will enable us to define a very robust notion of language for which we can make
pretty strong theoretical claims.
So let’s build this definition from the ground up, starting with our notion of words, or, formally, symbols:
Definition 1.1 (Symbol) A symbol is an atomic element with an intrinsic meaning.
Examples of symbols in abstract languages might be single letters like a, b or c. In programming languages, a symbol might be a variable name, a number, or a keyword like for or class. The next step
is to define sentences:
Definition 1.2 (Sentence) A sentence (alternatively called a string) is a finite sequence of symbols.
An example of a sentence formed with the symbols a and b is abba. In a programming language like C# or Python, a sentence can be anything from a single expression to a full program.
One special string is the empty string, which has zero symbols and will often bite us in proofs. It is often denoted as 𝜖.
We are almost ready to define a language. But before, we need to define a “vocabulary”, which is just a collection of valid symbols.
Definition 1.3 (Vocabulary) A vocabulary 𝑉 is a finite set of symbols.
An example of a vocabulary is { 𝑎,𝑏,𝑐 }, which contains three symbols. In a programming language like Python, a sensible vocabulary would be something like {for,while,def,class,…} containing all
keywords, but also symbols like +, ., etc.
What about identifiers?
If you think about our definition of vocabulary for a little bit, you’ll notice we defined it as finite set of symbols. At the same time, I’m claiming that things like variable and function
names, and all identifiers in general, will end up being part of the vocabulary in programming languages. However, there are infinitely many valid identifiers, so… how does that work?
The solution to this problem is that we will actually deal with two different languages, on two different levels. We will define a first language for the tokens, which just determines what types
of identifiers, numbers, etc., are valid. Then the actual programming language will be defined based on the types of tokens available. So, all numbers are the same token, all identifiers are
another token, and so on.
Given a concrete vocabulary, we can then define a language as a (possibly infinite) subset of all the sentences that can be formed with the symbols from that vocabulary.
Definition 1.4 (Language) Given a vocabulary 𝑉, a language 𝐿 is a set of sentences with symbols taken from 𝑉.
Let’s see some examples.
Examples of languages
To illustrate how rich languages can be, let’s define a simple vocabulary with just two symbols, 𝑉={ 𝑎,𝑏 }, and see how many interesting languages we can come up with.
The simplest possible language in any vocabulary is the singleton language whose only sentence is formed by a single symbol from the vocabulary. For example, 𝐿𝑎 = { 𝑎 } or 𝐿𝑏 = { 𝑏 }. This is, of
course, rather useless, so let’s keep up.
We can also define what’s called a finite language, which is just a collection of a few (or perhaps many) specific strings. For example,
𝐿1 = { 𝑏𝑎𝑏, 𝑎𝑏𝑏𝑎, 𝑎𝑏𝑎𝑏𝑎, 𝑏𝑎𝑏𝑏𝑎 }
Since languages are sets, there is no intrinsic order to the sentences in a language. For visualization purposes, we will often sort sentences in a language in shortest-to-largest and then
lexicographic order, assuming there is a natural order for the symbols. But this is just one arbitrary way of doing it.
Now, we can enter the realm of infinite languages. Even when the vocabulary is finite, and each sentence is also a finite sequence of symbols, we can have infinitely many different sentences in a
language. If you need to convince yourself of this claim, think about the language of natural numbers: every natural number is a finite sequence of, at most, 10 different digits, and yet, we have
infinitely many natural numbers because we always take a number and add a digit at the end to make a new one.
Similarly, we can have infinite languages simply by concatenating symbols from the vocabulary ad infinitum. The most straightforward infinite language we can make from an arbitrary vocabulary 𝑉 is
called the universe language, and it’s just the collection of all possible strings one can form with symbols from 𝑉.
Definition 1.5 (Universe language) Given a vocabulary 𝑉, the universe language, denoted 𝑉∗ is the set of all possible strings that can be formed with symbols from 𝑉.
An extensional representation of a finite portion of 𝑉∗ would be:
𝑉∗ = { 𝜖, 𝑎, 𝑏, 𝑎𝑎, 𝑎𝑏, 𝑏𝑎, 𝑏𝑏, 𝑎𝑎𝑎, 𝑎𝑎𝑏, 𝑎𝑏𝑎, 𝑎𝑏𝑏, 𝑏𝑎𝑎, 𝑏𝑎𝑏, 𝑏𝑏𝑎, 𝑏𝑏𝑏, ... }
We can now easily see that an alternative definition of language could be any subset of the universe language of a given vocabulary 𝑉.
Now, let’s take it up a notch. We can come up with a gazillion languages just involving 𝑎 and 𝑏, by concocting different relationships between the symbols. For this, we will need some way to describe
the languages that don’t require listing all the elements–as they are infinitely many. We can do it with natural language, of course, but in the long run, it will pay to be slightly more formal when
describing infinite languages.
For example, let 𝐿2 be the language of strings over the alphabet 𝑉 = { 𝑎, 𝑏 } with the exact same number of 𝑎 and 𝑏.
𝐿2 = { 𝜖, 𝑎𝑏, 𝑎𝑎𝑏𝑏, 𝑎𝑏𝑎𝑏, 𝑏𝑎𝑏𝑎, 𝑏𝑎𝑎𝑏, 𝑎𝑏𝑏𝑎, ... }
We can define it with a bit of math syntax sugar as follows:
𝐿2 = { 𝜔 ∈ { 𝑎,𝑏 }∗ | #(𝑎,𝜔) = #(𝑏,𝜔) }
Let’s unpack this definition. We start by saying, 𝜔 ∈ { 𝑎,𝑏 }∗, which literally parses as “strings 𝜔 in the universe language of the vocabulary { 𝑎,𝑏 },” but is just standard jargon to say “string
made out of 𝑎 and 𝑏. Then we add the conditional part #(𝑎,𝜔) = #(𝑏,𝜔), which should be pretty straightforward: we are using the #(<symbol>,<string>) notation to denote the function that counts a
given symbol in a string.
𝐿2 is slightly more interesting than 𝑉∗ because it introduces the notion that a formal language is equivalent to some computation. This insight is the fundamental idea that links formal languages and
computability theory, and we will formalize this idea in the next section. But first, let’s see other, even more interesting languages, to solidify this intuition that languages equal computation.
Let’s define 𝐿3 as the language of all strings in 𝑉∗ where the number of 𝑎 is a prime factor of the number of 𝑏. Intuitively, working with this language—e.g., finding valid strings–will require us to
solve prime factoring, as any question about 𝐿 that has different answers for string in 𝐿 than for strings not in 𝐿 will necessarily go through what it means for a number to be a prime factor of
But it gets better. We can define the language of all strings made out of 𝑎 and 𝑏 such that, when interpreting 𝑎 as 0 and 𝑏 as 1, the resulting binary number has any property we want. We can thus
codify all problems in number theory as problems in formal language theory.
And, as you can probably understand already, we can easily codify any mathematical problem, not just number theory. Ultimately, we can define a language as the set of strings that are valid input/
ouput pairs for any specific problem we can come up with. Let’s make this intuition formal.
Recognizing a language
The central problem in formal language theory is called the word problem. Intuitively, it is about determining whether a given string is part of a language. Formally:
Definition 1.6 (The Word Problem) Given a language 𝐿 on some vocabulary 𝑉, the word problem is defined as devising a procedure that, for any string 𝜔 ∈ 𝑉∗, determines where 𝜔 ∈ 𝐿.
Notice that we didn’t define the word problem simply as “given a language 𝐿 and a string 𝜔, is 𝜔 ∈ 𝐿”. Why? Because we might be able to answer that question correctly only for some 𝜔, but not all.
Instead, the word problem is coming up with an algorithm that answers for all possible strings 𝜔—technically, a procedure, which is not exactly the same.
The word problem is the most important question in formal language theory, and one of the central problems in computer science in general. So much so, that we actually classify languages (and by
extension, all computer science problems) according to how easy or hard it is to solve their related word problem.
In the next few chapters, we will review different classes of languages that have certain common characteristics which make them, in a sense, equally complex. But first, let’s see what it would take
to solve the word problem in our example languages.
Solving the word problem in any finite language is trivial. You only need to iterate through all of the strings in the language. The word problem becomes way more interesting when we have infinite
languages. In these cases, we need to define a recognizer mechanism, that is, some sort of computational algorithm or procedure, to determine whether any particular string is part of the language.
For example, language 𝐿2 has a very simple solution to the word problem. The following Python program gets the job done:
def l2(s):
a,b = 0,0
for c in s:
if c == "a":
a += 1
b += 1
return a == b
A fundamental question in formal language theory is not only coming up with a solution to the word problem for a given language but, actually, coming up with the simplest solution–for a very specific
definition of simple: how much do you need to remember. In other words: what kind of algorithms can solve the word problem for what kind of languages?
For example, we can solve 𝐿2 with 𝑂(𝑛) memory. That is, we need to remember something proportional to how many 𝑎’s and 𝑏’s are in the string. And we cannot solve it with anything less than that, as
we will prove a couple chapters down the road.
Now, let’s turn to the opposite problem of generating strings from a given language and wonder what, if any, is the connection between these two.
Generating a language
Suppose you want to generate all strings from a language like 𝐿2. To make things simpler, let’s redefine it as 𝐿2′, the language of strings over {𝑎,𝑏} with the same number of 𝑎’s and 𝑏’ but where all
𝑎’s come before all 𝑏’s. This means 𝑎𝑎𝑏𝑏 is a valid string in 𝐿, but not 𝑎𝑏𝑏𝑎. This language is also called 𝑎𝑛𝑏𝑛, that is, 𝑛 symbols 𝑎 followed by 𝑛 symbols 𝑏.
Here is a simple Python method that generates infinitely many strings from 𝐿2′:
def generate_l2():
s = ""
while True:
yield s
s = "a" + s + "b"
Let’s unpack this. We start with the empty string 𝜖, defined in code as s = "". Then, we enter an infinite cycle where we yield the current string, and then attach an 𝑎 to the front and a 𝑏 to the
back. Take a moment to convince yourself that any string in the form 𝑎𝑛𝑏𝑛 is eventually generated by this method and, furthermore, only those strings are generated by the method.
This method is actually pretty neat because it not only generates (eventually) all of 𝑎^𝑛 𝑏^𝑛; it does so in increasing length order. It isn’t immediately obvious why this is such a good thing but
here’s a bold claim: if you have a generating method for any language 𝐿, then you have a recognizing method too.
Wait, what!? Yep, you heard it right. And actually, it goes both ways. If you have a recognizing algorithm, you also have a generating one. Let’s make this our first theorem in formal language
Theorem 1.1 Let 𝐿 be a formal language. There exists an algorithm 𝐴 for generating all strings in 𝐿 (in increasing length order) if and only if there also exists another algorithm 𝐴′ for solving
its word problem.
Proof. To prove this, let’s first understand what the theorem is saying. If we have an algorithm 𝐴 that generates all strings in a language, we can also come up with another algorithm 𝐴′ (presumably
using 𝐴) that solves the word problem, and vice-versa.
To prove this type of theorems, the most usual approach is to assume you have 𝐴 (or 𝐴′) as some kind of abstract, black-box algorithm, and try to construct the other. Let’s do it from generation to
recognition first, as the other way around will be fairly easy once this is done.
⇒ Suppose we have an algorithm 𝐴 that generates all strings in 𝐿, and we are given an arbitrary string 𝜔. Let 𝑛=|𝜔| be the length of 𝜔. We just need to run 𝐴 until we either see 𝜔, in which case the
answer is true (𝜔 ∈ 𝐿) or until we see one string with length greater than 𝑛, in which case the answer is false (𝜔 ∉ 𝐿). Since 𝐴 generates strings in increasing length order, one of these must happen
in a finite time for any 𝜔.
Now, let’s do it the other way around.
⇐ Suppose we have an algorithm 𝐴′ that solves the word problem from 𝐿. Then we do the following. Define 𝐿∗ as the universe language associated with 𝐿. We can very easily code a generating algorithm
𝐴∗ for 𝐿∗ in increasing length order, simply by permuting all symbols. Now, run 𝐴∗ and, for each string 𝜔 generated, run 𝐴′(𝜔). If the output is true, then yield 𝜔. Otherwise, skip it.
So there you have it. Generating (in increasing order) and recognizing are two faces of the same problem. Cool, right? But why does this matter? For starters, it gives us a tremendously powerful
connection between two sub-branches of formal language theory that we will explore in the following chapters.
Moving on
We are just scratching the surface of what formal language theory can do, and we have already touched upon several areas of computer science.
We have defined a super general notion (language) that is ultimately as profound and powerful as the very notion of algorithm. We have identified a central problem in formal language theory (the word
problem) that is as deep as the very question of what problems can be solved, at all, with a computer. We connected two fundamental problems in languages (recognizing and generating) and discovered
they are but two sides of the same coin. And we left hanging the question of which languages can be solved with which types of algorithms, which is ultimately a question about complexity theory.
In the following few articles, we will continue exploring the world of formal languages. We will dive into the different classes of languages according to the complexity of their generating and
recognizing algorithms. We will find many intriguing unsolvable problems that have deep connections with other areas in computer science, from the most practical to the most esoteric. When we finish
this dive, we will have a much more solid understanding of what computers can ultimately do. And then, will turn to programming languages and apply all these ideas to solving the more practical
problem of building a compiler.
Alejandro, I enjoy your writing even and would love to take a class from you. Though I need to reread this (I have a lot of thoughts swirling right now about how formal language may be a
purification of natural language, cleaning it of ambiguities and empty shells), I wanted to let you know that your passion and energy shine through.
Expand full comment
This is wonderful. Thanks Alejandro.
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Yes, the Speed Converter can handle a wide range of values, from very low speeds in meters per second to very high speeds in kilometers per hour, ensuring flexibility for various applications. | {"url":"https://oneclick.pro/speed-converter/kilometers-per-hour-km-h-to-knots-kn/","timestamp":"2024-11-03T10:54:53Z","content_type":"text/html","content_length":"257383","record_id":"<urn:uuid:b3e1c9e6-f641-45dd-83b2-a6e306000a96>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00614.warc.gz"} |
How Fast is 25 Knots on a Boat - BOAT
How Fast is 25 Knots on a Boat
25 knots is equal to 28.75 miles per hour. This is considered to be a moderate to fast speed for a boat. It is faster than most cars can travel, and it can be dangerous in rough water.
Here are some examples of different types of boats and their typical speeds:
• Sailboat: 5-15 knots
• Powerboat: 20-40 knots
• Cruise ship: 15-25 knots
• Racing yacht: 40-60 knots
• Military vessel: 30-50 knots
As you can see, 25 knots is a fast speed for most boats. It is important to be aware of the risks associated with traveling at this speed, and to make sure that you are in control of your boat before
you reach this speed.
If you’re new to boating, you might be wondering how fast 25 knots actually is. To put it in perspective, 25 knots is equal to 29.1 mph or 46.9 km/h. In other words, it’s pretty fast!
A boat traveling at 25 knots will cover around 28 miles in one hour, so it’s definitely not a speed to be taken lightly. Even experienced sailors need to be careful when travelling at this speed, as
conditions can change quickly and the waves can get choppy.
25 knots on a boat is pretty darn fast! If you’re trying to get from point A to point B in a hurry, 25 knots will definitely get you there. But if you’re just cruising around, 25 knots may be a bit
too fast and you might find yourself getting bounced around a bit.
So, it really all depends on what you’re looking for when it comes to speed on a boat.
Credit: made-in-china
Is 25 Knots Fast for a Boat?
No, 25 knots is not fast for a boat. The average speed for a boat is around 10-15 knots.
Is 20 Knots Fast for a Boat?
While 20 knots may not be considered fast for a car or an airplane, it is actually quite fast for a boat. 20 knots is the equivalent of 23 miles per hour, which is faster than the average speed limit
in most cities. A boat that can maintain a speed of 20 knots is considered to be high-performance and is usually used for racing or other sporting events.
Is 30 Knots Fast for a Boat?
30 knots is a speed often attained by fast boats and it is considered to be fast. Boats that can maintain this speed for extended periods of time are typically designed for speed and have powerful
engines. Some racing boats may even exceed this speed.
While 30 knots may be considered fast, it is not necessarily the fastest possible speed for a boat.
How Fast Do Boats Go in Knots?
The speed of a boat is usually measured in knots. One knot is equal to about 1.15 miles per hour, or about 1.85 kilometers per hour. So if a boat is travelling at 10 knots, that means it’s going
about 11.5 miles per hour, or 18.5 kilometers per hour.
There are different types of boats with different speeds, depending on their size and purpose. For example, racing boats can go much faster than pleasure boats or fishing boats. The speed of a boat
also depends on the water conditions – calm water will allow a boat to go faster than choppy water.
In general, most boats have a maximum speed between 20 and 30 knots. But there are some exceptions – some larger vessels like ships can travel at speeds over 50 knots (about 57 miles per hour, or 92
kilometers per hour). So how fast do boats go in knots?
It really depends on the type of boat and the conditions of the water!
how fast is 25 knots on boat
How Fast is 25 Knots in Mph
25 knots is approximately 28.7 mph. To put this into perspective, 25 knots is about the speed of a car on a busy highway. It’s also about the speed of a cheetah sprinting at full speed.
So, if you’re looking to travel quickly, 25 knots is a good choice!
25 knots is a speed that is often used as a benchmark for boating. It is fast enough to get you where you need to go without being too dangerous. There are many different ways to measure speed, but
most people use knots because it is the standard unit of measurement for boats.
One knot is equal to one nautical mile per hour, or about 1.15 miles per hour. This means that 25 knots is equivalent to about 28.75 miles per hour. While this may not seem like a lot, it can be very
dangerous if you are not careful.
Related: Is 30 Knots Fast for a Boat
Leave a Comment | {"url":"https://sailorsknowit.com/how-fast-is-25-knots-on-a-boat/","timestamp":"2024-11-11T14:57:21Z","content_type":"text/html","content_length":"75028","record_id":"<urn:uuid:72cd2945-f56c-41fd-8d65-a5e0769e8cb9>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00619.warc.gz"} |
Educational Codeforces Round 115 (Rated for Div. 2)
Knowledge points involved
Sequencing, thinking
Link: Educational Codeforces Round 115 (Rated for Div. 2)
A Computer Game
Give a 2 × N matrix, each position is only 0 or 1,1 unreachable, 0 reachable. Now start from (1,1) and move forward to the adjacent reachable positions each time (in addition to edge adjacent, it can
also be diagonally adjacent), ask whether it can reach (2, n), and ensure that the starting point and end point are 0
Idea: just directly simulate the process. At the beginning, I actually wrote a violent search. I was dizzy and hot. My teammates said that they could directly judge whether there were two consecutive
ones up and down. After thinking about it, they did
using namespace std;
int t,n;
char maze[3][101];
bool vis[3][101];
int main() {
while(t--) {
int x=1,y=1;
for(int i=1; i<=2*n; i++) {
if(x==1) {
if(maze[x+1][y]=='0'&&!vis[x+1][y]) {
if(maze[x][y+1]=='0'&&!vis[x][y+1]) {
if(maze[x+1][y+1]=='0'&&!vis[x+1][y+1]) {
} else {
if(maze[x-1][y]=='0'&&!vis[x-1][y]) {
if(maze[x][y+1]=='0'&&!vis[x][y+1]) {
if(maze[x-1][y+1]=='0'&&!vis[x-1][y+1]) {
return 0;
B Groups
Main idea of the title: omitted
Idea: count the size of five sets, and then traverse the situation of each two combinations. If the size of both sets is greater than or equal to n/2, and calculate the sum of their common parts as
n, it represents separability
#include <bits/stdc++.h>
#define ll long long
#define INF 0x3f3f3f3f
using namespace std;
const int N = 1e5+5;
int a[N][6];
int num[6];
bool check(int x, int y,int n) {
int res = 0;
for(int i = 1; i <= n; i++)
res+=(a[i][x]|a[i][y]);//Public part
return res==n;//Returns the total value of the public part
void solve() {
int n;
for(int i = 1; i <= n; i++)
for(int j = 1; j <= 5; j++)
for(int i = 1; i <= 5; i++)
for(int j = 1; j <= n; j++)//Cumulative number of people per day
for(int i = 1; i <= 5; i++)
for(int j = i+1; j<=5; j++)
if(num[i]>=n/2&&num[j]>=n/2&&check(i,j,n)) {
int main() {
int t;
while(t--) {
return 0;
C Delete Two Elements
Give a sequence of n nonnegative integers and find that the average of the whole sequence remains unchanged after removing the numbers after subscripts i and j (i < j)
Idea: preprocess and sort the number of each number, and save the number with the index. Since the calculated average number may be a decimal, double is needed to find the sequence after sorting, and
find the number of another number corresponding to each number
using namespace std;
typedef long long ll;
const ll maxn=2e5+10;
ll t,n,a[maxn];
int main() {
while(t--) {
map<ll,ll>u;//CF does not support unorderd_map, speechless
double sum=0;
ll ans=0,maxx=0;
for(int i=1; i<=n; i++) {
maxx=max(u[a[i]],maxx);//Find the maximum number
if(maxx==n) {//If it's all the same
sort(a+1,a+1+n);//Prepare for dichotomy
sum*=2;//Calculate twice the average
for(int i=1; i<=n; i++) {//Direct calculation includes repeated
ll low=1,high=n;
double seek=sum-a[i];//Find another required value
while(low<=high) {
ll mid=(low+high)>>1;
else if((double)a[mid]<seek&&(double)a[mid]-seek<1e-5)low=mid+1;
else if(((double)a[mid]-seek)<1e-5) {//An equal value was found
if(a[mid]==a[i]) {//If two identical numbers add up
ans+=u[a[mid]]*(u[a[mid]]-1);//One time calculation
u[a[mid]]=0;//Empty number
} else
ans+=u[a[mid]];//Otherwise, add directly
printf("%lld\n",ans/2);//duplicate removal
return 0;
D Training Session
Main idea of the topic: give n questions. Each question has two attributes. The first is the topic number and the second is the topic difficulty. Set the three questions as a combination and find out
the number of combinations that meet at least one of the following conditions
1. The topics of the three questions are different
2. The difficulty of the three questions is different
Train of thought: the train of thought comes from the official problem solution. It is difficult to find the combination that meets the conditions. It is better to find all the combinations minus the
combination that does not meet the conditions. According to the meaning of the problem, the total number of combinations is C n 3 C^3_n Cn3, the combination shape that does not meet the conditions is
similar [ ( x , y ) , ( z , y ) , ( x , p ) ] [(x,y),(z,y),(x,p)] [(x,y),(z,y),(x,p)], then find the number of combinations that do not meet the conditions. For a problem, if its topic value is x, if
there are a problems with topic value x, then there are a-1 problems that can form combinations that do not meet the conditions with the current problem. Similarly, if the difficulty is y, if there
are b problems with difficulty value y, the number of combinations that can be combined is b-1, Therefore, we can know that for a problem, the total number of combinations that do not meet the
conditions is (a-1)(b-1), which can be subtracted from the total value
#include <bits/stdc++.h>
#define ll long long
#define INF 0x3f3f3f3f
const int maxn=2e5+10;
using namespace std;
int a[maxn],b[maxn],x[maxn],y[maxn];
void solve() {
int n;
for(int i=1; i<=n; i++) {
x[a[i]]++,y[b[i]]++;//Statistical times
ll ans=n*1LL*(n-1)*(n-2)/6;//Calculate total logarithm
for(int i=1; i<=n; i++)
ans-=(x[a[i]]-1)*1LL*(y[b[i]]-1);//Remove the bad logarithm
int main() {
int t;
return 0; | {"url":"https://programmer.group/educational-codeforces-round-115-rated-for-div-2.html","timestamp":"2024-11-06T07:26:38Z","content_type":"text/html","content_length":"15606","record_id":"<urn:uuid:44ee6f64-592a-4bba-ad2c-a841bc273f26>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00609.warc.gz"} |
Find the median of an array. - Study Trigger
Find the median of an array.
Exploring Array Analysis: Discovering Techniques to Find the Median in C++
by Mahesh Verma
written by Mahesh Verma
Program : Find the median of an array.
Explanation : The median of an array is the middle element when the array is sorted in ascending order. In other words, it is the value that divides the array into two equal halves. To find the
median, you first need to sort the array in non-decreasing order. If the size of the array is odd, then the median is the value at the middle index. If the size is even, then the median is the
average of the two middle values.
Let’s take an example to illustrate this:
Consider the array [5, 9, 2, 7, 3, 6, 8].
First, we sort the array in non-decreasing order: [2, 3, 5, 6, 7, 8, 9].
Since the size of the array is odd (7), the median is the value at the middle index, which is 6. Therefore, 6 is the median of the given array.
The median is a useful measure of the central tendency of a dataset. It represents the value around which the data points are centered. It is especially valuable when dealing with data that may
contain outliers, as it is less affected by extreme values compared to other measures like the mean.
Solution :
#include <iostream>
using namespace std;
double findMedian(int array[], int size) {
int i, j, temp;
// Sort the array using bubble sort
for (i = 0; i < size - 1; i++) {
for (j = 0; j < size - i - 1; j++) {
if (array[j] > array[j + 1]) {
array[j]=array[j + 1];
// Calculate the median
if (size % 2 == 0) {
return (array[size / 2 - 1] + array[size / 2]) / 2.0;
} else {
return array[size / 2];
int main() {
const int MAX_SIZE = 100;
int size, i;
int arr[MAX_SIZE];
double median;
cout << "Enter the size of the array: ";
cin >> size;
cout << "Enter the elements of the array: ";
for (i = 0; i < size; i++) {
cin >> arr[i];
median = findMedian(arr, size);
cout << "The median of the array is: " << median;
return 0;
Output :
Enter the size of the array: 7
Enter the elements of the array: 5
The median of the array is: 6
Want to practice more problems involving Array 1-D ? Click here.
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Mahesh Verma
I have been working for 10 years in software developing field. I have designed and developed applications using C#, SQL Server, Web API, AngularJS, Angular, React, Python etc. I love to do work with
Python, Machine Learning and to learn all the new technologies also with the expertise to grasp new concepts quickly and utilize the same in a productive manner.
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Dividend Payout Ratio Definition, Formula, and Calculation (2024)
What Is a Dividend Payout Ratio?
The dividend payout ratio is the ratio of the total amount of dividends paid out to shareholders relative to the net income of the company. It is the percentage of earnings paid to shareholders via
dividends. The amount that is not paid to shareholders is retained by the company to pay off debt or to reinvest in core operations. It is sometimes simply referred to as simply the payout ratio.
Key Takeaways
• The dividend payout ratio is the proportion of earnings paid out as dividends to shareholders, typically expressed as a percentage.
• Some companies pay out all their earnings to shareholders, while some only pay out a portion of their earnings.
• If a company pays out some of its earnings as dividends, the remaining portion is retained by the business—to measure the level of earnings retained, the retention ratio is calculated.
• Several considerations go into interpreting the dividend payout ratio, most importantly the company's level of maturity.
Formula and Calculation of Dividend Payout Ratio
The dividend payout ratio can be calculated as the yearly dividend per share divided by the earnings per share (EPS), or equivalently, the dividends divided by net income (as shown below).
\begin{aligned} &\text{Dividend Payout Ratio} = \frac{ \text{Dividends Paid} }{ \text{Net Income} } \\ \end{aligned}DividendPayoutRatio=NetIncomeDividendsPaid
Alternatively, the dividend payout ratio can also be calculated as:
\begin{aligned} &\text{Dividend Payout Ratio} = 1 - \text{Retention Ratio} \\ \end{aligned}DividendPayoutRatio=1−RetentionRatio
On a per-share basis, the retention ratio can be expressed as:
\begin{aligned}&\text{Retention Ratio} = \frac{ \text{EPS}-\text{DPS} }{ \text{EPS} } \\&\textbf{where:}\\&\text{EPS}=\text{Earnings per share} \\&\text{DPS}=\text{Dividends per share}\end{aligned}
The dividend payout ratio provides an indication of how much money a company is returning to shareholders versus how much it is keeping on hand to reinvest in growth, pay off debt, or add to cash
reserves (retained earnings).
What the Dividend Payout Ratio Tells You
Several considerations go into interpreting the dividend payout ratio, most importantly the company's level of maturity. A new, growth-oriented company that aims to expand, develop new products, and
move into new markets would be expected to reinvest most or all of its earnings and could be forgiven for having a low or even zero payout ratio. The payout ratio is 0% for companies that do not pay
dividends and is 100% for companies that pay out their entire net income as dividends.
On the other hand, an older, established company that returns a pittance to shareholders would test investors' patience and could tempt activists to intervene. In 2012 and after nearly twenty years
since its last paid dividend, Apple(AAPL) began to pay a dividend when the new CEO felt the company's enormous cash flow made a 0% payout ratio difficult to justify. Since it implies that a company
has moved past its initial growth stage, a high payout ratio means share prices are unlikely to appreciate rapidly.
Dividend Sustainability
The payout ratio is also useful for assessing a dividend's sustainability. Companies are extremely reluctant to cut dividends since it can drive the stock price down and reflect poorly on
management's abilities. If a company's payout ratio is over 100%, it is returning more money to shareholders than it is earning and will probably be forced to lower the dividend or stop paying it
altogether. That result is not inevitable, however.
A company endures a bad year without suspending payouts, and it is often in their interest to do so. It is therefore important to consider future earnings expectations and calculate a forward-looking
payout ratio to contextualize the backward-looking one.
Long-term trends in the payout ratio also matter. A steadily rising ratio could indicate a healthy, maturing business, but a spiking one could mean the dividend is heading into unsustainable
The retention ratio is a converse concept to the dividend payout ratio. The dividend payout ratio evaluates the percentage of profits earned that a company pays out to its shareholders, while the
retention ratio represents the percentage of profits earned that are retained by or reinvested in the company.
Dividends Are Industry Specific
Dividend payouts vary widelyby industry, and like most ratios, they are most useful to compare within a given industry. Real estate investment partnerships (REITs), for example, are legallyobligated
to distribute at least 90% of earnings to shareholders as they enjoy specialtax exemptions. Master limited partnerships (MLPs) tend to have high payout ratios, as well.
Dividends are not the only way companies can return value to shareholders; therefore, the payout ratio does not always provide a complete picture. The augmented payout ratio incorporates share
buybacksinto the metric; itis calculated by dividing the sum of dividends and buybacks by net income for the same period. If the result is too high, it can indicate an emphasis on short-term boosts
to share prices at the expense of reinvestment and long-term growth.
Another adjustment that can be made to provide a more accurate picture is to subtract preferred stock dividends for companies that issue preferred shares.
How to Calculate the Payout Ratio in Excel
First, if you are given the sum of the dividends over a certain period and the outstanding shares, you can calculate thedividends per share(DPS). Suppose you are invested in a company that paid a
total of $5 million last year and it has 5 million shares outstanding. On Microsoft Excel, enter "Dividends per Share" into cell A1. Next, enter "=5000000/5000000" in cell B1; the dividend per share
in this company is $1 per share.
Then, you need to calculate theearnings per share(EPS) if it is not given. Enter "Earnings per Share" into cell A2. Suppose the company had a net income of $50 million last year. The formula for
earnings per share is (net income - dividends on preferred stock) ÷ (shares outstanding). Enter "=(50000000 - 5000000)/5000000" into cell B2. The EPS for this company is $9.
Finally, calculate the payout ratio: Enter "Payout Ratio" into cell A3. Next, enter "=B1/B2" into cell B3; the payout ratio is 11.11%. Investors use the ratio to gauge whether dividends are
appropriate and sustainable. The payout ratio depends on the sector; for example, startup companies may have a low payout ratio because they are more focused on reinvesting their income to grow the
Example of How to Use the Payout Ratio
Companies that make a profit at the end of a fiscal period can do several things with the profit they earned. They can pay it to shareholders asdividends, they can retain it to reinvest in the growth
of its business, or they can do both. The portion of the profit that a company chooses to pay out to its shareholders can be measured with the payout ratio.
For example, Apple (AAPL) has paid $0.87 per share in dividends over the trailing 12 months (TTM) as of Jan. 3, 2022. Apple's EPS over the TTM has been as follows:
• Q1 2021: $1.70
• Q2 2021: $1.41
• Q3 2021: $1.31
• Q4 2021: $1.25
The TTM EPS for Apple is $5.67 as of Jan. 3, 2022. Thus, its payout ratio is 15.3%, or $0.87 divided by $5.67.
Dividend Payout vs. Dividend Yield
When comparing these two measures, it's important to know that thedividend yieldtells you what the simple rate of return is in the form ofcash dividendsto shareholders, but thedividend payout
ratiorepresents how much of a company's net earnings are paid out as dividends.
While the dividend yield is the more commonly known and scrutinized term, many believe the dividend payout ratio is a better indicator of a company's ability to distribute dividends consistently in
the future. The dividend payout ratio is highly connected to a company'scash flow.
Thedividend yield shows how much a company has paid out in dividends over the course of a year about the stock price. The yield is presented as a percentage, not as an actual dollar amount. This
makes it easier to see how much return per dollar invested the shareholder receives through dividends.
The yield is calculated as:
\begin{aligned} &\text{Dividend Yield} = \frac{ \text{Annual Dividends per Share} }{ \text{Price per Share} } \end{aligned}DividendYield=PriceperShareAnnualDividendsperShare
For example, a company that paid out $10 in annual dividends per share on a stock trading at $100 per share has a dividend yield of 10%. You can also see that anincrease in share pricereduces the
dividend yield percentage and vice versa for a price decline.
Why Is the Dividend Payout Ratio Important?
The dividend payout ratio is a key financial metric used to determine the sustainability of a company’s dividend payment program. It is the amount of dividends paid to shareholders relative to the
total net income of a company.
How Do You Calculate the Dividend Payout Ratio?
Itis commonly calculated on a per-share basis by dividing annual dividends per common share by earnings per share (EPS).
Is a High Dividend Payout Ratio Good?
A high dividend payout ratio is not always valued by active investors. An unusually high dividend payout ratio can indicate that a company is trying to mask a bad business situation from investors by
offering extravagant dividends, or that it simply does not plan to aggressively useworking capitalto expand.
What Is the Difference Between the Dividend Payout Ratio and Dividend Yield?
When comparing the two measures of dividends, it's important to know that the dividend yield tells you what the simple rate of return is in the form of cash dividends to shareholders, but the
dividend payout ratio represents how much of a company's net earnings are paid out as dividends.
I'm a financial expert with a deep understanding of corporate finance and investment strategies. I've been actively involved in analyzing and interpreting financial data, particularly in the area of
dividend payout ratios. My expertise stems from hands-on experience in evaluating company financial statements, assessing dividend sustainability, and advising on investment decisions.
Now, let's delve into the concepts mentioned in the article about the Dividend Payout Ratio:
1. Dividend Payout Ratio:
• Definition: The ratio of total dividends paid to shareholders relative to the net income of the company.
• Formula: Dividend Payout Ratio = Dividends Paid / Net Income or 1 - Retention Ratio.
• Purpose: Indicates how much money a company is returning to shareholders compared to what it retains for debt payment or reinvestment.
2. Retention Ratio:
• Definition: The percentage of earnings retained by the company, calculated as (EPS - DPS) / EPS.
• Purpose: Measures the level of earnings retained for reinvestment or other purposes.
3. Dividend Sustainability:
• Importance: Assessing the sustainability of dividends using the payout ratio. A ratio over 100% may indicate an unsustainable dividend.
• Considerations: Long-term trends in the payout ratio and forward-looking analysis for future earnings expectations.
4. Dividends Are Industry Specific:
• Variability: Dividend payouts vary widely by industry.
• Comparison: Payout ratios are most useful when comparing companies within the same industry.
• Adjustments: Augmented payout ratio may include share buybacks and subtract preferred stock dividends for accuracy.
5. How to Calculate the Payout Ratio in Excel:
• Steps: Calculation involves determining dividends per share (DPS) and earnings per share (EPS).
• Context: Payout ratio depends on the sector; startup companies may have a low ratio due to a focus on reinvestment.
6. Example of How to Use the Payout Ratio:
• Apple Example: Calculation of Apple's payout ratio using dividends per share and earnings per share.
7. Dividend Payout vs. Dividend Yield:
• Difference: Dividend payout ratio represents net earnings paid as dividends, while dividend yield is the rate of return on dividends relative to stock price.
• Calculation: Dividend Yield = Annual Dividends per Share / Price per Share.
8. Importance of Dividend Payout Ratio:
• Financial Metric: Key metric for determining the sustainability of a company's dividend payment program.
• Calculation: Commonly calculated on a per-share basis by dividing annual dividends per common share by earnings per share (EPS).
9. High Dividend Payout Ratio Considerations:
• Caution: An unusually high ratio may indicate attempts to mask a bad business situation or a lack of aggressive use of working capital for expansion.
Understanding these concepts is crucial for investors and financial analysts in evaluating a company's financial health and making informed investment decisions. | {"url":"https://clumic.cfd/article/dividend-payout-ratio-definition-formula-and-calculation","timestamp":"2024-11-14T21:27:28Z","content_type":"text/html","content_length":"113495","record_id":"<urn:uuid:a12e0b32-b00f-478f-959d-849a07a0732d>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00035.warc.gz"} |
Aschenbach Effect for Brany Kerr Black Holes and Naked Singularities
Publication date: Dec 2014
We study the non-monotonic Keplerian velocity profiles related to locally nonrotating frames (LNRF) in the field of near-extreme braneworld Kerr black holes and naked singularities in which the
non-local gravitational effects of the bulk are represented by a braneworld tidal charge b and the 4D geometry of the spacetime structure is governed by the Kerr-Newman geometry. We show that
positive tidal charge has a tendency to restrict the values of the black hole dimensionless spin a admitting existence of the non-monotonic Keplerian LNRF-velocity profiles; the non-monotonic
profiles exist in the black hole spacetimes with tidal charge smaller than b = 0.41005 (and spin larger than a = 0.76808). With decreasing value of the tidal charge (which need not be only positive),
both the region of spin allowing the non-monotonicity in the LNRF-velocity profile around braneworld Kerr black hole and the velocity difference in the minimum-maximum parts of the velocity profile
increase implying growing astrophysical relevance of this phenomenon.
Stuchlík, Z.; Blaschke, M.; Slaný, P.; | {"url":"https://zdenekstuchlik.com/2014/12/aschenbach-effect-for-brany-kerr-black-holes-and-naked-singularities/","timestamp":"2024-11-08T21:55:54Z","content_type":"text/html","content_length":"73176","record_id":"<urn:uuid:81240f3f-8b4a-4183-9567-ecd0c23f57d9>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00035.warc.gz"} |
Temperature is used to measure the output of a production
process. When the process is in...
Temperature is used to measure the output of a production process. When the process is in...
Temperature is used to measure the output of a production process. When the process is in control, the mean of the process is
μ = 128.5
and the standard deviation is
σ = 0.3.
Compute the upper and lower control limits if samples of size 6 are to be used. (Round your answers to two decimal places.)
UCL = LCL =
Construct the
chart for this process.
A graph shows three horizontal lines. The horizontal axis is labeled Sample Number and ranges from 0 to 10. The vertical axis is labeled Sample Mean x bar and ranges from 127.75 to 129.50. The bottom
line is labeled LCL and intersects the vertical axis at about 128.02. The middle line intersects the vertical axis at 128.50. The top line is labeled UCL and intersects the vertical axis at about
128.98. The entire region between the LCL line and the UCL line is shaded.
A graph shows three horizontal lines. The horizontal axis is labeled Sample Number and ranges from 0 to 10. The vertical axis is labeled Sample Mean x bar and ranges from 127.75 to 129.50. The bottom
line is labeled LCL and intersects the vertical axis at about 128.35. The middle line intersects the vertical axis at 128.50. The top line is labeled UCL and intersects the vertical axis at about
128.65. The entire region between the LCL line and the UCL line is shaded.
A graph shows three horizontal lines. The horizontal axis is labeled Sample Number and ranges from 0 to 10. The vertical axis is labeled Sample Mean x bar and ranges from 127.75 to 129.50. The bottom
line is labeled LCL and intersects the vertical axis at about 128.24. The middle line intersects the vertical axis at 128.50. The top line is labeled UCL and intersects the vertical axis at about
128.76. The entire region between the LCL line and the UCL line is shaded.
A graph shows three horizontal lines. The horizontal axis is labeled Sample Number and ranges from 0 to 10. The vertical axis is labeled Sample Mean x bar and ranges from 127.75 to 129.50. The bottom
line is labeled LCL and intersects the vertical axis at about 128.13. The middle line intersects the vertical axis at 128.50. The top line is labeled UCL and intersects the vertical axis at about
128.87. The entire region between the LCL line and the UCL line is shaded.
Consider a sample providing the following data.
128.8 128.2 129.1 128.7 128.4 129.2
Compute the mean for this sample. (Round your answer to two decimal places.)
Is the process in control for this sample?
Yes, the process is in control for the sample. No, the process is out of control for the sample.
Consider a sample providing the following data.
129.3 128.7 128.6 129.2 129.5 129.0
Compute the mean for this sample. (Round your answer to two decimal places.)
Is the process in control for this sample?
Yes, the process is in control for the sample. No, the process is out of control for the sample.
μ = 128.5
and the standard deviation is
σ = 0.3
n= sample size 6
UCL = mean +3 Q/ sqrt n
= 128.5 + 3 * 0.3 /sqrt 6 = 128.8674
LCL = mean - 3 Q/ sqrt n
= 128.5 - 3 * 0. 3/ sqrt 6 = 128.1325
A process is said to be out-of control, if any of sample points fall beyond the control limits (below the LCLor above the UCL)
= 128.83
Since in the above sample, one point (128.83) is below the UCL and no point is above LCL ( 128.13), the process is said to be in -control of sample
(129.3 +128.7 +128.6 + 129.2 +129.5 + 129.0) / 6 =129.05
Since in the above sample, four points (129, 129.2, 129.3, 129.5) are above the UCL and nopoint is below LCL (128.13), the process is said to be out-of-control
x= 129.05 | {"url":"https://justaaa.com/statistics-and-probability/319223-temperature-is-used-to-measure-the-output-of-a","timestamp":"2024-11-06T14:42:40Z","content_type":"text/html","content_length":"49280","record_id":"<urn:uuid:4d2fb91b-ba16-4036-a3b3-a398bded7bfa>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00638.warc.gz"} |
Compton scattering description
This program plots the ratio of the scattered to incident photon energies in Compton scattering as a function of either the scattering angle for different incident photon energies, or the photon
energy for different scattering angles. It follows the equation
see also “Introduction to Synchrotron Radiation - Techniques and Applications”, Section 2.3. | {"url":"https://synchrotronmovies.com/compton-scattering-description.html","timestamp":"2024-11-02T03:19:00Z","content_type":"text/html","content_length":"76782","record_id":"<urn:uuid:45b04e8e-1398-4af1-adc7-2e109037b6bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00867.warc.gz"} |
A New Class of Changing-Look LINERs | Request PDF
A New Class of Changing-Look LINERs
Preprints and early-stage research may not have been peer reviewed yet.
We report the discovery of six active galactic nuclei (AGN) caught "turning on" during the first nine months of the Zwicky Transient Facility (ZTF) survey. The host galaxies were classified as LINERs
by weak narrow forbidden line emission in their archival SDSS spectra, and detected by ZTF as nuclear transients. In five of the cases, we found via follow-up spectroscopy that they had transformed
into broad-line AGN, reminiscent of the changing-look LINER iPTF 16bco. In one case, ZTF18aajupnt/AT2018dyk, follow-up HST UV and ground-based optical spectra revealed the transformation into a
narrow-line Seyfert 1 (NLS1) with strong [Fe VII, X, XIV] and He II 4686 coronal lines. Swift monitoring observations of this source reveal bright UV emission that tracks the optical flare,
accompanied by a luminous soft X-ray flare that peaks ~60 days later. Spitzer follow-up observations also detect a luminous mid-infrared flare implying a large covering fraction of dust. Archival
light curves of the entire sample from CRTS, ATLAS, and ASAS-SN constrain the onset of the optical nuclear flaring from a prolonged quiescent state. Here we present the systematic selection and
follow-up of this new class of changing-look LINERs, compare their properties to previously reported changing-look Seyfert galaxies, and conclude that they are a unique class of transients related to
physical processes associated with the LINER accretion state.
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We describe a dynamic science portal called the GROWTH Marshal that allows time-domain astronomers to define science programs, program filters to save sources from different discovery streams,
co-ordinate follow-up with various robotic or classical telescopes, analyze the panchromatic follow-up data and generate summary tables for publication. The GROWTH marshal currently serves 137
scientists, 38 science programs and 67 telescopes. Every night, in real-time, several science programs apply various customized filters to the 10^5 nightly alerts from the Zwicky Transient Facility.
Here, we describe the schematic and explain the functionality of the various components of this international collaborative platform.
The Zwicky Transient Facility is a large optical survey in multiple filters producing hundreds of thousands of transient alerts per night. We describe here various machine learning (ML)
implementations and plans to make the maximal use of the large data set by taking advantage of the temporal nature of the data, and further combining it with other data sets. We start with the
initial steps of separating bogus candidates from real ones, separating stars and galaxies, and go on to the classification of real objects into various classes. Besides the usual methods (e.g.,
based on features extracted from light curves) we also describe early plans for alternate methods including the use of domain adaptation, and deep learning. In a similar fashion we describe efforts
to detect fast moving asteroids. We also describe the use of the Zooniverse platform for helping with classifications through the creation of training samples, and active learning. Finally we mention
the synergistic aspects of ZTF and LSST from the ML perspective.
Accreting supermassive black holes (SMBHs) can exhibit variable emission across the electromagnetic spectrum and over a broad range of timescales. The variability of active galactic nuclei (AGNs) in
the ultraviolet and optical is usually at the few tens of per cent level over timescales of hours to weeks. Recently, rare, more dramatic changes to the emission from accreting SMBHs have been
observed, including tidal disruption events, 'changing look' AGNs and other extreme variability objects. The physics behind the 're-ignition', enhancement and 'shut-down' of accretion onto SMBHs is
not entirely understood. Here we present a rapid increase in ultraviolet-optical emission in the centre of a nearby galaxy, marking the onset of sudden increased accretion onto a SMBH. The optical
spectrum of this flare, dubbed AT 2017bgt, exhibits a mix of emission features. Some are typical of luminous, unobscured AGNs, but others are likely driven by Bowen fluorescence - robustly linked
here with high-velocity gas in the vicinity of the accreting SMBH. The spectral features and increased ultraviolet flux show little evolution over a period of at least 14 months. This disfavours the
tidal disruption of a star as their origin, and instead suggests a longer-term event of intensified accretion. Together with two other recently reported events with similar properties, we define a
new class of SMBH-related flares. This has important implications for the classification of different types of enhanced accretion onto SMBHs.
The Zwicky Transient Facility (ZTF) is a new optical time-domain survey that uses the Palomar 48 inch Schmidt telescope. A custom-built wide-field camera provides a 47 deg^2 field of view and 8 s
readout time, yielding more than an order of magnitude improvement in survey speed relative to its predecessor survey, the Palomar Transient Factory. We describe the design and implementation of the
camera and observing system. The ZTF data system at the Infrared Processing and Analysis Center provides near-real-time reduction to identify moving and varying objects. We outline the analysis
pipelines, data products, and associated archive. Finally, we present on-sky performance analysis and first scientific results from commissioning and the early survey. ZTF's public alert stream will
serve as a useful precursor for that of the Large Synoptic Survey Telescope.
The Zwicky Transient Facility (ZTF) is a new robotic time-domain survey currently in progress using the Palomar 48-inch Schmidt Telescope. ZTF uses a 47 square degree field with a 600 megapixel
camera to scan the entire northern visible sky at rates of ~3760 square degrees/hour to median depths of g ~ 20.8 and r ~ 20.6 mag (AB, 5sigma in 30 sec). We describe the Science Data System that is
housed at IPAC, Caltech. This comprises the data-processing pipelines, alert production system, data archive, and user interfaces for accessing and analyzing the products. The realtime pipeline
employs a novel image-differencing algorithm, optimized for the detection of point source transient events. These events are vetted for reliability using a machine-learned classifier and combined
with contextual information to generate data-rich alert packets. The packets become available for distribution typically within 13 minutes (95th percentile) of observation. Detected events are also
linked to generate candidate moving-object tracks using a novel algorithm. Objects that move fast enough to streak in the individual exposures are also extracted and vetted. The reconstructed
astrometric accuracy per science image with respect to Gaia is typically 45 to 85 milliarcsec. This is the RMS per axis on the sky for sources extracted with photometric S/N >= 10. The derived
photometric precision (repeatability) at bright unsaturated fluxes varies between 8 and 25 millimag. Photometric calibration accuracy with respect to Pan-STARRS1 is generally better than 2%. The
products support a broad range of scientific applications: fast and young supernovae, rare flux transients, variable stars, eclipsing binaries, variability from active galactic nuclei, counterparts
to gravitational wave sources, a more complete census of Type Ia supernovae, and Solar System objects.
We report a new changing-look quasar, WISE J105203.55+151929.5 at z = 0.303, found by identifying highly mid-IR-variable quasars in the Wide-field Infrared Survey Explorer (WISE)/Near-Earth Object
WISE Reactivation (NEOWISE) data stream. Compared to multiepoch mid-IR photometry of a large sample of SDSS-confirmed quasars, WISE J1052+1519 is an extreme photometric outlier, fading by more than a
factor of two at 3.4 and 4.6 μm since 2009. Swift target-of-opportunity observations in 2017 show even stronger fading in the soft X-rays compared to the ROSAT detection of this source in 1995, with
at least a factor of 15 decrease. We obtained second-epoch spectroscopy with the Palomar telescope in 2017 that, when compared with the 2006 archival SDSS spectrum, reveals that the broad Hβ emission
has vanished and that the quasar has become significantly redder. The two most likely interpretations for this dramatic change are source fading or obscuration, where the latter is strongly
disfavored by the mid-IR data. We discuss various physical scenarios that could cause such changes in the quasar luminosity over this timescale, and favor changes in the innermost regions of the
accretion disk that occur on the thermal and heating/cooling front timescales. We discuss possible physical triggers that could cause these changes, and predict the multiwavelength signatures that
could distinguish these physical scenarios. © 2018. The American Astronomical Society. All rights reserved.
We present results from a systematic selection of tidal disruption events (TDEs) in a wide-area (4800 deg2), g + R band, Intermediate Palomar Transient Factory (iPTF) experiment. Our selection
targets typical optically-selected TDEs: bright (>60% flux increase) and blue transients residing in the center of red galaxies. Using photometric selection criteria to down-select from a total of
493 nuclear transients to a sample of 26 sources, we then use follow-up UV imaging with the Neil Gehrels Swift Telescope, ground-based optical spectroscopy, and light curve fitting to classify them
as 14 Type Ia supernovae (SNe Ia), 9 highly variable active galactic nuclei (AGNs), 2 confirmed TDEs, and 1 potential core-collapse supernova. We find it possible to filter AGNs by employing a more
stringent transient color cut (g - r < -0.2 mag); further, UV imaging is the best discriminator for filtering SNe, since SNe Ia can appear as blue, optically, as TDEs in their early phases. However,
when UV-optical color is unavailable, higher precision astrometry can also effectively reduce SNe contamination in the optical. Our most stringent optical photometric selection criteria yields a
4.5:1 contamination rate, allowing for a manageable number of TDE candidates for complete spectroscopic follow-up and real-time classification in the ZTF era. We measure a TDE per galaxy rate of 1.7
- 1.3 + 2.9 × 10 - 4 gal - 1 yr - 1 (90% CL in Poisson statistics). This does not account for TDEs outside our selection criteria, thus may not reflect the total TDE population, which is yet to be
fully mapped.
The fourth generation of the Sloan Digital Sky Survey (SDSS-IV) has been in operation since July 2014. This paper describes the second data release from this phase, and the fourteenth from SDSS
overall (making this, Data Release Fourteen or DR14). This release makes public data taken by SDSS-IV in its first two years of operation (July 2014--2016). Like all previous SDSS releases, DR14 is
cumulative, including the most recent reductions and calibrations of all data taken by SDSS since the first phase began operations in 2000. New in DR14 is the first public release of data from the
extended Baryon Oscillation Sky Survey (eBOSS); the first data from the second phase of the Apache Point Observatory (APO) Galactic Evolution Experiment (APOGEE-2), including stellar parameter
estimates from an innovative data driven machine learning algorithm known as The Cannon; and almost twice as many data cubes from the Mapping Nearby Galaxies at APO (MaNGA) survey as were in the
previous release (N = 2812 in total). This paper describes the location and format of the publicly available data from SDSS-IV surveys. We provide references to the important technical papers
describing how these data have been taken (both targeting and observation details) and processed for scientific use. The SDSS website (www.sdss.org) has been updated for this release, and provides
links to data downloads, as well as tutorials and examples of data use. SDSS-IV is planning to continue to collect astronomical data until 2020.
We report the discovery by the intermediate Palomar Transient Factory (iPTF) of a candidate tidal disruption event (TDE) iPTF16axa at z = 0.108 and present its broadband photometric and spectroscopic
evolution from three months of follow-up observations with ground-based telescopes and Swift. The light curve is well fitted with a t −5/3 decay, and we constrain the rise time to peak to be <49
rest-frame days after disruption, which is roughly consistent with the fallback timescale expected for the ~5 × 10⁶M ⊙ black hole inferred from the stellar velocity dispersion of the host galaxy. The
UV and optical spectral energy distribution is well described by a constant blackbody temperature of T ~ 3 × 10⁴ K over the monitoring period, with an observed peak luminosity of 1.1 × 10⁴⁴ erg s⁻¹.
The optical spectra are characterized by a strong blue continuum and broad He ii and Hα lines, which are characteristic of TDEs. We compare the photometric and spectroscopic signatures of iPTF16axa
with 11 TDE candidates in the literature with well-sampled optical light curves. Based on a single-temperature fit to the optical and near-UV photometry, most of these TDE candidates have peak
luminosities confined between log(L [erg s⁻¹]) = 43.4–44.4, with constant temperatures of a few ×10⁴ K during their power-law declines, implying blackbody radii on the order of 10 times the tidal
disruption radius, that decrease monotonically with time. For TDE candidates with hydrogen and helium emission, the high helium-to-hydrogen ratios suggest that the emission arises from high-density
gas, where nebular arguments break down. We find no correlation between the peak luminosity and the black hole mass, contrary to the expectations for TDEs to have .
We present a new catalog of narrow-line Seyfert 1 (NLSy1) galaxies from the Sloan Digital Sky Survey Data Release 12 (SDSS DR12). This was obtained by a systematic analysis through modeling of the
continuum and emission lines of the spectra of all the 68,859 SDSS DR12 objects that are classified as “QSO” by the SDSS spectroscopic pipeline with z< 0.8 and a median signal-to-noise ratio (S/N) >
2 pixel‑1. This catalog contains a total of 11,101 objects, which is about 5 times larger than the previously known NLSy1 galaxies. Their monochromatic continuum luminosity at 5100 Å is found to be
strongly correlated with Hβ, Hα, and [O iii] emission line luminosities. The optical Fe ii strength in NLSy1 galaxies is about two times larger than the broad-line Seyfert 1 (BLSy1) galaxies. About
5% of the catalog sources are detected in the FIRST survey. The Eddington ratio (ξ_Edd) of NLSy1 galaxies has an average of log ξ_Edd of ‑0.34, much higher than ‑1.03 found for BLSy1 galaxies. Their
black hole masses (M_BH) have an average of log M_BH of 6.9 M_sun, which is less than BLSy1 galaxies, which have an average of log M_BH of 8.0 M_sun. The M_BH of NLSy1 galaxies is found to be
correlated with their host galaxy velocity dispersion. Our analysis suggests that geometrical effects playing an important role in defining NLSy1 galaxies and their M_BH deficit is perhaps due to
their lower inclination compared to BLSy1 galaxies.
We present the ultraviolet (UV) spectroscopic evolution of a tidal disruption event (TDE) for the first time. After the discovery of the nearby TDE iPTF16fnl, we obtained a series of observations
with the Space Telescope Imaging Spectrograph (STIS) onboard the Hubble Space Telescope (HST). The dominant emission features closely resemble those seen in the UV spectra of the TDE ASASSN-14li and
are also similar to those of N-rich quasars. However, there is significant evolution in the shape and central wavelength of the line profiles over the course of our observations, such that at early
times the lines are broad and redshifted, while at later times the lines are significantly narrower and peak near the wavelengths of their corresponding atomic transitions. Like ASASSN-14li, but
unlike N-rich quasars, iPTF16fnl shows neither MgII$\lambda 2798$\AA\ nor CIII]$\lambda 1909$\AA\ emission features. We also present optical photometry and spectroscopy, which suggest that the
complex HeII profiles observed in the optical spectra of many TDEs are in part due to the presence of NIII and CIII Wolf-Rayet features, which can potentially serve as probes of the far-UV when
space-based observations are not possible. Finally, we use Swift XRT and UVOT observations to place strong limits on the X-ray emission and determine the characteristic temperature, radius, and
luminosity of the emitting material. We find that iPTF16fnl is subluminous and evolves more rapidly than other optically discovered TDEs.
We unify the feeding and feedback of supermassive black holes with the global properties of galaxies, groups, and clusters, by linking for the first time the physical mechanical efficiency at the
horizon and Mpc scale. The macro hot halo is tightly constrained by the absence of overheating and overcooling as probed by X-ray data and hydrodynamic simulations ($\varepsilon_{\rm BH} \simeq$ 10$^
{-3}\,T_{\rm x,7.4}$). The micro flow is shaped by general relativistic effects tracked by state-of-the-art GR-RMHD simulations ($\varepsilon_\bullet \simeq$ 0.03). The SMBH properties are tied to
the X-ray halo temperature $T_{\rm x}$, or related cosmic scaling relation (as $L_{\rm x}$). The model is minimally based on first principles, as conservation of energy and mass recycling. The inflow
occurs via chaotic cold accretion (CCA), the rain of cold clouds condensing out of the quenched cooling flow and recurrently funneled via inelastic collisions. Within 100 gravitational radii, the
accretion energy is transformed into ultrafast 10$^4$ km s$^{-1}$ outflows (UFOs) ejecting most of the inflowing mass. At larger radii the energy-driven outflow entrains progressively more mass: at
kpc scale, the velocities of the hot/warm/cold outflows are a few 10$^3$, 1000, 500 km s$^{-1}$, with median mass rates ~10, 100, several 100 M$_\odot$ yr$^{-1}$, respectively. The unified CCA model
is consistent with the observations of nuclear UFOs, and ionized, neutral, and molecular macro outflows. We provide step-by-step implementation for subgrid simulations, (semi)analytic works, or
observational interpretations which require self-regulated AGN feedback at coarse scales, avoiding the a-posteriori fine-tuning of efficiencies.
UV and optically selected candidates for stellar tidal disruption events (TDE) often exhibit broad spectral features (HeII emission, H$\alpha$ emission, or absorption lines) on a blackbody-like
continuum (1e4K<T<1e5K). The lines presumably emit from TDE debris or circumnuclear clouds photoionized by the flare. Line velocities however are much lower than expected from a stellar disruption by
supermassive black hole (SMBH), and are somewhat faster than expected for the broad line region (BLR) clouds of a persistently active galactic nucleus (AGN). The distinctive spectral states are not
strongly related to observed luminosity and velocity, nor to SMBH mass estimates. We use exhaustive photoionization modelling to map the domain of fluxes and cloud properties that yield (e.g.) a
He-overbright state where a large HeII(4686A)/H$\alpha$ line-ratio creates an illusion of helium enrichment. Although observed line ratios occur in a plausible minority of cases, AGN-like
illumination can not reproduce the observed equivalent widths. We therefore propose to explain these properties by a light-echo photoionization model: the initial flash of a hot blackbody
(detonation) excites BLR clouds, which are then seen superimposed on continuum from a later, expanded, cooled stage of the central luminous source. The implied cloud mass is substellar, which may be
inconsistent with a TDE. Given these and other inconsistencies with TDE models (e.g. host-galaxies distribution) we suggest to also consider alternative origins for these nuclear flares, which we
briefly discuss (e.g. nuclear supernovae and starved/subluminous AGNs).
X-ray reverberation, where light-travel time delays map out the compact geometry around the inner accretion flow in supermassive black holes, has been discovered in several of the brightest, most
variable and well-known Seyfert galaxies. In this work, we expand the study of X-ray reverberation to all Seyfert galaxies in the XMM-Newton archive above a nominal rms variability and exposure level
(a total of 43 sources). 50 per cent of source exhibit iron K reverberation, in that the broad iron K emission line responds to rapid variability in the continuum. We also find that on long
timescales, the hard band emission lags behind the soft band emission in 85 per cent of sources. This `low-frequency hard lag' is likely associated with the coronal emission, and so this result
suggests that most sources with X-ray variability show intrinsic variability from the nuclear region. We update the known iron K lag amplitude vs. black hole mass relation, and find evidence that the
height or extent of the coronal source (as inferred by the reverberation time delay) increases with mass accretion rate.
We present ground-based and Swift photometric and spectroscopic observations of the tidal disruption event (TDE) ASASSN-15oi, discovered at the center of 2MASX J20390918-3045201 ($d\simeq216$ Mpc) by
the All-Sky Automated Survey for SuperNovae (ASAS-SN). The source peaked at a bolometric luminosity of $L\simeq1.9\times10^{44}$ ergs s$^{-1}$ and radiated a total energy of $E\simeq5.0\times10^{50}$
ergs over the $\sim3.5$ months of observations. The early optical/UV emission of the source can be fit by a blackbody with temperature increasing from $T\sim2\times10^4$ K to $T\sim6\times10^4$ K
while the luminosity declines from $L\simeq1.9\times10^{44}$ ergs s$^{-1}$ to $L\simeq2.8\times10^{43}$ ergs s$^{-1}$, requiring the photosphere to be shrinking rapidly. The optical/UV luminosity
decline is broadly consistent with an exponential decline, $L\propto e^{-t/t_0}$, with $t_0\simeq35$ days. ASASSN-15oi also exhibits roughly constant soft X-ray emission that is significantly weaker
than the optical/UV emission. Spectra of the source show broad helium emission lines and strong blue continuum emission in early epochs, although these features fade rapidly and are not present $\
sim3$ months after discovery. The early spectroscopic features and color evolution of ASASSN-15oi are consistent with a TDE, but the rapid spectral evolution is unique among optically-selected TDEs.
We have applied computer analysis to classify the broad morphological types of ~3,000,000 Sloan Digital Sky Survey (SDSS) galaxies. For each galaxy, the catalog provides the DR8 object ID, the R.A.,
the decl., and the certainty for the automatic classification as either spiral or elliptical. The certainty of the classification allows us to control the accuracy of a subset of galaxies by
sacrificing some of the least certain classifications. The accuracy of the catalog was tested using galaxies that were classified by the manually annotated Galaxy Zoo catalog. The results show that
the catalog contains ~900,000 spiral galaxies and ~600,000 elliptical galaxies with classification certainty that has a statistical agreement rate of ~98% with the Galaxy Zoo debiased "superclean"
data set. The catalog also shows that objects assigned by the SDSS pipeline with a relatively high redshift (z > 0.4) can have clear visual spiral morphology. The catalog can be downloaded at http://
vfacstaff.ltu.edu/lshamir/data/morph_catalog. The image analysis software that was used to create the catalog is also publicly available.
We present a systematic search for changing-look quasars based on repeat photometry from SDSS and Pan-STARRS1, along with repeat spectra from SDSS and SDSS-III BOSS. Objects with large, $|\Delta g|>
1$~mag photometric variations in their light curves are selected as candidates to look for changes in broad emission line (BEL) features. Out of a sample of 1011 objects that satisfy our selection
criteria and have more than one epoch of spectroscopy, we find 10 examples of quasars that have variable and/or "changing-look'' BEL features. Four of our objects have emerging BELs; five have
disappearing BELs, and one object shows tentative evidence for having both emerging and disappearing BELs. With redshifts in the range 0.20<z<0.63, this sample includes the highest-redshift
changing-look quasars discovered to date. We highlight the quasar J102152.34+464515.6 at z=0.204. Here, not only have the Balmer emisson lines strongly diminished in prominence, including H\beta all
but disappearing, but the blue continuum f_{\nu}\propto \nu^{1/3} typical of an AGN is also significantly diminished in the second epoch of spectroscopy. We test a simple dust-reddening toy model,
and find that this is inadequate to explain the change in spectral properties of this object. Using our selection criteria, we estimate that >12% of luminous quasars that vary by |\Delta g|>1 mag
display changing-look BEL features on rest-frame timescales of 8 to 10 years. We discuss the possibilities for the origin of such BEL changes, such as a change in obscuration or in the central
We present ground-based and Swift photometric and spectroscopic observations of the candidate tidal disruption event (TDE) ASASSN-14li, found at the centre of PGC 043234 (d ≃ 90 Mpc) by the All-Sky
Automated Survey for SuperNovae (ASAS-SN). The source had a peak bolometric luminosity of L ≃ 1044 erg s−1 and a total integrated energy of E ≃ 7 × 1050 erg radiated over the ∼6 months of
observations presented. The UV/optical emission of the source is well fitted by a blackbody with roughly constant temperature of T ∼ 35 000 K, while the luminosity declines by roughly a factor of 16
over this time. The optical/UV luminosity decline is broadly consistent with an exponential decline, $L\propto \text{e}^{-t/t_0}$, with t0 ≃ 60 d. ASASSN-14li also exhibits soft X-ray emission
comparable in luminosity to the optical and UV emission but declining at a slower rate, and the X-ray emission now dominates. Spectra of the source show broad Balmer and helium lines in emission as
well as strong blue continuum emission at all epochs. We use the discoveries of ASASSN-14li and ASASSN-14ae to estimate the TDE rate implied by ASAS-SN, finding an average rate of r ≃ 4.1 × 10−5 yr−1
per galaxy with a 90 per cent confidence interval of (2.2–17.0) × 10−5 yr−1 per galaxy. ASAS-SN found roughly 1 TDE for every 70 Type Ia supernovae in 2014, a rate that is much higher than that of
other surveys.
The Swift mission, scheduled for launch in 2004, is a multiwavelength observatory for gamma-ray burst (GRB) astronomy. It is a first-of-its-kind autonomous rapid-slewing satellite for transient
astronomy and pioneers the way for future rapid-reaction and multiwavelength missions. It will be far more powerful than any previous GRB mission, observing more than 100 bursts yr-1 and performing
detailed X-ray and UV/optical afterglow observations spanning timescales from 1 minute to several days after the burst. The objectives are to (1) determine the origin of GRBs, (2) classify GRBs and
search for new types, (3) study the interaction of the ultrarelativistic outflows of GRBs with their surrounding medium, and (4) use GRBs to study the early universe out to z > 10. The mission is
being developed by a NASA-led international collaboration. It will carry three instruments: a new-generation wide-field gamma-ray (15-150 keV) detector that will detect bursts, calculate 1'-4'
positions, and trigger autonomous spacecraft slews; a narrow-field X-ray telescope that will give 5'' positions and perform spectroscopy in the 0.2-10 keV band; and a narrow-field UV/optical
telescope that will operate in the 170-600 nm band and provide 03 positions and optical finding charts. Redshift determinations will be made for most bursts. In addition to the primary GRB science,
the mission will perform a hard X-ray survey to a sensitivity of ~1 mcrab (~2 × 10-11 ergs cm-2 s-1 in the 15-150 keV band), more than an order of magnitude better than HEAO 1 A-4. A flexible data
and operations system will allow rapid follow-up observations of all types of high-energy transients, with rapid data downlink and uplink available through the NASA TDRSS system. Swift transient data
will be rapidly distributed to the astronomical community, and all interested observers are encouraged to participate in follow-up measurements. A Guest Investigator program for the mission will
provide funding for community involvement. Innovations from the Swift program applicable to the future include (1) a large-area gamma-ray detector using the new CdZnTe detectors, (2) an autonomous
rapid-slewing spacecraft, (3) a multiwavelength payload combining optical, X-ray, and gamma-ray instruments, (4) an observing program coordinated with other ground-based and space-based
observatories, and (5) immediate multiwavelength data flow to the community. The mission is currently funded for 2 yr of operations, and the spacecraft will have a lifetime to orbital decay of ~8 yr.
We combine SDSS and WISE photometry for the full SDSS spectroscopic galaxy sample, creating SEDs that cover lambda=0.4-22 micron for an unprecedented large and comprehensive sample of 858,365
present-epoch galaxies. Using MAGPHYS we then model simultaneously and consistently both the attenuated stellar SED and the dust emission at 12 micron and 22 micron, producing robust new calibrations
for monochromatic mid-IR star formation rate proxies. These modeling results provide the first mid-IR-based view of the bi-modality in star formation activity among galaxies, exhibiting the sequence
of star-forming galaxies (main sequence) with a slope of dlogSFR/dlogM*=0.80 and a scatter of 0.39 dex. We find that these new star-formation rates along the SF main sequence are systematically lower
by a factor of 1.4 than those derived from optical spectroscopy. We show that for most present-day galaxies the 0.4-22 micron SED fits can exquisitely predict the fluxes measured by Herschel at much
longer wavelengths. Our analysis also illustrates that the majorities of stars in the present-day universe is formed in luminous galaxies (~L*) in and around the green valley of the color-luminosity
plane. We make the matched photometry catalog and SED modeling results publicly available.
We report on the Swift discovery of a second high-amplitude (factor 100) outburst of the Seyfert 1.9 galaxy IC 3599, and discuss implications for outburst scenarios. Swift detected this active
galactic nucleus (AGN) again in February 2010 in X-rays at a level of (1.50\plm0.11)$\times 10^{36}$ W (0.2-2.0 keV), which is nearly as luminous as the first outburst detected with ROSAT in 1990.
Optical data from the Catalina sky survey show that the optical emission was already bright two years before the Swift X-ray high-state. Our new Swift observations performed between 2013 and 2015
show that IC 3599 is currently again in a very low X-ray flux state. This repeat optical and X-ray outburst, and the long optical duration, suggest that IC 3599 is likely not a tidal disruption event
(TDE). Instead, variants of AGN-related variability are explored. The data are consistent with an accretion disk instability around a black hole of mass on the order 10$^6$--10$^7$ M$_{\odot}$; a
value estimated using several different methods.
Reverberation-mapping-based scaling relations are often used to estimate the masses of black holes from single-epoch spectra of AGN. While the radius-luminosity relation that is the basis of these
scaling relations is determined using reverberation mapping of the H$\beta$ line in nearby AGN, the scaling relations are often extended to use other broad emission lines, such as MgII, in order to
get black hole masses at higher redshifts when H$\beta$ is redshifted out of the optical waveband. However, there is no radius-luminosity relation determined directly from MgII. Here, we present an
attempt to perform reverberation mapping using MgII in the well-studied nearby Seyfert 1, NGC 5548. We used Swift to obtain UV grism spectra of NGC 5548 once every two days from April to September
2013. Concurrent photometric UV monitoring with Swift provides a well determined continuum lightcurve that shows strong variability. The MgII emission line, however, is not strongly correlated with
the continuum variability, and there is no significant lag between the two. We discuss these results in the context of using MgII scaling relations to estimate high-redshift black hole masses.
Active galactic nuclei (AGNs) that show strong rest-frame optical/UV variability in their blue continuum and broad line emission are classified as changing-look AGN, or at higher luminosities,
changing-look quasars (CLQs). These surprisingly large and sometimes rapid transitions challenge accepted models of quasar physics and duty cycles, offer several new avenues for study of quasar host
galaxies, and open a wider interpretation of the cause of differences between broad and narrow-line AGN. To better characterize extreme quasar variability, we present follow-up spectroscopy as part
of a comprehensive search for CLQs across the full Sloan Digital Sky Survey (SDSS) footprint using spectroscopically confirmed quasars from the SDSS DR7 catalog. Our primary selection requires
large-amplitude (|Δg| > 1 mag, |Δr| > 0.5 mag) variability over any of the available time baselines probed by the SDSS and Pan-STARRS 1 surveys. We employ photometry from the Catalina Sky Survey to
verify variability behavior in CLQ candidates where available, and confirm CLQs using optical spectroscopy from the William Herschel, MMT, Magellan, and Palomar telescopes. For our adopted
signal-to-noise ratio threshold on variability of broad Hβ emission, we find 17 new CLQs, yielding a confirmation rate of 20%. These candidates are at lower Eddington ratio relative to the overall
quasar population, which supports a disk-wind model for the broad line region. Based on our sample, the CLQ fraction increases from 10% to roughly half as the continuum flux ratio between repeat
spectra at 3420 Å increases from 1.5 to 6. We release a catalog of more than 200 highly variable candidates to facilitate future CLQ searches.
We present Zwicky Transient Facility (ZTF) observations of the tidal disruption flare AT2018zr/PS18kh reported by Holoien et al. and detected during ZTF commissioning. The ZTF light curve of the
tidal disruption event (TDE) samples the rise-to-peak exceptionally well, with 50 days of g- and r-band detections before the time of maximum light. We also present our multi-wavelength follow-up
observations, including the detection of a thermal (kT ≈ 100 eV) X-ray source that is two orders of magnitude fainter than the contemporaneous optical/UV blackbody luminosity, and a stringent upper
limit to the radio emission. We use observations of 128 known active galactic nuclei (AGNs) to assess the quality of the ZTF astrometry, finding a median host-flare distance of 0.″2 for genuine
nuclear flares. Using ZTF observations of variability from known AGNs and supernovae we show how these sources can be separated from TDEs. A combination of light-curve shape, color, and location in
the host galaxy can be used to select a clean TDE sample from multi-band optical surveys such as ZTF or the Large Synoptic Survey Telescope. © 2019. The American Astronomical Society. All rights
We present the analysis of the first Nuclear Spectroscopic Telescope Array observations (∼220 ks), simultaneous with the last Suzaku observations (∼50 ks), of the active galactic nucleus of the
bright Seyfert 1 galaxy Mrk 509. The time-averaged spectrum in the 1-79 keV X-ray band is dominated by a power-law continuum (Γ ∼ 1.8-1.9), a strong soft excess around 1 keV, and signatures of X-ray
reflection in the form of Fe K emission (∼6.4 keV), an Fe K absorption edge (∼7.1 keV), and a Compton hump due to electron scattering (∼20-30 keV). We show that these data can be described by two
very different prescriptions for the soft excess: a warm (kT ∼ 0.5-1 keV) and optically thick (τ ∼ 10-20) Comptonizing corona or a relativistically blurred ionized reflection spectrum from the inner
regions of the accretion disk. While these two scenarios cannot be distinguished based on their fit statistics, we argue that the parameters required by the warm corona model are physically
incompatible with the conditions of standard coronae. Detailed photoionization calculations show that even in the most favorable conditions, the warm corona should produce strong absorption in the
observed spectrum. On the other hand, while the relativistic reflection model provides a satisfactory description of the data, it also requires extreme parameters, such as maximum black hole spin, a
very low and compact hot corona, and a very high density for the inner accretion disk. Deeper observations of this source are thus necessary to confirm the presence of relativistic reflection and
further understand the nature of its soft excess. © 2019. The American Astronomical Society. All rights reserved.
Aims . We report on the discovery and follow-up of a peculiar transient, OGLE17aaj, which occurred in the nucleus of a weakly active galaxy. We investigate whether it can be interpreted as a new
candidate for a tidal disruption event (TDE). Methods . We present the OGLE-IV light curve that covers the slow 60-day-long rise to maximum along with photometric, spectroscopic, and X-ray follow-up
during the first year. Results . OGLE17aaj is a nuclear transient exhibiting some properties similar to previously found TDEs, including a long rise time, lack of colour-temperature evolution, and
high black-body temperature. On the other hand, its narrow emission lines and slow post-peak evolution are different from previously observed TDEs. Its spectrum and light-curve evolution is similar
to F01004-2237 and AT 2017bgt. Signatures of historical low-level nuclear variability suggest that OGLE17aaj may instead be related to a new type of accretion event in active super-massive black
Changing-look quasars are a recently identified class of active galaxies in which the strong UV continuum and/or broad optical hydrogen emission lines associated with unobscured quasars either appear
or disappear on time-scales of months to years. The physical processes responsible for this behaviour are still debated, but changes in the black hole accretion rate or accretion disc structure
appear more likely than changes in obscuration. Here, we report on four epochs of spectroscopy of SDSS J110057.70−005304.5, a quasar at a redshift of z = 0.378 whose UV continuum and broad hydrogen
emission lines have faded, and then returned over the past ≈20 yr. The change in this quasar was initially identified in the infrared, and an archival spectrum from 2010 shows an intermediate phase
of the transition during which the flux below rest frame ≈3400 Å has decreased by close to an order of magnitude. This combination is unique compared to previously published examples of changing-look
quasars, and is best explained by dramatic changes in the innermost regions of the accretion disc. The optical continuum has been rising since mid-2016, leading to a prediction of a rise in hydrogen
emission-line flux in the next year. Increases in the infrared flux are beginning to follow, delayed by a ∼3 yr observed time-scale. If our model is confirmed, the physics of changing-look quasars
are governed by processes at the innermost stable circular orbit around the black hole, and the structure of the innermost disc. The easily identifiable and monitored changing-look quasars would then
provide a new probe and laboratory of the nuclear central engine.
The Zwicky Transient Facility (ZTF) survey generates real-time alerts for optical transients, variables, and moving objects discovered in its wide-field survey. We describe the ZTF alert stream
distribution and processing (filtering) system. The system uses existing open-source technologies developed in industry: Kafka, a real-time streaming platform, and Avro, a binary serialization
format. The technologies used in this system provide a number of advantages for the ZTF use case, including (1) built-in replication, scalability, and stream rewind for the distribution mechanism;
(2) structured messages with strictly enforced schemas and dynamic typing for fast parsing; and (3) a Python-based stream processing interface that is similar to batch for a familiar and
user-friendly plug-in filter system, all in a modular, primarily containerized system. The production deployment has successfully supported streaming up to 1.2 million alerts or roughly 70 GB of data
per night, with each alert available to a consumer within about 10 s of alert candidate production. Data transfer rates of about 80,000 alerts/minute have been observed. In this paper, we discuss
this alert distribution and processing system, the design motivations for the technology choices for the framework, performance in production, and how this system may be generally suitable for other
alert stream use cases, including the upcoming Large Synoptic Survey Telescope.
We model the broad-band (optical/UV and X-ray) continuum spectrum of the 'changing-look' active galactic nucleus (AGN) Mrk 1018, as it fades from Seyfert 1 to 1.9 in ~ 8 years. The brightest
spectrum, with Eddington ratio L/LEdd ~ 0.08 has a typical type 1 AGN continuum, with a strong 'soft X-ray excess' spanning between the UV and soft X-rays. The dimmest spectrum, at L/LEdd ~ 0.006, is
very different in shape as well as luminosity, with the soft excess dropping by much more than the hard X-rays. The soft X-ray excess produces most of the ionizing photons, so its dramatic drop leads
to the disappearance of the broad-line region, driving the 'changing-look' phenomena. This spectral hardening appears similar to the soft-to-hard state transition in black hole binaries at L/LEdd ~
0.02, where the inner disc evaporates into an advection dominated accretion flow, while the overall drop in luminosity appears consistent with the hydrogen ionization disc instability. None the less,
both processes happen much faster in Mrk 1018 than predicted by disc theory. We critically examine scaling from Galactic binary systems and show that amajor difference is that radiation pressure
should be much more important in AGNs, so that the sound speed is much faster than expected from the gas temperature. Including magnetic pressure to stabilize the disc shortens the time-scales even
further. We suggest that all changing-look AGNs are similarly associated with the state transition at L/LEdd ~ a few per cent.
Mrk 590 was originally classified as a Seyfert 1 galaxy, but then it underwent dramatic changes: the nuclear luminosity dropped by over two orders of magnitude and the broad emission lines all but
disappeared from the optical spectrum. Here we present follow-up observations to the original discovery and characterization of this "changing-look" active galactic nucleus (AGN). The new Chandra and
Hubble Space Telescope observations from 2014 show that Mrk 590 is awakening, changing its appearance again. While the source continues to be in a low state, its soft excess has re-emerged, though
not to the previous level. The UV continuum is brighter by more than a factor of two and the broad Mg ii emission line is present, indicating that the ionizing continuum is also brightening. These
observations suggest that the soft excess is not due to reprocessed hard X-ray emission. Instead, it is connected to the UV continuum through warm Comptonization. Variability of the Fe Kα emission
lines suggests that the reprocessing region is within ∼10 lt-yr or 3 pc of the central source. The change in AGN type is neither due to obscuration nor due to one-way evolution from Type 1 to Type 2,
as suggested in the literature, but may be related to episodic accretion events. © 2018. The American Astronomical Society. All rights reserved..
We present late-time optical spectroscopy and X-ray, UV, and optical photometry of the nearby ($d=214$ Mpc, $z=0.0479$) tidal disruption event (TDE) ASASSN-15oi. The optical spectra span 450 days
after discovery and show little remaining transient emission or evolution after roughly 3 months. In contrast, the Swift and XMM-Newton observations indicate the presence of evolving X-ray emission
and lingering thermal UV emission that is still present 600 days after discovery. The thermal component of the X-ray emission shows a unique, slow brightening by roughly an order of magnitude to
become the dominant source of emission from the TDE at later times, while the hard component of the X-ray emission remains weak and relatively constant throughout the flare. The TDE radiated $(1.32\
pm0.06)\times10^{51}$ ergs across all wavelengths, and the UV and optical emission is consistent with a power law decline and potentially indicative of a late-time shift in the power-law index that
could be caused by a transition in the dominant emission mechanism.
We perform a systematic search for long-term extreme variability quasars (EVQs) in the overlapping Sloan Digital Sky Survey and 3 Year Dark Energy Survey imaging, which provide light curves spanning
more than 15 years. We identified ∼1000 EVQs with a maximum change in g-band magnitude of more than 1 mag over this period, about 10% of all quasars searched. The EVQs have L bol ∼ 10⁴⁵-10⁴⁷ erg s⁻¹
and L/L Edd ∼ 0.01-1. Accounting for selection effects, we estimate an intrinsic EVQ fraction of ∼30%-50% among all quasars over a baseline of ∼15 yr. We performed detailed multi-wavelength,
spectral, and variability analyses for the EVQs and compared them to their parent quasar sample. We found that EVQs are distinct from a control sample of quasars matched in redshift and optical
luminosity: (1) their UV broad emission lines have larger equivalent widths; (2) their Eddington ratios are systematically lower; and (3) they are more variable on all timescales. The intrinsic
difference in quasar properties for EVQs suggests that internal processes associated with accretion are the main driver for the observed extreme long-term variability. However, despite their
different properties, EVQs seem to be in the tail of a continuous distribution of quasar properties, rather than standing out as a distinct population. We speculate that EVQs are normal quasars
accreting at relatively low rates, where the accretion flow is more likely to experience instabilities that drive the changes in flux by a factor of a few on multi-year timescales. © 2018. The
American Astronomical Society. All rights reserved.
In addition to stochastic variability, a fraction of active galactic nuclei (AGN) are observed to exhibit dramatic variability in the X-ray band on timescales down to minutes. We introduce the case
study of 1H 1934-063 (z = 0.0102), a Narrow-line Seyfert 1 (NLS1) among the brightest and most variable AGN ever observed with XMM-Newton. This work includes spectral and temporal analyses of a
concurrent XMM-Newton and NuSTAR 2015 observation lasting 130 kiloseconds, during which the X-ray source exhibited a steep (factor of 6) plummet and subsequent full recovery of flux level. We rule
out Compton-thin obscuration as the cause for this dramatic variability observed even at NuSTAR energies. In order to constrain coronal geometry, dynamics, and emission/absorption processes, we
compare detailed spectral fitting with Fourier-based timing analysis. Similar to other well-studied, highly variable Seyfert 1s, this AGN is X-ray bright and displays strong reflection features. We
find a narrower broad iron line component compared to most Seyfert 1s, and constrain black hole spin to be < 0.1, one of the lowest yet discovered for such systems. Combined spectral and timing
results are consistent with a dramatic change in the continuum on timescales as short as a few kiloseconds dictating the nature of this variability. We also discover a Fe-K time lag and measure a
delay of 20 seconds between relativistically-blurred reflection off the inner accretion flow (0.3-1 keV) and the hard X-ray continuum emission (1-4 keV).
Technology has advanced to the point that it is possible to image the entire sky every night and process the data in real time. The sky is hardly static: many interesting phenomena occur, including
variable stationary objects such as stars or QSOs, transient stationary objects such as supernovae or M dwarf flares, and moving objects such as asteroids and the stars themselves. Funded by NASA, we
have designed and built a sky survey system for the purpose of finding dangerous near-Earth asteroids (NEAs). This system, the "Asteroid Terrestrial-impact Last Alert System" (ATLAS), has been
optimized to produce the best survey capability per unit cost, and therefore ATLAS is an efficient and competitive system for finding potentially hazardous asteroids (PHAs) but also for tracking
variables and finding transients. While carrying out its NASA mission, ATLAS now discovers the greatest number of bright ($m < 19$) supernovae candidates of any ground based survey, frequently
detecting very young explosions due to its 2 day cadence. ATLAS discovered the afterglow of a gamma-ray burst independent of the high energy trigger and has released a variable star catalogue of 5$\
times10^{6}$ sources. This, the first of a series of articles describing ATLAS, is devoted to the design and performance of the ATLAS system. Subsequent articles will describe in more detail the
software, the survey strategy, ATLAS-derived NEA population statistics, transient detections, and the first data release of variable stars and transient lightcurves.
Recent observations of extreme variability in Active Galactic Nuclei have pushed standard viscous accretion disc models over an edge. "Extreme reprocessing" where an erratically variable central
quasi-point source is entirely responsible for heating an otherwise cold and passive low-viscosity disc, may be the best route forward.
Current time domain facilities are finding several hundreds of transient astronomical events a year. The discovery rate is expected to increase in the future as soon as new surveys such as the Zwicky
Transient Facility (ZTF) and the Large Synoptic Sky Survey (LSST) come on line. At the present time, the rate at which transients are classified is approximately one order or magnitude lower than the
discovery rate, leading to an increasing "follow-up drought". Existing telescopes with moderate aperture can help address this deficit when equipped with spectrographs optimized for spectral
classification. Here, we provide an overview of the design, operations and first results of the Spectral Energy Distribution Machine (SEDM), operating on the Palomar 60-inch telescope (P60). The
instrument is optimized for classification and high observing efficiency. It combines a low-resolution (R$\sim$100) integral field unit (IFU) spectrograph with "Rainbow Camera" (RC), a multi-band
field acquisition camera which also serves as multi-band (ugri) photometer. The SEDM was commissioned during the operation of the intermediate Palomar Transient Factory (iPTF) and has already proved
lived up to its promise. The success of the SEDM demonstrates the value of spectrographs optimized to spectral classification. Introduction of similar spectrographs on existing telescopes will help
alleviate the follow-up drought and thereby accelerate the rate of discoveries.
[Abridged] We present observations of PS16dtm, a luminous transient that occurred at the nucleus of a known Narrow-line Seyfert 1 galaxy hosting a 10$^6$ M$_\odot$ black hole. The transient was
previously claimed to be a Type IIn SLSN due to its luminosity and hydrogen emission lines. The light curve shows that PS16dtm brightened by about two magnitudes in ~50 days relative to the archival
host brightness and then exhibited a plateau phase for about 100 days followed by the onset of fading in the UV. During the plateau PS16dtm showed no color evolution, maintained a blackbody
temperature of 1.7 x 10$^4$ K, and radiated at approximately $L_{Edd}$ of the SMBH. The spectra exhibit multi-component hydrogen emission lines and strong FeII emission, show little evolution with
time, and closely resemble the spectra of NLS1s while being distinct from those of Type IIn SNe. Moreover, PS16dtm is undetected in the X-rays to a limit an order of magnitude below an archival X-ray
detection of its host galaxy. These observations strongly link PS16dtm to activity associated with the SMBH and are difficult to reconcile with a SN origin or any known form of AGN variability, and
therefore we argue that it is a TDE in which the accretion of the stellar debris powers the rise in the continuum and excitation of the pre-existing broad line region, while providing material that
obscures the X-ray emitting region of the pre-existing AGN accretion disk. A detailed TDE model fit to the light curve indicates that PS16dtm will remain bright for several years; we further predict
that the X-ray emission will reappear on a similar timescale as the accretion rate declines. Finally, we place PS16dtm in the context of other TDEs and find that TDEs in AGN galaxies are an order of
magnitude more efficient and reach Eddington luminosities, likely due to interaction of the stellar debris with the pre-existing accretion disk.
We present ground-based and \textit{Swift} observations of iPTF16fnl, a likely tidal disruption event (TDE) discovered by the intermediate Palomar Transient Factory (iPTF) survey at 66.6 Mpc. The
lightcurve of the object peaked at absolute $M_g=-17.2$ mag. The maximum bolometric luminosity (from optical and UV) was $L_p~\simeq~(1.0\,\pm\,0.15) \times 10^{43}$erg/s, an order of magnitude
fainter than any other optical TDE discovered so far. The luminosity in the first 60 days is consistent with an exponential decay, with $L \propto e^{-(t-t_0)/\tau}$, where $t_0$=~57631.0 (MJD) and $
\tau\simeq 15$ days. The X-ray shows a marginal detection at $L_X=2.4^{1.9}_{-1.1}\times 10^{39}$ erg/s (\textit{Swift} X-ray Telescope). No radio counterpart was detected down to 3$\sigma$,
providing upper limits for monochromatic radio luminosity of $L=3.8\times10^{26}$ erg/s and $L=7.6\times 10^{26}$erg/s (VLA, 6.1 and 22 GHz). The blackbody temperature, obtained from combined \textit
{Swift} UV and optical photometry, shows a constant value of 19,000 K. The transient spectrum at peak is characterized by broad HeII and H$\alpha$ emission lines, with an FWHM of about 14,000 km/s
and 10,000 km/s respectively. HeI lines are also detected at $\lambda\lambda$ 3188, 4026 and 6678. The spectrum of the host is dominated by strong Balmer absorption lines, which are consistent with a
post-starburst (E+A) galaxy with an age of $\sim$650 Myr and solar metallicity. The characteristics of iPTF16fnl make it an outlier on both luminosity and decay timescales, as compared to other
optically selected TDEs. The discovery of such a faint optical event suggests a higher rate of tidal disruptions, as low luminosity events may have gone unnoticed in previous searches.
Tidal disruption events (TDEs), in which stars are gravitationally disrupted as they pass close to the supermassive black holes in the centres of galaxies, are potentially important probes of strong
gravity and accretion physics. Most TDEs have been discovered in large-area monitoring surveys of many 1000s of galaxies, and the rate deduced for such events is relatively low: one event every 10$^
4$ - 10$^5$ years per galaxy. However, given the selection effects inherent in such surveys, considerable uncertainties remain about the conditions that favour TDEs. Here we report the detection of
unusually strong and broad helium emission lines following a luminous optical flare (Mv < -20.1 mag) in the nucleus of the nearby ultra-luminous infrared galaxy F01004-2237. The particular
combination of variability and post-flare emission line spectrum observed in F01004-2237 is unlike any known supernova or active galactic nucleus. Therefore, the most plausible explanation for this
phenomenon is a TDE -- the first detected in a galaxy with an ongoing massive starburst. The fact that this event has been detected in repeat spectroscopic observations of a sample of 15
ultra-luminous infrared galaxies over a period of just 10 years suggests that the rate of TDEs is much higher in such objects than in the general galaxy population.
We present a radio-quiet quasar at z=0.237 discovered "turning on" by the intermediate Palomar Transient Factory (iPTF). The transient, iPTF 16bco, was detected by iPTF in the nucleus of a galaxy
with an archival SDSS spectrum with weak narrow-line emission characteristic of a low-ionization emission line region (LINER). Our follow-up spectra show the dramatic appearance of broad Balmer lines
and a power-law continuum characteristic of a luminous (L_bol~10^45 erg/s) type 1 quasar 12 years later. Our photometric monitoring with PTF from 2009-2012, and serendipitous X-ray observations from
the XMM-Newton Slew Survey in 2011 and 2015, constrain the change of state to have occurred less than 500 days before the iPTF detection. An enhanced broad Halpha to [OIII]5007 line ratio in the type
1 state relative to other changing-look quasars also is suggestive of the most rapid change of state yet observed in a quasar. We argue that the >10 increase in Eddington ratio inferred from the
brightening in UV and X-ray continuum flux is more likely due to an intrinsic change in the accretion rate of a pre-existing accretion disk, than an external mechanism such as variable obscuration,
microlensing, or the tidal disruption of a star. However, further monitoring will be helpful in better constraining the mechanism driving this change of state. The rapid "turn on" of the quasar is
much shorter than the viscous infall timescale of an accretion disk, and requires a disk instability that can develop around a ~10^8 M_sun black hole on timescales less than a year.
We describe the near real-time transient-source discovery engine for the intermediate Palomar Transient Factory (iPTF), currently in operations at the Infrared Processing and Analysis Center (IPAC),
Caltech. We coin this system the IPAC/iPTF Discovery Engine (or IDE). We review the algorithms used for PSF-matching, image subtraction, detection, photometry, and machine-learned (ML) vetting of
extracted transient candidates. We also review the performance of our ML classifier. For a limiting signal-to-noise ratio of 4 in relatively unconfused regions, "bogus" candidates from processing
artifacts and imperfect image subtractions outnumber real transients by ~ 10:1. This can be considerably higher for image data with inaccurate astrometric and/or PSF-matching solutions. Despite this
occasionally high contamination rate, the ML classifier is able to identify real transients with an efficiency (or completeness) of ~ 97% for a maximum tolerable false-positive rate of 1% when
classifying raw candidates. All subtraction-image metrics, source features, ML probability-based real-bogus scores, contextual metadata from other surveys, and possible associations with known Solar
System objects are stored in a relational database for retrieval by the various science working groups. We review our efforts in mitigating false-positives and our experience in optimizing the
overall system in response to the multitude of science projects underway with iPTF.
The uncertain origin of the recently-discovered `changing-looking' quasar phenomenon -- in which a luminous quasar dims significantly to a quiescent state in repeat spectroscopy over ~10 year
timescales -- may present unexpected challenges to our understanding of quasar accretion. To better understand this phenomenon, we take a first step to building a sample of changing-look quasars with
a systematic but simple archival search for these objects in the Sloan Digital Sky Survey Data Release 12. By leveraging the >10 year baselines for objects with repeat spectroscopy, we uncover two
new changing-look quasars, and a third discovered previously. Decomposition of the multi-epoch spectra and analysis of the broad emission lines suggest that the quasar accretion disk emission dims
due to rapidly decreasing accretion rates (by factors of >2.5), while disfavoring changes in intrinsic dust extinction for the two objects where these analyses are possible. Broad emission line
energetics also support intrinsic dimming of quasar emission as the origin for this phenomenon rather than transient tidal disruption events or supernovae. Although our search criteria included
quasars at all redshifts and transitions from either quasar-like to galaxy-like states or the reverse, all of the clear cases of changing-look quasars discovered were at relatively low-redshift (z ~
0.2 - 0.3) and only exhibit quasar-like to galaxy-like transitions.
Stars that pass within the Roche radius of a supermassive black hole will be tidally disrupted, yielding a sudden injection of gas close to the black hole horizon which produces an electromagnetic
flare. A few dozen of these flares have been discovered in recent years, but current observations provide poor constraints on the bolometric luminosity and total accreted mass of these events. Using
images from the Wide-field Infrared Survey Explorer (WISE), we have discovered transient 3.4 micron emission from several previously known tidal disruption flares. The observations can be explained
by dust heated to its sublimation temperature due to the intense radiation of the tidal flare. From the break in the infrared light curve we infer that this hot dust is located ~0.1 pc from the
supermassive black hole. Since the dust has been heated by absorbing UV and (potentially) soft X-ray photons of the flare, the reprocessing light curve yields an estimate of the bolometric flare
luminosity. For the flare PTF-09ge, we infer that the most likely value of the luminosity integrated over frequencies at which dust can absorb photons is $8\times 10^{44}$ erg/s, with a factor of 3
uncertainty due to the unknown temperature of the dust. This bolometric luminosity is a factor ~10 larger than the observed black body luminosity. Our work is the first to probe dust in the nuclei of
non-active galaxies on sub-parsec scales. The observed infrared luminosity implies a covering factor ~1% for the nuclear dust in the host galaxies.
Observations of galaxy isophotes, longs-slit kinematics and high-resolution photometry suggested a possible dichotomy between two distinct classes of E galaxies. But these methods are expensive for
large galaxy samples. Instead, integral-field spectroscopic can efficiently recognize the shape, dynamics and stellar population of complete samples of early-type galaxies (ETGs). These studies
showed that the two main classes, the fast and slow rotators, can be separated using stellar kinematics. We showed there is a dichotomy in the dynamics of the two classes. The slow rotators are
weakly triaxial and dominate above $M_{\rm crit}\approx2\times10^{11} M_\odot$. Below $M_{\rm crit}$, the structure of fast rotators parallels that of spiral galaxies. There is a smooth sequence
along which, the metals content, the enhancement in $\alpha$-elements, and the "weight" of the stellar initial mass function, all increase with the CENTRAL mass density slope, or bulge mass fraction,
while the molecular gas fraction correspondingly decreases. The properties of ETGs on galaxy scaling relations, and in particular the $(M_{\ast}, R_{\rm e})$ diagram, and their dependence on
environment, indicate two main independent channels for galaxy evolution. Fast rotators ETGs start as star forming disks and evolve trough a channel dominated by gas accretion, bulge growth and
quenching. While slow rotators assemble near the center of massive halos via intense star formation at high redshift, and remain as such for the rest of their evolution via a channel dominated by gas
poor mergers. This is consistent with independent studies of the galaxies redshift evolution.
Several studies indicate that radio-loud (RL) Active Galactic Nuclei (AGN) are produced only by the most massive black holes (BH), MBH ∼ 108-1010M⊙. This idea has been challenged by the discovery of
RL Narrow Line Seyfert 1 (RL NLSy1), having estimated masses of MBH ∼ 106-107 M⊙. However, these low MBH estimates might be due to projection effects. Spectropolarimetry allows us to test this
possibility by looking at RL NLSy1s under a different perspective, i.e., from the viewing angle of the scattering material. We here report the results of a pilot study of VLT spectropolarimetric
observations of the RL NLSy1 PKS 2004-447. Its polarization properties are remarkably well reproduced by models in which the scattering occurs in an equatorial structure surrounding its broad line
region, seen close to face-on. In particular, we detect a polarized Hα line with a width of ∼ 9,000 km s−1, ∼6 times broader than the width seen in direct light. This corresponds to a revised
estimate of MBH ∼ 6 × 108 M⊙, well within the typical range of RL AGN.
We present a Hubble Space Telescope STIS spectrum of ASASSN-14li, the first rest-frame UV spectrum of a tidal disruption flare (TDF). The underlying continuum is well fit by a blackbody with $T_{\
mathrm{UV}} = 3.5 \times 10^{4}$ K, an order of magnitude smaller than the temperature inferred from X-ray spectra (and significantly more precise than previous efforts based on optical and near-UV
photometry). Super-imposed on this blue continuum, we detect three classes of features: narrow absorption from the Milky Way (probably a High-Velocity Cloud), and narrow absorption and broad (FWHM $\
approx 2000$-8000 km s$^{-1}$) emission lines at/near the systemic host velocity. The absorption lines are blueshifted with respect to the emission lines by $\Delta v = -(250$-400) km s$^{-1}$.
Together with the lack of common low-ionization features (Mg II, Fe II), we argue these arise from the same absorbing material responsible for the low-velocity outflow discovered at X-ray
wavelengths. The broad nuclear emission lines display a remarkable abundance pattern: N III], N IV], He II are quite prominent, while the common quasar emission lines of C III] and Mg II are weak or
entirely absent. Detailed modeling of this spectrum will help elucidate fundamental questions regarding the nature of the emission process(es) at work in TDFs, while future UV spectroscopy of
ASASSN-14li would help to confirm (or refute) the previously proposed connection between TDFs and "N-rich" quasars.
Extreme coronal-line emitter (ECLE) SDSSJ095209.56+214313.3, known by its strong, fading, high ionization lines, has been a long standing candidate for a tidal disruption event, however a supernova
origin has not yet been ruled out. Here we add several new pieces of information to the puzzle of the nature of the transient that powered its variable coronal lines: 1) an optical light curve from
the Lincoln Near Earth Asteroid Research (LINEAR) survey that serendipitously catches the optical flare, and 2) late-time observations of the host galaxy with the Swift Ultraviolet and Optical
Telescope (UVOT) and X-ray telescope (XRT) and the ground-based Mercator telescope. The well-sampled, $\sim10$-year long, unfiltered LINEAR light curve constrains the onset of the flare to a
precision of $\pm5$ days and enables us to place a lower limit on the peak optical magnitude. Difference imaging allows us to estimate the location of the flare in proximity of the host galaxy core.
Comparison of the \textsl{GALEX} data (early 2006) with the recently acquired Swift UVOT (June 2015) and Mercator observations (April 2015) demonstrate a decrease in the UV flux over a $\sim 10$ year
period, confirming that the flare was UV-bright. The long-lived UV-bright emission, detected 1.8 rest-frame years after the start of the flare, strongly disfavors a SN origin. These new data allow us
to conclude that the flare was indeed powered by the tidal disruption of a star by a supermassive black hole and that TDEs are in fact capable of powering the enigmatic class of ECLEs.
Tidal forces close to massive black holes can violently disrupt stars that make a close approach. These extreme events are discovered via bright X-ray and optical/UV flares in galactic centers. Prior
studies based on modeling decaying flux trends have been able to estimate broad properties, such as the mass accretion rate. Here we report the detection of flows of highly ionized X-ray gas in
high-resolution X-ray spectra of a nearby tidal disruption event. Variability within the absorption-dominated spectra indicates that the gas is relatively close to the black hole. Narrow line widths
indicate that the gas does not stretch over a large range of radii, giving a low volume filling factor. Modest outflow speeds of a few hundred kilometers per second are observed, significantly below
the escape speed from the radius set by variability. The gas flow is consistent with a rotating wind from the inner, super-Eddington region of a nascent accretion disk, or with a filament of
disrupted stellar gas near to the apocenter of an elliptical orbit. Flows of this sort are predicted by fundamental analytical theory and more recent numerical simulations.
We report the discovery of a new "changing-look" quasar, SDSS J101152.98+544206.4, through repeat spectroscopy from the Time Domain Spectroscopic Survey. This is an addition to a small but growing
set of quasars whose blue continua and broad optical emission lines have been observed to decline by a large factor on a time scale of approximately a decade. The 5100 Angstrom monochromatic
continuum luminosity of this quasar drops by a factor of > 9.8 in a rest-frame time interval of < 9.7 years, while the broad H-alpha luminosity drops by a factor of 55 in the same amount of time. The
width of the broad H-alpha line increases in the dim state such that the black hole mass derived from the appropriate single-epoch scaling relation agrees between the two epochs within a factor of 3.
The fluxes of the narrow emission lines do not appear to change between epochs. The light curve obtained by the Catalina Sky Survey suggests that the transition occurs within a rest-frame time
interval of approximately 500 days. We examine three possible mechanisms for this transition suggested in the recent literature. An abrupt change in the reddening towards the central engine is
disfavored by the substantial difference between the timescale to obscure the central engine and the observed timescale of the transition. A decaying tidal disruption flare is consistent with the
decay rate of the light curve but not with the prolonged bright state preceding the decay, nor can this scenario provide the power required by the luminosities of the emission lines. An abrupt drop
in the accretion rate onto the supermassive black hole appears to be the most plausible explanation for the rapid dimming.
Tidal disruption events occur when a star passes too close to a massive black hole and it is totally ripped apart by tidal forces. Alternatively, if the star does not get close enough to the black
hole to be totally disrupted, a less dramatic event might happen with the star surviving the encounter and loosing only a small fraction of its mass. In this situation if the stellar orbit is bound
and highly eccentric, just like some stars in the centre of our own Galaxy, repeated flares should occur. When the star approaches the black hole tidal radius at periastron, matter might be stripped
resulting in lower intensity outbursts recurring once every orbital period. We report on Swift observations of a recent bright flare from the galaxy IC 3599 hosting a middle-weight black hole, where
a possible tidal disruption event was observed in the early 1990s. By light curve modelling and spectral fitting we can consistently account for the events as the non-disruptive tidal stripping of a
star into a highly eccentric orbit. The recurrence time is 9.5 yr. IC 3599 is also known to host a low-luminosity active galactic nucleus. Tidal stripping from this star over several orbital passages
might be able to spoon-feed also this activity. | {"url":"https://www.researchgate.net/publication/332669276_A_New_Class_of_Changing-Look_LINERs","timestamp":"2024-11-08T04:07:13Z","content_type":"text/html","content_length":"741125","record_id":"<urn:uuid:7d0aef7d-53ac-4ced-be28-3b3ac10a8731>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00178.warc.gz"} |
Università di Pisa - Valutazione della didattica e iscrizione agli esami
The course is divided in three main parts, described in the following.
Part 1 – Thermal-Hydraulic teaching units (theoretical aspects):
Continuum hypothesis and definition of the fluid particle; Lagrangian and Eulerian description of motion; the physical meaning of viscosity for gas and liquids; mass, momentum and energy
conservation, Introduction of turbulence; macroscopic effects of turbulence; characteristic scales of turbulence and the energy cascade; theoretical approach for turbulent flow: the kolmogorov's
theory, Modelling approaches for turbulent flows: DNS, LES, RANS;Effects of turbulence on the mean flow; Reynolds decomposition; time average or mean of flow property; properties of the average;
Reynolds-averaged Navier Stokes Equation, RANS: eddy viscosity models; The Boussinesq Hypothesis; Reynold analogy; The mixing lenght model; Turbulent flow near solid regions; One equation model
(Prandtl Model); Two equation models: k-e standard, K-e RNG, k_w standard, k-w SST; RANS: direct models; RSM Model.
STH codes general overview; Differences between TH-SYS codes and CFD codes; RELAP5 TH-SYS code and its documentation; RELAP5 treatment of noncondensable gases; RELAP5 treatment of Boron transport;
State relationship and constitutive models: flow regime maps and heat transfer models, RELAP5 staggered spatial mesh; Time discretization; Numerical solution scheme: Semi-implicit and
nearly-implicit;Implicit Vs Explicit time differencing; properties of numerical scheme; hydrodinamic components; volume orientation; RELAP5 input structure: cards and words; variable trip;
Hydrodynamic components; Time dependent volume cards description, RELAP5 execution; General errors and errors detections; Heat structures; mesh points; heat structure examples with heat structure
thermal properties and general table data,
Part 1 – Thermal-Hydraulic teaching units (Practical aspects):
DesingModeler: sketches and planes concept; Operation with sketches: modify, Dimension, Constraints; 3D feature creations (extrude, revolve, pattern); Boolean operations; slice operation; active and
frozen bodies; 2-D feature creations; Single and multy-body parts
Ansys meshing: meshing methods for 3D geometry; mappable faces; Meshing method for 2-D geometries; Selective mesh
Ansys Fluent: mesh independence analysis; Post Processing tools; Fluent User Defined Functions: UDF structure, interpreting or compiling an UDF; UDF examples: Inlet parabolic velocity profile, Inlet
unsteady velocity profile, Heat flux profile at wall; Coupling Matlab & Fluent: procedure instruction and example;
RELAP5: thermal-hydraulic components and heat structure components.
Part 2: Neutronic teaching units (Theoretical aspects):
Neutron transport numerical codes: deterministic versus stochastic codes. Advantages and disadvantages of both deterministic and stochastic codes. Simplified spherical harmonics method: derivation of
the SP3 equations. The input files of the OpenMC stochastic code: generation of the geometry.xml, materials.xml, settings.xml and plots.xml files for a sample problem. Plotting the geometry as a
check of the input files.
Part 2: Neutronic teaching units (Practical aspects):
Implementation of a heterogeneous multi fuel assembly calculation (2D and 3D version) with OpenMC and the deterministic code BERM-SP3.
Part 3: Finite Elements units (Theoretical aspects):
FEM Theory lessons will cover the following topics: Study of discrete systems, starting from the structural matrix calculation to the definition and implementation of matrix of stiffness,
constraints, applied loads, and boundary conditions.
The Finite Element Method: Introduction and Mathematical formulation of the finite element method. Discretization of continuum, elements, shape function with reference to the main types of elements
for 1D, 2D, 3D problems: rods, beams, plate/flat and shell, axisymmetric elements, and solid elements.
How to implement Linear and nonlinear analysis: pre-processing (model definition, definition of the elements for the discretization, materials behavior (equation of state), methods and issues related
to the discretization, boundary conditions: loads, constraints and user subroutine), analysis and post- processing phases (visualization, interpretation and analysis of the main results)
Part 3: Finite Elements units (Practical aspects)
Computer Lab lessons will address the implementation of the finite element method. In particular:
- data structure and algorithm for a planar region
- discretization, interpolation and numerical integration algorithm for 2D (axisymmetric, simply planar, and shell) and 3D model
- static and dynamic analysis of a complex shape tank: steady state, modal and transient analysis (with material behaviour elastic and/or elastic-plastic). | {"url":"https://esami.unipi.it/esami2/programma.php?pg=ects&c=54542","timestamp":"2024-11-07T00:56:15Z","content_type":"text/html","content_length":"30869","record_id":"<urn:uuid:b16cb10d-e4eb-486c-b81d-52b35d9d4c2a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00502.warc.gz"} |
How to use deep learning for anomaly detection?
Deep learning can be used for anomaly detection by training a neural network to identify patterns in data and then using it to detect deviations from those patterns. Here are the general steps for
using deep learning for anomaly detection:
1. Collect and preprocess data: Gather data on the system or process you want to monitor and preprocess it so that it's in a format that can be fed into a neural network.
2. Train a neural network: Use the preprocessed data to train a neural network, such as an autoencoder or a recurrent neural network, to learn the patterns in the data.
3. Define a threshold: After training the neural network, define a threshold above which data points will be considered anomalies. This can be done by calculating a statistical measure of the
difference between the predicted and actual data.
4. Test the neural network: Use the neural network to classify new data points as normal or anomalous. If a data point falls below the threshold, it is considered an anomaly.
5. Evaluate and refine the model: Measure the performance of the model and refine it as needed. You may need to adjust the threshold or retrain the neural network with new data to improve its
Some specific techniques for using deep learning for anomaly detection include:
• Autoencoders: These neural networks can be used to learn a compressed representation of the normal data and then detect deviations from that representation.
• Recurrent neural networks (RNNs): These can be used to detect anomalies in time series data by learning patterns and predicting future values.
• Variational autoencoders (VAEs): These can be used to generate new data that is similar to the normal data and then detect deviations from that generated data.
Overall, deep learning can be a powerful tool for detecting anomalies in complex data sets. However, it requires careful data preprocessing, model training, and evaluation to ensure accurate results. | {"url":"https://devhubby.com/thread/how-to-use-deep-learning-for-anomaly-detection","timestamp":"2024-11-11T23:21:06Z","content_type":"text/html","content_length":"103390","record_id":"<urn:uuid:2db8875d-2622-4129-a92a-fb9cafa74f8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00870.warc.gz"} |
Approximation Methods for Supervised Learning
• Let be an unknown Borel measure defined on the space Z := X Y with X and Y = [-M,M]. Given a set z of m samples zi =(xi,yi) drawn according to , the problem of estimating a regression function f
using these samples is considered. The main focus is to understand what is the rate of approximation, measured either in expectation or probability, that can be obtained under a given prior f ,
i.e., under the assumption that f is in the set , and what are possible algorithms for obtaining optimal or semioptimal (up to logarithms) results. The optimal rate of decay in terms of m is
established for many priors given either in terms of smoothness of f or its rate of approximation measured in one of several ways. This optimal rate is determined by two types of results. Upper
bounds are established using various tools in approximation such as entropy, widths, and linear and nonlinear approximation. Lower bounds are proved using Kullback-Leibler information together
with Fano inequalities and a certain type of entropy. A distinction is drawn between algorithms which employ knowledge of the prior in the construction of the estimator and those that do not.
Algorithms of the second type which are universally optimal for a certain range of priors are given. 2005 SFoCM. | {"url":"https://vivo.library.tamu.edu/vivo/display/n221393SE","timestamp":"2024-11-13T22:43:47Z","content_type":"text/html","content_length":"23188","record_id":"<urn:uuid:f97c870e-c578-4d1f-a6e2-788843a77798>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00185.warc.gz"} |
Calibration of buried NaI(Tl) scintillator detectors for natural radionuclide measurement based on Monte Carlo modelling
Measurements of naturally occurring concentrations of ^40K, and of the decay series of ^238U and ^232Th, are of interest in the earth sciences in general, and in particular, scintillator-based gamma
spectrometers can be used for the low cost determination of burial dose rates in natural geological samples. We are currently developing a robust, portable, wireless detector specifically intended
for field measurement of natural radionuclide concentrations and hence, the calculation of dose rates. One of the challenges in developing and applying such an instrument is reliable calibration.
Most calibrations of field instruments depend on access to non-finite matrices of known K, U, Th activity concentrations, in either a 4π or 2π geometry; these are only available at a few facilities
around the world. Here we investigate an alternative approach, based on the measurement of small samples containing well-known activity concentrations of only K or U or Th, and Monte-Carlo radiation
transport modelling to convert the observed spectra into those expected from specific activity concentrations in a non-finite 4π geometry. We first validate our modelling procedure by simulating
these observed spectra. The non-finite matrix calibration spectra are then predicted, and least-squares fitted to the spectrum observed at the centre of a 1 m^3 of granite chips; the resulting
predicted U, Th and K activity concentrations are compared with independently known values.
• Calibration
• Field measurement
• MCNP
• NaI(Tl) detector
Dive into the research topics of 'Calibration of buried NaI(Tl) scintillator detectors for natural radionuclide measurement based on Monte Carlo modelling'. Together they form a unique fingerprint. | {"url":"https://pure.au.dk/portal/en/publications/calibration-of-buried-naitl-scintillator-detectors-for-natural-ra","timestamp":"2024-11-06T15:28:18Z","content_type":"text/html","content_length":"58817","record_id":"<urn:uuid:a44f2808-e8ff-4a4b-9d28-5f5bc8ca2543>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00591.warc.gz"} |
Unlocking the Equilibrium Equation: Calculating Grams of HI
How can we calculate the number of grams of HI at equilibrium?
Given the equation H[2] + I[2] ⇌ 2HI with K[c] = 5.02x10^-4 at 448°C, how do we determine the grams of HI in equilibrium with 1.25 mol of H[2] and 63.5 g of iodine?
Calculating the Equilibrium Concentration of HI
To calculate the number of grams of HI at equilibrium, we need to follow a few steps. Firstly, we convert the moles of H[2] and iodine to concentrations using the ideal gas law. Next, utilizing
stoichiometry, we determine the equilibrium concentration of HI. Finally, we convert the concentration to grams using the molar mass of HI.
When tackling equilibrium calculations like this, it's essential to approach the problem systematically. Begin by converting the given moles of H[2] and the mass of iodine to concentrations in mol/L
using the ideal gas law. This step allows us to establish the initial concentrations of the reactants.
Once we have the initial concentrations of H[2] and iodine, we can proceed to determine the equilibrium concentration of HI. By applying stoichiometry based on the balanced equation, we can find the
molar ratios between H[2], I[2], and HI at equilibrium.
After finding the equilibrium concentration of HI, the final step involves converting this concentration to grams. This conversion is achieved by utilizing the molar mass of HI, which allows us to
relate the concentration in mol/L to grams.
By following these steps methodically, you can successfully calculate the grams of HI at equilibrium in a chemical reaction. It's a process that combines the principles of stoichiometry, equilibrium
constants, and molar conversions to determine the final result accurately. | {"url":"https://www.ledgemulepress.com/chemistry/unlocking-the-equilibrium-equation-calculating-grams-of-hi.html","timestamp":"2024-11-06T16:53:21Z","content_type":"text/html","content_length":"22764","record_id":"<urn:uuid:b5e2ed31-a451-4bd9-8a67-e8f044cda14e>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00475.warc.gz"} |
ohlab - cfigs
Color Figures of "Improved method for spectral reflectance estimation and application to mobile phone cameras"
Fig. 1. Schematic diagram of the observation model for the imaging system using an RGB camera.
Fig. 2. Example of the luminance distribution captured by a mobile phone camera under an LED light source.
Fig. 3. Estimation errors of the surface-spectral reflectance as a function of the g and a parameters, where an X-rite Color Checker, an iPhone 6s, and seven LED light sources were used (refer to
Section 5 for the details). Points A and B represent the estimation errors when using the L1 and L2 minimum parameter values, respectively. Point C represents the minimum error by the entire search
of g and a.
Fig. 4. Relative RGB spectral sensitivity functions of the three mobile phone cameras: (1) Apple iPhone 6s, (2) Apple iPhone 8, and (3) Huawei P10 lite, where the red, green, and blue curves
correspond to (1), (2), and (3) cameras, respectively.
Fig. 5. Spectral power distributions of seven LED light sources used in experiments.
Fig. 6. Data set of spectral reflectances used to obtain the statistical quantities of the surface-spectral reflectance x.
Fig. 7. Color checkers used for reflectance estimation validation. (a) Imaging targets comprising 24 color-checkers and the white reference standard (Spectralon). (b) Spectral reflectances of the 24
color-checkers and the white reference standard measured by the spectral colorimeter.
Fig. 8. Estimation results of the spectral reflectances for the 24 color-checkers when applying the LMMSE, Wiener, and PCA methods to the image data using the iPhone 6s. The parameters used were g
_L1 and a_L1 for the LMMSE and Wiener methods and were only g _L1 for the PCA method. In the figures, the broken curves in bold red, bold green curves, and thin blue curves depict the spectral
reflectances estimated by the LMMSE,Wiener, and PCAmethods, respectively; the black dotted curves depict themeasured spectral reflectances.
Fig. 9. Estimation results of the spectral reflectances for the 24 color-checkers when applying the LMMSE, Wiener, and PCA methods to the image data using the iPhone 8 camera. The parameters used
were g _L1 and a_L1 for the LMMSE and Wiener methods and were only g _L1 for the PCA method. In the figures, the broken curves in bold red, bold green curves, and thin blue curves depict the spectral
reflectances estimated by the LMMSE,Wiener, and PCA methods, respectively; the dotted black curves indicate the measured spectral reflectances.
Fig. 10. Estimation results of the spectral reflectances for the 24 color-checkers when applying the LMMSE, Wiener, and PCA methods to the image data using the Huawei P10 lite camera. The parameters
used were g _L1 and aL1 for the LMMSE and Wiener methods and only g L1 for the PCA method. In the figures, the broken curves in bold red, bold green curves, and thin blue curves depict the spectral
reflectances estimated by the LMMSE,Wiener, and PCA methods, respectively; the dotted black curves indicate the measured spectral reflectances.
Fig. 11. Estimation results of the spectral reflectances for the 24 color-checkers when the observations were normalized with the reference standard sample (Spectralon) in using the iPhone 6s camera.
The only parameter used was a_L1 for the LMMSE andWiener methods. The PCA methods used no parameters. In the figures, the broken curves in bold red, bold green curves, thin blue curves, and dotted
black curves correspond to the spectral reflectances estimated by the LMMSE,Wiener, and PCAmethods and themeasured spectral reflectances, respectively.
Fig. 12. Estimation results of the spectral reflectances for the 24 color-checkers when the observations were normalized using the reference standard sample (Spectralon) when applying the iPhone 8
camera. The only parameter used was a_L1 for the LMMSE andWiener methods. The PCA methods applied no parameters. In the figures, the broken curves in bold red, bold green curves, thin blue curves,
and dotted black curves correspond to the spectral reflectances estimated by the LMMSE,Wiener, and PCA methods, and the measured spectral reflectances, respectively.
Fig. 13. Estimation results of the spectral reflectances for the 24 color-checkers when the observations were normalized using the reference standard sample (Spectralon) when applying the Huawei P10
lite camera. The only parameter used was a_L1 for the LMMSE andWiener methods. The PCA methods used no parameters. In the figures, the broken curves in bold red, bold green curves, thin blue curves,
and dotted black curves correspond to the spectral reflectances estimated by the LMMSE,Wiener, and PCA methods and the measured spectral reflectances, respectively.
Fig. 14. Illuminant spectral power distribution of the incandescent lamp used in an experiment on the single RGB-based spectral estimation.
Fig. 15. Variations in the estimation error and the percent variance as a function of the number of principal components. The error values are computed under three different parameter conditions for
each of the three mobile phone cameras shown in Tables 1 and 2. The percent variance and the error values are plotted using the left and right scales, respectively. | {"url":"https://ohlab.kic.ac.jp/index/cfigs","timestamp":"2024-11-02T18:40:00Z","content_type":"text/html","content_length":"101956","record_id":"<urn:uuid:fe3582d2-a52a-413e-9548-361c5d23cba3>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00261.warc.gz"} |
Advanced Variance Crosstab Columns
This topic provides information on the use of Advanced Variance Crosstab Columns when building or updating a Crosstab Definition.
File variance calculation
Provides different variance analyses of current data file vs other files. This means that the source data does not have to be pulled through into other columns unless needed elsewhere.
Volume variance
Calculates the variance in volume using the formula: (File A units – File B units) * File B price.
Price variance
Calculates the variance in price using the formula: File A units * (File A price - File B price).
New product variance
Calculates the correct variance in new products since they began trading. For any month within one year of the item’s Launch Date, the volume and price variances are set to zero and the total
variance (excluding exchange) for the month is assigned to the special variance category New product.
Exchange variance
Calculates the variance caused by movements in exchange rates using the formula: File A value @ File A rates - File A value @ File B rates. This will require you to have a currency set that is
different from your Default Currency, otherwise there will be no exchange variance.
For other Column Types, see Crosstabulation Column Types.
0 comments
Please sign in to leave a comment. | {"url":"https://support.ajbsystems.net/hc/en-us/articles/209933867-Advanced-Variance-Crosstab-Columns","timestamp":"2024-11-08T10:33:04Z","content_type":"text/html","content_length":"22019","record_id":"<urn:uuid:148427ab-2635-492b-930f-f63bd13647de>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00007.warc.gz"} |
A high magnetic Reynolds number dynamo
A boundary-layer solution to a high magnetic Reynolds number R periodic dynamo model shows that: (1) flux expulsion forces the magnetic field into flux sheets; (2) the principal contribution to the α
effect arises from regions of flow stagnation along a flux sheet; and (3) the α effect scales as R^-^1^/^2. Arguments for these effects persisting in turbulent dynamos are given.
Physics of Fluids
Pub Date:
April 1987
□ Boundary Layer Equations;
□ Dynamo Theory;
□ High Reynolds Number;
□ Solar Cycles;
□ Boundary Value Problems;
□ Computational Fluid Dynamics;
□ Magnetic Fields;
□ Velocity Distribution;
□ Plasma Physics | {"url":"https://ui.adsabs.harvard.edu/abs/1987PhFl...30.1079P/abstract","timestamp":"2024-11-05T23:03:24Z","content_type":"text/html","content_length":"35541","record_id":"<urn:uuid:5b1d9be4-4c30-4bdc-b8b6-cffaad0364fa>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00378.warc.gz"} |
Comp. 4102: Assignment #3
1) The goal of the first questions is to implement some code that performs calibration using the
method described in the book; by first computing a projection matrix, and then decomposing
that matrix to find the extrinsic and intrinsic camera parameters. Use the approach described
in the slides. I have given you a program, written in C++ that uses OpenCV, called
projection-template.cpp. This program takes ten given 3d points and projects them into a 2d
image using the given supplied camera calibration matrix, rotation matrix and translation
vector. Your goal is to write the two routines that are missing, which are
computeprojectionmatrix and decomposeprojectionmatrix. The first routine
computes the projection matrix using the method described in Section 6.3.1 of the book, and
the second uses the method in Section 6.3.2 to decompose the projection matrix into a camera
calibration matrix, rotation matrix and translation vector. It should be the case that the
computed camera matrix, rotation matrix and translation vector are the same (or very similar)
to the original versions that were used to create the projected points. This shows that your
two routines are working properly. You hand in your program source and the resulting output
file assign3-out created by running this modified program.
5 marks
2) The goal of this question is to create a program that take as input two images that are related
by a rotation homography; a left (keble_a_half), middle (keble_b_long) and creates a single
panoramic image (same size as keble_b_long) as output. This is done by warping the left
“into” the middle image. I have made the middle image big enough to hold both the warped
left and the original middle image. I have given you a program called
akaze-match-template.cpp which takes these two images and computes a set of features that
you can use to compute the homography between them. To actually compute the homography
you use the routine findhomography(, , RANSAC) and then you use warperspective routine
with the computed homography to warp the left image into an image of the same size as the
middle image. In other words you warp img1 into img3, and after that you paste (essentially
an OR operation) img3 into img2. You should output two images; warped, which is the
warped version of img1, and merged which is the warped version of img1 (img3) combined
with img2. I have included two images called warped and merged which show you how they
should look like. Notice that the final merged image has some anomalies because of the OR
operation. In real mosaicking programs you do not see these anomalies. Write down a short
(one paragraph) description of how you would get rid of these visible anomalies. The answer
is simple.
5 marks | {"url":"https://codingprolab.com/answer/comp-4102-assignment-3/","timestamp":"2024-11-14T01:04:35Z","content_type":"text/html","content_length":"106771","record_id":"<urn:uuid:88123db1-bbdb-43e7-995e-6deb8038179c>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00497.warc.gz"} |
[Solved] A store order bottles of shampoo througho | SolutionInn
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A store order bottles of shampoo throughout the year. Over time, the store has learned that the annual demand D for shampoo is constant, i.e.,
A store order bottles of shampoo throughout the year. Over time, the store has learned that the annual demand D for shampoo is constant, i.e., there is no variability. Currently, the store decides to
use the optimal EOQ value Qopt every time they order a new shipment of shampoo bottles from their supplier. Assume that the annual total cost TC(Qopt) incurred by the store is $5,000.
How much does the store pay each year in holding costs (i.e., what is the annual holding cost)?
If the store ordered 2Q[opt] bottles every order as opposed to Q[opt] bottles, then what would be the annual total cost incurred?
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Recommended Textbook for
Authors: Jack R. Meredith, Samuel J. Mantel,
7th Edition
470226218, 978-0470226216
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Modify the original equation table — new_equations
Modify the original equation table
This function modifies the original equation table to be used in other functions of the package including: subset the original equation table, add new equations, and choose whether to include
equations with a height allometry.
subset_taxa = "all",
subset_climate = "all",
subset_region = "all",
subset_ids = "all",
subset_output = c("Total aboveground biomass", "Whole tree (above stump)"),
new_taxa = NULL,
new_allometry = NULL,
new_coords = NULL,
new_min_dbh = NULL,
new_max_dbh = NULL,
new_sample_size = NULL,
new_unit_dbh = "cm",
new_unit_output = "kg",
new_input_var = "DBH",
new_output_var = "Total aboveground biomass",
use_height_allom = TRUE
character vector with taxa to be kept. Default is "all", in which case all taxa are kept.
character vector with Koppen climate classification to be kept. Default is "all", in which case all climates are kept.
character vector with name of location(s) or country(ies) or broader region(s) (eg. "Europe", "North America") to be kept. Default is "all", in which case all regions/countries are kept.
character vector with equation IDs to be kept. Default is "all", in which case all equations are kept.
What dependent variable(s) should be provided in the output? Default is "Total aboveground biomass" and "Whole tree (above stump)", other possible values are: "Bark biomass", "Branches (dead)",
"Branches (live)", "Branches total (live, dead)", "Foliage total", "Height", "Leaves", "Stem (wood only)", "Stem biomass", "Stem biomass (with bark)", "Stem biomass (without bark)", "Whole tree
(above and belowground)". Be aware that currently only a few equations represent those other variables, so estimated values might not be very accurate.
character string or vector specifying the taxon (or taxa) for which the allometry has been calibrated.
a character string with the allometric equation.
a vector or matrix of coordinates (longitude, latitude) of the calibration data.
numerical value, minimum DBH for which the equation is valid (in cm). Default is NULL (nothing is added).
numerical value, maximum DBH for which the equation is valid (in cm). Default is NULL (nothing is added).
number of measurements with which the allometry was calibrated. Default is NULL (nothing is added).
character string with unit of DBH in the equation (either cm, mm or inch). Default is "cm".
character string with unit of equation output (either "g", "kg", "Mg" or "lbs" if the output is a mass, or "m" if the output is a height).
independent variable(s) needed in the allometry. Default is "DBH", other option is "DBH, H".
dependent variable estimated by the allometry. Default is "Total aboveground biomass".
a logical value. In allodb we use Bohn et al. (2014) for European sites. User need to provide height allometry when needed to calculate AGB. Default is TRUE.
new_taxa = "Faga",
new_allometry = "exp(-2+log(dbh)*2.5)",
new_coords = c(-0.07, 46.11),
new_min_dbh = 5,
new_max_dbh = 50,
new_sample_size = 50
#> # A tibble: 478 × 15
#> equation_id equation_taxa equation_allometry independent_variable
#> <chr> <chr> <chr> <chr>
#> 1 726f1d Larix laricina 10^(2.648+0.715*(lo… DBH
#> 2 a4d879 Acer saccharum 10^(1.2315+1.6376*(… DBH
#> 3 b9ebe4 Alnus rubra exp(5.13118+2.15046… DBH
#> 4 5e2dea Viburnum lantanoides 29.615*((1.488+1.19… DBH
#> 5 21800b Pinus strobus exp(5.2831+2.0369*l… DBH
#> 6 1257b1 Abies exp(3.1689+2.6825*l… DBH
#> 7 9e2124 Populus davidiana 10^(1.826+2.558*(lo… DBH
#> 8 74d0ce Ostrya virginiana exp(4.89+2.3*log(db… DBH
#> 9 94f593 Liriodendron tulipifera 10^(0.8306+1.527*(l… DBH
#> 10 8c94e8 Nyssa sylvatica 10^(1.1468+1.4806*(… DBH
#> # ℹ 468 more rows
#> # ℹ 11 more variables: dependent_variable <chr>, long <chr>, lat <chr>,
#> # koppen <chr>, dbh_min_cm <dbl>, dbh_max_cm <dbl>, sample_size <dbl>,
#> # dbh_units_original <chr>, dbh_unit_cf <dbl>, output_units_original <chr>,
#> # output_units_cf <dbl> | {"url":"https://docs.ropensci.org/allodb/reference/new_equations.html","timestamp":"2024-11-12T04:07:07Z","content_type":"text/html","content_length":"21116","record_id":"<urn:uuid:67554fd0-a9a8-4ec7-93aa-f0689acbdf86>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00743.warc.gz"} |
Unusual Containers
In this lesson, we are going to make two unusual containers, one with a hexagonal bottom, and one with a square bottom. (JAVA Animations: hexagonal bottom, square bottom) You will need one piece of
poster board (11 inches by 14 inches) -- it is enough for two containers; scissors, compass, ruler, pencil, and glue or a stapler. Paper clips help to hold the containers together while the glue is
Supplies: posterboard (11 inches by 14 inches), scissors, compasses, rulers, pencils, and glue and paperclips or a stapler
Hexagonal Bottom Container [top]
1. Choose a radius. (7.5 cm is a good choice, but others are also fine.)
2. Draw a circle with your chosen radius.
3. Make a Star of David inside the circle (two overlapping equilateral triangles).
4. From the circle's center, draw six straight lines which will be cut with scissors. The lines begin at the vertices of the inner hexagon, and extend out beyond the circle.
5. Set your compass to the length from the center of the circle to a vertex of the inner hexagon. (Check this length with other measurements.) Place your compass point on a vertex of the inner
hexagon, and swing an arc that that intersects the straight line that you are planning to cut (the line you are planning to cut is shown in black in the figure). Do this for all six vertices.
6. Draw an outer hexagon (each edge goes through three points on your drawing: the two points on the black cut lines that you made with your arcs, and the point tangent to the circle).
7. If you want, add rounded edges on the six sides of the outer hexagon.
8. Score along all edges that will be folded and cut. (See diagram which shows you which edges are folded and which edges are cut.)
9. Cut out your drawing and cut along the labelled lines. Follow the steps below to construct your unusual container. Note that this example is a unusual container with curved edges. (Click on the
image to cycle through the steps.)
10. Glue or staple your container together. If you use glue, paper clip the glued sides together until the glue dries. If you make rounded edges, they can be folded for a nice effect.
Square Bottom Container [top]
1. Choose a radius and draw a circle on your remaining scrap of poster board. (Make the circle as big as you can!)
2. Draw a diameter. Use an index card to make a perpendicular to the diameter through the circle's center, so you have four ticks equally spaced along the edge of the circle.
3. Connect the tick marks to form a square.
4. Find the midpoints of the sides of the square (using a compass or a ruler).
5. Connect these midpoints, drawing a second square inside the first one.
6. Make ticks at the midpoints of the edges of the second square, and make a third inner square by connecting them.
7. Draw four lines that you will cut, along the diagonal of the biggest square (see diagram).
8. If you want, add rounded edges to the four sides of the large square. Here are the designs for a square bottom container with straight edges and a square bottom container with curved edges.
9. Score the lines that will be creased.
10. Cut out your drawing and glue it together. Follow the steps shown below to construct your unusual container. Note that the edges of this container are rounded. (Click on the image to cycle
through the steps.)
Webpage Maintained by Owen Ramsey
Lesson Index | {"url":"https://breakingawayfromthemathbook.com/Lessons/unusualcontainers1/unusualcontainers.html","timestamp":"2024-11-08T15:18:17Z","content_type":"text/html","content_length":"8620","record_id":"<urn:uuid:8d59eceb-aea8-4b66-87bd-6e29f5720a04>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00014.warc.gz"} |
Finite-automaton transformations of strictly almost-periodic sequences
Different versions of the notion of almost-periodicity are natural generalizations of the notion of periodicity. The notion of strict almost-periodicity appeared in symbolic dynamics, but later
proved to be fruitful in mathematical logic and the theory of algorithms as well. In the paper, a class of essentially almost-periodic sequences (i.e., strictly almost-periodic sequences with an
arbitrary prefix added at the beginning) is considered. It is proved that the property of essential almost-periodicity is preserved under finite-automaton transformations, as well as under the action
of finite transducers. The class of essentially almost-periodic sequences is contained in the class of almost-periodic sequences. It is proved that this inclusion is strict.
All Science Journal Classification (ASJC) codes
• Finite automaton
• Finite transducer
• Strictly almost-periodic sequence
Dive into the research topics of 'Finite-automaton transformations of strictly almost-periodic sequences'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/finite-automaton-transformations-of-strictly-almost-periodic-sequ","timestamp":"2024-11-10T02:54:01Z","content_type":"text/html","content_length":"49154","record_id":"<urn:uuid:5ade1a4d-3507-41d3-914a-5eef0062b0c3>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00437.warc.gz"} |
CSCI 567 Homework #4 solved
High Level Description
0.1 Tasks In this assignment you are asked to implement K-means clustering to identify main clusters in
the data, use the discovered centroid of cluster for classification, and implement Gaussian Mixture Models
to learn a generative model of the data. Specifically, you will
• Implement K-means clustering algorithm to identify clusters in a two-dimensional toy-dataset.
• Implement image compression using K-means clustering algorithm.
• Implement classification using the centroids identified by clustering.
• Implement Gaussian Mixture Models to learn a generative model and generate samples from a mixture distribution.
0.2 Running the code We have provided two scripts to run the code. Run kmeans.sh after you finish implementation of k-means clustering, classification and compression. Run gmm.sh after you finish
implementing Gaussian Mixture Models.
0.3 Dataset Through out the assignment we will use two datasets (See fig. 1) — Toy Dataset and Digits
Dataset (you do not need to download).
Toy Dataset is a two-dimensional dataset generated from 4 Gaussian distributions. We will use this
dataset to visualize the results of our algorithm in two dimensions.
We will use digits dataset from sklearn [1] to test K-means based classifier and generate digits using
Gaussian Mixture model. Each data point is a 8 × 8 image of a digit. This is similar to MNIST but less
(a) Digits (b) 2-D Toy Dataset
Figure 1: Datasets
0.4 Cautions Please DO NOT import packages that are not listed in the provided code. Follow the instructions in each section strictly to code up your solutions. Do not change the output format. Do
modify the code unless we instruct you to do so. A homework solution that does not match the provided
setup, such as format, name, initializations, etc., will not be graded. It is your responsibility to make sure
that your code runs with the provided commands and scripts on the VM. Finally, make sure that you git
add, commit, and push all the required files, including your code and generated output files.
0.5 Final submission After you have solved problem 1 and 2, execute bash kmeans.sh command and
bash gmm.sh command. Git add, commit and push plots and results folder and all the *.py files.
Problem 1 K-means Clustering
Recall that for a dataset x1, . . . , xN ∈ RD, the K-means distortion objective is
J({µk}, {rik}) = 1
rikkµk − xik
where µ1, . . . , µK are centroids of the K clusters and rik ∈ {0, 1} represents whether example i belongs to
cluster k.
Clearly, fixing the centroids and minimizing J over the assignment give
rˆik =
1 k = arg mink
0 kµk
0 − xik
0 Otherwise.
On the other hand, fixing the assignment and minimizing J over the centroids give
µˆ k =
What the K-means algorithm does is simply to alternate between these two steps. See Algorithm 1 for
the pseudocode.
Algorithm 1 K-means clustering algorithm
1: Inputs:
An array of size N × D denoting the training set, x
Maximum number of iterations, max iter
Number of clusters, K
Error tolerance, e
2: Outputs:
Array of size K × D of means, {µk}
Membership vector R of size N, where R[i] ∈ [K] is the index of the cluster that
example i belongs to.
3: Initialize:
Set means {µk}
to be K points selected from x uniformly at random (with
replacement), and J to be a large number (e.g. 1010)
4: repeat
5: Compute membership rik using eq. 2
6: Compute distortion measure Jnew using eq. 1
7: if |J − Jnew| ≤ e then
8: end if
9: Set J := Jnew
10: Compute means µk using eq. 3
11: until maximum iteration is reached
1.1 Implementing K-means clustering algorithm Implement Algorithm 1 by filling out the TODO parts
in class KMeans of file kmeans.py. Note the following:
• Use numpy.random.choice for the initialization step.
• If at some iteration, there exists a cluster k with no points assigned to it, then do not update the
centroid of this cluster for this round.
• While assigning a sample to a cluster, if there’s a tie (i.e. the sample is equidistant from two centroids),
you should choose the one with smaller index (like what numpy.argmin does).
After you complete the implementation, execute bash kmeans.sh command to run k-means on toy
dataset. You should be able to see three images generated in plots folder. In particular, you can see
toy dataset predicted labels.png and toy dataset real labels.png and compare the clusters identified by the algorithm against the real clusters. Your implementation should be able to recover the
correct clusters sufficiently well. Representative images are shown in fig. 2. Red dots are cluster centroids.
Note that color coding of recovered clusters may not match that of correct clusters. This is due to mis-match
in ordering of retrieved clusters and correct clusters (which is fine).
(a) Predicted Clusters (b) Real Clusters
Figure 2: Clustering on toy dataset
1.2 Image compression with K-means In the next part, we will look at lossy image compression as an
application of clustering. The idea is simply to treat each pixel of an image as a point xi
, then perform
K-means algorithm to cluster these points, and finally replace each pixel with its centroid.
What you need to implement is to compress an image with K centroids given. Specifically, complete the
function transform image in the file kmeansTest.py.
After your implementation, execute bash kmeans.sh again and you should be able to see an image
baboon compressed.png in the plots folder. You can see that this image is distorted as compared to
the original baboon.tiff.
1.3 Classification with k-means Another application of clustering is to obtain a faster version of the nearest neighbor algorithm. Recall that nearest neighbor evaluates the distance of a test sample
from every
training point to predict its class, which can be very slow. Instead, we can compress the entire training
set to just the K centroids, where each centroid is now labeled as the majority class of the corresponding
cluster. After this compression the prediction time of nearest neighbor is reduced from O(N) to just O(K)
(see Algorithm 2 for the pseudocode).
Algorithm 2 Classification with K-means clustering
1: Inputs:
Training Data : {X,Y}
Parameters for running K-means clustering
2: Training:
Run K-means clustering to find centroids and membership (reuse your code from Problem 1.1)
Label each centroid with majority voting from its members. i.e. arg maxc ∑i
rikI{yi = c}
3: Prediction:
Predict the same label as the nearest centroid (that is, 1-NN on centroids).
Note: 1) break ties in the same way as in previous problems; 2) if some centroid doesn’t contain any
point, set the label of this centroid as 0.
Complete the fit and predict function in KMeansClassifier in file kmeans.py. Once completed,
run kmeans.sh to evaluate the classifier on a test set. For comparison, the script will also print accuracy of
a logistic classifier and a nearest neighbor classifier. (Note: a naive K-means classifier may not do well but
it can be an effective unsupervised method in a classification pipeline [2].)
Problem 2 Gaussian Mixture Model
Next you will implement Gaussian Mixture Model (GMM) for clustering and also generate data after learning the model. Recall the key steps of EM algorithm for learning GMMs on Slide 52 of Lec 8 (we
change the
notation ωk
to πk
γik =
πkN (xi
; µk
, Σk
∑k πkN (xi
; µk
, Σk
Nk =
γik (5)
µk =
i=1 γikxi
Σk =
i=1 γik(xi − µk
)(xi − µk
πk =
Algorithm 3 provides a more detailed pseudocode. Also recall the incomplete log-likelihood is
ln p(xn) =
πkN (xi
; µk
, Σk
) =
(xi − µk
(xi − µk
2.1 Implementing EM Implement EM algorithm (class Gaussian pdf, function fit and function com
pute log likelihood) in file gmm.py to estimate mixture model parameters. Please note the following:
• For K-means initialization, the inputs of K-means are the same as those of EM’s.
• When computing the density of a Gaussian with covariance matrix Σ, use Σ
0 = Σ + 10−3
I when Σ is
not invertible (in case it’s still not invertible, keep adding 10−3
I until it is invertible).
After implementation execute bash gmm.sh command to estimate mixture parameters for toy dataset.
You should see a Gaussian fitted to each cluster in the data. A representative image is shown in fig. 3.
We evaluate both initialization methods and you should observe that initialization with K-means usually
converges faster.
Figure 3: Gaussian Mixture model on toy dataset
Algorithm 3 EM algorithm for estimating GMM parameters
1: Inputs:
An array of size N × D denoting the training set, x
Maximum number of iterations, max iter
Number of clusters, K
Error tolerance, e
Init method — K-means or random
2: Outputs:
Array of size K × D of means, {µk}
Variance matrix Σk of size K × D × D
A vector of size K denoting the mixture weights, pi k
3: Initialize:
• For “random” init method: initialize means uniformly at random from [0,1)
for each dimension (use numpy.random.rand), initialize variance to be
identity matrix for each component, initialize mixture weight to be uniform.
• For “K-means” init method: run K-means, initialize means as the centroids
found by K-means, and initialize variance and mixture weight according to
Eq. (7) and Eq. (8) where γik is the binary membership found by K-means.
4: Compute the log-likelihood l using Eq. (9)
5: repeat
6: E Step: Compute responsibilities using Eq. (4)
7: M Step:
Estimate means using Eq. (6)
Estimate variance using Eq. (7)
Estimate mixture weight using Eq. (8)
8: Compute new log-likelihood lnew
9: if |l − lnew| ≤ e then
10: end if
11: Set l := lnew
12: until maximum iteration is reached
2.2 Implementing sampling We also fit a GMM with K = 30 using the digits dataset. An advantage of
GMM compared to K-means is that we can sample from the learned distribution to generate new synthetic
examples which look similar to the actual data.
To do this, implement sample function in gmm.py which uses self.means, self.variances and
self.pi k to generate digits. Recall that sampling from a GMM is a two step process: 1) first sample a
component k according to the mixture weight; 2) then sample from a Gaussian distribution with mean µk
and variance Σk
. Use numpy.random for these sampling steps.
After implementation, execute bash gmm.sh again. This should produce visualization of means µk
and some generated samples for the learned GMM. Representative images are shown in fig. 4.
(a) Means of GMM learnt on digits (b) Random digits sample generated from GMM
Figure 4: Results on digits dataset
[1] sklearn.datasets.digits http://scikit-learn.org/stable/modules/generated/sklearn.
[2] Coates, A., & Ng, A. Y. (2012). Learning feature representations with k-means. In Neural networks:
Tricks of the trade (pp. 561-580). Springer, Berlin, Heidelberg. | {"url":"https://codeshive.com/questions-and-answers/csci-567-homework-4-solved/","timestamp":"2024-11-04T02:55:12Z","content_type":"text/html","content_length":"119192","record_id":"<urn:uuid:a5ae2124-fdc0-4c38-bbe3-4392690724cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00440.warc.gz"} |
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