content stringlengths 86 994k | meta stringlengths 288 619 |
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Math Games
Students entering the 7th grade continue to expand on concepts from their previous years. On-demand videos with teachers who explain the concepts and show students how to understand the
problem-solving process. Teachers go over rules, tips, and multiple problems helping students to be able to solve the problems themselves.
• Students learn about ratios, mixed properties, statistics and other seventh grade skills.
• Teachers incorporate the use of the scratchpad to give students a visual representation.
• Videos provide instant help for students who are struggling with their assignments. | {"url":"https://sol.mathgames.com/worksheets/grade6","timestamp":"2024-11-12T00:40:08Z","content_type":"text/html","content_length":"485216","record_id":"<urn:uuid:52ba949b-eaec-4669-91f0-c1ff615570f5>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00644.warc.gz"} |
Percentage Calculator
Percentage calculator
Percentage Calculator
Table of Contents:
Part: The portion you want to find the percentage of.
Whole: The whole or total amount.
New Value and Old Value: Values for calculating percentage change.
Percentage: The percentage to apply for discount/markup or calculate a portion of a number.
Number: The total number for calculating a percentage of it.
Measured Value and Actual Value: Values for calculating percentage error.
Rate: The rate for compound interest calculation.
Time: The time for compound interest calculation.
What Is Percentage Calculator
Would you like an easy way to figure out the percentage of a value without doing the math yourself? If yes, you can use our percentage calculator—it’s a simple tool designed to make calculating
percentages easy.
Our online percentage calculator is user-friendly, so whether you’re using a smartphone or a desktop, you can calculate percentages without any trouble. Best of all, it’s free! You won’t be charged
anything, no matter how often you use it.
Feel free to start using our online percentage finder today for all your percentage calculations!
How To Use
Understanding the Calculator Inputs:
• The calculator has various input fields, each labeled for a specific purpose. For example, “Part,” “Whole,” “New Value,” “Old Value,” “Percentage,” “Number,” “Measured Value,” “Actual Value,”
“Rate,” and “Time.”
• You can input values into these fields based on the type of percentage calculation you want to perform.
Performing Different Calculations:
• The calculator supports different types of percentage calculations, and there are buttons at the bottom for each type. The available calculations include Basic Percentage, Percentage Change,
Discount/Markup, Percentage of a Number, Percentage Error, and Compound Interest.
Using the Calculator Buttons:
• Click on the appropriate button based on the type of calculation you want to perform. For example, if you want to calculate the Basic Percentage, click on the “Basic Percentage” button.
Viewing the Result:
• The result of the calculation will be displayed below the buttons. It will show the calculated percentage result with two decimal places.
Resetting the Form:
• If you want to reset the calculator and clear all input values and results, you can click on the “Reset” button.
Tips for Inputs:
• Below the result section, there are tips explaining the purpose of each input field. It provides guidance on what each input represents for different types of calculations.
For a Basic Percentage calculation, you can enter the “Part” and “Whole” values and click on the “Basic Percentage” button to see the result.
Basic Percentage:
Suppose you want to find what percentage a certain value (part) is of another value (whole).
Part: 25
Whole: 100
Click on “Basic Percentage” button to get the result.
Percentage Change:
If you have an old value and a new value, and you want to find the percentage change.
Old Value: 50
New Value: 75
Click on “Percentage Change” button to get the result.
Calculate the final price after a certain percentage discount or markup.
Original Price: 80
Percentage: 20 (for a 20% markup)
Click on “Discount/Markup” button to get the result.
Percentage of a Number:
Find a certain percentage of a given number.
Percentage: 15
Number: 200
Click on “Percentage of a Number” button to get the result.
Percentage Error:
Calculate the percentage error between a measured value and an actual value.
Measured Value: 80
Actual Value: 100
Click on “Percentage Error” button to get the result.
Compound Interest:
Determine the compound interest based on principal, rate, and time.
Principal: 1000
Rate: 5
Time: 2
Click on “Compound Interest” button to get the result.
Code For Tool Implementation
<!DOCTYPE html>
<html lang="en">
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Percentage Calculator</title>
<link rel="stylesheet" href="styles.css">
<div class="container">
<label for="part">Part:</label>
<input type="text" id="part" placeholder="Enter part">
<label for="whole">Whole:</label>
<input type="text" id="whole" placeholder="Enter whole">
<label for="newVal">New Value:</label>
<input type="text" id="newVal" placeholder="Enter new value">
<label for="oldVal">Old Value:</label>
<input type="text" id="oldVal" placeholder="Enter old value">
<label for="percentage">Percentage:</label>
<input type="text" id="percentage" placeholder="Enter percentage">
<label for="number">Number:</label>
<input type="text" id="number" placeholder="Enter number">
<label for="measuredVal">Measured Value:</label>
<input type="text" id="measuredVal" placeholder="Enter measured value">
<label for="actualVal">Actual Value:</label>
<input type="text" id="actualVal" placeholder="Enter actual value">
<label for="rate">Rate:</label>
<input type="text" id="rate" placeholder="Enter rate">
<label for="time">Time:</label>
<input type="text" id="time" placeholder="Enter time">
<button onclick="calculateBasicPercentage()">Basic Percentage</button>
<button onclick="calculatePercentageChange()">Percentage Change</button>
<button onclick="calculateDiscountMarkup()">Discount/Markup</button>
<button onclick="calculatePercentageOfNumber()">Percentage of a Number</button>
<button onclick="calculatePercentageError()">Percentage Error</button>
<button onclick="calculateCompoundInterest()">Compound Interest</button>
<button class="reset" onclick="resetForm()">Reset</button>
<div id="result"></div>
<script src="script.js"></script>
This HTML code defines the structure of a web page for a Percentage Calculator. Let me break down the different sections:
1. <!DOCTYPE html>: This is the document type declaration, specifying the version of HTML (in this case, HTML5) that the document is using.
2. <html lang="en">: This is the root element of the HTML document, and it sets the language of the document to English.
3. <head>: This section contains meta-information about the HTML document, such as character set, viewport settings, title, and external stylesheet link.
☆ <meta charset="UTF-8">: Specifies the character encoding for the document as UTF-8.
☆ <meta name="viewport" content="width=device-width, initial-scale=1.0">: Sets the viewport configuration for responsive design.
☆ <title>Percentage Calculator</title>: Sets the title of the web page.
☆ <link rel="stylesheet" href="styles.css">: Links an external stylesheet named “styles.css” to apply styling to the HTML elements.
4. <body>: This is the main content of the HTML document.
☆ The content is organized within a <div> element with the class “container”. Inside this container, there are several sets of labels and input elements for different types of percentage
☆ Following the input fields, there are several <button> elements. Each button has an onclick attribute that triggers a specific JavaScript function when clicked.
☆ The JavaScript functions (e.g., calculateBasicPercentage(), calculatePercentageChange(), etc.) are expected to be defined in an external script file named “script.js”, which is linked at
the end of the document.
☆ There is also a <div> element with the id “result” where the calculated results will be displayed.
5. <script src="script.js"></script>: This script tag links an external JavaScript file named “script.js” to the HTML document. The actual logic for calculating percentages and handling button
clicks is expected to be defined in this script file.
Overall, this HTML code sets up a user interface for a percentage calculator and includes placeholders for user input.
body {
font-family: 'Arial', sans-serif;
background-color: #f5f5f5;
margin: 0;
justify-content: center;
align-items: center;
height: flex-start;
.container {
display: grid;
grid-template-columns: repeat(4, 0.2fr);
background: linear-gradient(to right, #00c6fb, #005bea);
border-radius: 15px;
box-shadow: 0 0 20px rgba(0, 0, 0, 0.2);
padding: 30px;
text-align: center;
max-width: 700px;
width: 100%;
margin: 10px;
/* Form Styles */
form {
display: flex;
flex-direction: column;
gap: 15px;
label {
display: block;
margin-top: 15px;
color: #000000;
font-size: 20px;
input {
width: flex-left;
padding: 9px;
margin-top: 10px;
border: 2px solid #000000;
border-radius: 8px;
outline: none;
text-align: center;
font-size: 8px;
transition: border-color 0.3s, transform 0.2s;
input:hover {
border-color: #27ae60;
transform: translateY(-2px);
button {
padding: 10px;
margin-bottom: 10px;
cursor: pointer;
background-color: #27ae60;
color: #ecf0f1;
font-size: 16px;
border: none;
border-radius: 8px;
transition: background-color 0.3s, transform 0.2s, box-shadow 0.3s;
box-shadow: 0px 6px 8px rgba(0, 0, 0, 0.2);
button:hover {
background-color: #219651;
transform: translateY(-3px);
box-shadow: 0px 8px 10px rgba(0, 0, 0, 0.3);
button:active {
transform: translateY(3px);
button.reset {
background-color: #e74c3c;
margin-left: 20px;
#result {
margin-top: 30px;
font-size: 24px;
color: #000000;
This CSS code provides styling rules for the HTML elements defined in the previous code snippet. Let’s break down the styles for different sections:
1. Body Styles:
☆ font-family: 'Arial', sans-serif;: Sets the font family for the entire document to Arial or a generic sans-serif font.
☆ background-color: #f5f5f5;: Sets the background color of the entire page to a light gray (#f5f5f5).
☆ margin: 0;: Removes any default margin on the body element.
☆ justify-content: center; align-items: center;: Centers the content both horizontally and vertically within the body.
☆ height: flex-start;: This seems incorrect; it should be min-height or height instead of height: flex-start;.
2. Container Styles:
☆ display: grid;: Sets the container to use the CSS Grid layout.
☆ grid-template-columns: repeat(4, 0.2fr);: Defines a grid with four columns of equal width.
☆ background: linear-gradient(to right, #00c6fb, #005bea);: Applies a linear gradient background from light blue to dark blue.
☆ border-radius: 15px;: Rounds the corners of the container.
☆ box-shadow: 0 0 20px rgba(0, 0, 0, 0.2);: Adds a subtle box shadow.
☆ padding: 30px;: Adds padding inside the container.
☆ text-align: center;: Centers the text within the container.
☆ max-width: 700px; width: 100%;: Sets a maximum width for the container and makes it responsive.
☆ margin: 10px;: Adds margin around the container.
3. Form Styles:
☆ display: flex; flex-direction: column; gap: 15px;: Styles the form elements as a flex container with a column layout and some spacing between elements.
4. Label Styles:
☆ display: block; margin-top: 15px; color: #000000; font-size: 20px;: Styles labels as block elements with a margin, color, and font size.
5. Input Styles:
☆ width: flex-left;: This is incorrect; it should be width: 100%; to make the input elements fill the available width.
☆ Various styling for padding, margin, border, border-radius, outline, text alignment, font size, and transitions.
☆ input:hover: Changes the border color and performs a slight upward translation on hover.
6. Button Styles:
☆ display: flex; padding: 10px; margin-bottom: 10px;: Styles buttons as flex containers with padding and margin.
☆ Various styling for cursor, background color, text color, font size, border, border-radius, transitions, and box shadow.
☆ button:hover: Changes the background color, performs a slight upward translation, and adjusts the box shadow on hover.
☆ button:active: Performs a slight downward translation when the button is clicked.
7. Reset Button Styles:
☆ Extends the general button styles and sets a different background color for the reset button.
8. Result Styles:
☆ margin-top: 30px; font-size: 24px; color: #000000;: Styles the result div with margin, font size, and color.
Overall, these styles aim to create a visually appealing and user-friendly interface for the percentage calculator, with a responsive layout and interactive button effects.
function calculateBasicPercentage() {
let part = parseFloat(document.getElementById('part').value);
let whole = parseFloat(document.getElementById('whole').value);
let percentage = (part / whole) * 100;
function calculatePercentageChange() {
let oldVal = parseFloat(document.getElementById('oldVal').value);
let newVal = parseFloat(document.getElementById('newVal').value);
let percentageChange = ((newVal - oldVal) / oldVal) * 100;
function calculateDiscountMarkup() {
let originalPrice = parseFloat(document.getElementById('part').value);
let percentage = parseFloat(document.getElementById('percentage').value);
let finalPrice = originalPrice * (1 + percentage / 100);
function calculatePercentageOfNumber() {
let percentage = parseFloat(document.getElementById('percentage').value);
let number = parseFloat(document.getElementById('number').value);
let result = (percentage / 100) * number;
function calculatePercentageError() {
let measuredVal = parseFloat(document.getElementById('measuredVal').value);
let actualVal = parseFloat(document.getElementById('actualVal').value);
let percentageError = ((measuredVal - actualVal) / actualVal) * 100;
function calculateCompoundInterest() {
let principal = parseFloat(document.getElementById('part').value);
let rate = parseFloat(document.getElementById('rate').value);
let time = parseFloat(document.getElementById('time').value);
let compoundInterest = principal * Math.pow(1 + rate / 100, time) - principal;
function displayResult(result) {
document.getElementById('result').innerHTML = `Result: ${result.toFixed(2)}%`;
function resetForm() {
document.getElementById('result').innerHTML = '';
document.querySelectorAll('input').forEach(input => input.value = '');
This JavaScript code defines a series of functions for a percentage calculator. Let’s break down each function:
1. calculateBasicPercentage:
☆ Retrieves the values of ‘part’ and ‘whole’ from the input fields in the HTML.
☆ Calculates the percentage using the formula: (part / whole) * 100.
☆ Calls the displayResult function with the calculated percentage.
2. calculatePercentageChange:
☆ Retrieves the values of ‘oldVal’ and ‘newVal’ from the input fields in the HTML.
☆ Calculates the percentage change using the formula: ((newVal - oldVal) / oldVal) * 100.
☆ Calls the displayResult function with the calculated percentage change.
3. calculateDiscountMarkup:
☆ Retrieves the values of ‘originalPrice’ and ‘percentage’ from the input fields in the HTML.
☆ Calculates the final price after applying a discount or markup using the formula: originalPrice * (1 + percentage / 100).
☆ Calls the displayResult function with the calculated final price.
4. calculatePercentageOfNumber:
☆ Retrieves the values of ‘percentage’ and ‘number’ from the input fields in the HTML.
☆ Calculates the result, which is a percentage of the given number, using the formula: (percentage / 100) * number.
☆ Calls the displayResult function with the calculated result.
5. calculatePercentageError:
☆ Retrieves the values of ‘measuredVal’ and ‘actualVal’ from the input fields in the HTML.
☆ Calculates the percentage error using the formula: ((measuredVal - actualVal) / actualVal) * 100.
☆ Calls the displayResult function with the calculated percentage error.
6. calculateCompoundInterest:
☆ Retrieves the values of ‘principal’, ‘rate’, and ‘time’ from the input fields in the HTML.
☆ Calculates compound interest using the formula: principal * Math.pow(1 + rate / 100, time) - principal.
☆ Calls the displayResult function with the calculated compound interest.
7. displayResult:
☆ Takes a result value as an argument.
☆ Sets the inner HTML of the ‘result’ div to display the result with formatting: "Result: ${result.toFixed(2)}%".
8. resetForm:
☆ Clears the result display by setting the inner HTML of the ‘result’ div to an empty string.
☆ Resets the values of all input fields by iterating through them and setting their values to an empty string.
These functions work together to perform various percentage calculations based on user input and display the results on the web page. The calculations cover basic percentage, percentage change,
discount/markup, percentage of a number, percentage error, and compound interest. The resetForm function is provided to reset the form and clear the result.
Steps For Implementation On WordPress:
To implement this Percentage calculator on a WordPress site, follow these steps:
1. Create a New Page:
□ Log in to your WordPress admin dashboard.
□ Navigate to “Pages” and click “Add New.”
□ Enter a title for your page, e.g., “Age Calculator.”
2. Switch to HTML Editor:
□ In the WordPress editor, switch to the HTML editor mode.
3. Copy HTML Content:
□ Copy the entire HTML code along with embedded CSS and JavaScript.
4. Paste Code:
□ Paste the copied code into the HTML editor of your WordPress page.
5. Update/Publish:
□ Click the “Update” or “Publish” button to save the changes.
6. View Page:
□ Visit the page on your WordPress site to see the Age Calculator in action.
Note: Depending on your WordPress theme and plugins, you may encounter styling conflicts. Adjustments might be needed to ensure proper integration.
How does our percent calculator operate?
Our percent calculator operates using intelligent algorithms to accurately determine percentages from input values. It simplifies the often tricky process of manually calculating percentages,
considering that not everyone is proficient at solving math equations. The tool is designed to be user-friendly, requiring only the submission of values to swiftly calculate percentages with a single
click. No registration hassles – you can use the percent calculator right away.
Why Use Percentage Calculator?
A percentage calculator proves invaluable in various fields for many common scenarios:
1. Calculate Sales Tax Percentage:
□ Determine the sales tax on purchased items, as it is typically expressed in percentage. Easily find out how much you’ll be paying in taxes for a specific product or service.
2. Income Percentage Calculator:
□ For those with a fixed income who wish to save a specific percentage, the percent calculator provides an easy solution. Quickly find the amount of income you want to save with accurate
3. Percentage Discount:
□ Use the percentage calculator to calculate discounts when shopping for products on sale. Understand the discounted amount and final price with ease.
4. Calculate Body Fat Percentage:
□ Maintain your fitness goals by using our online percentage calculator to determine the proportion of fat in your body relative to its weight.
5. Store Discounts:
□ When encountering store discounts presented in percentages, the percentage calculator helps you understand the final price after applying the discount.
6. Sales Tax:
□ Easily understand the percentage of sales tax you’re paying for the goods or services you enjoy by utilizing the online percentage calculator.
7. Interest Rate:
□ Before applying for a loan, check the interest rate and determine the repayable amount against the loan using the free online percentage calculator.
8. Statistics:
□ Students and statisticians in the field of statistics can benefit from the percentage calculator to enhance their understanding of finding percentages in a straightforward manner.
Q1: What is the definition of percentage?
A: Percentage is the ratio of a number expressed as a fraction of 100, typically denoted with the percent symbol, “%.”
Q2: How can I calculate the percentage between two numbers?
A: You can calculate the percentage between two numbers using a percentage calculator. Online tools are available, allowing you to input the two numbers and obtain the percentage result quickly.
Q3: What calculations can be performed with a percentage converter?
A: A percentage converter enables various calculations, such as determining the ratio of a student’s marks or assessing the amount paid in taxes. It’s a versatile tool for accurate percentage-related
Q4: Can a percentage exceed 100?
A: No, when identifying the percentage of a number, it cannot exceed 100. However, in percentage differences between two values, the result can be greater than 100.
Q5: Can a percentage be negative?
A: No, the percentage of a number itself cannot be negative. Negative percentages may arise when finding the percentage difference between two values, but the negative sign is typically ignored, and
the percentage is expressed in positive integers.
Q6: Can a percentage be represented as a decimal?
A: No, a percentage cannot be a decimal directly because it is obtained by multiplying the decimal value by 100.
Q7: Is it possible to calculate percentage without a percent calculator?
A: Yes, you can calculate percentage using the percentage calculation formula without an online calculator. However, for complex values, independent calculation may require additional assistance.
Q8: How is percentage calculated?
A: Percentage is calculated by taking the ratio of a number to 100 and expressing it as a fraction. The formula is (part/whole) * 100.
Q9: Can you provide an example of calculating percentage without a calculator?
A: Certainly, for simple cases, you can use the formula. For example, if you scored 75 out of 100, the percentage would be (75/100) * 100 = 75%.
Q10: What does the percent symbol, “%,” signify in percentage calculations?
A: The percent symbol signifies that the number is being expressed as a portion of 100, making it easier to understand and compare.
Q11: Why is it important to consider ignoring the negative sign in certain percentage calculations?
A: In certain contexts, like percentage differences, negative values may arise. Ignoring the negative sign ensures that the percentage is expressed in positive integers for clarity.
Q12: Can you provide an instance where a percentage difference might result in a value greater than 100?
A: Yes, in scenarios where comparing two values with a significant difference, the percentage difference might exceed 100, indicating a substantial change between the two values. | {"url":"https://trustofworld.com/percentage-calculator/","timestamp":"2024-11-06T01:44:09Z","content_type":"text/html","content_length":"213089","record_id":"<urn:uuid:c658e3fd-b2ba-4a27-a77e-1ed74d02e0f5>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00259.warc.gz"} |
Structure modal identification based on computer vision technology
Mobile phones have the potential to become useful tool in structural modal identification. In this paper, shaking table test videos of a 10-story steel structure captured by mobile phone is processed
using computer vision theory and then the modal parameters are identified. A signal processing method based on variational mode decomposition (VMD) is used to improve the accuracy of identification.
Using optical flow algorithm, the vibration data is extracted from the video, and then the response of the structure is obtained from the vibration data of selected feature points. Then, the
vibration data is processed by VMD and structural modal parameters (mode frequency and mode shapes) are identified using FFD. Finally, the identification results obtained from mobile phone and
professional sensors are compared to verify feasibility and accuracy of the proposed modal identification method.
• Mobile phones have the potential to become useful tool in structural modal identification.
• Shaking table test videos of a 10-story steel structure captured by mobile phone is processed using computer vision theory and then the modal parameters are identified.
• A signal processing method based on variational mode decomposition (VMD) is used to improve the accuracy of identification.
• Using optical flow algorithm, the vibration data is extracted from the video, and then the response of the structure is obtained from the vibration data of selected feature points.
• The vibration data is processed by VMD and structural modal parameters (mode frequency and mode shapes) are identified using FFD.
• The identification results obtained from mobile phone and professional sensors are compared to verify feasibility and accuracy of the proposed modal identification method.
1. Introduction
Structural modal identification is not only an important means to evaluate structural performance and safety but also a basic method for structural damage detection and health monitoring. Structural
system identification needs to obtain structural response, such as displacement, acceleration, and strain. Traditional wired sensors can capture structural behavior accurately, but the mass and
stiffness of the system may be affected, which will cause error. Because of the high cost and complex arrangements of the measuring points, non-contact structural vibration measurement based on
computer vision methods has gradually become a research focus in recent years. The vibration measurement based on computer vision theory show great potential, and the non-contact sensor for modal
identification is feasible [1]. At present, most of the dynamic detections based on vision uses high speed and high-resolution cameras [2, 3] as testing tools, which can obtain more response signal.
However, such cameras are non-portable and expensive. Smartphones are equipped with image sensors and portable, which has the potential to replace professional cameras. With the development of
imaging system, modal identification based on mobile camera has begun to take shape [4-6]. But related technologies are still under research, and further development in this field is needed. In this
paper, a method using optical flow algorithm and VMD is proposed for structural modal identification. The feasibility and accuracy of the method is verified by adopting it to the shaking table test
of a 10-story steel structure.
2. Methodology
2.1. Location of the camera and reference points
The flowchart of the method in this paper is shown in Fig. 1. The video of shaking table experiment was recorded by a mobile phone which was stabilized by a tripod. At the test point A (shown in Fig.
2) on the 1st, 3rd, 6th, 9th, and 10th floors of the structure, three professional sensors are placed for each floor which are respectively used to record two horizontal and one vertical acceleration
response of the structure and the mobile phone is perpendicular to the ground and the structure (shown in Fig. 2). The width of the bottom plate is measured to establish the reference point between
the image coordinate system and the real coordinate system.
Fig. 1Flowchart of the method
2.2. Vibration data extraction based on computer vision technology
The video of structure vibration is processed by using the optical flow algorithm to extract the velocity response of the structure. Firstly, the feature points for tracking need to be selected in
the vibration video. Therefore, the regions of interest (ROI) are manually selected in the first video frame, which is a circle with a radius of 5 pixels and the center of the circle (shown with the
reference points in Fig. 3) [7]. Then, the optical flow method is used to calculate the velocity of the feature points. The algorithm assumes that the gray levels of feature points in two consecutive
frames remain unchanged, which produces an image pyramid during processing (shown in Fig. 4), where the pyramid of 0th layer (at the bottom) is the original image. Every time the pyramid level goes
up, pixel per inch (PPI) of the image is reduced by half, so that the corresponding vibration displacement of the same pixel line segment in the upper layer can be compressed to 1/2 of its vibration
displacement in the lower layer.
Fig. 2Location of the camera
Fig. 3The structure and feature points
2.3. Modal identification
Before modal identification, the VMD method is used to process the vibration data which is aiming to reduce the noise cause by the optical flow method and ambient vibration. For each vibration data,
VMD is first used to decompose the data into several simple components, then the mode components which have the larger correlation coefficient with the original vibration data are summed up to
construct the new vibration data which are used in the subsequent modal identification. The vibration noise which is commonly has the smaller correlation coefficient with the original vibration data
is therefore reduced during above VMD process.
After the processed vibration data is obtained, the frequency domain (FFD) method is used for modal identification. FFD is based on the transfer function matrix or spectral matrix of the structure.
At a certain frequency, the value of the transfer function matrix or spectral matrix contributes the most by one or several modes, and the modal parameters of the system are determined by singular
value decomposition of the response spectral density function matrix. The procedure of the identification consists two steps, i.e., (1) identification of the natural frequency, and (2) identification
of the mode shapes. By analyzing the power spectrum of test model structure, the natural frequency can be determined according to the peak frequency point of the power spectrum. The auto and cross
power spectrum density of the data can be used to identify the mode shapes of the structure. The singular value decomposition of the power spectral density matrix ${S}_{pk}\left({\omega }_{i}\right)$
can be obtained by [8]:
${S}_{pk}\left({\omega }_{i}\right)={U}_{i}{\mathrm{S}}_{i}{U}_{i}^{H},$
where the matrix ${U}_{i}=\left[{u}_{i1},{u}_{i2},...,{u}_{im}\right]$ is a unitary matrix holding the singular vector ${u}_{ij}$ and ${S}_{i}$ is a diagonal matrix containing $m$ positive real
singular values arranged from largest to smallest. Near a peak corresponding to the $k$th mode in the spectrum this mode or may be a possible close mode will be dominating. If only the $k$th mode is
dominating here, the first singular vector is the estimation of the mode shape:
$\stackrel{^}{\phi }={u}_{i1},$
and the singular value is the power spectral density function of a single degree of freedom system.
In this paper, the average periodogram method is used to estimate the power spectrum density. This method divides the vibration signal data into several segments and calculates the power spectrum of
each segment of data separately and then averages them. The calculation of self-power spectral density function and cross-spectral density function is shown as:
${S}_{pp}\left({\omega }_{i}\right)=\frac{1}{M{N}_{FFT}}{\sum }_{j=1}^{M}{X}_{pj}\left({\omega }_{i}\right){X}_{pj}^{\mathrm{*}}\left({\omega }_{i}\right),$
${S}_{pk}\left({\omega }_{i}\right)=\frac{1}{M{N}_{FFT}}{\sum }_{j=1}^{M}{X}_{pj}\left({\omega }_{i}\right){X}_{kj}^{\mathrm{*}}\left({\omega }_{i}\right),$
where ${X}_{pj}\left({\omega }_{i}\right)$ is the Fourier transform of the $j$th data segment of the random vibration acceleration response at a certain test point; ${X}_{pj}^{*}\left({\omega }_{i}\
right)$ is the conjugate complex number of ${X}_{pj}\left({\omega }_{i}\right)$; ${N}_{FFT}$ is the data length of Fourier transform; $M$ is the average number of times [8].
3. Results and discussions
The shaking table test results of a 10-story steel model structure are taken to investigate the efficiency and accuracy of the proposed method. Test results in two cases (case 1 and case 2) with the
peak ground acceleration (PGA) of ground motion in $X$ direction (shown in Fig. 2) as 0.07 g and 0.19 g respectively are compared and analyzed. For each case, 3-direction (2 horizontal: $X$ and $Y$
directions, 1 vertical: $Z$ direction) ground motions are input at the bottom plate of the structure with PGA ratio as 1:0.85:0.65, and the structure response in $X$ direction are presented.
The time history curves of structure velocity response of the 6th and 10th floors for two cases are shown in Figs. 5 to 8 (the results of the 1st, 3rd and 9th floors are similar; therefore, they are
not presented here). For comparison purpose, the corresponding acceleration response and displacement response are also presented by time derivation or time integral. From visual inspection, the
structure response extracted from smartphone video are in well agreement with those recorded by professional sensors. Taking the data in Case 1 as an example, the peak velocity of the 6th floor and
10th floor extracted by video is 0.064 m/s and 0.101 m/s respectively, while the peak velocity of the 6th floor and 10th floor from professional sensors is 0.069 m/s and 0.108 m/s respectively. The
relative error of data from 6th floor and 10th floor is approximately 7.25 % and 6.48 % respectively.
With the structure response data, the modal parameters are identified using the FDD method mentioned above. In Table 1, the estimate of the natural frequency corresponding to the first two mode are
given. The natural frequency estimated from video and those recorded by professional sensors are close to each other, the discrepancy between them is negligible. However, the natural frequency in
Case 1 is slightly larger than the counterpart in Case 2 which may indicate that slight damage occurs in Case 2, since the ground motion intensity for Case 2 is corresponding to that of a moderate
earthquake according to the design condition of the model structure. Ambient vibration can cause measurement uncertainty. However, the frequency spectrum of ambient vibration changes little with time
during the video recording process, and it can be filtered out by VMD, which shows the characteristics and advantages of VMD.
Fig. 56th floor time history curves (Case 1)
Fig. 610th floor time history curves (Case 1)
Fig. 76th floor time history curves (Case 2)
Fig. 810th floor time history curves (Case 2)
Table 1Identification of natural frequency
1st mode 2nd mode 1st mode 2nd mode
(Case 1 0.07 g) (Case 1 0.07 g) (Case 2 0.19 g) (Case 2 0.19 g)
Computer vision 1.8513 Hz 6.8541 Hz 1.7007 Hz 6.5511 Hz
Professional sensors 1.8501 Hz 6.8403 Hz 1.7009 Hz 6.5516 Hz
The estimate of mode shapes for the first two modes are presented in Figs. 9 and 10. Like the estimate of natural frequency, the mode shapes estimated using the data from videos and professional
sensors are the same to each other. Since the professional sensors were only placed on 5 of the 10 floors of the model structure, the resulting mode shapes are piecewise lines, while the counterpart
using data from videos are smooth lines. The more feature points of interest selected, the more accurate and smooth mode shapes can be obtained, which is the potential merit of the proposed method
base on computer vision.
Fig. 9Mode shapes of the first two modes (Case 1)
Fig. 10Mode shapes of the first two modes (Case 2)
4. Conclusions
In this paper, videoing the shaking table test by mobile phone camera, a modal identification method based on computer vision and VMD is proposed. According to the comparison the analysis results of
the data extracted from smartphone video and the data recorded by professional sensors, the following conclusions are obtained.
1) Vibration data extracted from smartphone video using the computer vision technology can meet the requirements of measurement, which has high accuracy and precision. The structure modal
identification based on computer vision is feasible.
2) Processing structure response using VMD can obtain the effective decomposition components of vibration signal and can reduce the noise cause by the algorithm and ambient vibration.
3) Using the optical flow algorithm, the vibration data at any feature point of interest can be obtained, which is the potential merit of the proposed method. The limitation of the number of
professional sensors, the conventional test procedure is usually faced, therefore can be avoided.
4) Structural damage detection based on computer vision technology has potential to research.
• Han J., Zhang Y., Zhang H. Displacement measurement of shaking table test structure model based on computer vision. Earthquake Engineering and Engineering Vibration, Vol. 39, Issue 4, 2019, p.
22-29, (in Chinese).
• Chen J., Wadhwa N., Cha Y., et al. Structural modal identification through high speed camera video: motion magnification. Conference Proceedings of the Society for Experimental Mechanics Series,
Vol. 7, 2014, p. 191-197.
• Xiong W., Cheng Y. Non-contact identification algorithm of bridge vibration and modal based on high frame video analysis, (in Chinese). Journal of Southeast University (natural science edition),
Vol. 50, Issue 3, 2020, p. 433-439.
• Yoon H., Elanwar H., Choi H., Golparvar-fard M., Spencer B. F. Target‐free approach for vision-based structural system identification using consumer‐grade cameras. Structural Control and Health
Monitoring, Vol. 23, 2016, p. 1405-1416.
• Ozer E., Feng D., Feng M. Q. Hybrid motion sensing and experimental modal analysis using collocated smartphone camera and accelerometers, Measurement Science and Technology, Vol. 28, Issue 10,
2017, p. 105903.
• Chen T., Zhou Z. An improved vision method for robust monitoring of multi-point dynamic displacements with smartphones in an interference environment. Sensors, Vol. 20, Issue 20, 2020, p. 5929.
• Hosseinzadeh A. Z., Tehrani M. H., Harvey Jr P. S. Modal identification of building structures using vision-based measurements from multiple interior surveillance cameras. Engineering Structures,
Vol. 228, 2021, p. 111517.
• Brincker R., Zhang L., Andersen P. Output-only modal analysis by frequency domain decomposition. Proceedings of ISMA 25, Vol. 25, 2000.
About this article
Materials and measurements in engineering
modal identification
variational mode decomposition
optical flow algorithm
computer vision
Copyright © 2021 He Yuanjun, et al.
This is an open access article distributed under the
Creative Commons Attribution License
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Bootcamp Summer 2020 Week 2 — Neural Networks and Backpropagation
Previously, we covered gradient descent, an optimization method that adjusts the values of a set of parameters (e.g., the parameters of a linear regression model) to minimize a loss function (e.g.,
the mean squared error of a linear regression model network in predicting the value of a home given its zip code and other information). In this post, we introduce neural networks, which are
essentially hyperparameter function approximators that learns to map a set of inputs to a set of outputs.
To optimize a neural network model with respect to a given loss function, we could directly apply gradient descent to adapt the parameters of a neural network to learn a desired mapping of inputs to
outputs, this process would be inefficient. Due to the large number of parameters, performing symbolic differentiation as introduced in our gradient descent lesson would require a lot of redundant
computation and slow down the optimization process tremendously. Fortunately, there is a better way: the backpropagation algorithm. Backpropagation is a form of auto-differentiation that allows us to
more efficiently compute the derivatives of the neural network (or other model’s) outputs with respect to each of its parameters. Backpropagation is often overlooked or misunderstood as a simple
application of symbolic differentiation (i.e., the chain rule from Calculus), but it is much, much more.
This blog will cover
1. Neural Networks
2. Forward Propagation
3. Backpropagation
We will not take the time to discuss the biological plausibility of neural networks, but feel free to check out this paper.
Neural Networks
We will start by understanding what a neural network is. At its core, a neural network is a function approximator. Functions from algebra, such as $y=x^2$ and $y=2x+1$, can be represented by neural
networks. Neural networks can be used to learn a mapping given a set of data. For example, a neural network could be used to model the relationship of the age of a house, the number of bedrooms, and
the square footage with the value of a house (displayed in Figure 1).
A sample relationship a neural network can be expected to learn.
While the example in Figure 1 may appear trivial, neural networks can solve much more challenging problems. For example, neural networks have been applied to take as input Magnetic Resonance Imaging
(MRI) data and output diagnoses for the patient based upon what is present in the MRI. Neural networks have also had success in being applied to problems of “machine translation” in which one
attempts to translate one language to another (e.g., English to Chinese).
Typically, neural networks are given a dataset $\mathcal{D}=\{(x_k,y_k),\ k\in [n]\}$, where $(x_k,y_k)$ is the “example” number $k$, $x_k$ is the input to a model we want to train, and $y_k$ is
the “label” we want our model to learn to predict from $x_k$. We generalize the notation to $x$ being the input data, and $y$ being the set of ground-truth labels. A neural network can be represented
as a function $\hat{y} = f(x)$ (i.e., $f: X \rightarrow Y$). Here, we use “$\hat{y}$” as the output of the neural network instead of “$y$,” as $\hat{y}$ is our estimate of the ground-truth label,
$y$. Later, we will use the difference (i.e., error) between $\hat{y}$ and $y$, to optimize our neural network’s parameters.
If we look inside the function $f$, we see nodes and edges. These nodes and edges make up a directed graph, in which an input is operated on from left to right.
A more colorful inside look into a neural network. The output node in this example neural network uses a linear activation, denoted by the diagonal line (“/”).
A closer look inside a smaller neural network is shown in Figure 2. Figure 2 displays how an input, $x$ (with cardinality $|x|=d$), is mapped to an output, $\hat{y}$, a scalar. Before we move into
what is happening here, let’s will first explain the notation.
We have neural network weights, denoted by $w$, that weight the incoming value through multiplication. The subscripts represent the (to, from) nodes respectively, and the superscript denotes the
layer number resulting in the notation as $w^{(layer)}_{to,from}$. The terms with the number “1” inside are termed bias nodes and are represented in the notation “b_{to}”.
As an example, we present a simple neural network describing $\hat{y} = 2x + 4z +3$ in Figure 4.
An example neural network representing $\hat{y}=2x + 4z +3$
When an input is passed into a neural network, the input $x$ is multiplied by respective weights (denoted by $w$) and added to by a bias term $b$, resulting in a new value represented within a hidden
node. In our model, we set the number of nodes in each layer to $n$. Each node, along with its incoming edges, represents some inner function. As our neural network can have many nodes and many
layers, neural networks can be thought of as a composition of these inner functions to produce a more complex, outer function $f$ described above. The backslash in the most-right node represents a
summation, resulting in a scalar value output. Now we will discuss precisely how to produce an output $\hat{y}$ given an input $x$ through learning a mapping.
Forward Propagation
The final output $\hat{y}$ is computed from the node values and weights of the previous layer, in our case, layer 2. We can multiply the output of the hidden nodes in layer 2, denoted by $o^{(2)}_i$,
where $i$ represents a node index from $i$ to $n$ (i.e., $i \in \{1,2,\ldots,n\}$, by the weights connecting these nodes with the output node and add the bias term b^{(3)}_{i}. We display this
process in Equation 1.
\hat{y} = \sum_{i=1}^n w_{1,i}^{(3)} o^{(2)}_i + b^{(3)}_{1}
But, what is $o_i^{(2)}$ equal to? We display how $o_i^{(2)}$ is computed in Equation 2.
o_i^{(2)} = g(z_i^{(2)})
Here, $i$ is again the node index and $g$ is an activation function (i.e., $g: Z \rightarrow O$). Neural networks cannot represent nonlinear functions without utilizing nonlinear activation
functions. Activation functions help neural networks represent more complex functions by transforming node values by a nonlinear function. Common activation functions used include ReLU, tanh, sigmoid
, etc. Here, we will discuss and use the ReLU (Rectified Linear Unit) as our activation function.
For our example, we will use the ReLU activation function, as shown in Equation 3.
g(z) =\bigg\{\begin{aligned} &z \text{ if }z \geq 0\\ &0 \text{ otherwise}\end{aligned}\bigg\}
It is also helpful to know the gradient of the ReLU function, displayed in Equation 4.
g'(z) =\bigg\{\begin{aligned} &1 \text{ if }z \geq 0\\ &0 \text{ otherwise}\end{aligned}\bigg\}
Moving back to Equation 2, we solve for $z_i^{(2)}$ in Equation 5.
z^{(2)}_i = \sum_{j=1}^n w_{i,j}^{(2)} o^{(1)}_j + b^{(2)}_{i}
This procedure can be used throughout the neural network, computing the current layer using the previous layer’s output. For simplicity of algorithmic notation, you can consider the neural network’s
input, $x$, as $o^{(0)}$. Here, $o^{(0)}$ represents the hypothetical output of a hypothetical zeroth layer. In Algorithm 1, we display the complete forward propagation algorithm, expressing the
layer number as a variable for generalizability. This algorithm computes a predicted output, represented by $\hat{y}$, given input features, $x$.
We can now analyze the computational complexity of this algorithm. For the sake of simplifying the analysis, we assume the number of layers in the neural network, $l$, is equal to the number of nodes
in each layer, $n$, which we also set as equal to the size of the input, $|x|=d$. Thus, $n=l=w$. In Algorithm 1, Steps 3 and 5 represent a matrix multiplication combined with addition. This process
has a complexity of $O(n^2+n)$. As a reminder, for any matrix multiplication of two matrices of size $p \times q$ and $q \times r$, respectively, the computational complexity of this multiplication
is $O(pqr)$. Thus, since we are multiplying weight matrices of size $n \times n$ to a vector of size $n \times 1$, the complexity of the matrix multiplication alone is $O(n \times n \times 1) = O(n^
2)$. The addition operation adds complexity $O(n)$, resulting in $O(n^2+n)$ for Steps 2-6. Steps 8-10 have a complexity of $O(n)$ leading, resulting now in the complexity $O(n^2 + 2n)$. Since Step 1
iterates $n=l$ times, we arrive at a total complexity of $O(n(n^2 + 2n))=O(n^3)$ for Algorithm 1.
Now that we can perform a forward pass and know its complexity, we can discuss how to update our weights and attempt to learn a model that fits the data.
To optimize our neural network weights to more accurately represent our data, we first establish a loss function that quantifies our model’s error (i.e., how wrong its outputs are for a desired
task). Given this loss function, we could then compute the gradients of our loss function with respect to each weight. We could then perform gradient descent to optimize our model parameters as we
did in the gradient descent blog post. However, we will now do something a bit more sophisticated to make the optimization process more computationally efficient.
Let’s consider a common loss function, as shown in Equation 6. This loss function represents a mean-squared error loss between the true label (true y-value in the data) and the predicted value using
our neural network ($\hat{y}$).
L=\frac{1}{2} (y-\hat{y})^2 = \frac{1}{2} (y-f(x))^2
The derivative of this loss function with respect to $w_{1,1}^{(1)}$ is displayed in Equation 7 according to chain rule.
\frac{\partial L}{\partial w_{1,1}^{(1)}} = (y-\hat{y}) * \frac{\partial \hat{y}}{\partial w_{1,1}^{(1)}} \bigg|_x
So, how do we comput $\frac{\partial \hat{y}}{\partial w_{1,1}^{(1)}}$? There are 4 paths from the output node to $w_{1,1}^{(1)}$, which highlighted in blue in Figure 4. The upper-most path has the
gradient shown in Equation 8.
\frac{\partial \hat{y}}{\partial w_{1,1}^{(1)}} = \frac{\partial \hat{y}}{\partial o^{(3)}} \frac{\partial o^{(3)}}{\partial z^{(3)}} \frac{\partial z^{(3)}}{\partial o^{(2)}_1} \frac{\partial o^
{(2)}_1}{\partial z^{(2)}_1} \frac{\partial z^{(2)}_1}{\partial o^{(1)}_1} \frac{\partial o^{(1)}_1}{\partial w_{1,1}^{(1)}}
The rule for computing the “total derivative” encompassing all of these paths can be seen through an example here.
A depiction of neural network dependencies on $w_{1,1}^{(1)}$.
Computing a single partial derivative is computationally expensive, and computing each weight’s partial derivative individually (e.g., (e.g., $\frac{\partial \hat{y}}{\partial w_{1,1}^{(1)}}$ and
then $\frac{\partial \hat{y}}{\partial w_{1,2}^{(1)}}$) and so forth all the way to $\frac{\partial \hat{y}}{\partial w_{1,n}^{(l)}}$) would be terribly inneficient.
If we instead look at $\frac{\partial \hat{y}}{\partial w_{1,2}^{(1)}}$, you can see that it shares many of the same nodes as $\frac{\partial \hat{y}}{\partial w_{1,1}^{(1)}}$. In Figure 5, we
display a depiction of gradient computations required for $w_{1,1}^{(1)}$, $w_{2,1}^{(1)}$, and the overlap in the computations.
A depiction of neural network dependencies on $w_{1,1}^{(1)}$ (left), neural network dependencies on $w_{2,1}^{(1)}$ (middle), and the overlap between depedencies.
We would like to take advantage of this overlap in computation, which is exactly where the Backpropagation algorithm comes in. Backpropagation is a form of reverse-mode automatic differentiation that
allows us to eliminate redundant computation in applying the chain rule to compute the gradient of our neural network.
Applied to neural networks, the backpropagation algorithm starts from the last layer of the neural network, computes an error term (denoted as $\delta$), and multiplies this error term by the
gradient. The algorithm computes each node’s respective gradient in memory and reuses these variables as the algorithm continues, backpropagating the information for efficiency. We display the
algorithm for Backpropagation in Algorithm 2. The output of the Backpropagation algorithm is gradients for each set of weights and biases.
Looking at the computational complexity of this algorithm, we see Step 4 has a complexity of $O(1)$. Step 6 has a complexity of $O(n^2 + n)$ or $O(n^2)$. Step 10 has a complexity of $O(n)$. Steps 12
and 14 both have complexities of $O(n^2)$. Since we iterate over the number of layers, equal to $n$ for our analysis, the total complexity is $O(n^3)$. If we did not use the Backpropagation
algorithm, we would end up redundantly computing each component’s derivatives multiple times. Using the previous assumptions setting the cardinality of $|x|$, the number of nodes, and the number of
layers to $n$, we would be iterating over $O(n^3 + n^2)$ weight derivatives resulting in a total complexity of $O(n^6)$.
The last step in optimizing our neural network model to minimize our loss function is to update weights and biases. Equations 9-10 show one step of gradient descent using the derivatives from our
backpropagation procedure in Algorithm 2. Note that Equations 9-10 represent only one step of gradient descent, so one would need to iteratively perform the forward pass, then backward pass, then
parameter updates (Equations 9-10) until the model converges to a local minimum, as described in our previous post on gradient descent.
w^{(l)} \to w^{(l)} + \alpha \Delta w^{(l)}
b^{(l)} \to b^{(l)} + \alpha \Delta b^{(l)}
We now have the ability to train neural networks in a computationally efficient manner, which, in turn, allows us to do more training iterations per unit time, which should help us train more
accurate neural networks!
Key Points throughout this Blog:
1. Neural networks can be used to represent almost any function.
2. Forward propagation is the process of taking input features, $x$, and computing an output in the form of a model’s estimate, $\hat{y}$.
3. We can use the difference in the output, $\hat{y}$, of the neural network and our ground-truth data, $y$, to compute a loss, and then use this loss value to compute gradients in the
backpropagation algorithm.
4. Backpropagation is a special case of reverse-mode automatic differentiation and allows for efficient gradient computation. The result is that the training of extremely large neural networks is | {"url":"https://core-robotics.gatech.edu/2021/01/18/bootcamp-summer-2020-week-2-backpropagation/","timestamp":"2024-11-07T13:53:44Z","content_type":"text/html","content_length":"53141","record_id":"<urn:uuid:14185a9d-c9a5-4233-abc2-a81bcbd4fd23>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00027.warc.gz"} |
Question by S M Fazle Alahe - ACS Doubts
When a circular wheel turns once on a straight road, it covers a distance equal to its circumference, and inside that circular wheel there is another small circle which covers a distance equal to the
circumference of the big circle ... but the circumference of the small circle is small, so it covers a distance equal to the circumference of the big circle. How do you do?? | {"url":"https://acsdoubts.com/question/6668a3ab877e9fd05acb","timestamp":"2024-11-08T04:36:09Z","content_type":"text/html","content_length":"21836","record_id":"<urn:uuid:7e2dddab-1ba4-4234-b4b6-cf29f5ef5c79>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00611.warc.gz"} |
Advanced Simulation of Motorsport Fuel Cell Dynamics - ENGYS
In fluid dynamics, sloshing refers to the movement of fluid inside a solid object, which can also be under motion. Some examples of this application are the fuel cell (tank) of racing cars (like in
Formula 1 cars) [1] and the propellant slosh in spacecraft tanks and rockets [2].
In these applications, the fluid sloshing created by the large g-forces due to acceleration/braking/cornering can produce stability problems in the overall performance of the vehicle. In most of
these applications, every drop of the fuel must be consumed and must be well distributed and controlled inside the tank. Being able to reliably predict the dynamics of fuel motion are an extremely
important design points.
To influence fuel motion during dynamic events, anti-slosh devices such as solid baffles and trap-doors are widely used to limit the slosh effect inside the tanks. In the following applications, we
will use one-way trap-doors (Figure 1) for “baffling”. The doors open based on the signed pressure difference between the two sides and the opening threshold value depends on the attached spring
type, which typically is given by the manufacturer.
Figure 1: Trap-door examples. Pictures are taken from [1][7].
From the modelling perspective, we’ll use the helyxVoF solver, which is a Volume-of-Fluid [3] solver using the Flux-Corrected-Transport (FCT) [4] for the transport of the phase-field of the two
fluids (air-fuel). Usually, the starting position of the fuel is predefined and the governing equations are solved to predict the fuel motion. For acceleration/braking modelling, the time-dependent
g-force acceleration is provided by the user as input. One could use actual track data and model the fuel sloshing of an F1 car for a whole race lap.
General Approach for Trap-Door Modelling
Under the assumption that a moving trap-door does not significantly influence the flow simulation, there are multiple approaches to model the trap-door system:
• Porous Media Approach: The trap-door is represented as a thin porous media with high and zero values to model a solid and a pass through trap-door, respectively. The disadvantage of this approach
is the reduced accuracy since the modelling of the wall using a porous media (via a force term in the momentum equation) is never as accurate as a real boundary, where the flow is blocked via
boundary conditions (BCs).
• Baffled Boundary Approach: The trap-door as a baffle-boundary which behaves as cyclic or wall (no-slip) boundary. This is accomplished by special BCs where their contributions to the matrix
assembly switch from cyclic to wall. From the programming point of view, this approach is not clean since every boundary condition of each field must be rewritten, leading to code duplication.
Moreover, exceptions must be added in many places, such as in wall distance computation which is required from the turbulence models.
• Moving Trap-Door Boundary: A moving boundary option requires performing a re-mesh when a trap-door changes state, which is time-consuming since mesh-generation and solution mapping are
computationally demanding.
• Generalized Internal Boundary (GIB): No topology changes, no re-meshing required, and easy treatment of physics in a single boundary. GIB is computationally efficient and general enough to use
for many different problems.
With all the aforementioned options, the best option is to use the approach of Generalized Internal Boundaries (GIB).
Trap-Door Modelling Using GIB
The Generalised Internal Boundary Method (GIB) [5, 6] allows prescribing boundary conditions on internal faces of the domain on-the-fly. The GIB method consists of three main stages. Assume that we
have a background mesh and an interface, as it is presented in Figure 2,
Figure 2: The three stages of GIB (left) identify closest points, (middle) impose BCs, and (right) apply contributions to transport equations
• First, the closest points of the background mesh from the interface are identified and projected onto the interface. This is similar to the “snapping” stage of the helyxHexMesh mesh generator.
After this stage, some of the internal faces are located exactly on the interface. These internal faces constitute the GIB boundary.
• Second, we impose BCs for the fields on each side of the interface. Here, any existing BC is compatible with the GIB (for instance fixedValue for Dirichlet, zeroGradient for zero Neumann etc).
• Third, our main goal is to keep the computational cost as low as possible. For this reason, the BCs on the GIB must behave as a standard boundary and at the same time, the mesh topology should
remain unchanged. Otherwise, data structure updates can slow down the performance of the simulation. To solve this problem, we modify the finite-volume implicit (fvm::) and explicit (fvc::)
operators to inject on-the-fly the contributions of the aforementioned BCs into the matrix. All operators are modified and are now compatible with GIB.
To better understand the operator modifications, the computation of the gradient at a cell for the quantity ξ is given in Figure 3. Our goal is to compute the gradient of the quantity ξ. Firstly, the
gradient expression is transformed from the volume to the surface integral using the Gauss divergence theorem. Based on the above formula, the gradient is equal to the sum of the ξ at the surrounding
faces of the cell multiplied with the surface area. Assume now that we want to add a GIB on face 4. The contributions of the GIB boundaries to the matrix (C[GIB]) contain the contribution of the
boundary condition (ξ[5]S[5j]) and one additional term (-ξ[4]S[4j]) which cancels out the already added contributions of the internal faces that collocate with the GIB (face 4). Based on the final
expression, the middle cell behaves exactly (machine accuracy) as a standard (external) boundary of the domain without changing the mesh topology.
Figure 3: GIB and the gradient operator
Now that you have a brief explanation of how GIB is implemented, a few relevant examples will show how elegant this approach can be when applied to sloshing simulation.
Simple Application: 2D SLOSHING
The first application is a simple 2D geometry with 3 fuel tanks presented in the following Figure 4.
Figure 4: Two-dimensional sloshing example with GIB trap-doors
Three fuel tanks “communicate” via two ducts. Each duct contains a trap-door (red and green patch), which allows the fuel to flow from the side tanks to the middle one. The trap-doors are controlled
using flow characteristics. In this case, we use the signed pressure difference between the two sides of the trap-door. Based on this control mechanism, the trap-door is either open or closed.
Finally, the blue patches represent the atmospheric outlet boundaries to provide a fixed pressure for the flow solver.
Initially, the fuel is positioned in the side tanks (black regions) and, after the simulation starts, acceleration is applied to mimic cornering. The helyxVoF solver is selected to solve the
governing VoF equations and the simulation time is 10 seconds.
The transient simulation results are show in the following video. As the fuel is sloshing, the trap-doors change state (close/open) in order to move the fuel from the side to the middle tank.
Figure 5: Transient sloshing results for a two-dimensional test case
Two-dimensional simulations provide tremendous insight into the dynamic behavior of the baffles on a simple system. A more challenging test would be to apply similar GIB-based trap-door mechanics to
a three-dimensional VOF simulation.
Advanced Application: 3D FUEL TANK
The second application is the fuel sloshing of a 3D geometry composed by two side tanks (left and right). The tanks communicate via two trap-doors, which are modelled using GIBs. The red GIB allows
flow from the right to the left tank, whereas the green one from the left to the right tank. Each tank has an atmospheric boundary (dark blue). The computational mesh is generated using the
hex-dominant mesh generator helyxHexMesh and it has approximately 1e+6 cells.
Figure 6: Three-Dimensional Fuel tank Domain with trap-Doors
The fuel is represented with cyan colour and its initial position is only in the left tank. The GIBs switch from active (no-slip wall modelling) to inactive (pass-through/invisible) based on the
signed pressure drop of the two sides of the GIB. The helyxVoF is selected as the multiphase flow solver and the k-ωSST model is chosen for turbulence modelling. When the trap-door is closed, the GIB
inherits all the characteristics of a no-slip wall. This means that a) turbulent wall-function boundary conditions are imposed on the GIB and that b) the GIBs are considered in the wall-distance
computation, which is required by the turbulence models. Additional g-force acceleration (Figure 7) is again added to model cornering over a chicane.
Figure 7: Gravitational forces imposed on the sloshing system vs time
The transient simulation results of the fuel sloshing are presented in the video in Figure 8. The fuel starts moving from the left to the right tank via the green trap-door, and, then, at the second
corner (t=5-7s), the fuel moves back to the left tank from the red trap-door. The case is decomposed in 32 sub-domains (processors) and the overall computational cost is 3.5 hours (wall clock-time)
on one AMD Epyc 7351 node.
Figure 8: Transient multiphase sloshing results using a GIB trap-door model
Overall, the case predicts the dynamic motion of the fuel free-surface as well as the trap-door mechanism to control the fuel motion during dynamic events.
To sum up, a new method for modelling trap-doors in fuel tanks is presented. Though these examples are based on a motorsport application, plenty of aerospace or automotive applications related to
fuel management analysis could use the same approach. Furthermore, it’s worth mentioning that the GIB method is very versatile and general (hence the name “generalized” in GIB), which means that the
presented applications are just one use of the GIB. The same mechanism can be used in other applications governed by other flow equations, for instance modelling windows in fire modelling
If you are interested to learn more or think this functionality can help you in your design process, the method is readily available in HELYX. Feel free to contact us or drop a comment below.
1. Technical F1. fuel tank, 1999.
2. Hubert, C. “Behavior of Spinning Space Vehicles with Onboard Liquids“, NASA GSFC Symposium, 2003.
3. Hirt, C.W. and Nichols, B.D. “Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries“, Journal of Computational Physics, pp. 201-225, 1981.
4. Boris, J.P. and Book, D.L. “Flux-corrected transport, I: SHASTA, a fluid transport algorithm that works“, Journal of Computational Physics, pp. 38-67, 1973.
5. Karpouzas, G. “A Hybrid Method for Shape and Topology Optimization in Fluid Mechanics”. PhD Thesis, NTUA, 2019.
6. Karpouzas, G. and de Villiers E. “Generalized Internal Boundaries (GIB)” , arXiv.org, 2017. | {"url":"https://engys.com/blog/advanced-simulation-of-motorsport-fuel-cell-dynamics/","timestamp":"2024-11-06T02:37:50Z","content_type":"text/html","content_length":"158818","record_id":"<urn:uuid:2c0e78ed-b7c1-4b21-bf41-f5f9526bcb34>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00227.warc.gz"} |
New upper limit on the total neutrino mass from the 2 Degree Field Galaxy Redshift Survey
We constrain f(nu)equivalent toOmega(nu)/Omega(m), the fractional contribution of neutrinos to the total mass density in the Universe, by comparing the power spectrum of fluctuations derived from the
2 Degree Field Galaxy Redshift Survey with power spectra for models with four components: baryons, cold dark matter, massive neutrinos, and a cosmological constant. Adding constraints from
independent cosmological probes we find f(nu)<0.13 (at 95% confidence) for a prior of 0.1<Omega(m)<0.5, and assuming the scalar spectral index n=1. This translates to an upper limit on the total
neutrino mass m(nu,tot)<1.8 eV for "concordance" values of Omega(m) and the Hubble constant.
• POWER-SPECTRUM
• MATTER
• OSCILLATIONS
• SUPERNOVAE
• CLUSTERS
• DENSITY
Dive into the research topics of 'New upper limit on the total neutrino mass from the 2 Degree Field Galaxy Redshift Survey'. Together they form a unique fingerprint. | {"url":"https://research-portal.st-andrews.ac.uk/en/publications/new-upper-limit-on-the-total-neutrino-mass-from-the-2-degree-fiel","timestamp":"2024-11-06T01:32:01Z","content_type":"text/html","content_length":"57821","record_id":"<urn:uuid:4f6b3f77-d855-411c-bb64-90406cea0308>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00393.warc.gz"} |
Yanai Waves and the Classification of Westward Propagating Waves on the Rotating Spherical Earth
Monday, 26 June 2017: 2:15 PM
Salon F (Marriott Portland Downtown Waterfront)
The elegant and seminal linear wave theory developed in Matsuno (1966) on the equatorial β-plane is the foundation of our understanding of equatorial wave theory in the ocean and atmosphere. In
Matsuno’s theory Planetary (Rossby) waves and Inertia-Gravity (Poincaré) waves are derived as three distinct wave types from solutions of an eigenvalue (Schrödinger) equation for the meridional
velocity, that are obtained from Linear Rotating Shallow Water Equations (LRSWE, hereafter) by eliminating the zonal velocity and height. The waves’ phase speeds are the three roots of a cubic in
which one of the coefficients is the discrete eigenvalue (i.e. energy level) of the Schrödinger equation and two of these roots are negative i.e. they describe westward propagating waves.
Accordingly, the mode-numbers of the three wave types are determined by the index of the corresponding discrete eigenvalues. The modes with n=0 are a special case in that one of the two westward
propagating roots is associated with an unphysical zonal velocity i.e. one of the roots of the cubic equation is not a physically acceptable solution of the LRSWE. The other n=0 westward propagating
wave is a mixed-mode, AKA Yanai wave, that changes from an Inertia-Gravity mode at long zonal wavenumbers to a Rossby mode at short zonal wavenumbers. Despite its wide-spread implications on the
dynamics in the ocean, atmosphere and climate on earth, Matsuno’s equatorial wave theory has never been extended to spherical coordinates. With the recent successful formulation of eigenvalue
equations for Planetary waves and Inertia-Gravity waves on the entire sphere that are analogous to the single planar equation derived by Matsuno on a plane we were able to derive explicit expressions
for Planetary (Rossby) waves and Inertia-Gravity (Poincaré) waves on a sphere including the meridional amplitude structure and dispersion relations. In particular we were able to confirm the
existence of Yanai wave on a sphere where the dynamical reason for the existence of the Yanai wave on a sphere is not the singularity of the zonal velocity. Instead, the explicit expressions for one
of the n=0 westward propagating mode ceases to be valid at a zonal wavenumber where its phase speed approaches the gravity wave phase speed. In my talk I will present the various eigenvalue equations
on a sphere and reconcile the ambiguity encountered when two of them overlap. In addition, I will also demonstrate how the disappearance of the superfluous westward propagating mode gives rise to the
formation of the mixed-mode.
Supplementary URL: http://www.earth.huji.ac.il/people/paldor.asp | {"url":"https://ams.confex.com/ams/21Fluid19Middle/webprogram/Paper318697.html","timestamp":"2024-11-13T08:12:18Z","content_type":"text/html","content_length":"11782","record_id":"<urn:uuid:51214bf8-795c-4f1a-8432-9f9ab0c377f9>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00341.warc.gz"} |
Cryptography and Number Theory
The Art of the Hidden Message: The role of number theory and prime numbers in online security
Online security presents new challenges for security. Let’s see how some discoveries in mathematics unexpectedly made the age of secure online transactions possible. Some of the mathematics in the
section on prime numbers and Fermat’s Little Theorem, and the encryption scheme known as RSA, are on the level of a typical intermediate level university mathematics course; if you have a few hours
and want to understand exactly how this encryption scheme works, you can work through the mathematics of this. Otherwise, feel free to read through the beginnings of these sections, and skim
through the more technical parts and head to the final section for a brief outline of the mathematics involved.
Introduction: how ‘pure’ mathematics can have unexpected applications
One of the amazing things about theoretical mathematics – mathematics done for its own sake, rather than out of an attempt to understand the “real world” – is that sometimes, purely theoretical
discoveries can turn out to have practical applications. This happened, for example, when non-Euclidean geometries described by the mathematicians Karl Gauss (pictured below) and Bernard Riemann
turned out to provide a model for the relativity between space and time, as shown by Albert Einstein.
(Image from scienceworld.wolfram.com).
So oftentimes, theoretical results in mathematics turn out to have completely unexpected applications. This has happened with a number of results in the field of mathematics known as number theory.
Number theory studies the properties of the natural numbers: 1, 2, 3,… You might also think of them as the “counting numbers”. While they’re deceptively easy to conceive and understand, their study
has gained its reputation as the “queen of mathematics”, and many of the greatest mathematicians have devoted study to their properties, which are not as simple as you might expect.
That’s weird. What could be more simple than the natural numbers 1, 2, 3,…?
You’re right! They’re easy to conceptualize, but their sometimes mind-boggling properties actually have crucial applications in the modern world, which we’ll see in this article. In the age of the
internet, when we use the internet to conduct business transactions, check our bank account, use credit cards to buy things online, and take cash out of an ATM, security technologies such as armed
guards, armored trucks, and X-ray machines simply don’t work. The danger with modern internet transactions is that you must send private data through a public network in order to reach its
destination, such as your bank. The problem with this is that anyone can intercept read the message you have sent.
An internet-based economy therefore requires a new kind of security technology in order to protect information people send online. Actually, the field of information protection – known as
“cryptography”, or “hidden writing” – isn’t new. It dates back several thousand years.
The key ideas behind message encryption
Our first example: the Caesar cipher
To illustrate the idea behind information protection, let’s look at a simple way to protect a message that’s being sent: an encryption mechanism known as the shift cipher. It works like this. Suppose
you want to send a message to a friend, but don’t want anyone other than your friend to be able to read the message. One way to do this is to agree with your friend on a way to encode or disguise the
message you will be sending, and then send the message, so that your friend, and only your friend, will know how to decode it.
The shift cipher is a simple way to encode a message: to use it, we simply shift each letter in the message by a certain, predetermined number of letters. For example, the cipher used by Roman
emperor Julius Caesar to communicate with his generals was a shift cipher created by shifting each letter by three letters.
Caesar’s cipher would encode A as D, B as E, C as F, and so forth. It would encode W as Z, X as A, Y as B, and Z as C, as well. So, for example, we would encode the message, “Meet me at the
restaurant” as “Phhw ph dw wkh uhvwdxudqw”, simply by taking each letter in the message and encoding it as the third letter after it.
You can probably see how to decode such a message once it is received – simply shift each letter in the message back by three letters. Got it?
I understand, but it seems so simple that it wouldn’t work. Someone could just guess how the encoding works, right?
You’re right. The problem with the shift cipher is that it is fairly easy to break. If someone captures your message before it is received by your intended recipient, they will find it easy to break,
especially using a computer.
One way they could figure out how you encoded your message is by looking at the frequencies of the letters in your encoded message. The 7 most common letters, according to a study completed at
Cornell University, are e, t, a, o, i, n, s, and so on – see the graph below.
By analyzing which letters appear most frequently in the encoded message, it’s pretty easy to break a shift cipher. Notice, for example, in the encoded version of our message, “Phhw ph dw wkh
uhvwdxudqw”, the letters ‘h’ and ‘w’ appear frequently – because they stand for ‘e’ and ‘t’, the two most commonly used letters in the English alphabet.
Cryptographers have worked to find better methods for encoding messages – and cryptanalysts have been able to analyze or break every single method of cryptography developed in the past two millennia.
A famous example of this occurred during World War II, when British mathematicians, lead by the influential computer scientist Alan Turing, were able to break the German code-making machine known as
You can watch a documentary about this produced by National Geographic TV.
Prime numbers and Fermat’s Little Theorem
The computer age has presented new challenges to cryptographers due to the computational power a computer possesses. The computer can quickly and easily break code-making techniques developed prior
to the electronic age. This is where the work by mathematicians such as Fermat has come into play. A team of mathematicians working in the 1970’s were able to use discoveries made several hundred
years before to produce a powerful method of encryption that is very difficult for even the most powerful supercomputers to break.
To see how this method, known as the RSA algorithm, works, we need to first look at some basic results of number theory, the study of the natural numbers 1, 2, 3, etc. Let’s specifically examine the
subset of the natural numbers known as the prime numbers. The prime numbers are those natural numbers which have no divisors other than 1 and themselves. For example, 2, 3, and 5 are prime, while 4
and 15 are not prime, since 2 is a divisor of 4 and 3 is a divisor of 15. Here’s a table of the first 100 natural numbers. The prime numbers are in the white squares.
(image from http://technomaths.edublogs.org/category/number/)
The prime numbers are easy to define and understand, but also turn out to have some surprisingly complex properties. The ancient Greeks were able to prove that the number of primes is infinite – that
there is always another prime number. More precisely, they were able to prove that, given any finite list of prime numbers, there is always another prime number not on that list.
Mathematicians such as Fermat, Euler and Gauss, have devoted a considerable portion of their work to studying the properties of the primes, such as what’s known as the Prime Number Theorem, which
tells how to estimate the number of prime numbers smaller than one million, or one billion, or any other number. More recent work has been devoted to finding large prime numbers . The largest known
prime number has close to 13 million digits! It was discovered in 1008 on UCLA computers, and its finders were awarded a $100,000 prize for their work!
$100,000 for finding a prime number? That seems like a ridiculous waste of money!
Well, there’s a reason that large prime numbers are considered so valuable: for one, they can be used for internet security purposes, as we shall see. Also, showing such a large number is actually
prime is a feat of computer engineering that relies on networking many computers together. So, by doing this work, computer scientists are showing how the combined power of many different computers
can be linked together to solve a problem.
That makes sense then. So how do large prime numbers help in ensuring computer security?
Great question. To see how prime numbers can be used to ensure internet security, let’s discuss a few basic properties about prime numbers. One of the central results in number theory pertains to the
properties of prime numbers, and is known as Fermat’s Little Theorem. There’s a more famous theorem of Fermat’s, known as Fermat’s Last Theorem, which received a lot of media attention due to the
amount of work required to prove it, but Fermat’s Little Theorem is probably more important in our day-to-day lives because it was a crucial step in the development of the RSA algorithm, which
enables us to make secure transactions via the internet or ATMs.
So first let’s state Fermat’s Little Theorem, and then try to understand what it means. Keep in mind that for mathematicians, a theorem is simply a true statement. It’s something that mathematicians
know to be true. Fermat’s Little Theorem deals with a fundamental property of prime numbers:
If p is a prime number, and a is any natural number, then the following holds:
Fermat’s Little Theorem states that for any prime number p and any natural number a, the above formula holds. Another way to state this is:
is always a multiple of p, whenever p is prime and a is any natural number.
OK, so what are all those funny looking symbols?
is an expression of modular arithmetic. It is not concerned with equality (which is represented by two lines, one on top of the other), but instead is concerned with the remainders left over after
performing division.
The left-hand side is read as “a to the p power” – this represents the number we get by multiplying a by itself for p iterations:
Inside the parentheses on the right, the “mod” is short for “modulo”, and refers to what happens when we divide the term on the left by p. The a to the right of the three lines represents the
remainder left over when we divide by p.
So this entire statement simply says that when we take a, multiply it by itself p times, and divide this number by p, we always will end up with a remainder of a; in other words,
is always a multiple of p.
If you’re interested in how to prove this fact, check this page for a few different proofs of Fermat’s Little Theorem.
OK, but I’m still not convinced. Can you show an example of what you’re talking about?
Sure – let’s try this out with a few examples. Suppose we take p = 3, a prime number, and a=4. Then
simplifies to
In this case, Fermat’s Little Theorem implies that 60 is a multiple of p=3, which you can verify on your own.
Similarly, let’s take p = 11, a prime number, and a=6. Then
simplifies to
Once again, Fermat’s Little Theorem guarantees that this number is a multiple of p = 11, and you can check this on your own.
Now, Fermat was able to prove, or demonstrate the truth of, his Little Theorem during his lifetime. At this point, however, there were no obvious real-world applications for his work.
Fermat’s Little Theorem and the RSA Algorithm
It was only in the middle of the 20th century that Fermat’s Little Theorem became useful. At this point, relatively primitive computers were being used – they sometimes took up an entire room, and
generated huge amounts of heat. Check out the picture of the early computer known as UNIVAC, below. If you get the chance to speak with scientists who were working at this time, they’ll be able to
tell you stories about how time-consuming it was to run computations with these early computers. Even in these days, though, the computers were able to break primitive forms of ciphers.
image from http://www.computermuseum.li/Testpage/UNIVAC-1-FullView-B.htm
With the advent of the computer age, mathematicians began to develop new types of cryptography that would be strong enough to resist being broken by a computer. One of the most useful types of
cryptography developed is known as public-key cryptography, which is used in the present day for internet security. Public-key cryptography involves both a public key – known to both the sender, the
receiver, and anyone who intercepts the message in between – and a private key, known only to the sender and the receiver.
In the 1970’s, a trio of mathematicians, Rivest, Shamir, and Adelman, at Massachusetts Institute of Technology, were able to use Fermat’s Little Theorem to come up with the public-key encryption
scheme known as the RSA algorithm, which is the standard encryption method used for online transactions today.
The RSA algorithm consists of (1) an encryption procedure, which is publicly known, and (2) a decryption procedure, which is only known to the sender and the receiver.
Here’s how it works. When you send information via electronic means, such as at the ATM, the computer converts your information – the message you want to send – into a single number. You might try to
think of a way to do this on your one, but to start you might simply designate a as 01, b as 02, and so forth. Call the number representing your message m.
Now – here’s the key idea behind the RSA scheme – we want to find numbers e and d (for encryption and decryption) such that
(recall this means that if we subtract m from either of the quantities on the left, what’s left is a multiple of n). If we can find such numbers, then raising our message, m, to the e power – that
is, multiplying m by itself e times (and finding the remainder modulo n) – will correspond to encoding our message, and raising the encoded message to the d power (modulo n), corresponds to decoding
the message.
To carry out the RSA encryption, we then use two large prime numbers; prime numbers of around 100 digits should be sufficient. Call these primes p and q. Form the product of the two primes, and call
this number n, so that n = p * q. Also form the number z = (p-1)*(q-1). (The number z is important because it represents how many numbers between 1 and n do not share factors with n).
To find such numbers, we can recall Fermat’s Little Theorem, which suggests that, for any prime number p,
If we pick e to be any prime number less than z that is not a divisor of z, then we are able to find a number d such that
We can find such a number using a process known as Euclid’s algorithm. A bit of arithmetic, using Fermat’s Little Theorem, will show that once we have found d and e in this manner, then
So, we encode the message using the encryption number e, send the encoded message, together with e and n, over the public domain, while keeping d known to only the sender and the receiver. This
ensures the security of the message since a hacker would calculate the factors of n, the product of two large prime numbers, in order to find d, which would take a modern computer many decades to
complete. The prime numbers p and q used to calculate n are changed sufficiently often to ensure that the RSA algorithm is secure.
Here’s a diagram representing the process used in RSA for encrypting and decrypting our message:
To illustrate this, let’s carry RSA out for a real “message”. Suppose we want to “send” the number 23 in an encrypted fashion.
First, we pick two large prime numbers. A real-life RSA encryption scheme might use prime numbers with 100 digits, but let’s keep it simple and use relatively small prime numbers. Take p=47 and q=43.
Now we form the product n=p*q=47*43=2021, and the number z=(p-1)*(q-1)=46*42=1932.
At this point we’re ready to find our actual encoding and decoding schemes. We want to find numbers e and d such that
First, let’s try to find a prime number e that is not a divisor of 1932. Let’s take e=17.
Now we compute d using the Euclidean algorithm. We use this algorithm to find a number d such that
We find that d=341 will work. You can check on your own that 341*17=5797, while 1932*3 = 5796, so 341*17 leaves a remainder of 1 (modulo 1932).
So, to encode our message m=23, we simply send the “message” attained by encrypting 23 using e:
So our encrypted message is 1863, and we also also send e=17 and the number n=2021 with the encrypted message, while maintaining the privacy of d=341 to ourselves and the receiver. Now we come to the
key point. If a hacker wants to decrypt the message, s/he must calculate d, which in this case would entail factoring 2021 to p and q. This is easy for a computer to complete in this example, but
very time consuming for large choices of p and q.
The receiver, however, knows d, which allows him/her to perform the following operation to recover the initial message:
To sum it all up…
Now, we’ve covered some fairly heavy-duty mathematics. The basic idea behind RSA algorithm is: first, we convert a message we want to send (such as bank account or credit card information) to a
number, designated as m. We then pick two large prime numbers, p and q, and from these numbers we calculate (based on Fermat’s Little Theorem) two numbers e and d, respectively the encryption and
decryption schemes. We use e to encode m, then send this encoded message over the public domain. We keep d private, known only to ourselves and our intended destination. Now, someone who
intercepts the encoded message and tries to decode it will have to factor a very large number formed by the product of p and q. For large values of p and q, such as prime numbers with 100 digits,
this might take a modern computer 100 years. Our message is therefore, in practice, secure.
If you’ve made it through all this – congratulations. This is university-level material. If you’re interested in reading more of the heavy-duty mathematics behind the RSA algorithm, check out this
paper the mathematician Yevgeny Milanov.
I’ve chosen this as a first article to write here on science4all.org because it serves as a good example of how a topic in pure mathematics can turn out to have a completely unexpected application –
the type of unexpected connection that often results in scientific discoveries and is known as serendipity. In this case, discoveries about prime numbers culminating in Fermat’s Little Theorem gives
us a way to encrypt a message and send it through a network securely, so that only its intended target, who has a decryption mechanism, can read the message.
Recall that the main reason the RSA algorithm is so strong is the fact that, in order to break it, it’s necessary to find the prime factors of a gigantic number, a number with around 200 digits. This
might take a supercomputer 100 years to complete!
However, computer scientists are busy working on a quantum computer, a computer that would use the states of an electron as its computing components, rather than the ‘0’ and ‘1’ bits current
computers use.
If they are successful in implementing a quantum computer, it will be very easy to find the prime factors used in the RSA algorithm. So a much stronger algorithm will be necessary to maintain network
security in an age with quantum computers.
Amazingly, mathematicians working in the topic area known a “knot theory”, which makes a formal study of the knots we use to tie shoelaces or ropes, have developed an algorithm that uses an analogy
of prime numbers, called “prime knots”, to develop a public-key cryptography method that is immune to quantum eavesdroppers. Amazing! | {"url":"http://www.science4all.org/article/cryptography-and-number-theory/","timestamp":"2024-11-03T21:41:10Z","content_type":"text/html","content_length":"77583","record_id":"<urn:uuid:fba6d987-530c-4750-aa9f-8e91531be40c>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00124.warc.gz"} |
Hogwarts Solve Herodianas Puzzle
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They Can Be Found Inside Of The Astronomy.
960k views 1 year ago.
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Taming the LM3886: Power Supply Design | Neurochrome
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Power Supply DesignÂ
The power supply design appears to be causing some confusion in the DIY community as well. I figured I'd shed some light on the topic by providing the math needed for the power calculations.
As mentioned on the Thermal Design page, the power drawn from the power supply, including the quiescent current, for a given output swing can be calculated as,
where P[S] is the total supply power, I[bias] is the quiescent or bias current, V[OUT[peak]] is the peak output voltage, R[L] is the load impedance, and V[CC] is the supply voltage. It is assumed
that the amplifier operates from a symmetric power supply, i.e. V[CC] = -V[EE]. The maximum output swing is determined by the supply voltage and the output dropout voltage of the LM3886, which is
found in the data sheet, as excerpted below.
The LM3886 clips asymmetrically with the negative swing being clipped before the positive. Thus, the peak undistorted swing possible for the LM3886 is 2.5 V less than the supply voltage. Similarly,
the quiescent current is found in the data sheet. I will use the typical number rather than the worst case for this example.
Example: Supply voltage: ±25 V; Load resistance: 4 Ω. The power drawn from the power supply at the full undistorted output power can be calculated as:
The supply power for a range of common supply voltages and load impedances is tabulated below.
VCC V[OUTpeak] R[L] P[S]
±25 V 22.5 V 4 Ω 92.0 W
±25 V 22.5 V 8 Ω 47.3 W
±28 V 25.5 V 4 Ω 116 W
±28 V 25.5 V 8 Ω 59.6 W
±35 V 32.5 V 8 Ω 94.0 W
P[S]Â is the total power drawn by the amplifier from the power supply. From this power, the VA rating of the power transformer needs to be determined. Had the amplifier presented a purely resistive
load, this would have been a simple task, however, the load presented by a full-wave rectifier is by no means a "nice" load. Rather, the current through the rectifier is a pulse train. It is
possible to find an analytical solution for this, but the math gets rather involved. For those interested in the math, I suggest consulting Blencowe, who suggests using a conversion factor of 1.5 to
convert from resistive power to the VA rating of the transformer. I.e. for the ±25 V, 4 Ω example above, a transformer with a VA rating of 1.5 × 92.0 W = 138 VA should be specified.
To verify this rule of thumb, I simulated the class AB load current using LTspice. The sim sheet is shown below. The power transformer model is an Antek AS-2222 toroidal transformer.
The resulting VA rating of the power transformer can be found as the sum of the reactive power (V-A products) of the two secondary windings. This is plotted below.
The extreme power at the beginning of the simulation is caused by the current needed to charge the reservoir capacitors to the full supply voltage. After about 100 ms, the supply voltage is within 90
% of its final value and it has fully settled by 250 ms. This extreme power is no cause for alarm. It is, however, a good argument for using an in-rush limiter or soft-start circuit, in particular
in supplies using high efficiency transformer types, such as toroids.
The VA product is a somewhat distorted, half-wave rectified, sine wave with a peak value of 400 VA. The average value works out to 141 VA - pretty darn close to the 138 VA obtained by Blencowe's rule
of thumb. Note that the reactive power does depend on the size of the reservoir capacitor. The bigger the capacitor, the larger the conversion factor between resistive power (W) and reactive power
(VA). Blencowe's rule of thumb appears to hold up well for reasonable values of reservoir capacitors of, say, 4700 µF - 22000 µF.
Conclusion: An LM3886 running on a ±25 V supply, delivering the largest possible undistorted sine wave into a 4 Ω load will need a 140 VA transformer. For a stereo amplifier a 280 VA transformer
should be used. I would round up to the nearest available standard size, which, typically, is 300 VA.
The Crest Factor, Revisited
Anyone who's ever taken apart a commercially available amplifier will comment that none of the 65 W rated amplifiers they have looked at have contained transformers capable of supplying 300 VA.
What's the deal...? It's the crest factor again (see the thermal design section for a more thorough treatment of the subject).
Assuming the amplifier is to be used for music reproduction rather than the reproduction of sine waves, the power transformer can be undersized quite a bit. Extremely compressed music, such as some
heavy metal, has a crest factor of 5~6 dB. Classical music, lands at the other end of the spectrum with a crest factor of about 20 dB. This means the peak power of classical music is 100Ă— higher
than the RMS power. In an analysis of 4500 tracks performed by Sound on Sound Magazine, an average crest factor of 14 dB was found. The supply power and resulting power transformer VA ratings (per
LM3886 channel) are tabulated below for a range of common crest factors.
Crest Factor (dB) P[L] RMS P[L] peak VCC R[L] P[S] Power Transformer VA Rating
3 (sine wave) 63.3 W 127 W ±25 V 4 Ω 92.1 W 138 VA
6 31.8 W 127 W ±25 V 4 Ω 66.0 W 98.9 VA
10 12.7 W 127 W ±25 V 4 Ω 42.5 W 63.8 VA
14 5.04 W 127 W ±25 V 4 Ω 27.8 W 41.6 VA
20 1.27 W 127 W ±25 V 4 Ω 15.2 W 22.7 VA
As seen in the table, a 100 VA transformer can actually be specified for a stereo LM3886 amplifier, assuming the crest factor remains around 12~14 dB. A transformer providing 20~22 V RMS, such as the
Antek AS-1220 would be suitable.
Section 7: Rectification & Snubbers
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Attacking rooks
Execution time limit is 1 second
Runtime memory usage limit is 128 megabytes
Chess inspired problems are a common source of exercises in algorithms classes. Starting with the well known 8 - queens problem, several generalizations and variations were made. One of them is the n
-rooks problem, which consists of placing n rooks in an n by n chessboard in such a way that they do not attack each other.
Professor Anand presented the n-rooks problem to his students. Since rooks only attack each other when they share a row or column, they soon discovered that the problem can be easily solved by
placing the rooks along a main diagonal of the board. So, the professor decided to complicate the problem by adding some pawns to the board. In a board with pawns, two rooks attack each other if and
only if they share a row or column and there is no pawn placed between them. Besides, pawns occupy some squares, which gives an additional restriction on which squares the rooks may be placed on.
Given the size of the board and the location of the pawns, tell Professor Anand the maximum number of rooks that can be placed on empty squares such that no two of them attack each other.
The first line contains an integer n (1 ≤ n ≤ 100) representing the number of rows and columns of the board. Each of the next n lines contains a string of n characters. In the i-th of these strings,
the j-th character represents the square in the i-th row and j-th column of the board. The character is either "." (dot) or the uppercase letter "X", indicating respectively an empty square or a
square containing a pawn.
Output a line with an integer representing the maximum number of rooks that can be placed on the empty squares of the board without attacking each other.
Submissions 603
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Answer true or false to each of the
Answer true or false to each of the following statements and explain your answers. Polynomial regression equations are useful for modeling more complex curvature in regression equations than can be
handled by using the method of transformations.
Answered question
Answer true or false to each of the following statements and explain your answers. Polynomial regression equations are useful for modeling more complex curvature in regression equations than can be
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On the Augmentation of Ring Recovery Data with Field Information
Freeman, Stephen N., Morgan, Byron J. T., Catchpole, Edward A. (1992) On the Augmentation of Ring Recovery Data with Field Information. Journal of Animal Ecology, 61 (3). pp. 649-657. ISSN 0021-8790.
(doi:10.2307/5620) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided) (KAR id:22573)
The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided.
Official URL:
1. It is well-known that the Cormack-Seber model (Cormack 1970; Seber 1971) for ring-recovery data from birds ringed as nestlings does not provide unique maximum-likelihood estimates of the
parameters without the addition of a constraint. Lakhani & Newton (1983) pointed out that the use of a constraint might yield misleading estimates. Unique estimates do result from the addition of
separate information on one of the parameters; Lakhani (1987) suggested the possibility of the survival of first-year birds being based on observed deaths of radio-marked birds. 2. We show here that
the estimate of the survival probability for first-year birds, obtained from radio-marked birds alone, may not correspond to the overall maximum-likelihood estimate of this probability. 3. We explain
how such a discrepancy may arise, and show that, in this case, one of the other parameters in the model is estimated on a boundary to the parameter space, i.e. is either 0 or 1. 4. A small simulation
study is carried out to explore the implications of this finding for the bias of the parameter estimators. We see that if the radio-marking does not affect first-year survival, then the most serious
bias is for the survival probability phi(k), where k is the number of years' duration of the ringing study. For other parameters, bias will be lower, and possibly of the order of 10%, though this
depends on the size of the radio-tracking study. This bias might well be less than the possibly unknown bias in the absence of any augmentation, as claimed by Lakhani (1987). 5. If radio-marked birds
experience a reduced probability of survival then more serious bias may extend to all parameters. In this situation, bias increases as the size of the radio-tracking study increases. 6. When a
discrepancy of the kind described in (2) above arises, repeating the radio-tracking experiment is advisable only if survival is not expected to be impaired by radio-marking, and if a large
radio-marking experiment may be carried out. 7. Lakhani (1987, 1990) has suggested further improvements to the simple radio-tracking scheme examined here. Morgan & Freeman (1989) have discussed a
model with first-year variation in survival rates which allows the estimation of model parameters, without the need of any constraints. The possibility of combining this modelling development with
further field information is discussed.
Item Type: Article
DOI/Identification number: 10.2307/5620
Uncontrolled keywords: Bias; Boundary Estimation; Cormack-Seber Model; Radio-Tracking; Ring-Recovery Data
Subjects: Q Science > QH Natural history > QH541 Ecology
Divisions: Divisions > Division of Natural Sciences > Biosciences
Depositing User: P. Ogbuji
Date Deposited: 05 Sep 2009 12:02 UTC
Last Modified: 05 Nov 2024 10:01 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/22573 (The current URI for this page, for reference purposes)
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Iterator index querying
Array Locator Methods
Locator methods iterate over the array elements, which are then used to evaluate the expression specified by the with clause. The iterator argument optionally specifies the name of the variable used
by the with expression to designate the element of the array at each iteration. If it is not specified, the name item is used by default. The scope for the iterator name is the with expression.
The following locator methods are supported (the with clause is mandatory) :
• find(): returns all the elements satisfying the given expression
• find_index(): returns the indexes of all the elements satisfying the given expression
• find_first(): returns the first element satisfying the given expression
• find_first_index(): returns the index of the first element satisfying the given expression
• find_last(): returns the last element satisfying the given expression
• find_last_index(): returns the index of the last element satisfying the given expression
For the following locator methods, the with clause (and its expression) can be omitted if the relational operators (<, >, ==) are defined for the element type of the given array. If a with clause is
specified, the relational operators (<, >, ==) must be defined for the type of the expression.
• min(): returns the element with the minimum value or whose expression evaluates to a minimum
• max(): returns the element with the maximum value or whose expression evaluates to a maximum
• unique(): returns all elements with unique values or whose expression is unique
• unique_index(): returns the indexes of all elements with unique values or whose expression is unique
string SA[10], qs[$];
int IA[*], qi[$];
// Find all items greater than 5
qi = IA.find( x ) with ( x > 5 );
// Find indexes of all items equal to 3
qi = IA.find_index with ( item == 3 );
// Find the first item equal to Bob
qs = SA.find_first with ( item == "Bob" );
// Find the last item equal to Henry
qs = SA.find_last( y ) with ( y == "Henry" );
// Find the index of the last item greater than Z
qi = SA.find_last_index( s ) with ( s > "Z" );
// Find the smallest item
qi = IA.min;
// Find the string with a largest numerical value
qs = SA.max with ( item.atoi );
// Find all unique strings elements
qs = SA.unique;
// Find all unique strings in lower-case
qs = SA.unique( s ) with ( s.tolower );
Array ordering methods
Array ordering methods can reorder the elements of one-dimensional arrays or queues.
The general prototype for the ordering methods is:
function void ordering_method ( array_type iterator = item )
The following ordering methods are supported:
• reverse(): reverses all the elements of the array (packed or unpacked). Specifying a with clause shall be a compiler error.
• sort(): sorts the unpacked array in ascending order, optionally using the expression in the with clause.
The with clause (and its expression) is optional when the relational operators are defined for the array element type.
• rsort(): sorts the unpacked array in descending order, optionally using the expression in the with clause.
The with clause (and its expression) is optional when the relational operators are defined for the array element type.
• shuffle(): randomizes the order of the elements in the array. Specifying a with clause shall be a compiler error.
string s[] = { "hello", "good", "morning" };
s.reverse; // s becomes { "morning", "good", "hello" };
logic [4:1] a = 4’bXZ01;
a.reverse; // a becomes 4’b10ZX
int q[$] = { 4, 5, 3, 1 };
q.sort; // q becomes { 1, 3, 4, 5 }
struct { byte red, green, blue } c [512];
c.sort with ( item.red ); // sort c using the red field only
c.sort( x ) with ( x.blue << 8 + x.green ); // sort by blue then green
Array reduction methods
Array reduction methods can be applied to any unpacked array to reduce the array to a single value. The expression within the optional with clause can be used to specify the item to use in the
The prototype for these methods is:
function expression_or_array_type reduction_method (array_type iterator = item)
The method returns a single value of the same type as the array element type or, if specified, the type of the expression in the with clause. The with clause can be omitted if the corresponding
arithmetic or boolean reduction operation is defined for the array element type. If a with clause is specified, the corresponding arithmetic or boolean reduction operation must be defined for the
type of the expression.
The following reduction methods are supported:
• sum(): returns the sum of all the array elements, or if a with clause is specified, returns the sum of the values yielded by evaluating the expression for each array element.
• product(): returns the product of all the array elements, or if a with clause is specified, returns the product of the values yielded by evaluating the expression for each array element.
• and(): returns the bit-wise AND ( & ) of all the array elements, or if a with clause is specified, returns the bit-wise AND of the values yielded by evaluating the expression for each array
• or(): returns the bit-wise OR ( | ) of all the array elements, or if a with clause is specified, returns the bitwise OR of the values yielded by evaluating the expression for each array element
• xor(): returns the logical XOR ( ^ ) of all the array elements, or if a with clause is specified, returns the XOR of the values yielded by evaluating the expression for each array element.
byte b[] = { 1, 2, 3, 4 };
int y;
y = b.sum ; // y becomes 10 => 1 + 2 + 3 + 4
y = b.product ; // y becomes 24 => 1 * 2 * 3 * 4
y = b.xor with ( item + 4 ); // y becomes 12 => 5 ^ 6 ^ 7 ^ 8
Iterator index querying
The expressions used by array manipulation methods sometimes need the actual array indexes at each iteration, not just the array element. The index method of an iterator returns the index value of
the specified dimension.
The prototype of the index method is:
function int_or_index_type index ( int dimension = 1 )
The slowest variation is dimension 1. Successively faster varying dimensions have sequentially higher dimension numbers. If the dimension is not specified, the first dimension is used by default. The
return type of the index method is an int for all array iterator items except associative arrays, which returns an index of the same type as the associative index type.
For example:
int arr[]
int mem[9:0][9:0], mem2[9:0][9:0];
int q[$];
//Finds all items equal to their position (index)
q = arr.find with ( item == item.index );
//Finds all items in mem that are greater than the corresponding item in mem2
q = mem.find( x ) with ( x > mem2[x.index(1)][x.index(2)] ); | {"url":"https://vlsisource.com/tag/iterator-index-querying/","timestamp":"2024-11-03T15:17:54Z","content_type":"text/html","content_length":"58563","record_id":"<urn:uuid:98ce531c-760a-4512-b9d4-c036f5267fb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00833.warc.gz"} |
170+ Maths Graduation Captions For Instagram
Are you looking for the perfect way to sum up the culmination of your mathematical journey on Instagram? As you gear up for your graduation, let your mathematical prowess shine with these catchy and
clever Maths Graduation Captions for Instagram.
Whether you aced calculus or conquered complex equations, these captions are tailored to encapsulate the essence of your academic achievement. Get ready to add a touch of numerical flair to your
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Maths Graduation Captions For Instagram
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Funny Maths Graduation Captions For Instagram
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Short Maths Graduation Captions For Instagram
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Clever Maths Graduation Captions For Instagram
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Maths Graduation Quotes For Instagram
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River tides
From Coastal Wiki
The general characteristics of tidal rivers are described in the article Tidal rivers. This article is concerned with the characteristics of tidal propagation.
Why river tides matter
The issue of tidal propagation is important for several reasons:
• The tide often yields a major contribution to high water levels and to the associated risk of flooding. This is especially the case when the mean water level is raised by an exceptional river
discharge or by a storm surge at the estuarine mouth. The tide is therefore an important factor in the design and maintenance of flood defenses.
• The tide often plays a major role in processes of sedimentation and erosion. It therefore influences the navigable depth and the management strategies (e.g. dredging) to maintain the fairway.
• The tide enables the exchange of water, dissolved and particulate material and organisms between the river and adjacent wetlands. The tide therefore contributes to improving water quality and to
ecological diversity of the fluvial ecosystem, by offering a unique habitat for many species.
If the downstream estuary is not closed off by a weir or a dam, the tidal wave penetrates much further inland than the seawater, see the article Tidal rivers. The penetration length depends primarily
on the river discharge and the tidal amplitude at the river mouth (i.e. the head of the estuary). Frictional damping is the main process limiting tidal intrusion. Other factors limiting tidal
intrusion are:
• the presence of a weir or a dam where the tidal wave is reflected,
• a runoff speed which is higher than the tidal wave propagation speed.
In practice, a tidal wave generally does not propagate upstream beyond the location where the river bed level exceeds the mean water level at the river mouth. This is not because the waves cannot run
up the slope, but rather because friction is always strong in the upstream part of the tidal river ^[1].
River channel deepening
Many tidal rivers have seaports located far inland. Modern maritime traffic requires deep fairways, which are created by dredging of the natural tidal channels. Shoals are removed, channels are
excavated and maintained at necessary depth by dredging and by structures such as training walls and groins.
These interventions have several consequences:
• Increase in tidal range, illustrated in Fig. 1 for several dredged tidal rivers in Europe. Deepening and smoothing the fairway, removing sills, beams and other turbulence-creating bedforms
reduces frictional damping of the tidal wave. Frictional damping is further reduced as greater water depth promotes salinity and mud stratification of the water column (see Estuarine turbidity
• Increased risk of flooding when storm surge set-up coincides with springtide. The increase of tidal range mainly corresponds to a lowering of the LW tide level, as illustrated in Fig. 2 for the
Elbe tidal river. The increase of the HW tide level is therefore modest in comparison to the downward shift of the LW level. Besides, channel deepening reduces the sensitivity of high water
levels in the tidal river to high river runoff events^[2]. In microtidal environments, this can lead to a lower risk of flooding in case of extreme river discharge^[3].
• Increased seawater intrusion; the salinity limit is moved inland. The river runoff velocities are decreased due to the greater depth while tidal current velocities are strengthened due to reduced
frictional damping. The greater tidal excursion and stronger estuarine circulation bring seawater further inland.
• Trapping of fine sediment further upstream in the tidal river as river runoff velocities are reduced and tidal currents strengthened. Moreover, flood dominance associated with tidal asymmetry is
enhanced because channel deepening increases the tidal asymmetry component M4 to a similar extent as the tidal range^[2]. Fluid mud layers can form during periods of neap tide and slack water.
The positive feedback between tidal amplification and mud stratification can lead to hyperturbidity, as observed in the Ems tidal river^[4]. In many dredged tidal rivers (e.g. Ems, Gironde,
Humber), the turbidity maximum moves far upstream into the freshwater zone during periods of low river discharge^[5]^[6]. More detailed explanations are given in the article Estuarine turbidity
Tide-induced water level set-up
Several characteristics of tidal wave propagation in tidal rivers are illustrated in Fig. 3 that refers to the St. Lawrence River in Canada. Remarkable features are:
• the huge distance along the river over which the tide has a significant influence on water levels,
• the correlation between the tidal amplitude at the river mouth and the water level set-up upstream,
• the upstream distortion of the tide, with a fast tidal rise and a slow fall.
The St. Lawrence river follows a seismically active rift valley that surfaced after the last glacial period. The great tidal intrusion length is primarily due to the small bed slope of only a few
meters per 100 km in the downstream river reach and the substantial water depth that exceeds 10 m in many places.
The tide plays an important role in the upstream net water level set-up. During spring tide, much of the set-up is due to the nonlinear nature of frictional momentum dissipation (i.e. the second term
in the right-hand side of Eq. (A1), which is not a linear function of the discharge). The nonlinear frictional momentum dissipation transfers tidal momentum to a residual pressure gradient that
creates a water level slope. Transfer of tidal momentum to a residual pressure gradient also occurs due to the nonlinear nature of momentum advection in the tidal momentum balance (second term in the
left-hand side of Eq. (A1)). These two nonlinear flow processes are responsible for the smaller net water level set-up at the gauge station Trois-Rivières during neap tide compared to the larger
set-up during spring tide. A more detailed explanation is given in appendix A. Part of the water level set-up is further related to the water level slope needed to transmit the river runoff trough
the tidal river.
Tidal asymmetry
The tidal records displayed in Figs. 3 and 4 show that the sinusoidal character of the tide is progressively lost when moving upstream along the tidal river. Tidal rise is faster and tidal fall is
slower. This means that the flood component of the total discharge is a strong short pulse in comparison to the ebb component. The development of this type of tidal asymmetry is a common feature of
tidal propagation in tidal rivers. It is related to the faster propagation of the high water (HW) crest of the tidal wave compared to the slower propagation of the low water (LW) trough of the tidal
wave. An explanation is given in appendix C. A similar process also occurs in estuaries, see for example Tidal asymmetry and tidal inlet morphodynamics.
Tidal asymmetry is an important phenomenon for sediment transport in tidal rivers, especially in the downstream part of the tidal river during spring tides, when the maximum flood discharge exceeds
the river discharge^[11]. The strong current associated with the short flood pulse can mobilize and carry more sediment than the weaker ebb current, even in the presence of residual downstream
runoff. In many cases this results in a sediment convergence zone downstream of the site where the sediment transport capacity of the flood current is overturned by the transport capacity of the
combined ebb current and river discharge. This sediment convergence zone corresponds to the so-called estuarine turbidity maximum and is generally located around the transition between estuary and
tidal river. In periods of low river discharge and spring tide, fine sediment particles can be transported upstream over a long distance by tidal asymmetry. For example, fine sediments of marine
origin have been found in the docks of Rouen, 80 km upstream from the seawater intrusion limit^[12]. Deposits of fine marine sediments have also been found in the fresh water reach of the tidal
Scheldt river^[13].
Appendix A: Influence of friction on tidal propagation
We consider tidal rivers that are not bounded by a weir or a dam. The tide can propagate upstream till it has faded by frictional damping. Important characteristics of the tidal influence on the
water level variation in tidal rivers can be derived from the equations describing the tidal wave propagation. These equations express the momentum balance
[math]acceleration + inertia = pressure \; gradient - friction[/math]
and the volume (or mass) balance
[math]volume \; change = water \; transport \; gradient [/math] .
The corresponding formulas are
[math]\Large\frac{\partial Q}{\partial t}\normalsize + u \, \Large\frac{\partial Q}{\partial x}\normalsize = -gA \, \Large\frac{\partial \zeta}{\partial x}\normalsize - \Large\frac{c_D}{HA}\
normalsize Q |Q| \qquad (A1) \quad[/math] and [math]\quad b \, \Large\frac{\partial \zeta}{\partial x}\normalsize = - \Large\frac{\partial Q}{\partial x}\normalsize . \qquad (A2)[/math]
Meaning of the symbols (see Fig. A1):
[math]Q(x,t)[/math] is the water discharge at time [math]t[/math] through a river cross-section with area [math]A(x,t)=b(x)H(x,t)[/math] at distance [math]x[/math] from the seaward boundary, [math]b
(x)[/math] is the average cross-section width and [math]H(x,t)=h(x)+\zeta(x,t)[/math] is the water depth, [math]\zeta(x,t)[/math] is the water level with respect to a horizontal datum, [math]u(x,t)=Q
/A[/math] is the average current velocity in the cross-section, [math]c_D[/math] is the friction coefficient (typically of order 0.003) and [math]g[/math] is the gravitational acceleration. The
equations (A1) et (A2) are simplified expressions; major assumptions are: the cross-sectional variation of the velocity [math]u[/math] can be ignored, and the shape of the cross-section [math]A[/
math] can be approximated by a rectangle.
In tidal rivers, the terms in the right-hand side of Eq. (A1) are usually much larger than the terms in the left-hand side^[9]. The momentum balance can then be approximated by
[math] gA \, \Large\frac{\partial \zeta}{\partial x}\normalsize =- \Large\frac{c_D}{HA}\normalsize Q |Q| . \qquad (A3)[/math]
We consider a horizontal tide [math]Q(x,t)[/math] with a single component of radial frequency [math]\omega[/math],
[math]Q(x,t) = -Q_0 (x) + Q_1 (x) \cos \theta, \quad \zeta (x,t) = \zeta_0 (x) + \zeta_1 (x) \cos (\theta + \psi(x)) \, + h.h. \, , \quad \theta = \omega t + \phi(x) , \qquad (A4)[/math]
where [math]h.h.[/math] stands for terms of higher harmonic order. There is no weir or dam where the tidal wave is reflected. Sufficiently far upstream (large positive [math]x[/math]), the fluvial
discharge [math]Q_0[/math] is larger than the maximum tidal discharge [math]Q_1[/math]. In this case the friction term is given by
[math] - \Large\frac{c_D}{HA}\normalsize Q |Q| = \Large\frac{c_D}{HA}\normalsize Q^2 = \Large\frac{c_D}{HA}\normalsize \Big( Q_0^2 + \frac{1}{2} Q_1^2 - 2 Q_0 Q_1 \cos \theta + \frac{1}{2} Q_1^2 \cos
2 \theta \Big) = gA \, \Large\frac{\partial \zeta}{\partial x}\normalsize . \qquad (A5)[/math]
In the case of strong friction, due to the [math]\frac{1}{2} Q_1^2 [/math]-term, a steep water level slope [math]\Large\frac{\partial \zeta_0}{\partial x}\normalsize [/math] is required to enable the
net runoff [math]Q_0[/math]. This implies that the mean height of the upstream water level [math]\zeta_0(x)[/math] is positively correlated with the tidal amplitude at the river mouth.
The term [math]-2 Q_0 Q_1 \cos \theta[/math] produces a negative gradient [math]\Large\frac{\partial \zeta_1}{\partial x}\normalsize [/math] and therefore dampens the tidal component of the upstream
water level. Equation (A5) also shows that the friction generates a higher harmonic tidal component through the term [math]\frac{1}{2} Q_1^2 \cos(2 \theta)[/math].
A similar tidal influence on riverine water levels occurs when [math]Q_1 \gt Q_0[/math]. The mathematical development of the friction term is more complicated in this case, see Appendix B. The
contributions of the friction term to the mean upstream water level (through the mean water level slope [math]\Large\frac{\partial \zeta_0}{\partial x}\normalsize [/math]), to the principal tidal
component [math]\zeta_1[/math] and to higher tidal components are shown in Fig. A1 as a function of the relative strength of the fluvial discharge, [math]Q_0/Q_1[/math]. The friction and pressure
terms are developed in harmonic components of successive order, and Eq. (A3) is rewritten as
[math]Q |Q| = Q_1^2 \big( f_0 + f_1 \cos \theta + f_2 \cos 2 \theta + f_3 \cos 3 \theta \big) = - \Large\frac{gHA^2}{c_D}\normalsize \, \Large\frac{\partial \zeta}{\partial x}\normalsize = - \Large\
frac{gHA^2}{c_D}\normalsize \, \Big(\Large\frac{\partial \zeta_0}{\partial x}\normalsize + \Large\frac{\partial}{\partial x}\normalsize (\zeta_1 \cos( \theta +\psi)) + h.h. \, \Big). \qquad (A6)[/
For a given value of the maximum tidal discharge [math]Q_1[/math], the term [math]f_1[/math] represents the influence of the fluvial discharge [math]Q_0[/math] on the damping of the upstream tidal
amplitude [math]\zeta_1[/math]. Figure A3 shows that for values of [math]Q_0 \gt \frac{1}{2} Q_1[/math], the damping increases linearly with [math]Q_0[/math]. In contrast, no strong higher harmonic
tides [math]\zeta_2[/math] and [math]\zeta_3[/math] are generated by the fluvial discharge.
For a given fluvial discharge [math]Q_0[/math], the mean water level slope [math]\Large\frac{\partial \zeta_0}{\partial x}\normalsize [/math] is proportional to [math]-f_0 Q_1^2/Q_0^2[/math]. This
proportionality is shown in Fig. A2 as a function of [math]Q_0/Q_1[/math]. This figure shows that the upstream water level increase is strongly enhanced when the tidal discharge [math]Q_1[/math]
increases compared to the fluvial discharge [math]Q_0[/math].
In conclusion, whilst the tidal amplitude is strongly damped in the case of large fluvial discharge, this decrease of the tidal amplitude goes along with a substantial increase of the mean water
level in the upstream tidal river.
The expressions (A5) and (A6) show that the water level slope [math] \left\vert \Large{ \partial \zeta \over \partial x }\normalsize \right\vert [/math] is smaller during maximum flood flow [math]\
theta=0[/math] than during maximum ebb flow [math]\theta=\pi[/math] . Assuming that the propagation speed of the tidal wave is the same at maximum flood flow and maximum ebb flow, then an observer at
a fixed location will see a relatively slower water level rise at maximum flood flow and a relatively faster water level fall at maximum ebb flow. Water level records display the opposite behavior,
as illustrated in Fig. 2: a faster water level rise (steep dipping wave front [math]\Large\frac{\partial \zeta}{\partial x}\normalsize \lt 0[/math]) and a slower water level fall (gently rising level
behind the wave front [math]\Large\frac{\partial \zeta}{\partial x}\normalsize \gt 0[/math]). This behavior is due to the faster propagation (diffusion) speed of the high water (HW) wave crest
relative to the low water (LW) wave trough. Two responsible effects are:
1. The frictional retardation of the propagation speed is smaller at HW than at LW because the water depth [math]h[/math] is higher and the cross-sectional area [math]A[/math] greater at HW relative
to LW;
2. The HW propagation speed is enhanced by advection with the positive flood flow, while the LW propagation speed is retarded due to the opposing ebb flow.
These features are not included in the simplified equations (A2) and (A3). An explanation of the higher propagation speed of the HW tidal wave crest relative to the LW trough is given in appendix C
and in the article Tidal asymmetry and tidal inlet morphodynamics.
Appendix B: Development of the friction term
The development of the friction term for [math]Q_1 \gt Q_0[/math] was given by Dronkers (1964^[14]). In the case of a single harmonic tide [math]Q = -Q_0 + Q_1 \cos \theta[/math], the friction term
is written as
[math]Q|Q| = Q_1^2 (\cos \alpha + \cos \theta) |\cos \alpha + \cos \theta|, \quad \cos \alpha = - Q_0/Q_1 \equiv -y. \qquad (B1)[/math]
The right-hand side can be developed into a sum of Chebyshev polynomials. A high degree of accuracy is achieved with only 4 terms. The final result is
[math] \Large\frac{ Q |Q| }{Q_1^2}\normalsize = f_0 + f_1 \cos \theta + f_2 \cos 2 \theta + f_3 \cos 3 \theta , \qquad (B2) [/math]
[math]f_0=d_0+ \frac{1}{2}d_2 - (d_1+ \frac{3}{2}d_3)y + d_2 y^2-d_3y^3, \; f_1=d_1 + \frac{3}{4}d_3 -2d_2y+3d_3y^2, \; f_2=\frac{1}{2}d_2-\frac{3}{2} d_3 y, \; f_3=\frac{1}{4}d_3. \qquad (B3)[/math]
The coefficients [math]d_0, d_1, d_2, d_3[/math] are given by
[math]d_0 = -\frac{7}{120} \sin 2 \alpha + \frac{1}{24} \sin 6 \alpha - \frac{1}{60} \sin 8 \alpha[/math]
[math]d_1 = \frac{7}{6} \sin \alpha - \frac{7}{30} \sin 3 \alpha - \frac{7}{30} \sin 5 \alpha + \frac{1}{10} \sin 7 \alpha[/math]
[math]d_2 = \pi - 2 \alpha + \frac{1}{3} \sin 2 \alpha + \frac{19}{30} \sin 4 \alpha - \frac{1}{5} \sin 6 \alpha[/math]
[math]d_3 = \frac{4}{3} \sin \alpha -\frac{2}{3} \sin 3 \alpha + \frac{2}{15} \sin 5 \alpha [/math]
Appendix C: Tidal asymmetry
Friction-dominance and [math]Q_0 \gt Q_1[/math] are assumed. According to (A1, A5) the tidal equations can be written as:
[math]b\Large\frac{\partial \zeta}{\partial t}\normalsize + \Large\frac{\partial Q}{\partial x}\normalsize = 0 , \quad Q^2 = \Large\frac{g \, b^2 (h+\zeta)^3}{c_D} \frac{\partial \zeta}{\partial x}\
normalsize \normalsize . \qquad (C1)[/math]
Derivation of the second equation and substitution in the first equation gives
[math]\Large\frac{\partial \zeta}{\partial t}\normalsize = - \Large\frac{1}{b}\frac{\partial Q}{\partial x}\normalsize = D \, \Biggl(\Large\frac{\partial^2 \zeta}{\partial x^2}\normalsize + \Large\
frac{3}{h+\zeta}\normalsize \Bigl(\Large\frac{dh}{dx}\normalsize +\Large\frac{\partial \zeta}{\partial x}\normalsize \Bigr) \Large\frac{\partial \zeta}{\partial x}\normalsize \Biggr) , \quad D = \
Large\frac{g \, b \, (h+\zeta)^3}{2c_D \left \vert Q \right \vert}\normalsize. \qquad (C2)[/math]
Far upstream the variation of the water depth [math]h+\zeta[/math] along the tidal river is small ([math]\Large\frac{\partial \zeta}{\partial x}\normalsize \sim - \Large\frac{dh}{dx}\normalsize [/
math]) if [math]Q_0[/math] is close to the mean river discharge. In this case the upstream propagation of HW and LW into the tidal river is approximately described by a diffusion equation with
time-varying diffusion coefficient,
[math]\Large\frac{\partial \zeta}{\partial t}\normalsize = D \, \Large\frac{\partial^2 \zeta}{\partial x^2}\normalsize . \qquad (C3)[/math]
[The case where [math]Q_0[/math] is very different from the mean discharge is discussed in Kästner et al. (2019^[1])].
Around high water (HW), the tidal component of the discharge is directed upstream, and around low water (LW), downstream. The total discharge [math]|Q|[/math] is thus larger around low water than
around high water. This implies that the diffusion coefficient [math]D[/math] defined in Eq. (C2) is larger around high water than around low water. High water will thus diffuse faster upstream into
the tidal river than low water, shortening the flood period relative to the ebb period. Since the ebb and flood tidal prisms are equal, the tidal component of the current must be on average stronger
during flood than during ebb.
Assuming [math]|Q_1|\lt \lt |Q_0|[/math] and [math]a\lt \lt h[/math], Eq. (2) can be simplified to [math]D \approx \Large\frac{\omega}{2 k^2}\normalsize (1+ 3 \Large\frac{\zeta}{h}\normalsize)[/math]
, with [math]k = \sqrt{\Large\frac{c_D \omega Q_0}{g b h^3}\normalsize}[/math]. If we further assume that the mean depth [math]h[/math] is uniform, Eq. (C2) can be solved analytically up to order
[math]\zeta (x,t)=a e^{-kx} \cos(\omega t -kx) + \Large\frac{3 a^2}{4h}\normalsize \Big( e^{-\sqrt{2}kx} \cos(2 \omega t - \sqrt{2}kx) - e^{-2kx} \cos(2 \omega t -2kx) \Big) . \qquad (C4)[/math]
We consider the part of the tidal river where [math]kx \lt \lt 1[/math]. By expanding the expression (C4) around the phases of HW and LW, where [math]\partial \zeta / \partial t =0[/math], the
upstream propagation speeds [math]c^+[/math] and [math]c^-[/math] can be derived of HW and LW respectively,
[math]c^{\pm} = \Large\frac{\omega}{k}\normalsize \Big(1 \pm \Large\frac{3a}{2h}\normalsize (2- \sqrt{2}) \Big) . \qquad (C5)[/math]
This expression also indicates flood dominance due to the higher propagation speed of HW compared to LW. Flood dominance also occurs in the downstream part of the tidal river and in the estuary,
where the tidal component of the discharge is much larger than the fluvial component, see Tidal asymmetry and tidal basin morphodynamics. In fact, dominance of the maximum tidal discharge during
flood over the maximum tidal discharge during ebb occurs over the entire range of tidal rivers where tidal propagation is strongly influenced by bottom friction.
Related articles
1. ↑ ^1.0 ^1.1 Kästner, K., Hoitink, A. J. F. T., Torfs, P. J. J. F., Deleersnijder, E. and Ningsih, N. S. 2019. Propagation of tides along a river with a sloping bed. Journal of Fluid Mechanics
872: 39–73
2. ↑ ^2.0 ^2.1 Pareja-Roman, L. F., Chant, R. J. and Sommerfield, C. K. 2020. Impact of historical channel deepening on tidal hydraulics in the Delaware Estuary. Journal of Geophysical Research:
Oceans 125, e2020JC016256
3. ↑ Talke, S. A., Familkhalili, R. and Jay, D. A. 2021. The influence of channel deepening on tides, river discharge effects, and storm surge. Journal of Geophysical Research: Oceans 126,
4. ↑ Winterwerp, J.C., Vroom, J., Wang, Z.B., Krebs, M., Hendriks, E.C.M., van Maren, D.S., Schrottke, K., Borgsmüller, C. and Schöl, A. 2017. SPM response to tide and river flow in the hyper-turbid
Ems River. Ocean Dyn. 67:559–83
5. ↑ Uncles, R. J., Stephens, J. A. and Law, D. J. 2006. Turbidity maximum in the macrotidal, highly turbid Humber Estuary, UK: Flocs, fluid mud, stationary suspensions and tidal bores. Estuarine,
Coastal and Shelf Science 67: 30–52
6. ↑ Burchard, H., Schuttelaars, H. M. and Ralston, D. K. 2018. Sediment trapping in estuaries. Annual Review of Marine Science 10: 371–395
7. ↑ Winterwerp, J.C. 2013. On the response of tidal rivers to deepening and narrowing. Report for the Flemish-Dutch Scheldt Committee research program 'LTV Safety and Accessibility'.
8. ↑ ABP mer (Whitehead, P.A. ed.) 2011. River Elbe River Engineering and Sediment Management Concept Review of sediment management strategy in the context of other European estuaries from a
morphological perspective. Hamburg Port Authority (HPA) and Federal Waterways and Shipping Administration (WSA) Report R.1805
9. ↑ ^9.0 ^9.1 LeBlond, P. H. 1979. Forced fortnightly tides in shallow waters. Atmos.-Ocean 17: 253– 264
10. ↑ LeBlond, P.H. 1978. On tidal propagation in shallow rivers. J. Geophys. Res. 83: 4714-4721
11. ↑ Dronkers, J. 2017. Dynamics of Coastal Systems, 2nd ed. Advanced Series on Ocean Engineering 41. World Scientific Publ. Co., Singapore. 753 pp.
12. ↑ Guezennec, L., Lafite, R., Dupont, J-P., Meyer, R. and Boust, D. 1999. Hydrodynamics of suspended particulate matter in the tidal freshwater zone of a microtidal estuary. Estuaries 22: 717-727
13. ↑ Salomons, W. and Mook, W.G. 1981. Field observations of isotopic composition of particulate organic carbon in the Southern North Sea and adjacent estuaries. Mar.Geol. 41: 11-20
14. ↑ Dronkers, J.J. 1964. Tidal computations in rivers and coastal waters. North Holland Publ. Co., Amsterdam, 518 pp.
The main author of this article is Job Dronkers
Please note that others may also have edited the contents of this article.
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Suppose a food scientist wants to determine whether two experimental preservatives result in different mean shelf lives for bananas. He treats a simple random sample of 15 bananas with one of the preservatives.
Suppose a food scientist wants to determine whether two experimental preservatives result in different mean shelf lives for bananas. He treats a simple random sample of 15 bananas with one of the
preservatives. He then collects another simple random sample of 20 bananas and treats them with the other preservative. As the bananas age, the food scientist records the shelf life of all bananas in
both samples. The food scientist does not know the population standard deviations. What test should the food scientist run in order to determine if the two experimental preservatives result in
different mean shelf lives for bananas
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Deep Learning Quantum Computing
Quantum computing has emerged as a promising technology that has the potential to revolutionize various industries, including artificial intelligence. Deep learning, a subfield of machine learning,
has gained significant attention in recent years due to its ability to analyze complex data and make accurate predictions. When combined with quantum computing, deep learning algorithms can leverage
the unique properties of quantum systems to enhance computational performance and enable faster and more efficient computations.
Key Takeaways
• Deep learning and quantum computing have the potential to revolutionize various industries.
• Combining deep learning algorithms with quantum computing can enhance computational performance.
• Quantum systems offer unique properties that can enable faster and more efficient computations.
**Deep learning** is a subfield of machine learning that focuses on training artificial neural networks to process and analyze complex data. It has been successful in various applications, such as
image recognition, natural language processing, and speech recognition. Deep learning algorithms rely on massive computational power to train complex models, and this is where quantum computing comes
into play.
**Quantum computing**, unlike classical computing, leverages the principles of quantum mechanics to encode and process information. Quantum systems, represented by quantum bits or qubits, can exist
in multiple states simultaneously. This property, known as superposition, allows quantum algorithms to perform parallel computations and potentially solve certain problems significantly faster than
classical computers.
*Quantum computing holds the promise of tackling computational challenges that are currently beyond the capabilities of classical computers.
Deep Learning Quantum Algorithms
Researchers have been exploring the integration of deep learning algorithms with quantum computing. By using quantum circuits to process data, deep learning algorithms can take advantage of quantum
effects to improve their computational efficiency. *Machine learning tasks can be further accelerated by mapping them onto quantum algorithms designed for specific purposes.
One significant application of deep learning quantum algorithms is the enhancement of **optimization problems**. Quantum computers can efficiently solve optimization problems, which have widespread
applications in various fields, including finance, logistics, and energy optimization. Deep learning techniques can be used to design quantum algorithms that find optimal solutions faster and more
accurately than classical methods.
*Combining deep learning with quantum computing shows promise in improving the performance of generative models, such as **generative adversarial networks** (GANs). GANs are used for tasks like image
generation and data synthesis. Quantum algorithms can enhance GANs by offering more expressive models and faster training times.
Advantages of Deep Learning Quantum Computing
Advantages Explanation
Speedup Quantum computing can offer significant speedup for certain computational tasks compared to classical computing.
Enhanced Learning Capability Quantum computing can provide more powerful computational models that enable deeper analysis of complex data.
Parallelism Quantum systems can perform parallel computations, enabling faster training of deep learning models.
Quantum computing combined with deep learning offers several advantages over classical computing for various applications.
1. **Speedup**: Quantum computers have the potential to solve certain computational problems significantly faster than classical computers, leading to faster model training and prediction times.
2. **Enhanced Learning Capability**: Quantum systems can provide more powerful computational models that enable deeper analysis of complex data, allowing for more accurate predictions.
3. **Parallelism**: Quantum computing enables parallel computations, which in turn allows for faster training of deep learning models and more efficient processing of large datasets.
Future Outlook
Deep learning quantum computing is an exciting field with immense potential. As both deep learning and quantum computing continue to advance, the synergy between the two fields is expected to unlock
new frontiers in artificial intelligence and computational science. We can anticipate groundbreaking discoveries and significant advancements in areas such as drug discovery, optimization, and
pattern recognition. The progress made in this field will pave the way for a future where complex problems can be solved efficiently and accurately using quantum-enabled deep learning algorithms.
With the integration of deep learning and quantum computing, we are on the verge of a technological leap that will redefine the boundaries of computational capabilities. As researchers continue to
push the limits of what is possible, the potential impact of deep learning quantum computing is truly staggering.
Common Misconceptions
Deep Learning and Quantum Computing
There are several common misconceptions about the intersection of deep learning and quantum computing that often lead to misunderstandings. One misconception is that deep learning and quantum
computing are competing technologies that cannot be combined. In reality, deep learning and quantum computing can work together to enhance each other’s capabilities and solve complex problems.
• Deep learning and quantum computing are complementary technologies.
• The combination of deep learning and quantum computing can lead to more efficient algorithms.
• Deep learning algorithms can benefit from quantum-inspired techniques.
Another misconception is that quantum computing can completely replace deep learning. While quantum computing has the potential to revolutionize many areas of computation, including machine learning,
it does not render deep learning obsolete. Deep learning models still excel in many areas and can be used in conjunction with quantum computing for improved performance and accuracy.
• Quantum computing can enhance deep learning models but not replace them entirely.
• Deep learning is still the preferred approach in many practical scenarios.
• Combining quantum computing and deep learning can provide a more powerful and versatile solution.
One common misconception is that quantum computers will make deep learning models exponentially faster. While quantum computers have the potential to solve certain problems faster than classical
computers, they do not necessarily speed up deep learning training processes. The benefits of quantum computing in deep learning primarily lie in their ability to tackle complex optimization tasks
and analyze large datasets more efficiently.
• Quantum computers do not necessarily speed up the training of deep learning models.
• Quantum computing can help with optimization and large-scale data analysis in deep learning.
• The true advantage of quantum computing lies in specific tasks rather than overall speed improvements.
Some people believe that quantum computers will instantly solve all problems in deep learning, making it easier for everyone. This is a misconception as quantum computers present new challenges and
require specialized knowledge to utilize effectively. Quantum algorithms and their implementation are still under development, and understanding them requires a deep understanding of both quantum
mechanics and deep learning algorithms.
• Quantum computing introduces new challenges and requires specialized knowledge.
• Quantum algorithms and their implementation are still being developed and are not yet widely accessible.
• Achieving successful integration of quantum computing in deep learning requires expertise in both quantum mechanics and deep learning concepts.
Lastly, there is a misconception that deep learning models can be directly transferred to quantum computers without any modifications. In reality, quantum computers operate on a fundamentally
different paradigm and utilize qubits instead of classical bits. Deep learning models need to be adapted to suit the strengths and limitations of quantum computing, such as utilizing quantum-inspired
algorithms and rethinking the representation and processing of data.
• Deep learning models need to be adapted to leverage the unique characteristics of quantum computers.
• Quantum-inspired algorithms can enhance the performance of deep learning on quantum computers.
• Data representation and processing need to be reconsidered to fully utilize quantum computing in deep learning.
Comparing Deep Learning and Quantum Computing
Deep learning and quantum computing are two cutting-edge technologies that have the potential to revolutionize various fields. The following tables highlight key elements and features of both
technologies to better understand their capabilities and applications.
Deep Learning Algorithms
Deep learning algorithms are a fundamental component of deep learning models. The tables below showcase some commonly used algorithms and highlight their strengths and applications.
Quantum Computing Architectures
Quantum computing architectures form the basis for harnessing the power of quantum systems. The following tables illustrate different types of architectures and their unique characteristics, enabling
us to perform complex computations more efficiently.
Deep Learning Applications
Deep learning finds applications in numerous fields. The tables below provide insight into some of the key areas where deep learning techniques have been successfully implemented, leading to
significant advancements and breakthroughs.
Quantum Computing Applications
Quantum computing has the potential to transform sectors that rely heavily on computational power. The tables presented below showcase a range of application domains where quantum computing can
provide significant advantages over classical computing methodologies.
Deep Learning Hardware
Specific hardware components and accelerators are designed to optimize deep learning tasks. The tables below provide details on various hardware technologies and their respective characteristics,
contributing to faster and more efficient deep learning processing.
Quantum Computing Quantum Bits (Qubits)
Quantum bits, or qubits, are the fundamental units of quantum information. The tables below explore different types of qubits and their unique properties, essential for quantum computing operations.
Deep Learning Challenges
Despite the remarkable progress, deep learning encounters challenges that researchers continue to tackle. The tables below highlight some of the key obstacles faced in deep learning model design and
Quantum Computing Challenges
Quantum computing faces several challenges that need to be addressed for wider adoption and practical utilization. The tables below outline the key challenges associated with quantum computing,
emphasizing the areas of active research and development.
Comparison of Deep Learning and Quantum Computing
Deep learning and quantum computing, although distinct technologies, share some similarities and differences. The tables below provide a comprehensive comparison between these two paradigms, helping
to elucidate their contrasting characteristics and potential synergies.
Deep learning and quantum computing represent significant advancements in the fields of machine learning and computation. While deep learning excels in processing vast amounts of data for complex
pattern recognition, quantum computing harnesses the power of quantum mechanics to solve computationally infeasible problems. These technologies have immense potential to drive innovation and propel
us towards new frontiers. By further understanding their unique attributes and exploring their applications, we unlock a world of possibilities.
Frequently Asked Questions
1. What is Deep Learning?
Deep learning is a subfield of machine learning that focuses on algorithms inspired by the structure and function of the human brain’s neural networks. It involves training artificial neural networks
with multiple layers to learn and make predictions from large amounts of data.
2. What is Quantum Computing?
Quantum computing is an area of computing that utilizes the principles of quantum physics to perform computations. Unlike classical computers that store and process information using bits, quantum
computers use quantum bits or qubits, which can exist in multiple states simultaneously, enabling exponential parallelism and the potential to solve complex problems more efficiently.
3. How do Deep Learning and Quantum Computing relate?
Deep learning and quantum computing are two distinct fields that can complement each other. Deep learning techniques can be applied to analyze and extract insights from quantum computational results
and quantum data. Similarly, quantum computing can potentially enhance deep learning algorithms by providing more efficient computations and improved optimization.
4. What are the potential applications of Deep Learning in Quantum Computing?
Deep learning can find applications in various aspects of quantum computing, including quantum error correction, quantum state tomography, and quantum circuit design optimization. It can assist in
making sense of the vast amounts of data generated by quantum systems and extract patterns and meaningful information to drive advancements in quantum technologies.
5. How can quantum computing enhance Deep Learning algorithms?
Quantum computing can potentially improve deep learning algorithms by accelerating training processes through quantum speedup. It can also aid in solving complex optimization and combinatorial
problems, which are at the heart of many deep learning tasks. Quantum annealing, for example, can be utilized to optimize hyperparameters and improve overall model performance.
6. Are there any quantum-inspired approaches to Deep Learning?
Yes, there are quantum-inspired approaches to deep learning, such as quantum-inspired neural networks, variational quantum algorithms, and quantum-inspired optimization techniques. These approaches
attempt to capture some of the advantageous features of quantum computing without the need for actual quantum hardware, bringing potential benefits in certain deep learning scenarios.
7. What are the challenges in combining Deep Learning and Quantum Computing?
Combining deep learning and quantum computing faces several challenges. Quantum hardware is still in its early stages of development, and scalability is a significant hurdle. Additionally, mapping
complex deep learning architectures onto limited qubit resources and managing noise in quantum computations are ongoing research challenges that need to be addressed to achieve effective integration.
8. Are there any real-world examples of the integration of Deep Learning and Quantum Computing?
While still in the early stages, there are promising real-world examples of the integration of deep learning and quantum computing. For instance, researchers have used deep learning to assist in the
discovery of new quantum materials and to analyze data generated by quantum simulators. These intersections hold great potential for advancing both fields in the future.
9. How can I get started with Deep Learning and Quantum Computing?
To get started with deep learning and quantum computing, it is recommended to have a foundational understanding of both fields. Familiarize yourself with concepts in machine learning, neural
networks, and quantum physics. There are online courses, tutorials, and resources available to help you learn and experiment with deep learning libraries and quantum development frameworks like
TensorFlow Quantum or Qiskit.
10. What is the future outlook for the integration of Deep Learning and Quantum Computing?
The integration of deep learning and quantum computing holds immense promise for solving complex problems in various domains. It has the potential to revolutionize industries such as pharmaceuticals,
weather forecasting, optimization, and cryptography. As both fields continue to advance, we can expect to witness novel synergies and breakthroughs that propel scientific, technological, and societal | {"url":"https://getneuralnet.com/deep-learning-quantum-computing/","timestamp":"2024-11-10T05:03:36Z","content_type":"text/html","content_length":"64364","record_id":"<urn:uuid:80822611-78ff-4e82-ab36-9b05d10b3aab>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00731.warc.gz"} |
{-# LANGUAGE BangPatterns #-}
{-| Utility functions for statistical accumulation. -}
Copyright (C) 2014 Google Inc.
All rights reserved.
Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are
1. Redistributions of source code must retain the above copyright notice,
this list of conditions and the following disclaimer.
2. Redistributions in binary form must reproduce the above copyright
notice, this list of conditions and the following disclaimer in the
documentation and/or other materials provided with the distribution.
module Ganeti.Utils.Statistics
( Statistics
, TagTagMap
, AggregateComponent(..)
, getSumStatistics
, getStdDevStatistics
, getMapStatistics
, getStatisticValue
, updateStatistics
) where
import qualified Data.Foldable as Foldable
import Data.List (foldl')
import qualified Data.Map as Map
-- | Type to store the number of instances for each exclusion and location
-- pair. This is necessary to calculate second component of location score.
type TagTagMap = Map.Map (String, String) Int
-- | Abstract type of statistical accumulations. They behave as if the given
-- statistics were computed on the list of values, but they allow a potentially
-- more efficient update of a given value.
data Statistics = SumStatistics Double
| StdDevStatistics Double Double Double
-- count, sum, and not the sum of squares---instead the
-- computed variance for better precission.
| MapStatistics TagTagMap deriving Show
-- | Abstract type of per-node statistics measures. The SimpleNumber is used
-- to construct SumStatistics and StdDevStatistics while SpreadValues is used
-- to construct MapStatistics.
data AggregateComponent = SimpleNumber Double
| SpreadValues TagTagMap
-- Each function below depends on the contents of AggregateComponent but it's
-- necessary to define each function as a function processing both
-- SimpleNumber and SpreadValues instances (see Metrics.hs). That's why
-- pattern matches for invalid type defined as functions which change nothing.
-- | Get a statistics that sums up the values.
getSumStatistics :: [AggregateComponent] -> Statistics
getSumStatistics xs =
let addComponent s (SimpleNumber x) =
let !s' = s + x
in s'
addComponent s _ = s
st = foldl' addComponent 0 xs
in SumStatistics st
-- | Get a statistics for the standard deviation.
getStdDevStatistics :: [AggregateComponent] -> Statistics
getStdDevStatistics xs =
let addComponent (n, s) (SimpleNumber x) =
let !n' = n + 1
!s' = s + x
in (n', s')
addComponent (n, s) _ = (n, s)
(nt, st) = foldl' addComponent (0, 0) xs
mean = st / nt
center (SimpleNumber x) = x - mean
center _ = 0
nvar = foldl' (\v x -> let d = center x in v + d * d) 0 xs
in StdDevStatistics nt st (nvar / nt)
-- | Get a statistics for the standard deviation.
getMapStatistics :: [AggregateComponent] -> Statistics
getMapStatistics xs =
let addComponent m (SpreadValues x) =
let !m' = Map.unionWith (+) m x
in m'
addComponent m _ = m
mt = foldl' addComponent Map.empty xs
in MapStatistics mt
-- | Obtain the value of a statistics.
getStatisticValue :: Statistics -> Double
getStatisticValue (SumStatistics s) = s
getStatisticValue (StdDevStatistics _ _ var) = sqrt var
getStatisticValue (MapStatistics m) = fromIntegral $ Foldable.sum m - Map.size m
-- Function above calculates sum (N_i - 1) over each map entry.
-- | In a given statistics replace on value by another. This
-- will only give meaningful results, if the original value
-- was actually part of the statistics.
updateStatistics :: Statistics -> (AggregateComponent, AggregateComponent) ->
updateStatistics (SumStatistics s) (SimpleNumber x, SimpleNumber y) =
SumStatistics $ s + (y - x)
updateStatistics (StdDevStatistics n s var) (SimpleNumber x, SimpleNumber y) =
let !ds = y - x
!dss = y * y - x * x
!dnnvar = (n * dss - 2 * s * ds) - ds * ds
!s' = s + ds
!var' = max 0 $ var + dnnvar / (n * n)
in StdDevStatistics n s' var'
updateStatistics (MapStatistics m) (SpreadValues x, SpreadValues y) =
let nm = Map.unionWith (+) (Map.unionWith (-) m x) y
in MapStatistics nm
updateStatistics s _ = s | {"url":"https://docs.ganeti.org/docs/ganeti/2.16/api/hs/Ganeti/Utils/Statistics.html","timestamp":"2024-11-08T09:06:06Z","content_type":"text/html","content_length":"25287","record_id":"<urn:uuid:55ecd4a6-293f-42db-8382-03a9fbeddfc1>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00533.warc.gz"} |
Acres Per Hour Calculator - Treeier
This acres per hour calculator helps you determine the speed and time required to complete agricultural tasks on your land. Even if you own a garden, you can use it as a lawn mowing time calculator.
Read on to learn more about the formulas behind this tool, along with suggestions for other useful calculators for your agricultural or gardening needs.
Acres per Hour Calculator
Tool’s Width (cm):
Speed (km/h):
Area per Hour (hectares/hour):
Plot Area (hectares):
Time to Complete (hours): hrs min
Overlap (%):
How to Use the Acres Per Hour Calculator
Whether you’re plowing, seeding, fertilizing, or mowing your lawn, the acres per hour calculator is easy to use for any task.
To calculate how many acres per hour you can cover, you’ll need to input:
• The effective width of your tool (e.g., mower, plow, combine harvester)
• The average speed of your machine
If you want to calculate how long it will take to complete the work on your field, lawn, or garden, provide:
• The effective width of your tool
• The average speed of work
• The area of your plot (in acres, hectares, or any other unit)
🔎 If you’re unsure about the size of your plot, you can use our area calculator to find it, whether it’s a regular or irregular shape.
Formulas Behind the Mowing Time Calculator
The calculator uses two formulas. The first formula calculates the area you can cover per hour:
aps=tw × ts × (1−o)
• aps: Area per second
• tw: Tool width (e.g., mower width)
• ts: Machine speed
• o: Overlap percentage
🙋 Machines typically don’t use the full effective width due to overlap. By default, the calculator assumes a 10% overlap, but you can adjust this to 0% if overlap isn’t needed. You can modify this
in the “Overlapping Area” section.
The second formula estimates the time required to complete the task:
• t: Time needed to finish the task
• a: Plot area
• aps: Area per second, calculated using the first formula
These simple formulas allow you to accurately estimate the time and speed needed for your agricultural or lawn maintenance tasks.
Factors Affecting Mowing Efficiency
There are several factors that influence how many acres can be mowed in a given time. These include:
1. Tool Width: The wider your mower or agricultural tool, the more ground you can cover in one pass. Tool width is typically measured in centimeters (cm). Larger mowers are able to cover more area
in a shorter amount of time.
2. Speed: The faster your machine moves, the more ground it covers per hour. Speed is usually measured in kilometers per hour (km/h). However, faster speeds can sometimes lead to less precise work,
so speed must be balanced with the quality of mowing or plowing.
3. Overlap: Overlap refers to the amount of tool width that overlaps with the previous pass to ensure even coverage. A common overlap percentage is 10%, meaning that only 90% of the tool’s width is
effectively used. Reducing overlap increases efficiency but can risk uneven mowing.
4. Plot Area: The size of the land being worked on is crucial for determining how long the work will take. Larger plots will naturally require more time to complete.
Example: Understanding the Acres Per Hour Calculator
Let’s walk through a real-world example to understand how the acres per hour calculator works.
You own a field that needs mowing, and you have the following information:
• Tool width: 150 cm (1.5 meters)
• Speed of mower: 6 km/h
• Overlap: 10%
• Plot size: 5 hectares
Step 1: Calculate Effective Width
The first step is to adjust the tool width based on the overlap. Since the overlap is 10%, it means that only 90% of the tool’s width is used effectively.
Effective width = Tool width × (1 – Overlap)
Effective width = 1.5 m × (1 – 0.10) = 1.35 meters
So, 1.35 meters is the actual width used for mowing.
Step 2: Calculate Area Per Hour
Next, you can calculate the number of acres you can mow per hour. Since you’re moving at 6 km/h and the effective width is 1.35 meters, the formula is:
The division by 10 converts the result into hectares per hour.
This means that with the given parameters, you can mow 0.81 hectares per hour.
Step 3: Calculate Total Time to Mow the Plot
The next step is to determine how long it will take to mow the entire plot, which is 5 hectares in size. To do this, divide the total plot size by the area covered per hour:
So, it will take approximately 6 hours and 10 minutes to mow the entire 5-hectare plot.
Step 4: Calculate Mowing Efficiency Over Different Hours
Once you know how many hectares you can mow in one hour, you can easily calculate how many hectares can be mowed in multiple hours. Let’s say you want to know how much area you can mow in 2, 3, 4,
and 5 hours:
• 1 hour: 0.81 hectares
• 2 hours: 0.81 × 2 = 1.62 hectares
• 3 hours: 0.81 × 3 = 2.43 hectares
• 4 hours: 0.81 × 4 = 3.24 hectares
• 5 hours: 0.81 × 5 = 4.05 hectares
You can mow approximately 4.05 hectares in 5 hours.
Summary of the Calculator’s Usage
This calculator is extremely helpful for farmers and gardeners who need to plan their agricultural activities. By inputting the tool’s width, speed, and overlap, users can estimate how much area
can be covered per hour. Additionally, it allows you to predict how long a job will take based on the size of your plot. | {"url":"https://treeier.com/acres-per-hour-calculator/","timestamp":"2024-11-08T11:04:04Z","content_type":"text/html","content_length":"289952","record_id":"<urn:uuid:c24ca711-9816-4a12-8625-a2bfde50acc2>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00615.warc.gz"} |
The thickness of the outermost telescoping leg segment (the rectangular sides of a cross section) determines the leg’s pivoting range limit. For a thicker leg the pivoting range limit is smaller, and
for a thinner leg the pivoting range limit is larger. The tradeoffs for a thicker leg include the following features:
1. more strength against leg bending strain or damage
2. more space for the mechanical system to telescope the leg
3. more leg segments for telescoping the leg
For both Figures 21a and 21b the variable a and the module core assembly base component ratios are based on the calculations from the Optimal Ratio Calculation for Module Core Assembly Base
Component article.Figure 21a is a diagram of a lengthwise cross section of the base of the module that forms the module core assembly base component as a black outline and a representation of the
boundary of the leg that can enter the well as a blue line. The lengthwise well perimeter is indicated by the red vertical lines. The variable L represents half the lengthwise thickness of the leg,
and the variable θL represents the lengthwise plane of rotation angle for the leg from a perpendicular position relative to the face of the cell cube.
The following equations and calculations are derived from the geometrical information provided by Figure 21a:
Figure 21b is a diagram of a widthwise cross section of the base of the module that forms the module core assembly base component as a black outline and a representation of the boundary of the leg
that can enter the well as a blue line. The widthwise well perimeter is indicated by the red vertical lines. The variable W represents half the widthwise thickness of the leg, and the variable θW
represents the widthwise plane of rotation angle for the leg from a perpendicular position relative to the face of the cell cube.
The following equations and calculations are derived from the geometrical information provided by Figure 21b:
Graph #1 Graph #2
Graph # 1 shows a plot of the angle versus the rectangular lenghwise thickness factor (to ratio Graph # 2 shows a plot of the angle versus the rectangular widthwise thickness factor (to ratio
variable a) of the leg. Green is for an angle of 45 degrees, and red is for an angle of 60 degrees. variable a) of the leg. Green is for an angle of 45 degrees, and red is for an angle of 60 degrees.
Beyond 60 degrees the decrease is almost linear. Beyond 60 degrees the decrease is almost linear.
Graphs # 1 and # 2 were generated using the following Matlab code: matlab.txt
Figure 21c shows a view of the face of the module core assembly base component with the perimeter of the well as a blue rectangular line. The well perimeter and the surface plane of the face that it
intersects with forms the well perimeter edge. The green rectangle represents the leg thickness and rectangular perimeter as a cross section of a leg that can pivot up to 45 degrees, and the red
rectangle represents the leg thickness and rectangular perimeter as a cross section of a leg that can pivot up to 60 degrees.
In reality, the intermediate universal joint-like component between the module core assembly base component and the first segment of the leg will affect either the shape of the rectangle on the
widthwise side of the leg or the shape of the well, because of the cylindrical twisting effect introduced by the universal joint-like component. Also, the universal joint-like component can also
affect the lengthwise range, since it pivotally mounts to the module core assembly base component in the region the leg would enter in the well; this might be a limitation that can be minimized or
eliminated with an appropriate design for the structure of this universal joint-like component. | {"url":"https://selfreconfigurable.com/?p=451","timestamp":"2024-11-03T00:20:01Z","content_type":"text/html","content_length":"46942","record_id":"<urn:uuid:ae1f5041-9265-49f1-a98c-03723a3e3bb6>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00200.warc.gz"} |
I Love These Puzzles
Maureen Dowd delivers an interview with Trump titled "The Mogul And The Babe". Sarah Palin is not mentioned and I am left suspecting the "babe" in question might somewhat improbably be Ms. Dowd.
OTOH, the column concludes with a throwaway anecdote about Babe Ruth, so maybe MoDo is engaging in the sort of subtle double entendre for which she earns the big bucks.
Whatev. This has not been my day for puzzles. The Times crossword du jour (which was a holdover from earlier in the week) had as one clue "2016 campaigner". We had four letters with [blank] [blank] U
Z and I was mumbling "Jindal"? Perry? Huckleberry? WTF"?
A dark moment when we finally got it. Talk about blocking.
SPOILER: No, not Kasich either.
I stopped at Maureen D...
those words they have been using,
Ignatz --
I am sure that the editors of the NYT would be strongly against a president who had anything to do with a womanizer.
trolling everyday, is their motto,
A Senator Sasse from Nebraska
Made an ass of himself when he asked a
Convention show of hands
they replied "Reprimand"
Next election he'll dread Sarah from Alaska.
Dostoevsky's The Idiot; I swear he wrote parts of this under pressure of paying off gambling debts. I mean it's still very good but Brothers Karamazov was *much* more coherent.
Long as we're talking Russian Lit and I'm linking about "Missing Nose" syndrome:
some actual insight into the current schemes,
this is delivering the cluebat to salon, not that they would feel it,
Gogol is outstanding, his short stories and Dead Souls.
I picked some annotated pushkin's some month back,
I know this is supposed to be a bad thing,
It used to be. "Just vote for us to win the House and we'll do what u voters want us to do". Followed after the election victory by
" we need the Senate, if only we had the Senate plus the House then we'd be able to do what you voters want us to do." So we gave them the Senate.
Then it was "We can't do what u want us to do because we only have the House and the Senate. If only we had the Presidency, plus the House and the Senate, then we could do what u voters want is to
So now we're trying our best to give them just what they asked for, A President, the House, and the Senate, and all we're hearing from them
Is "You're giving is the wrong President. This guy might win, and then we couldn't blame not doing what U wanted us to do on not having theHouse, the Senate, or the Presidency, so quick, let's throw
the Comvention and rally around a 3rd Party tomato can."
ugh, one puzzle is who greenlighted terminator genesys, I'm wagering skynet, to destroy the franchise,
Yep daddy.
Lots of people have taken the red pill this election, on both sides of the aisle. The usual suspects have no idea.
The GOP had the Presidency and Congress and spent money like total [Redacted] idiots. They've squandered every bit of trust they were given. I think the entire party base is as furious with the lies
as I am.
they had no intention of following through, they never considered a guerilla campaign with free media, could gain the 'commanding heights,' why would they what purpose would the roves, the
daysprings, the murphy's of the world, serve.
Maureen Dowd is the way over the hill lush on the bar stool at the end of the Oak Bar in the Plaza Hotel hoping that some young guy will take her home and rattle her Grandma Cougar bones. She looks
in the mirror and sees a "Babe!" The guys in the bar look at her and see a slightly inebriated old bag. Charity carnal activity is not on their agenda tonight.
oh alright then,
A fair point,
Down at the FBO at KPAE, there was a shiny 76 so new that they hadn't hung the engines on her, yet. All white, except for a little purple around the hinges on the rudder blade. The FBO manager says
that it has to be a 'reject' in that the tail colors normally get finished as part of the initial paint job and normally the engines get hung before then. Any idea why a certain freight op would turn
one away? He thought you guys were still sucking up any that got turned out.
Did you see the wonderful article JMH put together for you on the last or maybe it was the thread before that?
You missed a gem..
Beats me. I just went thru our website trying to see what I could learn about our 767 Fleet goals, but it's so full of acronyms that I can't make heads or tails of any of it. Maybe they're
intentionally trying to keep us in the dark---I wish they'd simply use English.
I'm already in a sour Company mood today in that my new Company issued iPad for the plane arrived and Momma and I have spent about 8 hours so far trying to get it to download/upload proper, but its
not doing what it's supposed to do, and trying to get it sorted out via calling "Tech-guys" in wherever is like dealing with the IRS, so flames are shooting out of Momma's exasperated head and as a
consequence my normal good nature is a good deal less good natured than normal:(
In the old days all we had to memorize were numbers and Emergency procedures, but in these new days I've also got to be able to remember half a dozen passwords just to turn on all this crap, and once
you get access then you have to decipher strings of acronyms out the ying-yang. Arggggh!
Blank blank U...Z???
Perhaps JEUZ?? or SNUZ???
Maureen Dowd does resemble The Babe, and I bet she hits left handed too!
Dowd's readers aren't happy she isn't ripping Trump to shreds.
Well here's an unexpected bummer. My favorite Bollywood Actress, Priety Zinta, owner of the Kings XI Punjabi Cricket Team and a huge Cricket fan, just got secretly married to some California Finance
guy for a Hydroelectric company.
I always thought that Priety had the nicest "dimples" in the business.
Oh well. I'm in Bangalore next week, so I guess I'll have to have my Hindu Bartenders point out some other cricket-loving, Bollywood Bombshell to fall for. Karma, neh?
Daddy, Piety Zinta is gorgeous. Can she emote, or is that the question? :)
This comment flows back to the previous thread: many thanks to those with kind words for the kid. I am happy to be in the mix once again. Y'all are a bit addicting, though, so I will be regulating
(restricting) myself a bit (I think).
And . . . rich@gmu? Formerly rich@uf I presume?
Clarice's latest:
Question, do you try out different curries (restaurants) while in Bangalore? And, not being catty or racist, but while there, do you notice people have a curry body odor? Only asking cause my sister
was taking a supplement with curcumin, etc.and she began to have a bit of a spicy scent. Overwhelming, in fact. The stuff is suposed to be wonderful for your joints, cartilage, etc.
Another interesting.
Though it's as if French just woke up to this.
Clarice, Love your column today.
I did see JMH's excellent post. Momma and I have been to Maui a few times and agree with all the bits she wrote regarding the places we have been. Thanks for mentioning her post again. Part of the
deal of getting Momma to move to Alaska from the Bay Area was the promise that in the middle of Winter we would take a month off and go hang in the islands. We did that consistently until the girls
hit 7th Grade, at which point we had to start worrying about grades and attendance for College, but up until then we were gone every winter and really enjoyed the islands. We did Maui twice, but
generally we stayed at the Military resorts on Kauai, the Big Island, and Oahu. Great times, great memories.
My mention of should I go to Maui to my Chakra friends house, was more a joke about going back again into the lions den of a guy who now has a spiritual guru and believes there are Sumerian Pyramids
in Antarctica erected when they arrived in spaceships. I was joking that if I went I might get converted and roped in to the journey, and come back pregnant like Katie Holmes, and married to a
Scientologist type:)
Sorry to have made JMH do all that work but it was good for everyone to read, and you could tell she really enjoyed writing it.
So, should I risk it?
Priety Zinta is gorgeous. Can she emote, or is that the question? :)
She could emote like George Patton when her Cricket Team was playing like bush-leaguers.
I avoid the fruit and I always try the curries. The curries I am told are usually cooked in such a way that I won't get sick and I never have, whereas the water and ice I avoid as it's so easy for
them, or for things washed in the water, to get me loose quickly.
As for a curry smell, I haven't noticed it. I really notice the smell of kimchee in Korea on everyone, and even out of the faucet in the bathrooms there, and then in the Mid-East you notice some
terrible BO. I think Kimchee has permeated everything in Korea. I'll pay attention this time in Bangalore. There are an awful lot of smells in India obviously, but hat one doesn't stand out in my
It is the best of times, but alas, optimism has become uncool and so we ignore the many ways in which Obama's reforms have made things better and instead fall prey to Trump's demagoguery.
And this is a shame, because optimism would lead to MORE reforms being implemented. So start cheering up so we can work on our next set of reforms!
Good job airing some hypocrite laundry Clarice!
But then you do live in a target rich environment.
Wretchard posted this link on FB with this lead in - "In 2013 they described it as a miracle"
Hugo Chavez’s economic miracle
The Venezuelan leader was often marginalized as a radical. But his brand of socialism achieved real economic gains
Excerpts from Clarence Thomas's commencement speech at Hillsdale College
Speaking of gorgeous, RIP Madeline LeBeau, of "Vive La France!" fame in "Casablanca," the last surviving cast member.
daddy @ 11:41 Congress does "So Much Whining We Are Sick of It"
Here's NPR's solution ...
from Feb. 2016 -
Facing Severe Food Shortages, Venezuela Pushes Urban Gardens
starving? societal collapse? no electricity? no medicine?
Urban gardens!!
"They were on a mission to find dirt for their gardens, which they keep on balconies, rooftops and small plots of their homes. After digging up the fresh earth, they lugged it back down the
I see the Clinton Dead listis surfacing again (people who died under mysterious circumstances).
It's quite a list of murders, suicides, and iffy accidents and medical trauma.
I always wondered about that, as the number of people in my circle of friends and extended who have been murdered or committed suicide is ZERO.
Joan, not to be "crude" but Priety Zinta smells like ROSES not Curry, when I make sweet sweet luv to her. She can cook, but I shouldn't talk about "private stuff".
Still waiting for Carlos Slim's and Bezos's barely trained chimps to discover Slick's multiple trips to Epstein Island. C'mon Bob Woodward, do you have to be handed everything by a bitter liar to
"get" the story? Absolutely pathetic.
I hope you're up in time to watch this, because I know you don't want to miss it!
owardKurtz @HowardKurtz 11m11 minutes ago
On #Mediabuzz at 11 ET, @megynkelly on her Trump truce, the impact of the attacks on her life & career, and grappling with the price of fame
Capn hate, are you confusing Epstein Island with Billy Jeff's "charity work" in Haiti.
Oh c'mon, Gus, Tony "Yes we're all fatasses in this family" Rodham has eliminated poverty in Haiti with taking nary a penny for his efforts. There's been a flood of people from the Dominican Republic
trying to get into this paradise.
Every personal acquaintance I ever made in Moscow sooner or later took me out to visit Gogol's residence in the city (it has been made into a sort of museum and library know as "The Gogol House", and
it is one of the few residences in the inner city from before the fire that has survived more or less in tact ).
The people who have taken me to this shrine--for that is what it is--include people with no literary inclinations whatsoever--even people that have little education. This is such a constant
occurrence that I mark it as a ritual of the beginnings of friendship.
The Russians, or at least the Muscovites, are Gogol fanatics, and rightly so. They take a great, fierce pride in him.
It is quite moving to see.
This is a picture of Donald and Melania Trump with JOHN MILLER, the supposedly imaginary publicist.
This is what the "Paper of Record" has sunk to:
They've really lost it, haven't they?
Plus, putting that picture on the front page sort of looks like he's surrounded by beautiful, admiring women, which is not the message they hoped to send.
Miss Marple, those "women" aren't supposed to CHOOOOOSE that lifestyle. I'll bet each and every one of them uses the "ladies room".
Oh c'mon, jimmyk; after Holocaust minimizing, denial of starvation in the Ukraine and refusal to disown Duranty's Pulitzer this doesn't even qualify for anything major in the Shitberger family.
I wonder how Greta, Lou, Judge Janine and others who've been interviewing Mr. Trump, his kids and supporters from day one, feel professionally about Queen Kelly Megyn now poaching their territory.
My offering for your enjoyment, via Drudge:
I figure Trump gave her the interview as a favor to Ailes. She so damaged her brand with the constant snarking, that a bunch of people have turned her off.
I'd like to see ratings numbers from prior to the start of the first debate compared with where her show is today.
Not just her show, but the others and the News Hour too, MM. I think I am down to about 5 minutes a day of Fox News.
Was it here that somebody yesterday mentioned a contempt for an audience?
Old Lurker,
True. I watch Bill Hemmer in the morning and occasionally Outnumbered.
Have lost interest in The Five because of Gutfeld monopolizing the talk and being nasty about Trump. Sometimes I watch Greta.
Far cry from having it on from early morning until bedtime, which is what I used to do.
If the same pattern governs the general campaign as it did the primaries, I expect his poll numbers to go up after every hit piece, the worse the dirt, the higher his polls numbers. The number of his
twitter followers will also continue to increase daily.
Even if I wasn't a political junkie it would be hard to take that NYT story seriously with a Clinton in the race.
"Crossing the Line: How Donald Trump Behaved With Women in Private"
Really??? Did Trump put cigars in their vaginas?
Those are pageant contestants are they not as for the demure miss lane not so much.
How about that driving performance from young (18 y.o.) Max Verstappen to win the Spanish Grand Prix today?
Youngest ever F1 winner. And he did it without one practice or test lap in the car. Pretty impressive. Even if Rosberg and Hamilton crashed out.
Only caught the last few laps after coming home from church hoping to see the BPL on NBCsn.
I hope you're up in time to watch this, because I know you don't want to miss it!
I will check my rat traps instead.
MM you asked about ratings. This was published May 7th at TVNewser:
"While each of the top-10 cable news programs in total viewers fell from their March 2016 average, many of them were up year-over-year: The O’Reilly Factor (+9 percent), The Kelly File (+7 percent),
Special Report with Bret Baier (+1 percent) and most notably, Hannity (+35 percent) to name a few."
As you can see, Megyn's show is up over last year. Hannity's show which is almost entirely devoted to Trump is up the most - so that should cheer you.
Yes Janet, she actually kept the JIPE STAINED blue dress for "Posterity"
Crossing the line? Oh, my. :(
Carlos' dancing chimps have a very thick membrane on their bubble. Heck, this is water under Trump's wheels - he got his picture on the front of the paper with a bunch of babes, and now he gets to
drive the cycle, re: the hypocrisy of Slick Willie's adventures. Again. They're making it too easy for him, but they can't understand that paradigm.
Trump is enjoying every bit of this.
While each of the top-10 cable news programs in total viewers fell from their March 2016 average, many of them were up year-over-year:
What does that mean?
I thought it said "March 2015."
Oh, as Mrs. JiB has just pointed out, Verstappen is Belgian of the Flemish variety (born in Hasselt and carries a Belgian passport) but drives under the Dutch flag since that is where he spent most
of his time learning the art of racing by karting. Father is Dutch, Mother Belgian.
He lives in Bree, Belgium.
Happy Pentecost! Can't catch up until after church, but don't miss the wheels coming off the Dem Nevada convention yesterday. My apologies if already linked.
this doesn't even qualify for anything major in the Shitberger family.
You have a point, Cap'n. If only they'd given the concentration camps that kind of front page coverage.
Where is this story? -
"Crossing the Line: How LGBT activist leaders Behaved With children in Private"
an expose on Terry Bean - known for co-founding several national LGBT rights organizations, including the Human Rights Campaign, the Gay & Lesbian Victory Fund
"Bean, 67, of Portland and former boyfriend Kiah Loy Lawson, 25, were charged with having sex with a 15-year-old boy at a Eugene hotel in 2013. Each faced two counts of third-degree sodomy, a felony,
and third-degree sexual abuse, a misdemeanor.
A civil compromise is legal under Oregon law. Defendants may compensate the victims of certain crimes and have the charges dismissed if a judge approves the deal.
An offer to settle a child sex abuse case through civil compromise is unprecedented, prosecutors said."
Chris Wallace has lost his shit as he grills Reince Priebus about Trump's "women problem". Does anybody still trust this trash network?
Janet, he doesn't smoke.
Hannity was pretty fair and balanced, and when Cruz accused him of favoring Trump, that made Hannity ripping mad.
I've read many Trump supporters post that at pro Cruz sites like Right Scoop all pro Trump comments were immediately deleted.
I guess if Trump got any positive coverage at all, that was seen as "entirely devoted." It is statements like that that caused Mr. Trump to start calling Cruz "Lyin' Ted".
Their was a time when posting positive Trump comments here was frowned upon, cheerleader.
Not anymore.
Priebus seems to get it about why people gravitate to Trump. Wallace is yammering about how "nobody knows who Trump is". Please continue to stick with this idiot, Ailes.
Priebus calls a "conservative third party run a suicide mission for the country" and states that Kristol and Romney need to hash out their differences with Trump man to man.
I'm starting to understand Mr. Trump's pre-emptive strikes on Hillary for enabling Bill's "women problems." He must have seen this coming.
Tim Huelscamp joins in Wallace's whining about de wimmenz problem; Noot asks where's the Epstein island coverage. Huelscamp keeps complaining about Trump not being a conservative; Noot says look at
the alternative in Rodham. Dear God, Huelscamp keeps whining about his kids and soccer moms. Noot understands why people are pissed off. Huelscamp seems to believe nobody can win unless they're 100%
pro life; where has he been this century? Newt says Trump gets in trouble by speaking off the cuff for long periods of time and gets into situations where he backtracks on positions. Huelscamp must
have really loved Romney in 2012; how'd that work out, Tim? Noot plays coy on whether he'd be a VP candidate.
What I saw of Newt in Key Biscayne back in New Year's he is not physically fit to do any hard campaigning. I'd be surprised its him or Christie.
Thanks for the truthful ratings wrt the ratings of the different Fox programs.
On May 17th watch those Meghan Kelly ratings soar with her one on one with the Repub candidate.
Chris Wallace was his usual asshat self today grilling Priebus.
Good to know NYT hit piece won't stick.
Interesting article on Wisconsin Republican Party Meeting.
Newt is a good spokesperson for our Party.
Meghan Kelly is doing just fine despite the chagrin of some people.
George Will:
I support nominee but I don't endorse him is what some Repubs say.
Tim is from Kansas but I still think with a little persuading he can be won over.
Bob Woodward says the Amazon Daily Worker will investigate Rodham with an "appropriate number of reporters". Hume joins the MFM bandwagon on covering Trump but is extremely skeptical that Rodham will
be scrutinized. Amy Walter says wimmenz are more worried about his judgement as President but still arent in love with Rodham. Will says a lot of GOP people are engaged in semantic summersaults.
Walter says that Trump seems popular enough with the base to marginalize the party insiders.
maryrose, You are either naive or stupid. Maybe both.
old luker 2
You are definitely stupid.
The tagteam is in the ring at the same time?
Will says 404's decree will help private schools and the lies about what the language of the law on "gender" are being revealed. Walter says rule by fiat will continue until Congress pushes back.
Hume says this is something that came out of nowhere and is completely absurd to be reasonably understood. Woodward shows that he's confused about everything.
Will says the TSA has been doing security at campaign rallies which have nothing to do with Trasportation. Walter says airports have to redesign how they do security.
And that's a wrap.
You mean you and lurker?
The problem is they lost the prized 18-45 demographic.
On May 17th watch those Meghan Kelly ratings soar with her one on one with the Repub candidate.
If Cruz was still in the race, imagine how well Magpie would be doing.
Thanks, CH.
Suspicious package found in the North stands at Old Trafford in Manchester, UK. Controlled evacuation of stadium. ManU v. Bournemouth game postponed until further notice.
No idicaiton if terror related.
maryrose is certainly not stupid, but trolls who think they can disparage her here with affect are.
Thousands standing around, have done little of use.
By contrast, they had a very effective Mexican American spokesman for trump, on ramos's show.
Capn, why didn't WOODTURD vet Rodham when she ran against President Obamster. Was he short on reporters back then.
Wait a second, the power player of the week is Schmucky Schooooomer and how he's meddled in his staffer's lives as a matchmaker. Does this have to be listed as a campaign contribution.
And the last thing is a plug for the Magpie's special.
Now that's really a wrap.
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Comment on MegaAmpere to MegaGauss- Using laser fluctuations to measure gravitational potential turbulence
This intro added 10 Jul 2022 in update to “Solar System Gravimetry and Gravitational Engineering ”
Comment on “MegaAmpere to MegaGauss” – Using laser fluctuations to measure gravitational potential turbulence
To generate intense fields to control the motion of matter. To emulate gravitational acceleration so precisely there is almost no error in position or other measure, should be possible with the
resources and abilities in groups today. If they would work together, not all go their separate ways. I wrote this yesterday and do not know how to point to comments on ResearchGate, so I am copying
this here since it relates to making compact, low cost, three axis, high sampling rate gravitational detectors. Precisely generated and monitored laser channels in vacuum or matter can be used as
gravitational detectors. Essentially solid state (no moving parts), reliable, lasting for decades or longer.
Pierre-Alexandre, P-A Gourdain
I started writing this, but cannot finish just now to my satisfaction. I wanted to give you specifics for a laser, or particle beam, vector gravitational detector. Will try to do that soon. Will post
this, but will have to try again tomorrow. I think I can estimate the fluctuation spectrum for a detector stabilized near 380 Tesla. The average magnetic field is independent of the total power, and
only depends on the power density..
I am reading your Mega-ampère to mega-gauss paper. You mention 172 Tesla. Just wanted to check that I am using the same energy density as you. That should be B^2/(2 mu0) = 172^2/(2*1.25663706212E-6)
= 1.1771E10 Joules/meter^3. Unless the cavity is filled.
I have been tracking experiments and papers where anyone is looking beyond 100 Tesla, static or dynamic, for about 40 years now. The energy density of the gravitational field at the earth surface is
roughly equivalent to a static magnetic field of 379 Tesla. And I have been looking for a way to see if that is true. The gravitational potential field must be fairly uniform and stable second by
second. But its variations should show up in high energy density experiments of many sorts. Not total power, energy density or power density.
The expression I found is one that Robert Lull Forward (Univ MD College Park, worked with Joe Weber) used. Joe recommended I follow what Robert had done, and that energy density seemed an odd item. I
tried every possible model to try to visual and model what might be the basis of something like that. The expression he used is g^2/(8 pi G), where g is the local gravitational acceleration (about
9.8 m/s2) and G is the SI value 6.67430E-11 (m^3/s^2)/kg. There are cases where this is valid. Then g = B/sqrt(8 pi G/2 mu0) = B/38.70768. So a value of 172 Tesla should connect with an acceleration
of (172/38.70768) = 4.444 m/s^2.
Measuring the energy density directly is difficult. But it might be possible, if a magnetic experiments of those energy densities is run consistently for different times over days and weeks or years.
If I am right, then a magnetic field maintained near that value should interact with the gravitational energy density field. The energy density is the gravitational potential times a density. That
should be the density of the cavity, the mass in the cavity where the magnetic field is. Possibly with an offset (scalar field).
I just wanted to check if you calculate the energy density for these dynamic fields the same way.
When I use the laser (theoretically) for generating intense fields, the power per unit area is just c * g^2/(8 pi G) or about 1.716E19 Watts/meter^2 for a 9.8 m/s^2 field. Since the acceleration
varies with time, the energy density and the equivalent magnetic field does as well. I am trying to keep track of all the laser vacuum papers, because it is not an empty vacuum people will be
measuring, but a gravitational potential field with a corresponding energy density. Yes, the sun’s gravitational potential matters. Not only for changes in the rate of clocks, but there might also be
a measurable threshold for magnetic fields.
I looked at fusion experiments a lot, because I felt that a natural gravitational field would be grainy and vary. So that anyone trying to build a static fusion model would run into trouble. And, for
measuring gravitational effects, it might be that any continuously operating fusion reactor would also be an outstanding gravitational and magnetic field sensor for natural fields.
It does not depend on the current, rather on the energy density. With femtosecond pulses, the energy required for a high energy density pulse could be quite small. They running it continuously, the
variations should correlate with sun moon tidal signal, the same as measured at a superconducting gravimeter (or other type of gravimeter) station.
I took a break writing this, to think about how to make a three axis laser detector, or a multi-axis particle beam detector to track the sun and moon, and also separate out the magnetic field
components, and not make it too complicated or expensive.
The earth’s magnetic potential and the gravitational potential are probably the same thing underneath. A pure potential field only has to give the values for the potential at every point and fit the
measurements for all gradient measurements over time, by frequency. But, in my view, the total potential is made up of spatial and temporal variations. Somewhat like an oscillating cloud that is also
moving and changing size. The turbulence spectrum is what drives whether something is “magnetic” or “gravitational”. Somewhat like how we separate parts of the electromagnetic field by frequency. But
in a larger model where you also carefully trace the spatial spectrum of sources. Imagine five wind driven clouds, each a different color. They are like galaxies where intersecting clouds of stars do
not have many actual collisions of the stars. So these intersecting clouds pass right through each other. That helps me visualize. But what I am also thinking in the background – the frequencies of
any signal are independent. If each cloud has a different frequency, they do not interfere with each other. The signal is “linear”. The signals can be superposed in a simple way. The reason I think
that, is because the vector sun moon tidal signal I have worked with for almost two decades now is pure “Newtonian”. At the sampling rates and sensitivities used in detectors, the field of the sun
and the field of the moon are quite distinct. They each have a location and, I am fairly certain, a unique signature.
Three lasers of high energy density, be as small power levels as possible.
Three electron beams
Three ion beams.
An electrochemical flow
A neutral plasma
A charged plasma
Solid state flows of many sorts
I am getting over the flu and a bit tired. I will try to come back and finish this comment. The main thing is that the tidal gravitational is directional and very stable over time. It can be exactly
calculated using the Newtonian approximation using JPL solar system ephemeris and fairly simple instruments for local verification. Because it is so precise, correlations are simple. If the sampling
rate goes up into “electromagnetic frequencies” like GHz or THz or higher, then in the gravitational sensors people will talk about Gsps, Tsps and other samples per second. The sensitivities are well
within reach of many experiments. Many have already reached the necessary sensitivity, they just need a reference signal or common source, or collaboratively chosen target.
The fluctuations of the gravitational potential, and the fluctuations of the magnetic potential of the earth should fit together as a single potential with overlap. Some of the signals called
“gravitational” are partly “magnetic”. Some “magnetic” are partly gravitational. That is why I keep recommending that groups go to high enough sampling rates to use time of flight and angle of
arrival for locating and mapping sources. The correlations between sensors for a given point in space and time as a source element are characteristic of that source. And many many kinds of machine
learning, and AI algorithms can take that and use it for imaging.
I cannot finish this just now. I try to work an 18 hour day, but today I just cannot go past 16. LOL!
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Lecture Slides: Calculating the Pearson r Correlation (Study Hours and GPA)
The goal of this PowerPoint is to work through the step-by-step process of hand calculating a Person Correlation. The slides start with the formula involved and then a sample problem is presented. It
is designed for you to have the class solve the problem along with you as you advance through the slides and animation. Students will calculate the correlation coefficient, describe it, and determine
if it is significant.
This presentation will take approximately 20 minutes to complete if you have the students work along with you.
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ONS methodology working paper series number 12 – a comparison of index number methodology used on UK web scraped price data
Office for National Statistics (ONS) has been looking into alternative sources of data to supplement or replace existing data collection methods such as the use of surveys.
Currently the majority of ONS statistics are derived from carefully designed surveys. The research into other data sources is part of a programme to innovate and transform official statistics. These
datasets fall into two categories, administrative data and found data.
Administrative data are data collected by governmental departments for non-statistical purposes, for example, ONS has been investigating using VAT turnover data to compile the UK National Accounts,
(Edwards [2017]).
Found data are data whose main purpose is not necessarily the purpose it was created for. One example of this is the use of satellite imagery. This was originally created to aid mapping but now it is
also being used for land use statistics.
These administrative and found data are also useful in the construction of price statistics. Administrative data have been used in the Producer Price Index (for example, prices for home-produced
foods come from the Department for Environment, Food and Rural Affairs (ONS [2014b]), and in the Services Producer Price Index (for example, a Business Rail fares index is supplied to ONS by Office
of Rail and Road (ONS [2015]). On the consumer prices side there are two found data sources that can be used in the compilation of Consumer Prices Indices. These two sources are transactions or
scanner data and web scraped.
Scanner data are the data on sales of products bought by consumers at the point of sale. These data then contain both the quantities of the good bought and the prices at which the customers bought
the good. These data aren't collected for the purpose of price statistics; it is often kept for stock-keeping reasons. Scanner data could also contain other information that is useful, such as
characteristics of the products.
Scanner data also has the added benefit of allowing the use of weights at the lowest level of aggregation, the so-called elementary aggregate level. Currently in the Consumer Prices Index including
owner occupiers’ housing costs (CPIH) at the elementary aggregate level there are no weights used to aggregate individual product price relatives together. For example, the total household
expenditure on apples in the UK is used in the CPIH to weight the prices of apples together with other items in the basket of goods and services to form the aggregate measure. However, when measuring
the price changes for apples specifically, the expenditure on Pink Lady apples bought in Cardiff from a multiple shop^1 is not known, and therefore all apple products are weighted equally together at
the elementary aggregate level. Scanner data would allow for this information, if the retailer keeps it at this level of disaggregation.
Scanner data also gives us more detail than just quantities; it gives us a larger set of products and possibly a larger geographic coverage than the current CPIH data collection method, which prices
a sample of around 700 representative items at a sample of outlets from a sample of 140 locations across the country. Scanner data also has the potential to be of a higher frequency than the current
collection, as weekly or possibly daily prices and quantities could be observed, giving more information to be used in the CPIH.
However, other aspects of coverage may be an issue for scanner data. It is likely that scanner data would only be obtained for larger stores and wouldn't be available for small stores such as corner
shops. In addition, not all the areas of the CPIH basket would be covered by scanner data. For example, the fees charged by plumbers, electricians, carpenters and decorators are collected as part of
the CPIH collection. It wouldn’t be possible to collect scanner data for this item as most of the providers of these services wouldn’t collect these data and there would be too many providers to ask
to get a reasonable coverage.
Currently, ONS does not have access to scanner data. However, other national statistics institutes (NSIs) have been using it to compile their Consumer Prices Indices for a number of years. Statistics
New Zealand, (Bentley and Krsinich [2017]), have been using scanner data in the technological goods part of the basket. The Australian Bureau of Statistics has since 2014 included scanner data for
25% of their CPI. Statistics Netherlands has been using scanner data since 2002, (de Haan and van der Grient [2009]), and now make use of it for many areas of their basket. As of 2017, it is to the
best of the author’s knowledge that the following European countries use scanner data in their CPIs: Norway, (Johansen and Nygaard [2012]), Sweden, (Sammar et al. [2012]), Switzerland, (Müller
[2010]), Iceland, (Guđmundsdótti and Jónasdóttir [2014]), and Belgium, (Statistics Belgium [2016]). INSEE, the NSI of France, have plans to implement scanner data after a change of legislation
(Rateau [2017]).
Another found data source that can be used in the compilation of consumer price statistics is web scraped data. Web scrapers can be defined as software tools, which are used for extracting data from
web pages. Web scraped prices data are collected by scraping retailers’ websites to get price and other attribute information on products.
Web scraped data has similar advantages to scanner data in terms of product coverage and frequency, but also has a number of disadvantages as well. One disadvantage is that products with very low
expenditure may still be scraped (for example, products which have not been bought recently by consumers). This means that products could be included in the index which aren't representative of
consumer spending. Web scraping is also only restricted to those retailers with an online presence. These are usually medium to large retailers, not small retailers; a similar constraint to the use
of scanner data. Even for medium to large retailers, not all will have a website; for example, some discount retailers do not have a retail presence on their websites.
ONS has been exploring the use of web scraped data since 2014, and has published a number of research articles on the use of web scraped grocery data (Breton et al. [2015] and Breton et al. [2016]).
The web scraped data are often of a higher frequency and volume than traditionally collected price data. For example, the ONS traditional collection for the 33 CPIH items that are currently included
in the grocery web scraping pilot is 6,000 price quotes a month whereas the web scrapers collect 8,000 price quotes per day.
Other countries have also been investigating the use of web scraping in the calculation of consumer prices statistics. For example, the Netherlands, (Chessa and Griffioen [2016]), for clothing data;
New Zealand, (Bentley and Krsinich [2017]), for varied areas of the basket; Italy, (Polidoro et al. [2015]), for airfares and consumer electronics; and Germany for flights, hotels, mail order selling
(mainly clothing and footwear), mail order pharmacies, hire cars, train travel and city breaks, (Brunner [2014]). The Federal Statistics Office of Germany has also been looking into web scraping to
examine dynamic pricing (Blaudow and Burg [2017]).
From this ongoing research into these alternative data sources, it has become clear that traditional methods of compiling consumer price statistics may not be able to cope with both the frequency and
volume of these new data, therefore new methodology is required. Chessa and Griffioen [2016], have compared indices for scanner data. Breton et al. [2016] details some of these methodologies (with a
particular focus on web scraped data) and Metcalfe et al. [2016] describes new methodology to compile price indices from these data.
This article will assess these new methods for use on web scraped data using the criteria of the different approaches to index number theory. In index number theory there are many ways to derive and
assess which methodology is suitable, three of those approaches will be used here. These are as follows:
• the axiomatic approach (see International Labour Organisation (ILO) consumer price index manual chapter 16) – the index is tested against some desirable properties
• the economic approach (see ILO consumer price index manual chapter 18) – the index is ranked against whether or not it approximates or is exact for a Cost of Living index
• the statistical (Stochastic) approach (see ILO consumer price index manual chapter 16) – each price change is an observation of population value of inflation and the index is the point estimate
of inflation
Notes for: Introduction
1. A multiple shop is a shop which has 10 or more stores in a region.
Back to table of contents
3. Price index methods for use on web scraped prices data
3.1 Introduction
In index number theory there are many different price indices that have been formulated to measure price changes. These can be split into weighted and un-weighted formulae, depending on whether
quantities have been observed with the prices. For example, the Jevons index is an unweighted index whereas the Laspeyres index is a weighted index.
Price indices can also be divided into bilateral and multilateral indices. Bilateral indices only compare two periods with each other to measure price change while multilateral indices compare three
or more periods. For web scraped data, this means that only indices that are classed as un-weighted should be assessed (although weight proxies could be used if weighted indices are desired; this is
left to future research), and both bilateral and multilateral indices can be included.
The following indices have been used to calculate price indices from web scraped data:
• bilateral indices
□ Fixed Base Jevons (this was called the “unit price” index in our previous releases)
□ Chained Bilateral Jevons (this was called the “daily chained” index in our previous releases)
□ Unit Value
□ Clustering Large datasets into Price Indices (CLIP)
• multilateral indices
□ the GEKS family of indices; all the GEKS methods in this article use Jevons indices as an input into the GEKS procedure (GEKS-J)^1
□ the Fixed Effects index with a Window Splice (FEWS)
□ the Gheary-Khamis index
All of these indices, with the exception of the Fixed Based Jevons, aim to overcome the inherent problem in both web scraped and scanner data of product churn (Definition 3.1). This is done mainly
through frequent chaining. The FEWS index also tries to adjust for the quality changes (Definition 3.2), which occur due to product change.
Definition 3.1 [Product churn]
Product churn is the process of products leaving and/or entering the sample.
Products can leave or enter the sample in one of five ways:
1. product goes out of stock, temporally leaves the sample
2. product is restocked, and re-enters the sample
3. product is discontinued and permanently leaves the sample
4. product is new to the market
5. product is rebranded
Example 3.1 [Product churn]
Figure 1: Product churn example
Source: Office for National Statistics
Download this image Figure 1: Product churn example
.png (23.3 kB) .xlsx (9.8 kB)
Figure 1 shows an example of product churn. Only one product remains in the sample for all 10 periods, product 3. Three products start in stock then go out of stock and then return in stock at
different frequencies. Product 2 goes out of stock permanently. Products 7, 8, 9 and 10 are new to the market, but 8 goes out of stock to be replaced by product 9. Product 10 appears for one period
Figure 2: Product churn in apples from June 2014 to March 2017 coloured by store
Source: Office for National Statistics
1. Areas of where all products have no values is due to scraper breaks not product churn.
Download this image Figure 2: Product churn in apples from June 2014 to March 2017 coloured by store
.png (384.1 kB) .xlsx (5.8 MB)
Figure 2 shows the product churn in apples data collected by Office for National Statistics (ONS) web scrapers. There is no product that exists in all time periods in the dataset, even if scraper
breaks^2 are discounted.
Definition 3.2 [Quality change]
Products only exist in an unchanged form for a period of time. When a product is discontinued, often it is replaced with a "new and improved" model. This improvement is called a quality change.
Example 3.2
If a laptop is discontinued and replaced with a new model with a larger hard drive in the next time period then there has been a quality change between the two products. Therefore, any price change
observed between the two periods may be attributed to quality change, rather than pure price change, which is what a price index is intended to capture. It is therefore important to account for
quality change when constructing price indices.
3.2 Fixed Based Jevons
The Fixed Based Jevons fixes the base period to the first period in the dataset, and matches the products common to all periods. This is the main method used in constructing the elementary aggregates
in the Consumer Prices Index including owner occupiers’ housing costs (CPIH). It compares the current period price back to the base period. The formula is defined as follows:
where p^t[j] is the price of product j in time period t, S* is the set of all products that appear in every period and n*=∣ S*∣ is the number of products common to all periods.
The Fixed Based Jevons index will suffer in markets where there is high product churn. This is because when you get further away from the base period, the less likely it is that products are going to
remain in the sample. A practical example is women's coats where at the end of each "season" the whole stock can be phased out in certain retailers and replaced with the new season's stock, (Payne
[2017]). For cases when this occurs, that is, S*=∅ then an index cannot be calculated. For this reason, the base period is changed each January to allow for changes in the products available to
consumers and for more matches to occur. These within-year indices are then linked together. This happens in both the web scraped data and in the CPIH.
The advantages of the Fixed Based Jevons include the fact that there is a direct comparison from the base period to the current period and that it tracks individual products across time. It is also
relatively straightforward and easy to explain, and follows the methodology that is currently used in CPIH.
3.3 Chained Bilateral Jevons
The Chained Bilateral Jevons index involves constructing Bilateral Jevons indices between period t and t-1 and then chaining them together. The formula is defined as follows:
where P[J]^i-1,i is the Jevons index between the current period and the previous period, p[j]^i is the price of product j at time i, S^i-1,i is the set of products observed in both period i and i-1,
and n^i-1,i is the number of products in S^i-1,i.
The Chained Bilateral Jevons index uses more data than the Fixed Based Jevons because products only need to exist in contiguous periods to be included in the index. It has other advantages in a
production environment as it only needs the current and previous period data to calculate the index, compared with some of the other methodologies, which require more of the historical data. The
index is also computationally straight forward, and easy to explain. Disadvantages to the Chained Bilateral Jevons index include the fact that the index isn't being directly compared to the base
period, and there is a possibility of chain drift. Using the notation of Clews et al. [2014], chain drift is defined as follows:
Definition 3.3 [Chain drift]
Let f(a,b) be an arbitrary index formula. Then a direct index Λ between 0 and 2 is defined as follows:
A linked index Λ* is defined as:
Then chain drift is defined as the difference between the chained index and the direct index, that is:
This can also be restated as the chained index not equalling the direct index.
Proposition 3.1
The Chained Bilateral Jevons index exhibits chain drift if there exists product churn in the dataset.
Let f in Definition 2.1 be the Jevons index, then Λ is the Fixed Based Jevons index and Λ* is the Chained Bilateral Jevons. Then there are four conditions depending on product churn, which are as
1. If S^0,1 = S^1,2 = S* (that is, there is no product churn) then:
Therefore, there is no chain drift under this condition.
2. If S^0,1 ≠ S^1,2, n^0,1 = n^1,2 and S^0,1 ∩ S^1,2 = S* ≠∅, (that is, there is product churn but the amount of items that have disappeared are replaced with the same amount of new products, and
not all products have disappeared), then:
Therefore, if there is product churn that keeps the number of products on the market constant over time then there exists some chain drift.
3. If S^0,1 ≠ S^1,2, n^0,1 ≠ n^1,2 and S^0,1 ∩ S^1,2 = S^* ≠∅ (that is, products are churning but the number of products going out of stock aren't being replaced by the same amount of new products),
Therefore, in this type of product churn there is chain drift.
4. Finally, if S^0,1 ≠ S^1,2 and S^0,1 ∩ S^1,2 =∅ (that is, all products are replaced) then:
This means that as none of the prices cancel out, so there is product churn.
For a proof of all four conditions please see Appendix A.
Figure 3: Chain drift for the different types of product churn in Proposition 2.1
Source: Office for National Statistics
Download this chart Figure 3: Chain drift for the different types of product churn in Proposition 2.1
Image .csv .xls
Figure 3 shows the chain drift for different conditions. These are the averages of multiple simulations of prices randomly generated as follows:
For condition 2, the percentage overlap is the amount of products that appear in all three periods; for condition 3 the percentage indicates how much larger or smaller the number of products is in
the second comparison compared with the first. It can be seen that the direction and magnitude of the chain drift is dependent on the different conditions.
3.4 Unit Value index
The Unit Value index is normally defined as the ratio of the unit value^3 in the current period to the unit value in the base period. However, there is no quantity data in the web scraped data so a
true Unit Value index can't be calculated. Instead the ratio of geometric means of unmatched sets of products will be used, that is:
where S^0 is the set of products in period 0, and n^0 is the number of products in S^0, S^t is the set of products in period t, and n^t is the number of products in S^t. The geometric mean has been
used so that it is consistent with the other indices presented in this article. This is instead of an arithmetic mean, which would have fallen out if all the products had quantities equal to 1
(equivalent to a Dutot index).
3.5 Clustering Large dataset Into Price indices (CLIP)
A recent price index developed by ONS is the CLIP, (Metcalfe et al. [2016]). The CLIP takes a slightly different approach to forming an index in that instead of following individual products over
time like the previous indices, it follows a group of products over time instead. This method attempts to overcome the product churn issue mentioned earlier when tracking individual products over
As the CLIP tracks groups of products over time, it allows for more of the data to be used as individual products do not need to be matched over time. This means that even if a product only appears
in one period then it will still be included in the index.
The groups are found in a two-step process. First, the data from the base period is clustered to form the initial groups. The clustering method used is the mean shift algorithm. This fits a kernel
over the characteristics, (such as product name, shop, offer and price), and then finds the nearest local maximum to the current point to be clustered. Once the clusters have been found, they are
fixed and a decision tree is used to find the underlying rules that make up each cluster (here price is removed as a characteristic). New data are then parsed through the tree when it is observed in
each new period and put into a particular cluster.
Following this, the geometric mean of the prices for each product in that cluster is taken to be the price for that cluster. The current price is then compared back to the base price for that cluster
and then the resulting price relatives for each cluster are aggregated together arithmetically using weights calculated as follows:
where w[i]^0 is the weight for cluster i and C[i] is cluster i.
An advantage of using the clustering is that it allows for substitution when products go out of stock. For example; for apples, if in the base period Royal Gala, Pink Lady and Braeburn apples are
placed in the same cluster, and in the next period Pink Lady apples go out of stock, the consumer might substitute to another item within the cluster. This is because the products within each cluster
will have similar characteristics and the consumers might be indifferent to those characteristics. The CLIP therefore captures the price change for similar products as a group whereas other methods
capture individual price change (and therefore can be more affected when these individual products leave the data).
3.6 The Gini, Eltetö, Köves and Szulc (GEKS) family of indices
The GEKS index was devised as a possible solution to calculating multilateral indices over domains that have no natural order. It is often used when calculating purchasing power parities (PPPs),
where the domain of interest is countries; countries don’t have a natural ordering. The basic premise of a GEKS index is that it is an average of the bilateral indices for all combinations of
comparisons of the domain units. Using PPPs as an example, if the countries of interest are {UK, USA, Germany} then the combinations are UK-Germany, UK-USA and Germany-USA.
3.6.1 The GEKS-J index
The GEKS index was adapted for the time domain by Ivancic et al. [2011]. As the authors had access to scanner data, they were able to use Fisher indices to calculate the bilateral component indices.
However, as web scraped data does not have expenditure information, this article uses Jevons indices instead. This means that the GEKS-J price index for period t with period 0 as the base period is
the geometric mean of the chained Jevons price index between period 0 and period t with every intermediate point ( i =1,..., t -1 ) as a link period. The formula is defined as follows:
A product is included in the index if it is in the period i and either period 0 or period t.
There are some problems with using a GEKS-J index. In later periods, there is a loss of characteristicity^4 when the index involves price movements over large periods. That is, if a product is
available in the base period and the comparison period i such that i is 2 years or more from the base period, then this price movement will be included in the index; this price movement is less
relevant than a price movement from the previous month or week to the comparison period. The GEKS also does not adjust for the quality changes available on new products, or improvements on existing
3.6.2 The Rolling Year GEKS-J index (RYGEKS-J)
The Rolling Year GEKS-J or RYGEKS-J is an extension of GEKS-J that accounts for this loss of characteristicity, and was first devised by de Haan and van der Grient [2009]. For a RYGEKS-J, after the
initial window period, recent movements are then chained on to the previous movements as follows:
Here d is the window length. The optimal window length is still subject to discussion in the international community. For example, de Haan and van der Grient [2009] suggest 13 months should be used
for monthly data, as this allows for the comparison of products that are strongly seasonal. The Australian Bureau of Statistics has adopted a 9-quarter window as their work shows that this is a
better solution for seasonality, (Kalisch [2016]).
RYGEKS is preferable to a GEKS index due to this window. This is because it means that not all of the time series is required to calculate the current period index, and it also deals with the loss of
characteristicity. The RYGEKS does lose transitivity but the effect of this is negligible.
3.6.3 Imputation Törnqvist RYGEKS - ITRYGEKS
As new products are introduced on the market and old products disappear, an implicit quality change may occur, for example, this often happens in technological goods. Hence, there is an implicit
price movement which isn't captured in the standard RYGEKS method because it doesn’t account for quality change. There is an implicit price change when these goods are introduced, and if the
consumption of these goods increased then these implicit movements need to be captured.
De Haan and Krsinich [2012] propose using an imputed Törnqvist as the base of the RYGEKS. An imputed Törnqvist is a hedonically adjusted Törnqvist index, where the prices of new or disappeared
products are imputed using a hedonic regression in the base or current period respectively. A hedonic regression assumes that the price of a product is uniquely defined by a set of K characteristics.
The imputed Törnqvist index is defined as follows:
where w[j]^0 is the expenditure share of item j at time 0, w[j]^t is the expenditure share for item j at time t, p[j]^t is the estimated price for a missing product at time t, S^0,t is the set of
products observed in both periods, S[N(0)]^t is the set of new products at time t but weren't available at time 0, and S[D(t)]^0 is the set of products at time 0 that have disappeared from the market
at time t. De Haan and Krsinich [2012] suggest three different imputation methods, these are:
1. The linear characteristics model:
This method estimates the characteristic parameters using a separate regression model for each period. The imputed price is calculated as follows:
where α^t is the estimate of the intercept, β[k]^t is the estimate of the effect characteristic k has on the price, and z[jk] is the value of characteristic k for product j.
2. The weighted time dummy hedonic method:
This method assumes parameter estimates for characteristics don't change over time, and includes a dummy variable, D[j]^i, for in which period the product was collected. In this method the
imputed price is calculated by:
3. The weighted time product dummy method:
This method can be used when detailed characteristic information is not available, and a dummy variable, D[j], for the product is created. The missing price is then estimated using:
where γ[j] is the estimate of the product-specific dummy, and the N^th product is taken as the reference product. This method assumes that the quality of each distinct product is different to the
quality of other products to a consumer. It is a reasonable assumption as the number of potential characteristics is large and not all of them are observable.
For each of these methods, a weighted least squares regression is used, with the expenditure shares as the weights. A disadvantage to using the ITRYGEKS especially with the linear characteristics and
the time dummy methods is that they will give different results depending on what characteristics are placed in the model. De Haan et al. [2016] compare the differences between Time Dummy and Time
Product Dummy indices. The results indicate that a time product dummy model has too many parameters, fits outliers and unduly raises the R^2 compared with the underlying hedonic model. This then
biases the out of sample predictions, the imputations in this case, meaning that the Time Product Dummy index is susceptible to quality change bias.
De Haan et al. [2016] decomposed the ratio between the time dummy and the time product dummy in terms of the residuals into three components as follows:
Here w[D(t)]^0 is the aggregate weight of products that have disappeared by the current periods, w[N(0)]^t is the aggregate weight of the new products, w[M]^t is the aggregate weight of the items in
both periods. u^-0[D(t)(TDH)] is the weighted arithmetic mean of the residuals for the disappearing products using the time dummy model in period 0. u^-0[D(t)(TPD)] being the equivalent for the time
product dummy model. u[M(TDH)]^-0(t) is a weighted arithmetic mean of the period 0 residuals but using the period t weights.
The first and second components are driven by the difference in the weighted average residuals in the disappearing and new products respectively, so if the difference is larger (in absolute value)
there is a bigger disparity between the models. It also depends on the weights that each product has, so if there is a higher weight to the products that appear in both periods than to those that
only appear in one period, the disparity would be smaller.
The third component depends on the normalised expenditures, and will generally be different from 1, this is because as a result of using the weighted least squares (WLS) time dummy technique are
model-dependent. This term has the possibility of being larger when the weights of the matched product differ significantly, and will be equal to 1 in the situation where weights are constant over
time. If an ordinary least squares (OLS) regression is used this term would always be equal to 1.
Due to weights not being observed at an individual product level, the WLS regressions discussed can’t be used and therefore OLS regression would be used to estimate the missing prices. In the ONS web
scraped data there is limited characteristic data to produce a time dummy model therefore only the time product dummy method would be used. The Törnqvist would also be replaced with a Jevons due to
the lack of weights, creating an IJRYGEKS (a Jevons version of the ITRYGEKS).
3.6.4 Intersection GEKS-J (IntGEKS-J)
The IntGEKS was devised by Lamboray and Krsinich [2015], to deal with an apparent flattening of RYGEKS under longer window lengths, although this was found later to be an error in applying the
weights. It removes the asymmetry in the matched sets between periods 0 and i and between periods i and t, by including products in the matched sets only if they appear in all three periods, the set
S^0,i,t. The formula is defined as follows:
If there is no product churn then the IntGEKS-J reduces to the standard GEKS-J. The IntGEKS-J has more chance of "failing", that is, not being able to calculate an index, than a standard GEKS-J as
the products need to appear in more periods. Using a Jevons as the base in the IntGEKS-J means that it then simplifies to an average of Jevons indices over differing samples. This is because the
prices of the intermediate period i would cancel out in the calculation.
3.7 The Fixed Effects index with a Window Splice (FEWS)
The FEWS produces a non-revisable and fully quality-adjusted price index where there is longitudinal price and quantity information at a detailed product specification level, (Krsinich [2016]). It is
based around the Fixed Effects index which is defined as follows:
where γ^0 is the average of the estimated fixed effects regression coefficient at time 0. The Fixed Effects index is equivalent to a fully interacted^5 Time Dummy index if the characteristics are
treated as categorical. So if the product identifier, such as a barcode, changes when a price determining characteristics changes the fixed effects will equal a Time Dummy index (Krsinich [2016]).
The Fixed Effects index is also equivalent to a Time Product Dummy index. Using a similar methodology to RYGEKS, the new series is spliced onto the current series for subsequent periods after the
initial estimation window; this is called a window splice. The window splice is done as follows:
where P[[0,d]]^0,1 is the index from period 0 to period 1 using the estimation window [ 0 , d ].
This approach has the advantage of incorporating implicit price movements of new products at a lag. However, this means that there is a trade-off between the quality of the index in the current
period and in the long-term. Over the long-term, the FEWS method will remove any systematic bias due to not adjusting for the implicit price movements of new and disappearing items.
3.8 The Geary-Khamis index
The Geary-Khamis (GK) index was developed for PPPs, but unlike the GEKS, which compares each country to each other, the GK index compares each country to a standard country. It is an implicit price
index that divides a value index by a weighted quantity index. Using similar notation to Chessa [2016] it is defined as:
where the weights v[i] are as follows:
The denominator essentially adjusts for quality, which is why this form of the Geary-Khamis index is also called a quality adjusted Unit Value index.
For the following analysis, the IntGEKS and the IJRYGEKS have been excluded. This is due to the former being an average of Fixed Based Jevons indices over different samples, and would have both the
properties of the GEKS-J. The latter is just a different input index to a RYGEKS and should therefore have the same properties. The Geary-Khamis index is also excluded due to the current lack of a
quantity proxy.
Notes for: Price index methods for use on web scraped prices data
1. The IntGEKS-J was not included in the comparisons as it uses the same structure as GEKS-J but just includes a different set of products. The IJRYGEKS (a Jevons version of the ITRYGEKS) was also
not included due to the lack of characteristics to perform the imputations required.
2. A scraper break occurs when no data appears for all products due to a website redesign or a lab outage. For more information, please see Box 1 in Breton et al 2016.
3. Unit value is defined as
4. Characteristicity is the extent to which an index is based on relevant data.
5. Fully interacted means that all possible interaction terms are included in the model, for example y=a+b+c is not fully interacted, neither is y=a+b+c+ab, which includes an interaction term, only
y=a+b+c+ab+ac+bc+abc is fully interacted.
Back to table of contents
4. The test/axiomatic approach
The test/axiomatic approach to index numbers assesses the index number formulae against some desirable functional properties (axioms) an index should have. An index will either pass or fail each
axiom. If an index passes all of the axioms, then it is deemed to perform well. However, it is important to note that different price statisticians may have different ideas about what axioms are
important, and alternative sets of axioms can lead to the selection of different “best” index number formula. There is no universal agreement on what is the best set of reasonable axioms and
therefore, the axiomatic approach can lead to more than one best index number formula.
The following definitions will be used throughout this section.
Let p^0 and p^t be the vector of prices observed at the period 0 and t respectively and let P ( p^0 , p^t ) be an arbitrary price index.
Axiom 1 [Positivity Test]
An index should be positive:
P ( p^0 , p^t )>0
All of the formulae satisfy Axiom 1. This is easy to show because all of the formulae involve a geometric mean, either of the prices or price relatives. The geometric mean is defined as:
Given that measured prices are always positive, the domain is restricted to R[+]^n. This means that the co-domain is restricted to R[+].
Further consideration is required of the Fixed Effects index with a Window Splice (FEWS) index because of the quality adjustment term. However, although the regression coefficients, their arithmetic
mean and the difference between the two periods could all be negative, taking the exponential of the resulting difference makes it positive. This is because the exponential function maps R to R[+].
Axiom 2 [Continuity Test]
P ( p^0 , p^t ) is a continuous function of its arguments
Axiom 2 is satisfied by all of the formulae. This is because all of the input functions (that is, arithmetic and geometric means and the exponential) are continuous, the sums and products of
continuous functions are also all continuous, and therefore all of the price indices are continuous.
Axiom 3 [Identity/constant prices Test]
P ( p , p )=1
If the prices are constant then the index is 1
If there is no product churn in the data then all of the indices satisfy Axiom 3. However, if there is product churn then the FEWS, Unit Value and the Clustering Large datasets into Price Indices
(CLIP) indices fail this axiom. It is worth noting that the CLIP only fails with respect to products rather than clusters (this is because the price of each cluster is constant).
The reason that the Jevons-based indices pass this axiom is that they require matched products to be included in the index and therefore unmatched products are excluded from the index. The FEWS, Unit
Value and CLIP use unmatched products and therefore are affected by product churn. To see this, let S[0] = { p[1] , p[2] , p[3] } and S[1] = { p[1] , p[3] } be the prices observed in period 0 and
period 1 respectively. For the CLIP, Unit Value and the non-quality adjustment part of the FEWS, the following is obtained:
In general this is true but there are two special cases where the ratio is equal to 1:
1. p[1] = p[2] = p[3]
2. p[2] = √ p[1] p[3].
For the quality adjustment factor part of the FEWS index, the difference of the arithmetic means of the fixed effects coefficients needs to equal zero for the Identity Test to be satisfied. However,
when a product churns out of the dataset, this is unlikely to occur, as:
It is entirely possible that when a product churns out of the dataset, the coefficients for the other products will adjust to take account of this missing product. For the FEWS to pass this axiom
both parts of the index must either be equal to 1 individually or they can be not equal to 1 but their product must be.
Figure 4 shows the effect of product churn on FEWS when all of the prices are constant. For each period 5% of the products were removed from the sample, taking a simple random sample without
replacement. For most of the periods this removal of products decreases the index because the geometric mean of the current period will be smaller than for the base period. However, for some periods
this increases the index, due to the quality adjustment factors over-adjusting for the change in quality due to the product churn.
Figure 4: The effect of product churn on the FEWS in the Identity Test, showing only the first 130 periods
Source: Office for National Statistics
Download this chart Figure 4: The effect of product churn on the FEWS in the Identity Test, showing only the first 130 periods
Image .csv .xls
Axiom 4 [Proportionality in Current Prices Test]
If all current prices are multiplied by λ then the new price index is λ times the original index, that is:
P ( p^0 , λ p^t ) = λ P ( p^0 , p^t )
Axiom 5 [Inverse proportionality in Base Prices Test]
If all base prices are multiplied by λ then the new price index is the original index divided by λ, that is:
P ( λ p^0 , p^t ) = P ( p^0 , p^t) / λ
All indices satisfy both Axioms 4 and 5; this is shown in the following proof for the Jevons index.
Let P ( p^0 , p^t ) be the Jevons index. Then it follows:
The same logic can be applied for base prices.
For the other indices, the same logic applies, for example in the CLIP the λ would be multiplied by itself ∣ C[i]^t ∣ times within a cluster but then the ∣ C[i]^t ∣ th root would be taken in each
cluster and the λ would be taken out as a factor. The same happens for the Unit Value and the FEWS. This λ factor wouldn’t affect the quality adjustment factors as it would be absorbed into the
coefficient of the time dummy for the current period, making it logλ larger.
For the GEKS, from each of the bilateral Jevons there is a λ, this is raised to the power of t+1 because of the t+1 comparisons from intermediate period i to current period t but since the geometric
mean of all these comparisons is taken in the GEKS this becomes λ. In summary, product churn does not have an impact on this axiom, as it only affects n.
Axiom 6 [Commodity Reversal Test]
If the ordering of the products changes the price index stays the same, that is:
where A is a permutation matrix .
Due to the commutative property of addition and multiplication over the real numbers all of the indices pass the Commodity Reversal Test.
Axiom 7 [Invariance to the change in units/Commensurability Test]
P ( Dp^0 , Dp^t ) = P ( p^0 , p^t )
where D is a diagonal matrix with the conversion factors on the diagonal.
The CLIP, Unit Value and FEWS indices fail this test in the presence of product churn because the conversion factors do not cancel out in the unmatched means. Using the same two observation vectors
that were used in the Identity Test (Axiom 3) along with the following conversion matrix, D = diag ( d[1~,d~2~,d~3] ) the following happens:
As with Axiom 3, there are special cases where the geometric means of the conversion factors happen to be equal and where all the conversion factors are equal.
Axiom 8 [Time Reversal Test]
If the data for the base and current periods are interchanged then the resulting index is the reciprocal of the original one.
The RYGEKS, FEWS, and CLIP indices fail this test^1. The RYGEKS index fails it due to the initial window before the chaining starts. This is because if there is price change in the window, the
following happens:
GEKSJ^0,d ≠ GEKSJ^t,t-d.
For the CLIP, there are two reasons. The first reason is that the clusters are weighted together arithmetically, so that:
The second reason is that the clusters that would be formed if the current period was used as a base for the clustering may be different to those formed when clustering on the base period. This is
not a fault of clustering, as consumers would have chosen different products given what was available on the market at that point. Overall, if the clusters were geometrically aggregated then the CLIP
would pass this axiom given that the clusters are equivalent in the base and current period.
There are two potential reasons why the FEWS fails this test; either the fixed effect regressions provide different answers when running the regressions reversing time, or the splicing factors are
Simulations can be run to test which factor is more important. First, an artificial dataset is generated, simulated from the following formulae:
p[i]^t = p[i]^0 + β[i] t
where the initial prices are a set of random numbers drawn from the gamma distribution with shape parameter 20 and rate parameter 2:
p[i]^0 ∼ Γ (20,2)
and the gradient is drawn from a standard uniform distribution, that is:
β[i] ∼ U [0,1].
Figure 5: Fixed Effects with a Window Splice (FEWS) index from simulated data
Source: Office for National Statistics
Download this chart Figure 5: Fixed Effects with a Window Splice (FEWS) index from simulated data
Image .csv .xls
An Index calculated from this data can be seen in figure 5. From this data, P^0,1000 =43.06169 and the reciprocal of P^1000,0 is 43.06169. On this level of precision, the axiom appears to hold,
however when the difference between the two indices are taken the difference is equal to 1.588987×10^-10. The FEWS can be decomposed as the product of the Fixed Effects index and the splicing
factors. For P^0,1000 this decomposition is 1.011574 for the Fixed Effects index and 42.5698 for the splicing factors. For the reciprocal of P^1000,0, the Fixed Effects index is 1.634896 and the
splicing factors is 26.339094. This difference in decomposition is the reason that the indices are different. The main contributors to this difference are the fixed effect index factors, which might
be because the indices are calculated on different estimation windows.
Axiom 9 [Circularity Test]
P ( p^0 , p^s ) P ( p^s , p^t ) P ( p^t , p^0 ) = 1
If an index is calculated from period 0 to an intermediate period s, from s to t, from t back to 0 and then chained, the resulting index should not show any change.
All indices that fail the Time Reversal Test fail the Circularity Test due to the reversal of the time periods in the final chain.
Axiom 10 [Monotonicity in Current Prices Test]
P ( p^0 , p^t ) < P ( p^0 , p )if p^t < p
Axiom 11 [Monotonicity in Base Prices Test]
P ( p^0 , p^t ) > P ( p , p^t )if p^0 > p
All indices pass these axioms as all the functions are monotonic. This can be shown as follows.
Let p > p^t (in other words p[i] > p[i]^t ∀ i), then for all indices that are based on price relatives the following is obtained:
p[i] > p[i]^t ∀i
divide by the base price for each product
taking the geometric mean of both sides
Since the geometric mean of the price relatives using the current prices ( p^t ) is less than using the larger prices ( p ) this means that P ( p^0 , p^t ) < P ( p^0 , p ). For price indices that use
the ratio of averages the proof is the same (that is, taking the average of the prices then dividing by the average base price).
For Monotonicity in base prices you get the following:
p[i] > p[i]^0 ∀i
divide into the current price for each product
taking the geometric mean of both sides
Since the geometric mean of the price relatives using the base prices is greater than using the larger prices this means that P ( p , p^t ) < P ( p^0 , p^t ). For price indices that use the ratio of
averages the proof is the same (that is, taking the average of the prices then dividing by the average base price). For the quality adjustments in the FEWS they would be the same for both p and p^t
as the increase in the price would be absorbed into the time dummy coefficients.
Axiom 12 [Price Bounce Test]
The price index should not change if prices are rearranged and then returned to their original order:
All indices except the Fixed Based Jevons fail the Price Bounce Test in the presence of product churn. This is due to the unmatched products that have to be removed from the calculations.
Example 4.1
Table 1 contains a set of prices for three different products where the prices are bouncing. After the third period, products start churning.
Table 1: Example prices for three products
Product\Time 0 1 2 3 4 5 6
A 1 3 1 3 1 1
B 2 1 2 1 2 1 2
C 3 2 3 3 2
Source: Office for National Statistics
Download this table Table 1: Example prices for three products
.xls (26.1 kB)
Table 2 contains the corresponding price relatives for these prices (taking the previous period as the base period). Please note that it is slightly sparser than Table 1 because of the compounding
factor of the missing prices.
Table 2: Example price relatives for three products
Product\Time 0 1 2 3 4 5 6
A 1 3 0.33 3 0.33
B 1 0.5 2 0.5 2 0.5 2
C 1 0.66 1.5 0.66
Source: Office for National Statistics
Download this table Table 2: Example price relatives for three products
.xls (18.4 kB)
A Chained Bilateral Jevons index can be calculated from these price relatives in Table 2. When the products are not churning, you can see that the Chained Bilateral Jevons is not affected by price
bounce. However, when the product goes out of stock, the Chained Bilateral index is affected by price bounce. The same happens for the GEKS-J index but the calculations are more complicated.
Table 3: Bilateral Jevons indices P[J]^t-1,t and Chained Bilateral Jevons indices P[CBJ]^0,t for the three products
Time P[J]^t-1,t P[CBJ]^0,t
3 1.22 1.22
4 0.82 1
5 0.58 0.58
6 2 1.16
Source: Office for National Statistics
Download this table Table 3: Bilateral Jevons indices P~J~^t-1,t^ and Chained Bilateral Jevons indices P~CBJ~^0,t^ for the three products
.xls (26.1 kB)
4.1 Summary of axiomatic approach
Table 4 shows whether the indices under consideration passed or failed each of the axioms considered.
Table 4: Summary of index performance under the axiomatic approach
Source: Office for National Statistics
Download this image Table 4: Summary of index performance under the axiomatic approach
.png (33.5 kB)
In summary, only the Fixed Based Jevons index passes all of the axioms. However, this is not suitable for a monthly production index because it relies on products existing in every month. Figure 6
uses ONS web scraped data for strawberries to demonstrate how the Fixed Based Jevons fails when products are not consistently in the dataset each month.
Figure 6: Comparison of indices based on web scraped data for strawberries, Index June 2014 =100
UK, June 2014 to March 2017
Source: Office for National Statistics
Download this chart Figure 6: Comparison of indices based on web scraped data for strawberries, Index June 2014 =100
Image .csv .xls
For 2016, there is no product that exists in every month, in particular, from September onwards there are no products on the market that were sold in the first 8 months of the year. Therefore, a
Fixed Basket Jevons index cannot be calculated as the set of matching products is empty, that is, ( S ^* =∅.
The GEKS-J and the Chained Bilateral Jevons index pass all axioms other than price bounce. Therefore, under the axiomatic approach, the GEKS-J index and the Chained Bilateral Jevons index also
perform well. According to Ivancic, L. [2011] the GEKS methodology eliminates chain drift. Therefore, if a method that does not suffer from chain drift is desired then the GEKS-J method is
Notes for: The test/axiomatic approach
1. This is one of the tests that may be deemed less relevant as two commonly used price indices, the Laspeyres index and the Paasche index, fail this test as well.
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The economic approach to index numbers assumes that prices and quantities are interdependent. This means that the quantity of a product bought depends on its prices; q = q ( p ), that is, the
quantities are functions of prices. This allows the consumer to adjust their spending depending on how relatively cheap or expensive the product becomes. In the context of consumer prices, the
economic approach usually requires the chosen index formulae to be some kind of Cost of Living index (COLI).
This approach considers the indices against the COLI. However, it is important to note that this is not appropriate in the context of choosing a methodology that is suitable for use in the Consumer
Prices Index including owner occupiers’ housing costs (CPIH).
This is because the CPIH measures the change in prices of goods and services purchased to be consumed; this is different to a COLI, which measures the change in the cost of obtaining a certain level
of utility given the different prices. In the CPIH, the consumption (that is, demand) is fixed, whereas a COLI allows for demand to change as prices change. Therefore, even if an index approximates
or is exact for a COLI, it might not be a good index to use in the context of the CPIH. However, the results of assessing each index against the economic approach are presented here to provide useful
reference to other national statistical institutes (NSIs) who produce a COLI alongside their other consumer price statistics.
The economic approach is based on consumer preferences. A consumer chooses the combination of products that they most prefer out of all that they can afford. It is easy to define what is meant by
“afford” by defining the budget set (Definition 5.1).
Definition 5.1 [Budget set]
For a set of products q with prices p and total budget x, then the budget set B is defined as:
A budget is exhausted when
A consumer can afford a combination of products if that combination is within the budget set. Defining preferences is slightly less easy to do, but this is managed through the utility function. A
utility function can only be used when the consumer’s preferences are "well-behaved"^1. The utility function assigns a number to a set of products and preserves preferences, for example, if the
consumer prefers a set of good X to a set of good Y, that is, X≺Y, then u(X) < u(Y) where u is the utility function.
A utility function allows us to summarise a large amount of data in a few values. In this context, the actual values of the utility function are not significant; we are just focusing on the ordering
of preferences.
To see this, let u[1] and u[2] be two utility functions and X and Y be two consumption bundles such that u[1](X)=5, u[1](Y)=10, u[2](X)=20 and u[2](Y)=25. Both utility functions show that X≺Y so the
actual utility functions chosen don’t matter. However, it still allows us to identify sets of goods, which consumers are indifferent to (Definition 5.2).
Definition 5.2 [Indifference]
For two sets of goods X and Y, a consumer is indifferent between them if u(X)=u(Y). This is shown as X∼Y
Figure 7 presents some example indifference curves, which correspond to the utility function u ( q ) = 2ln ( q[1] ) + 2ln ( q[2] ). The indifference curve labelled u = 12 gives all combinations of
good one (X) and good two (Y), which give the consumer a utility of 12.
Figure 7: A utility function with three indifference curves labelled
Source: Office for National Statistics
Download this image Figure 7: A utility function with three indifference curves labelled
.png (20.5 kB)
What economists mean by "prefer" and "afford" in the following statement: "A consumer chooses the combination of products which they most prefer out of all they can afford" can be rewritten as:
"Consumers get on to the highest indifference curves, subject to their budget constraint".
This is shown in Figure 8. The consumer’s budget constraint, q[1] + 594 q[2] = 594, is the straight line and their budget set B is the shaded triangle. In this example, two utility curves fall within
the budget set: one where the utility is equal to 8 and the other when the utility is equal to 10. The consumer can therefore buy any set of products on the u = 8 line and the u = 10 line. As the
consumer is assumed to want to maximise their utility they will buy the set of goods where the budget line touches their highest indifference curve ( u = 10 ). Buying this set of products also
exhausts their budget.
Figure 8: A utility function with a budget constraint
Source: Office for National Statistics
Download this image Figure 8: A utility function with a budget constraint
.png (64.6 kB)
Mathematically, this process of the consumer choosing their most preferred set of products out of those that they can afford is a constrained maximisation problem, which can be defined as:
Solving this constrained maximisation gives us the result that quantities depend on prices because the solutions are of the form:
that is, the quantities bought are a function of the prices observed and the consumer budget. This solution is called the indirect utility function, v(p,x) and gives the maximum utility that the
consumer can reach given the prices faced and the available budget.
For the example utility function (u( q ) = 2ln ( q[1] ) + 2ln ( q[2] )), the quantities that maximise it given the budget constraint are
In this situation, the quantities are dependent on the price (if the price of a good doubles then the quantity bought halves).
The indirect utility function can then be derived as:
The inverse of the indirect utility function is called the cost function, c ( p , u ), and represents the minimum cost of reaching a given level of utility, given the prices that the consumer faces.
The cost function essentially says that if a consumer is given a choice of different sets of products to which they are indifferent, then the consumer will choose the least expensive. For this
indirect utility function, the cost function is:
This means that when the prices are p[0] then the consumer will choose to buy q[0] of the products. If prices change to p[1], then the consumer may not want to buy q[0] (the quanity that is assumed
in the fixed basket approach, such as the Fixed Based Jevons). Instead, they might change to another combination of products q^* where u ( q[0] ) = u ( q^* ). This allows for the substitution of
products that have become relatively expensive towards those products whose relative price has fallen.
A Cost of Living index (COLI) allows for this behaviour to be taken into account in the calculation of the index (Definition 5.3).
Definition 5.3 [Cost Of Living index]
A Cost of Living index P[K] is the ratio of the cost of achieving a given utility at current period prices to the cost of achieving a given utility at the base period prices:
Figure 9: Demonstrating a Cost of Living index
Source: Office for National Statistics
Download this image Figure 9: Demonstrating a Cost of Living index
.png (90.9 kB)
Figure 9 illustrates a COLI. At period 0 prices the budget constraint is line A and the highest indifference curve that can be obtained with that constraint is u0 at the point E0. E0 is the cost of
attaining utility u0 at base prices. When the price of good one increases in the next period, the budget set moves down to line B. This means that the original indifference curve can't be reached.
Instead a new indifference curve is reached, which meets u1 at the point E1. E1 is the cost of attaining the utility u1 at current prices.
A COLI index can either use the cost of attaining the utility u0 at current prices or the cost of attaining the utility u1 at base prices. This can be achieved by moving the respective budget
constraint to become tangential to the desired utility function. This is what the lines C and D represent in Figure 9.
Line C is the budget constraint at period 1 prices shifted to attain the original utility. The distance that it has been shifted is called the compensated variation; the amount that the consumer’s
income needs to be increased in order to offset inflation. From this, it can be seen that the cost of achieving the original utility is the point E2, which is different to the cost at base prices, E0
, in this example. As prices changed, the consumer then substituted from the quantities consumed at E0 to the quantities consumed at E2.
In contrast, if utility u1 is used instead, the budget constraint at base prices, line A, is shifted down and becomes line D. The distance by which it is shifted is called the equivalent variation;
the amount of consumer’s income that needs to be taken away at the base price level, to have the same impact on their utility as inflation between the two periods. Another cost is calculated, the
point E3, which is the cost of attaining the current utility at base prices. Therefore, if the consumer faced period 0 prices on this utility curve they would substitute to E3.
In summary, there are two cost of living indices, one for each utility level. In general, P[K] ( p[0] , p[1] , u[0] ) ≠ P[K] ( p[0] , p[1] , u[1] ), which means that a COLI depends on the level of
However, if the cost function can be written as c ( p, u ) = a (p) b ( u ) (which comes about if the utility function is homothetic), then the COLI doesn't depend on the level of utility. Using the
example given, the cost function is already written in the form c ( p , u ) = a ( p ) b ( u ) where a ( p ) = √( p[1] p[2] ) and
This means that the COLI is:
This utility function was chosen because the consumer preferences that it represents mean that the COLI can be written as being equal to a Fixed Based Jevons and a Unit Value. This demonstrates how
the economic approach can be used as a way of evaluating the indices; the aim is to identify a particular set of preferences that mean that the indices can be classified into one of three groups:
• approximate
• exact
• superlative
An index approximates a COLI if it appears in a Taylor expansion of the COLI index (for example, Paasche and Laspeyres indices are approximate). An index is exact for a COLI if under a certain form
of consumer preferences the resulting COLI index is equal to a known index formula. A superlative index is an index that is exact with a cost function that has a flexible functional form^2. The
Törnqvist and Fisher are both exact and superlative indices. For this work, the exact class of indices is going to be used.
There are two caveats to using this consumer preference theory: the consumer is assumed to be rational, and the consumer aims to maximise their utility when making their choice. There have been many
experiments to test these assumptions, for example Harbaugh et al. [2001] tested this on children aged 7 and 11, and on economics undergraduates. They found that 26%, 62%, and 65% respectively were
"rationalisable". Beatty and Crawford [2011] looked at the Spanish Continuous Family Expenditure Survey from 1985 to 1997 and found that 95.7% of those in the survey were consistent with the theory.
There are many different types of preferences that economists have developed to explain consumer behaviour, two of which are useful for this work: Cobb-Douglas preferences and Stone-Geary
Definition 5.4 [Cobb-Douglas preferences]
A consumer has Cobb-Douglas preferences if their preferences can be explained by the following utility function:
which has the following cost function:
Definition 5.5 [Stone-Geary preferences]
A consumer has Stone-Geary preferences if their preference can be explained by the following utility function:
which has the following cost function:
From Cobb-Douglas preferences the following result is found:
where w[j] = m ( w[j]^0, w[j]^t ) with m a function such that m (x,x) = x. The arithmetic mean is a function that has this property. This means that any geometric index is exact under Cobb-Douglas
preferences. Since the indices under consideration involve a geometric mean, they may be exact indices.
Let’s take the Jevons index:
This is equivalent to the geometric index formulae if the weights are set equal to 1/n. This means that the Jevons index is an exact index under Cobb-Douglas preferences.
The Unit Value index doesn't look like the form that is required for the geometric index, although it can be rewritten to show that it fits in this form:
The weights are defined as:
In this solution, the first case is when a product disappears, and the last is when a product is introduced. Using this, the Unit Value index is also exact for Cobb-Douglas preferences. This might
show a downside to the economic approach in that a type of preference does not give a unique solution.
Further research is required to find a set of preferences for which an arithmetically aggregated CLIP is exact or if it approximates a COLI. However, a geometrically aggregated CLIP mentioned in the
axiomatic approach would be exact. This can be shown by augmenting the unit value weights to incorporate the clusters. The extra case is required in the event that a product moves between clusters in
the different time periods:
Neary and Gleeson [1997] show that Stone-Geary preferences lead to a GEKS index, and therefore a GEKS index is exact. Balk [2001] also looks to find preferences for GEKS, in the context of
international comparisons. Using a Stone-Geary preference structure for the GEKS makes sense in that the demand function depends on all prices whereas the demand function for Cobb-Douglas only
depends on the price of the product in question.
The Fixed Effects index with a Window Splice (FEWS) index, more specifically the fixed effects part of the index, is a hedonic index. It adjusts the index to account for the change in quality of the
new set of products in the index. The hedonic indices also come from economic theory and can be placed in the utility maximisation, consumer theory paradigm. The dual problem to the utility
maximisation is the following minimisation:
In the hedonic space, the quantities of the goods aren’t important; rather it is the characteristics that the goods have that affect the regression. Therefore, the minimisation problem changes to be
over the characteristics, that is, the minimisation then becomes in the hedonic space:
where h is the hedonic function and U[h] is the utility function in the hedonic space. This can then be used to calculate COLIs depending on the choice of hedonic function. In this case the hedonic
function is the time product dummy. For more information please see Triplett [2006] about hedonics and Diewert [2003] on hedonics in relation to consumer theory.
In summary the economic approach gives little insight into which formulae to use because the indices are all exact under a certain set of preferences, and therefore no clear conclusions can be drawn.
Notes for: The economic approach
1. “Well-behaved preferences” mean that they are complete, reflexive, transitive, monotonic, convex and continuous, and there exists a continuous function that describes those preferences, (Debreu
2. A flexible functional form f is a functional form that has enough parameters in it so that f can approximate an arbitrary, twice continuously differentiable function f* to the second order at x*.
Thus f must have enough free parameters to satisfy the following 1+N+N(N+1)/2 equations. f ( x* ) = f* ( x* ); ∇ f ( x* ) = ∇ f* ( x* ) ; ∇^2 f ( x* ) = ∇^2 f* ( x* );
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6. The statistical approach
The statistical approach to index number theory treats each price relative as an estimate of a common price change. Hence, the expected value of the common price change can be derived by the
appropriate averaging of a random sample of price relatives drawn from a defined universe. These are effectively models for inflation rates.
For example, the most simplistic model of price change is the following model:
Here π is the common inflation rate and ε[i] is the error for product i. This essentially assumes that price changes are normally distributed. The Ordinary Least Square (OLS) estimator or maximum
likelihood estimator (MLE) for this model is the Carli index:
If instead it is assumed that price changes are log-normally distributed, then the model that prices follow is:
where θ=logπ is the log inflation rate. The OLS and MLE estimates of θ is:
and therefore the estimate of π is:
which is the Jevons price index.
These examples show that different models will give different inflation rate estimates. These estimators can also be derived from a design-based survey sampling methodology.
As well as this form of price change (that is, a constant term π plus an error term ε[i]), there are different forms that can be used. A second set are models of the form:
Models of this form are known as ratio estimators.
The Dutot index can be directly derived from the above, as:
If log prices are used in the place of prices, a ratio estimator of the log inflation rate is obtained:
This gives a Jevons index in the case of a matched sample, (Silver and Heravi [2006]). In the case of unmatched samples, which are more likely to be the case given the use of web scraped data, a Unit
Value index can be derived. This shows that the Unit Value index is a geometric equivalent of the Dutot index.
The Unit Value index is sensitive to heterogeneity in the products, as is the Dutot index. Silver and Heravi [2006] suggest hedonically adjusting to control for this heterogeneity, but this is
difficult without characteristic information. However, the way that the CLIP is designed should overcome this heterogeneity. This is because the CLIP stratifies a Unit Value index by defining the
strata as the local maxima of the kernel density estimates, using a clustering algorithm. These strata should then be homogeneous. This stratification improves the precision of the estimate, as
elements within strata would have similar values, and between strata should have different values, (Lohr [1999]).
As an example, Figure 10 shows the price distributions of tea bag prices for Office for National Statistics (ONS) web scraped data and the published Consumer Prices Index including owner occupiers’
housing costs (CPIH) microdata. Both modes of collection show that there is some heterogeneity in the prices, and therefore a unit value would be sensitive to this. The CLIP methodology should help
to overcome this. In the CPIH, the heterogeneity may also be reduced by the current stratification in the compilation of the index; for example, tea is stratified by region and shop type.
Figure 10: The distribution of tea bag prices in the published CPIH micro-data (blue) and the ONS web scraped data (pink), January 2015
Source: Office for National Statistics
Download this image Figure 10: The distribution of tea bag prices in the published CPIH micro-data (blue) and the ONS web scraped data (pink), January 2015
.png (67.7 kB)
The weights, w[h], used to aggregate the CLIP can be interpreted as the probability of a product being in that cluster or strata in the base period. These weights can be used to aggregate the
estimates of the inflation rate or to aggregate the estimates of the log inflation rates give two different versions of the CLIP. If the weights are applied after estimating π[h], (that is, the
stratum-specific inflation rates using prices), the Arithmetic CLIP (ACLIP) is obtained. If the weights are applied after estimating θ[h] (that is, the stratum-specific inflation rates using log
prices) then the Geometric CLIP (GCLIP) is obtained:
The next set of models combine the two previous forms. That is, the variables that are used in the model are price relatives, but the estimators are ratio estimators. Let’s suppose that the model is:
where j∈[0, t ] are the intermediate periods, and ε[i]^j is the error for product i when going through intermediate period j. This gives us a ratio estimator, which equates to the Chained Jevons part
of the GEKS-J index. If a 1/t weight is then given to each intermediate estimate of the log inflations, this gives a GEKS-J index, as follows:
The final set of models decompose price into constituent parts. These are what hedonic indices do. There are three types of hedonic models:
1. Linear characteristics:
where α^j is the intercept at time j, β[k]^j is the effect characteristic k has on the price at time j, and z[ik] is the value of characteristic k for product i. The linear characteristics model
assumes that the characteristic effects change over time.
2. The time dummy hedonic method:
This method assumes that the characteristic effects don't change over time, and includes a dummy variable, D[i]^j, for which period the product was collected:
where δ^t is the time-specific effect.
3. The time product dummy method:
This method can be used when detailed characteristic information is not available, and a dummy variable, D[i], for the product is created:
where γ[i] is the product-specific effect and the N^th product is taken as the reference product. This method assumes that the quality of each distinct product is different to the quality of
other products to a consumer. It is a reasonable assumption as the number of potential characteristics is large and not all of them are observable. The time product dummy is also equivalent to a
fully interacted time dummy method.
The method that was used in this work was the time product dummy (TPD) method. A TPD index is the same as a Fixed Effect index (FE), which is the basis of the FEWS.
Now that all of the indices have a statistical model associated with them, the statistical approach becomes a model-fitting exercise. This involves selecting the model which “best fits'' the data. In
this context, best fit means that the model is chosen that minimises the information lost from using this model to fit the data. The Akaike Information Criterion (AIC), (Akaike [1974]), was used to
measure this fit. The AIC is defined as:
where k is the number of parameters in the model, L is the likelihood function for that model, and which is evaluated at the maximum likelihood estimator (MLE) of the parameter θ.
The models for the Jevons, Unit Value, GEKS-J and FE/TPD were fitted to a sample of items from the grocery collection, clothing data and the “central collections”^1 project. For the grocery and
clothing data, daily prices were used to fit the models. Two months of data were used; one being from a period when the CPIH 12-month growth rate was around 1% (June 2014) and the second from a
period when the CPIH 12-month growth rate was around 0% (June 2015). This was done to test whether different inflationary environments caused a change in “best" model. The centrally collected items
are collected monthly and the whole time series was used.
Table 5 presents a summary of index performance under the statistical approach. It is worth noting that for the groceries and the clothing data, the FE/TPD overfits the data. This overfitting is due
to the large number of product dummy parameters in the estimation, and confirms the problems with the TPD index that De Haan et al. [2016] present in their research.
Table 5: Summary of index performance under the statistical approach
Group Best Fit
Groceries Fixed Based Jevons
Clothing Unit Value
Central Collection – Chart Fixed Based Jevons
Central Collection – Technological Fixed Effects
Source: Office for National Statistics
Download this table Table 5: Summary of index performance under the statistical approach
.xls (26.1 kB)
From the statistical approach, the Unit Value index was identified as the preferred index to use on clothing items. Stratification improves the precision of the estimate if the sampling units are
homogeneous within a stratum and heterogeneous between strata. As the CLIP is a stratified Unit Value index, it follows that it should be a better approach to use for this category.
Assessing the indices against the statistical approach provides evidence to suggest that no single index is suitable to cover the full range of items in the CPIH basket of goods and services that
could be sourced from web scraped data.
Notes for: The statistical approach
1. Prices for these centrally collected items are obtained manually by ONS price collectors through websites, phone calls, emails, CDs and brochures. Centrally collected items cover approximately
26% of the CPIH basket. As part of its research into alternative data sources, ONS is currently piloting alternative ways of collecting these data through “point and click" web scraping software.
Back to table of contents
7. Summary and recommendations
The axiomatic and statistical approaches to index number theory identify different indices as being more appropriate for particular categories of items. This section considers from a methodological
viewpoint how these indices could be incorporated in the Consumer Prices Index including owner occupiers’ housing costs (CPIH). The economic approach was not considered here because even if an index
approximates or is exact for a Cost of Living Index (COLI), it might not be appropriate in the context of the CPIH (Section 5).
7.1 Grocery prices
When considering both the axiomatic and statistical approaches, the GEKS-J is the most appropriate index to use for groceries within the CPIH basket. This is due to its axiomatic properties and
because the GEKS-J is designed to limit chain drift, (de Haan and van der Grient [2006]).
The statistical approach also suggested that for groceries a Fixed Based Jevons index may also be appropriate. It is common practice to periodically chain indices to allow for new products entering
the market. While the Chained Bilateral Jevons does chain the indices to allow for new products entering the market, the presence of product churn results in the index displaying chain drift.
In the construction of the GEKS-J, the Bilateral Jevons between base and intermediate period is chained with the bilateral Jevons between the intermediate and current period. This reduces any chain
drift that would have occurred when calculating a Chained Bilateral Jevons between the base and current period. However, chain drift is also observed when the GEKS-J index is chained, a full proof of
which is provided in Appendix A:
The Australian Bureau of Statistics has also used the GEKS index with a Törnqvist (GEKS-T) base in their CPI (Kalisch [2016]), with a mean splice as an extension method (Kalisch [2017]). Extension
methods extend the index beyond the initial observation window, with a choice of which period to chain the index onto. There are a number of extension methods that can be used:
• the window splice is where the index is spliced in the start of the window
• the half splice is where the index is spliced in the middle of the window
• the movement splice is where the index is spliced at the end of the window
• the mean splice is where every period is used as a splicing period and the geometric mean of the periods is used as the splicing factor; for a detailed description of the method see Kalisch
These extension methods help to overcome the traditional problem associated with formulating a GEKS index, which is that it is open to revisions. This is because it involves using price movements
from the current period to future periods, which is overcome if these splicing methods are used. These extension methods also reduce the loss of characterisiticity when using multilateral methods on
long time series.
Office for National Statistics (ONS) developed another solution to this problem: an ”In Period GEKS-J", where only data observed up to the current period are used in the calculation of the index.
This means that when a new period's data is collected, the new data is only used to calculate the current period index, and is not used to recalculate the historical series. The “In Period GEKS-J”
has been used throughout this review.
Another potential problem with the GEKS index is that it is responsive to atypical prices. Chessa et al. [2017] demonstrates that the GEKS-T is sensitive to these prices, mainly those associated with
products that have been “reduced to clear”, referred to by Chessa as “dump prices”. Rewriting equations 13 to 15 of Chessa et al. [2017] in the notation introduced earlier, the second part of the
GEKS, from the intermediate period to the current period is: | {"url":"https://www.ons.gov.uk/methodology/methodologicalpublications/generalmethodology/onsworkingpaperseries/onsmethodologyworkingpaperseriesnumber12acomparisonofindexnumbermethodologyusedonukwebscrapedpricedata","timestamp":"2024-11-12T10:11:52Z","content_type":"text/html","content_length":"1049596","record_id":"<urn:uuid:c378e767-0b62-4766-82e0-dd3593e02b9d>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00267.warc.gz"} |
Auto-calibrated colloidal interaction measurements
Auto-calibrated colloidal interaction measurements with extended optical traps
We describe an efficient technique for measuring the effective interaction potential for pairs of colloidal particles. The particles to be tested are confined in an extended optical trap, also known
as a line tweezer, that is projected with the holographic optical trapping technique. Their diffusion along the line reflects not only their intrinsic interactions with each other, but also the
influence of the line's potential energy landscape, and inter-particle interactions mediated by scattered light. We demonstrate that measurements of the particles' trajectories at just two laser
powers can be used to correct explicitly for optically-induced forces and that statistically optimal analysis for optically-induced forces yields auto-calibrated measurements of the particles'
intrinsic interactions with remarkably few statistically independent measurements of the particles' separation.
pacs: 82.70.Dd, 87.80.Cc, 42.40.Jv, 05.40.Jc
Colloidal interactions tend to be diminutive, often no greater than a few femtonewtons, and typically are masked by vigorous Brownian motion. Nevertheless, they govern the microscopic stability and
macroscopic properties of colloidal dispersions. Monitoring these interactions therefore is useful for understanding and controlling the many natural and industrial processes governed by colloidal
This Article introduces an efficient and accurate method for measuring the interactions between a pair of colloidal particles that minimizes the measurement duration by optimizing the use of data.
Combining optical micromanipulation (1), digital video microscopy (2); (3); (4); (5) and a new analytical scheme based on adaptive kernel density estimation (6), this method requires just a few
thousand measurements of the inter-particle separation to characterize the pair potential of micrometer-scale particles in water. It also avoids experimental artifacts identified in previous studies
of colloidal interactions and automatically separates the measured pair potential into intrinsic and optically-induced contributions.
Section I reviews methods for measuring colloidal interactions with an emphasis on the practical considerations that have limited their widespread adoption. This section also highlights some of the
benefits and challenges of confining colloidal particles to one dimension using extended optical traps known as line tweezers. Section II briefly describes our holographic implementation of line
traps, which have been described in detail elsewhere (1); (7). The principal contributions of this Article are presented in Sec. III, which addresses the statistical mechanics of interacting
colloidal particles on a line trap. This discussion develops a statistically optimal analysis of trapped particles' trajectories that yields accurate results for the pair potential with exceedingly
small data sets. We apply these methods to a well-studied model system in Sec. IV to demonstrate that just 4,000 statistically independent samples of two particles' trajectories can suffice to
measure their pair potential to within
§ I. Measuring colloidal interactions
Most methods for measuring colloidal interactions use digital video microscopy (2); (4); (8) to track particles' motions. They differ in how the particles are handled during the measurement and in
how the pair potential is recovered from the measured trajectories. For instance, colloids' interactions can be inferred from the pair distribution function of dispersions in equilibrium. Imaging
measurements of the distribution function (9); (10); (11) involve large numbers of particles with sufficiently uniform properties that interpreting the many-particle statistics in terms of an
effective pair potential is meaningful. This approach is limited, therefore, to measuring interactions among identical particles and cannot be applied to heterogeneous samples. Acquiring sufficient
statistics to measure interactions at small separations requires large data sets and long experimental runs (3). Maintaining sufficiently uniform conditions over the courses of such a measurement can
be challenging (12); (13). Increasing the particles' concentration to obtain results more quickly introduces many-body correlations that can obscure the pair potential (14); (15). Even imaging an
equilibrium dispersion poses challenges because high-resolution microscopes have a limited depth of field (16); (17), three-dimensional imaging techniques can be too slow to acquire snapshots of the
particle distributions, and confining the particles to a plane can modify their interactions (18); (19); (5). The images themselves can be subject to artifacts, identified in Ref. (4), that must be
addressed with care to obtain meaningful results (4); (5).
Many of the limitations and much of the time and difficulty involved in equilibrium interaction measurements can be avoided by using optical tweezers (20) to arrange pairs (21) or clusters (15) of
particles into appropriate configurations. Colloidal interaction measurements based on optical tweezer manipulation generally fall into two categories: measurements performed with intermittent or
blinking traps, and those performed with continuously illuminated traps. In the former case, particles positioned by optical tweezers are released by extinguishing the traps (21); (18); (22); (23)
and the resulting nonequilibrium trajectories can be analyzed with a Fokker-Planck formalism (21); (23) to yield the equilibrium pair potential. This approach has the benefit that the particles'
interactions are measured while the tweezers are extinguished, ensuring that the results are not contaminated by light-induced phenomena (21). It also lends itself to measurements of dissimilar
particle pairs (18). “Blinking tweezer” measurements also require large data sets, however, and only work if the relaxation to equilibrium is free from kinematic effects, such as hydrodynamic
coupling (24); (25); (26); (27). Demonstrating the absence of such artifacts is difficult.
Both long sampling times and nonequilibrium effects can be avoided by tracking the motions of particles trapped in optical tweezers. Accurate measurements of dynamic interactions, such as
hydrodynamic coupling, can be extracted from observations of the coupled diffusion of particles individually trapped in optical tweezers (28); (29). Fast pair potential measurements can be realized
by replacing the discrete optical tweezers with extended optical line traps (30); (31); (32); (33); (34); (35); (36); (1), which allow trapped objects freedom of motion in one dimension.
Appropriately sculpting the trap's one-dimensional force landscape optimizes statistical sampling (35). Previous reports of line-trap interaction measurements have relied on separate calibrations of
the lines' longitudinal potential energy landscape (37), and have accounted for light-induced interactions by extrapolating measurements at multiple laser powers to obtain the zero-power limit (34);
(38); (35). These calibrations and background measurements can be time-consuming and exacting, particularly if optical forces cannot be described simply, or if measuring optically-induced
interactions is one of the goals.
Using holographic methods to project line traps (1); (7); (39) and optimal statistical methods (6) to analyze the particles' trajectories addresses all of these issues. In particular, this
combination eliminates the need for single-particle calibrations altogether and explicitly distinguishes particles' intrinsic interactions from one- and two-particle optically-induced interactions.
The result is a reliable, robust and, above all, rapid method for measuring colloidal interactions.
§ II. Holographic line traps
We project extended line tweezers using shape-phase holography (1) in the optimized (40) holographic optical trapping configuration (41); (42). Our system is built around an inverted optical
microscope (Nikon TE2000U) with a (1), this system brings the beam of light to a diffraction-limited focus as an anastigmatic conical wedge. The three-dimensional intensity distribution for such a
trap is shown as a volumetric reconstruction (7) in Fig. 1(a). The line's image in the focal plane, shown in Fig. 1(b), has a half-width of 200 nm. The axial half-width is roughly 3 times larger,
which also is consistent with diffraction-limited focusing. These intensity gradients establish the extended three-dimensional potential energy well within which colloidal particles can be captured.
The image of 1.5 1(c) demonstrates the trap's ability to hold particles in three dimensions.
The line appears less bright at its ends because it is designed to have a parabolic intensity profile. Shaping the light's intensity along the focal line is useful for tuning the line tweezer's
trapping characteristics (1); (39). Control over the line's intensity profile also can be used mitigate imperfections due to aberrations and other defects in the optical train (43). The stray light
evident in the lower left corner of Fig. 1(a) results from such practical limitations.
Holographic line traps also can be combined with point-like holographic optical tweezers to select particular particles for measurement and to prevent others from intruding.
§ III. Statistical mechanics of colloidal particles on a line trap
The potential energy landscape (34); (36), optical binding forces (30); (44), and optically-induced changes in the particles' intrinsic interactions. It is reasonable to assume that these optical
contributions to the system's free energy depend linearly on the laser power,
Once particles are trapped on the line, they diffuse in the line's potential energy well with autocorrelation times set by viscous relaxation (40); (29), which typically is less than a second for
micrometer-diameter spheres. This also contrasts with measurements based on many-particle dynamics, which require long periods of equilibration (3).
The interacting particles' dynamics are dominated by random thermal fluctuations. Rather than studying their detailed trajectories, therefore, we measure the joint probability
through the Boltzmann distribution
where (2), taking care (5) to avoid imaging artifacts at small separations (4). Inverting Eq. (2) then yields the total potential,
Extracting the intrinsic interaction, (34) extrapolates measurements of
Alternatively, the single-particle contributions to (37). The resulting single-particle probability distribution, 2 were obtained from one-dimensional projections of the measured single-particle
probability distribution, (34); (36), others have found them to be negligibly weak (37) and have ignored them.
Rather than relying on extrapolations or calibrations to correct for light-dependent contributions to
Adequately sampling the two-dimensional distributions, 3) over
Equation (4) is useful only if an efficient method can be found to compute the integrals. Our approach is to treat each measurement (6),
which should converge to (6). Consequently, we adopt
using a width,
Using adaptive non-parametric density estimators to compute (45). The potential, which scales as the logarithm of
The integrals in Eq. (4) usually have to be computed numerically. Given their dimensionality and the computational cost of evaluating the density estimator, Monte Carlo integration is a natural
choice (46). This approach is inherently more accurate than computing histograms of the particle positions because every data point contributes to the estimate for
Although the numerical integrals in Eq. (4) are computationally intensive, they reduce the analysis to a one-dimensional form and thus greatly reduce the number of data points required to sample 4)
and (5) are thus both optimally parsimonious with data and auto-calibrating.
§ IV. Experimental demonstration
We demonstrate our procedure by measuring the well-understood electrostatic interactions between micrometer-scale charge-stabilized colloidal silica spheres dispersed in deionized water. In this
case, the electrostatic pair potential for two spheres of radius (47); (18); (48) has the form (49)
where (21); (2); (18); (16) have confirmed that Eq. (8) accurately describes the interactions between pairs of highly charged colloidal spheres provided they are kept far enough away from charged
surfaces (18); (19); (12); (13); (5) or other spheres (14); (15).
We performed measurements on two silica spheres of nominal diameter (50) reveals the mean diameter of the spheres in this sample to be
A holographic line trap (1). Figure 2 shows the measured potential energy profile, which differs from the design by roughly 20%. Such variations would pose challenges if an accurate profile were
required for our analysis. Because none of the spurious local potential energy wells is deep enough to trap a particle against thermal forces, however, deviations from the designed profile do not
affect our measurement.
The curvature of the line's potential energy well was adjusted to bring the particles into proximity while still allowing them freedom of motion. Three half-hour data sets were obtained at laser
powers of (1). The total power projected onto each sphere at the highest power is of the order of 3 mW, which is comparable to conditions in conventional point-like optical tweezers.
Thermal forces cause particles to wander away from the projected line. Although transverse in-plane root-mean-square (rms) fluctuations were no larger than 100 nm near the center of the line trap,
and grew to no more than 200 nm at the ends, axial rms fluctuations were as large as 200 nm near the center and larger than 500 nm at the comparatively dim ends of the line. Equation (4) can be
generalized to incorporate averages over the extra dimensions, with the appropriate redefinition of the inter-particle separation
In future studies, off-line excursions can be minimized by using uniformly bright line traps whose force profiles are tailored with phase gradients (39). This would greatly increase data retention
rate and correspondingly reduce the measurement time required to acquire adequate statistics. Still further improvements in accuracy and efficiency could be obtained with the use of video holographic
microscopy for precise three-dimensional particle tracking (50). Acquiring data through conventional bright-field imaging on a parabolic line trap therefore should be considered a challenging test of
the analytical methods that are the principal contributions of this work.
The pruned data were analyzed with Eqs. (4), (5) and (7) to obtain estimates for the intrinsic pair potential, which are plotted as points in Fig. 3. The results, plotted as circles in Fig. 3, are
consistent with an energy resolution of
Because the energy resolution at a given separation scales roughly linearly with the number of data points acquired at that separation (45) the time required to attain a desired resolution depends on
the rate at which particles explore the available phase space along the line. This, in turn, depends on the particles' viscous relaxation time in the longitudinal trapping potential. By reducing the
number of statistically independent data points required to achieve a given resolution, optimal statistical analysis reduces the number of viscous relaxation times that a measurement requires, and
therefore can substantially reduce the duration of a measurement.
By making full use of the available data, optimal statistical analysis also eliminates the need for statistical oversampling to reduce round-off errors encountered in binning and bandwidth
limitations encountered in power spectral analysis. Consequently, the present approach does not benefit particularly from high-speed data acquisition, and can be implemented with lower-cost
The inset to Fig. 3 shows the measured colloidal pair potential plotted for easy comparison with the prediction of Eq. (8). The observed linear trend is consistent with the anticipated screened
Coulomb repulsion, and thus with previous measurements on similar colloidal particles under similar conditions (21); (16); (18). The best-fit slope of this plot suggests a Debye-Hückel screening
length of
Based on the dissociation of terminal silanol groups with an estimated surface coverage of (3) to be no larger than (18) charge renormalization (47) result,
relates the effective charge number to the effective surface potential, 3 is the prediction of Eq. (8) for these values.
If we assume that there are no light-induced interactions, we can use the calibrated line profile to compute 3.
The difference between 4) is the one-dimensional projection of
which provides at least a rough estimate for the light-induced interaction between the two spheres. For 4). Such an optically-induced repulsive interaction is consistent with previous studies of
multiple colloidal particles on extended optical traps (34). Presumably, light scattered by one sphere impinges on its neighbor and gives rise to radiation pressure. In this interpretation, the range
of the repulsion is set by the non-trivial angular distribution of the Lorenz-Mie scattered light (51).
Peaks in 2 and might therefore explain the peaks in the estimate for 3.
Alternatively, structure in 4 would constitute new experimental evidence for longitudinal optical binding (30); (44). It should be emphasized, however, that these features are barely resolved over
the estimated
Distinguishing optical binding from power-dependent artifacts is made difficult in this data set by the line trap's parabolic intensity profile. Measurements in uniformly bright line traps with
phase-gradient longitudinal potential wells are under way and will be reported elsewhere.
Regardless of the interpretation of
The apparent absence of light-induced interactions between particles trapped on scanned line tweezers (37) may be ascribed to the smaller size of the silica particles in that study. Whereas the Mie
scattering pattern for 1.5 (51); (50). No optically-induced interaction should be expected unless spheres scatter light toward their neighbors.
The electrostatic interactions between isolated pairs of colloidal spheres far from surfaces are very well understood. The agreement between experiment and theory in this model system demonstrates
that the protocol described above can be applied with reasonable confidence to systems whose underlying interactions are less well understood.
§ V. Conclusion
We have described and demonstrated a method for measuring colloidal pair interactions based on particles' equilibrium statistics in an extended optical trap. This method is self-calibrating in the
sense that no a priori information regarding the trap's effective potential energy landscape is required to measure trapped particles' interactions. This offers an advantage in both time and effort
over previously described methods, which require separate single-particle calibrations of the trapping potential.
Our method makes good use of the flexible reconfigurability of holographic trap projection through shape-phase holography. The same analytical technique also can be applied to line tweezers created
with cylindrical lenses, or through rapid scanning.
Optimizing the transverse stiffness of the trap, particularly in the axial direction, can substantially improve data retention efficiency and thereby reduce measurement duration. The ultimate limit
on measurement speed is set, however, by the particles' viscous relaxation rate in the line tweezer. This also can be optimized through holographic control over the potential energy well's shape. The
use of optimal statistical analysis then minimizes the number of viscous relaxation times required for adequate statistical sampling.
Combining optical micromanipulation, digital video microscopy and optimal statistical analysis offers an efficient and effective method to probe colloidal interactions. The method described here is
easily generalized for dissimilar pairs of particles. Even more appealing is the possibility of performing multiple simultaneous measurements by projecting multiple holographic line traps. This opens
up the possibility of using colloidal interaction measurements for process control and quality assurance testing.
This work was supported by the National Science Foundation through Grant Number DMR-0606415 and through a support of the Keck Foundation.
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Driving motors, lights, etc. from an Arduino output pin
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Posted Nick Gammon Australia (23,097 posts) Bio Forum Administrator
Fri 30 Jan 2015 03:25 AM (UTC)
Amended on Tue 03 Feb 2015 02:43 AM (UTC) by Nick Gammon
Arduino output pins are generally rated at 20 mA continuous, with an absolute maximum rating of 40 mA. Therefore your circuit designs should not attempt to rely on driving a device needing
more than 20 mA. Also, there is a limit to the total amount that can be driven at once. For example, if you are driving 20 output pins you cannot drive 20 x 20 mA (400 mA) because (depending
on the exact chip) you may be limited to a total of 200 mA, and groups of pins are limited to 100 mA for the group.
MOSFET driver
A commonly-used technique is to use a MOSFET to drive loads, and in particular an N-channel MOSFET. An N-channel MOSFET is used as a "low side" driver, that is, it is designed to "sink"
The advantage of a low-side driver is that you can control more than the 5V on the Arduino output pin, without extra components. If you want to switch "high side" (that is, to source current)
then an extra transistor is required, as described below.
MOSFET efficiency
A MOSFET is quite efficient, when turned fully on or fully off. When fully off, it presents a very high resistance between drain and source. When fully on, it presents a very low resistance
Because of the low resistance when turned on, there is only a small voltage drop between drain and source, and thus only a low amount of power is consumed in the transistor (and thus turned
into heat).
For use with 5V circuits you need a so-called logic-level MOSFET which lets it be turned on largely or fully by 5V between Gate and Source (this is called V[GS] on the datasheet).
Gate resistor
A resistor between the output pin and the MOSFET gate limits the surge of current into the gate when the MOSFET is turned on by program logic. There is a very lengthy discussion about this on
the Arduino forum: Myth: "You must have a gate resistor".
In the example given in the schematic, the 150 ohm resistor (R1) would limit current to 33 mA, even when presented with a short-circuit:
I = E / R
I = 5 / 150
I = 0.033 (33 mA)
Too large a resistor, however, would slow down the turning on or off the MOSFET, and thus it would be operating in the non-saturated region for longer, which increases the R[DS] value,
causing the MOSFET to get hotter.
According to my calculations the turn-on time would be 2.2RC and thus using an example of 1350 pF input capacitance, and 150 ohm gate resistor, it would take:
2.2 * 1350e-12 * 150 = 4.455e-007 (0.0000004455) = 446 nS
See Wikipedia - RC time constant
Thus it would turn on in 446 nS which is probably acceptable.
Message Pull-down resistor
In the schematic above R2 (10k) keeps the Gate at zero volts if the Arduino pin is not configured for output, this keeps the MOSFET off if the processor is booting and has not yet set the
pins correctly.
Fly-back diode
In the schematic above D1 is wired across the coil of the motor to prevent reverse EMF from damaging the MOSFET. For non-inductive loads (eg. LEDs) this diode would not be necessary. A fast
switching diode (eg. 1N4001) or Schottky diode would be appropriate here.
Possible alternatives would be:
Heat calculations
To work out if the MOSFET can handle the current you are planning to use (and whether or not to use a heat-sink) first you calculate the power used, based on R[DS]On for the V[GS] voltage
that you are planning to use.
For example, R[DS]On at V[GS] = 5V, might be quoted as 0.035 ohms.
The formula for power is
Power = I^2R
So, let's say the motor draws 2 amps, it would be:
Power = (2)^2 * 0.035 = 0.140 (140 mW)
Now we need to look on the datasheet for the "Thermal Resistance Junction to Ambient" (R[ØJA]). In my example (RFP30N06LE) it is given as 80 °C/W.
So we multiply that by the number of watts, like this:
0.140 * 80 = 11.2 (degrees)
If we assume ambient is 25 °C then that means the junction temperature will be:
25 + 11.2 = 36.2 °C
This must be less than the maximum quoted junction temperature (TJ[max]), which if not quoted we can assume to be around 150 °C.
Check voltages
We would also check that we are operating within quoted voltage ranges. For example, V[DS] is the drain to source voltage. In the case of the RFP30N06LE that is 60V, so using it with a 12V
supply is well within spec.
We would also check the continuous drain current (I[D]). In this case the MOSFET is rated at 30A, and we are planning to use 2 amps, so this is OK as well.
- Nick Gammon
www.gammon.com.au, www.mushclient.com
Posted Nick Gammon Australia (23,097 posts) Bio Forum Administrator
Reply #1 on Fri 30 Jan 2015 03:25 AM (UTC)
Amended on Fri 31 Dec 2021 12:27 AM (UTC) by Nick Gammon
High side driver
The circuit above was for a low-side driver, one that would "sink" current. At times, though, we want to "source" current, which requires a high-side driver. This is easy enough if we only
want to control 5V, in which case we can connect the MOSFET up like this:
Things are backwards however, as we are using a P-channel MOSFET. It is off when the Gate is the same voltage as the Source (V[GS] = 0), which means that we have to output 5V (HIGH) from the
Arduino to turn the MOSFET off. To turn it on we need -5V at the Gate (relative to the Source) so we need to output 0V (LOW) from the Arduino to turn the MOSFET on.
Once again there is a current-limiting resistor (R1) of 150 ohms, to limit the Arduino output pin current, and a pull-up resistor (R2) to make sure that the MOSFET is off if the Arduino pin
is not configured as an output.
Things get more complicated if we need to source more than 5V, because to turn the MOSFET on we have to have a V[GS] of -5V or more (the Gate needs to be -5 relative to the Source, which is
the opposite compared to an N-channel MOSFET). To turn it off we need to be able to set V[GS] to zero. So, for example, if we need to source 12V then to turn the MOSFET off we have to present
12V at the Gate. This can't be done with the Arduino output pin. To achieve this, we need a "helper" transistor, like this:
The transistor switches the 12V from our power supply on or off, based on the Arduino output pin. The 1k resistor (R1) limits the base current to the transistor.
There would be about a 0.95V drop over the base/emitter junction (V[BE(sat)]), so the current through base (I[BE]) would be:
(5 - 0.95) / 1000 = 0.00405 (4.05 mA)
The current flow of 4.05 mA is acceptable to both the Arduino output pin and the transistor. We will reckon on a gain of around 10 at saturation so that means it can sink 40.5 mA through R2.
In order for R2 to drop 12V the current through it must be:
12/1000 = 0.012 (12 mA)
We calculated above that the transistor can sink 40.5 mA so this is acceptable.
Next we need to calculate the power dissipated by the transistor which will be the sum of the base-emitter power and the collector-emitter power.
power = (V[BE(sat)] * I[BE]) + (V[CE(sat)] * I[CE])
V[BE(sat)] = 0.95 (from datasheet)
5V (at pin) - V[BE(sat)] / 1k = (5 - 0.95) / 1000 = 0.00405 (4.05 mA)
V[CE(sat)] = 0.2 (from datasheet)
Sink 12V through the 1k resistor = 12 / 1000 = 0.012 (12 mA)
R[ØJA] = 200 °C/W (from datasheet)
Power = (0.95 * 0.00405) + (0.2 * 0.012) = 0.00625 (6.25 mW)
Heat gain = 0.00625 * 200 = 1.25 °C
Once again we have a flyback diode to protect the MOSFET if it is driving an inductive load.
To turn the MOSFET on we output a HIGH signal from the Arduino, which means the transistor conducts, and sinks the Gate of the MOSFET to ground, effectively making it -12V compared to the
To turn the MOSFET off, we output a LOW signal from the Arduino, thus Q1 does not conduct, and therefore the 1k resistor (R2) pulls the Gate to 12V. Now Gate and Source are at 12V so the
MOSFET is off (V[GS] = 0).
Switching higher voltages
Things get more complicated again if we want to switch more than about 12 V, because we need to keep V[GS] within the specified range of V[GS(max)] for the MOSFET. Check the datasheet, but V
[GS(max)] may be in the order of ± 15 V, possibly ± 25 V.
To do this we need to clamp the V[GS] voltage with a 10 V zener diode, shown below as D2:
We know that we will have around 4.3 V on the emitter of Q1 (5 V - 0.7 V drop from base to emitter) thus the current through the emitter must be:
4.7 / 330 = 14.2 mA
If the main supply voltage is off, then that entire 14.2 mA will be provided by the Arduino output pin, which is within tolerance for it.
However if the main supply (shown above as 24 V) is on, then most of the current will flow from the collector. This is known as the "alpha" ratio, caculated as:
α = β / (1 + β)
Since β will be about 100, then we have:
α = 100 / 101 = 0.99
In that case the base current will be 142 µA and the collector current will be 14.06 mA.
That should be enough current to "activate" the zener diode, causing it to maintain a voltage drop of 10 V over V[GS].
The 1 k resistor (R2) is to pull the gate quickly back up to the supply voltage if the transistor is off.
Switching time
It is important that the MOSFET be switched on and off quickly, as it has low power dissipation when fully off (ie. no current flowing) or fully on (when there is a low resistance between
gate and drain). At other times the power dissipation will be higher, and the part may get hot.
This won't be an issue for low-switching speeds (eg. turning garden lights on and off, or a fish feeder) but becomes an issue when doing PWM (pulse-width modulation) where you may be
switching on and off many times a second.
The analogWrite function on the Atmega328-based Arduinos runs by default at 490 Hz. Tests indicate that the circuit above will take about 1 µS to switch on and 8 µS to switch off (if using
the high-side driver) and 1 µS to switch both on and off if using the low-side driver.
This is one of the reasons for choosing a reasonably low value for R2 in the high-side driver. We rely on R2 to pull the gate of the MOSFET high when the transistor is turned off, and we need
that done as quickly as reasonably possible. We could potentially use a lower value resistor for R2 (eg. 470 ohms) to increase the switching speed, but then Q1 (the driver transistor) has to
switch higher currents, and we need to watch that this transistor (Q1) does not get too hot.
Where does the motor power come from?
Another point which can be initially confusing is expecting the Arduino board to provide enough power to run a motor, or lots of LEDs. It won't, basically. The power supply on the Arduino is
rated with enough capacity to drive the chips on the board itself, with a little over for extra chips like clocks, SD cards, etc.
It is not intended to drive a 3 amp electric motor.
In this case you have a separate power supply (maybe a battery, maybe a plug-pack) which is the "+5V" or "+12V" on the circuits above.
However it is important to connect the grounds. That is, connect the ground wire from the battery or plug-pack to the ground wire on the Arduino. That way they have a common "reference
- Nick Gammon
www.gammon.com.au, www.mushclient.com
Posted Nick Gammon Australia (23,097 posts) Bio Forum Administrator
Reply #2 on Fri 30 Jan 2015 03:35 AM (UTC)
Amended on Sat 31 Jan 2015 12:08 AM (UTC) by Nick Gammon
More information
AddOhms' YouTube video about MOSFETs: MOSFETs and How to Use Them
Mike Cook's tutorial on power: Power & Heat
Wikipedia: MOSFET
Afrotechmods: Transistor / MOSFET tutorial and What is PWM? Pulse Width Modulation tutorial!
Also see the links above regarding flyback protection and snubber diodes.
Voltages, currents and resistances on datasheets are often given like this: V[GS]
That means V (voltage) between G (gate) and S (source).
They may also have a condition attached, eg.
In that case this means R (resistance) between D (drain) and S (source) when the MOSFET is turned on. The datasheet may specify different values for different conditions, eg. R[DS(on)] when V
[GS] = 5V. In other words, the resistance between Drain and Source when there is 5V between Gate and Source.
- Nick Gammon
www.gammon.com.au, www.mushclient.com
Posted Nick Gammon Australia (23,097 posts) Bio Forum Administrator
Reply #3 on Tue 03 Feb 2015 02:48 AM (UTC)
Amended on Fri 07 Aug 2015 04:29 AM (UTC) by Nick Gammon
BJT transistor driver
Moving on from MOSFETs, let's look at using BJTs (Bipolar Junction Transistors) as switches.
The advantages of BJTs are that they are small and cheap. The disadvantages are that they are current-driven rather than voltage-driven like MOSFETs, and in general are more suited to
low-power applications, like driving a few LEDs, a small motor or a relay.
This circuit is similar to the "low side" MOSFET driver described above. The emitter of the transistor is connected to Ground, so it "sinks" current. The transistor is an NPN transistor. A
similar and common part would be the BC546/547/548 transistors.
Two important calculations here are the two resistors. R[B] - the Base resistor, and R[L] - the Load resistor.
Some loads (like motors) will not need a load resistor as they will be naturally current limited (however they may require a fly-back diode as described above).
Calculate the load resistor
Let's assume that we want to drive the LED with 20 mA, and we establish that the LED has a forward voltage drop of 1.7V (from the datasheet, or by measurement).
We can see from the datasheet for our transistor that (in this case) that the Collector−Emitter Saturation Voltage V[CE(sat)] is around 0.4V with a load of 15 mA.
Thus the voltage drop over the resistor will be:
V = 5 - 1.7 - 0.4 = 2.9 V
And since we want 20 mA to flow we can use Ohm's Law to calculate R[L]:
RL = 2.9 / 0.020 = 145 ohms
So we'll use a standard value of 150 ohms for R[L].
Power dissipated by this resistor will be I^2R, which would be:
PL = 0.020^2 * 150 = 0.060 (60 mW)
Thus a 1/8 watt resistor would be fine.
Calculate the base resistor
We want to fully "saturate" (turn on) the transistor, so a fairly substantial amount of current will be required. The datasheet quotes the "Collector−Emitter Saturation Voltage" as being one
when I[C] is 150 mA and I[B] is 15 mA, thus we can deduce that (at those current levels) the transistor will only have a gain of 10 (that is, 150/15) when saturated, thus we would require a
current of 2 mA to flow through the Base to the Emitter (that is, 20mA / 10).
The datasheet says that the Base−Emitter Saturation Voltage V[BE(sat)] is around 1.3V with a load of 15 mA, thus the voltage drop over the resistor will be:
V = 5 - 1.3 = 3.7 V
Since we want 2 mA to flow we can use Ohm's Law to calculate R[B]:
RB = 3.7 / 0.002 = 1850 ohms
Thus a 2k resistor should be satisfactory.
Power dissipated by this resistor will be I^2R, which would be:
PB = 0.002^2 * 1600 = 0.008 (8 mW)
Thus a 1/8 watt resistor would be fine here too.
Transistor power dissipation
Next we need to calculate the power dissipated by the transistor which will be the sum of the base-emitter power and the collector-emitter power.
power = (V[BE(sat)] * I[BE]) + (V[CE(sat)] * I[CE])
V[BE(sat)] = 1.3 (from datasheet)
I[BE] = 0.002 (2 mA - as calculated above)
V[CE(sat)] = 0.4 (from datasheet)
I[CE] = 0.020 (20 mA - as calculated above)
R[ØJA] = 200 °C/W (from datasheet)
Power = (1.3 * 0.002) + (0.4 * 0.020) = 0.0106 (10.6 mW)
Heat gain = 0.0106 * 200 = 2.12 °C
Calculations in Lua
-- standard values for 5% resistors
standardValues = {
1.0, 1.1, 1.2, 1.3, 1.5, 1.6, 1.8, 2.0, 2.2, 2.4, 2.7, 3.0,
3.3, 3.6, 3.9, 4.3, 4.7, 5.1, 5.6, 6.2, 6.8, 7.5, 8.2, 9.1,
10.0 }
-- turn a resistance into the next highest standard value (5%)
function makeStandardValue (value)
local exponent = math.floor (math.log10 (value))
local fraction = 10^(math.log10 (value) - exponent)
for k, v in ipairs (standardValues) do
if v > fraction then
return v * 10^exponent
end -- if
end -- for
error "Shouldn't get here"
end -- function makeStandardValue
-- parameters
-- the transistor (from the datasheet)
VBEsat = 1.3 -- volts (Base-Emitter Saturation Voltage)
VCEsat = 0.4 -- volts (Collector-Emitter Saturation Voltage)
RthetaJA = 200 -- °C/W (Thermal Resistance, Junction to Ambient)
-- the load
Vload = 5 -- volts (load supply voltage on the motor / LED)
LEDforward = 1.2 -- LED forward voltage (zero if no LED)
ICE = 0.02 -- current through load (collector) in amps
-- how we are turning on the transistor
Venable = 5 -- volts to enable the transistor (eg. Arduino pin)
-- calculate RL (load resistor)
VL = Vload - LEDforward - VCEsat
RL = makeStandardValue (VL / ICE)
PL = ICE^2 * RL
print (string.rep ("-", 10) .. " Load resistor " .. string.rep ("-", 10) )
print ("Voltage over RL =", VL .. " V")
print ("RL =", RL .. " ohms")
print ("RL power =", PL .. " watts (" .. math.floor (PL * 1000) .. " mW)")
-- calculate RB (base resistor)
print (string.rep ("-", 10) .. " Base resistor " .. string.rep ("-", 10) )
VB = Venable - VBEsat
IBE = ICE / 10 -- rule of thumb, use 1/10 of collector current for full saturation
RB = makeStandardValue (VB / IBE)
PB = IBE^2 * RB
print ("RB =", RB .. " ohms")
print ("RB power =", PB .. " watts (" .. math.floor (PB * 1000) .. " mW)")
-- calculate transistor power
print (string.rep ("-", 10) .. " Transistor power " .. string.rep ("-", 10) )
Power = (VBEsat * IBE) + (VCEsat * ICE)
print ("Power =", Power * 1000, "mW")
Heatgain = Power * RthetaJA
print ("Heat =", Heatgain, "degrees C")
The net result here is that we are using 2 mA from the Arduino output pin to control a 20 mA LED. Now, an Arduino can directly drive 20 mA, however the workings above should let you calculate
resistors for other loads.
- Nick Gammon
www.gammon.com.au, www.mushclient.com
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Examine whether you can construct ΔDEF such that EF = 7.2 cm, m ∠ E = 110 o and m ∠ F = 80 o . Justify your answer.
According to the question,
It is given that,
∠E = 110° and ∠F = 80°
That shows that ∠E + ∠F = 110 + 80
= 190°
This is greater than 180°
And, we know that,
The sum of interior angles of triangle is 180°
The given measurements cannot form a triangle.
We have to draw figure using following steps of construction:
Step 1: Draw a line segment EF of 7.2 cm
Step 2: Now, from point E draw a ray EX making an angle of 110° from EF.
Step 3: From F, draw a ray FY from EF making 80° angle
Now, we can observe that EX and FY does not intersect.
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The Stacks project
Lemma 29.6.3. Let $f : X \to Y$ be a morphism of schemes. Let $Z \subset Y$ be the scheme theoretic image of $f$. If $f$ is quasi-compact then
1. the sheaf of ideals $\mathcal{I} = \mathop{\mathrm{Ker}}(\mathcal{O}_ Y \to f_*\mathcal{O}_ X)$ is quasi-coherent,
2. the scheme theoretic image $Z$ is the closed subscheme determined by $\mathcal{I}$,
3. for any open $U \subset Y$ the scheme theoretic image of $f|_{f^{-1}(U)} : f^{-1}(U) \to U$ is equal to $Z \cap U$, and
4. the image $f(X) \subset Z$ is a dense subset of $Z$, in other words the morphism $X \to Z$ is dominant (see Definition 29.8.1).
Comments (2)
Comment #4285 by Dario Weißmann on
typo: $\mathcal{I}=\text{Ker}(\mathcal{O}_Y\to \mathcal{O}_{X'})$ is missing an $f'_{\ast}$
Comment #4449 by Johan on
Thanks very much and fixed here.
There are also:
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Inferno's REGEXP(6
A regular expression specifies a set of strings of characters. A member of this set of strings is said to be matched by the regular expression. In many applications a delimiter character,
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| e1 e2
e0: e1
| e0 '|' e1
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Lectures on Analytic Differential Equationssearch
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Lectures on Analytic Differential Equations
Yulij Ilyashenko : Cornell University, Ithaca, NY and Moscow State University, Moscow, Russia and Steklov Institute of Mathematics, Moscow, Russia and Independent University of Moscow, Moscow, Russia
Hardcover ISBN: 978-0-8218-3667-5
Product Code: GSM/86
List Price: $135.00
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AMS Member Price: $108.00
eBook ISBN: 978-1-4704-2116-8
Product Code: GSM/86.E
List Price: $85.00
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Hardcover ISBN: 978-0-8218-3667-5
eBook: ISBN: 978-1-4704-2116-8
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Click above image for expanded view
Lectures on Analytic Differential Equations
Yulij Ilyashenko : Cornell University, Ithaca, NY and Moscow State University, Moscow, Russia and Steklov Institute of Mathematics, Moscow, Russia and Independent University of Moscow, Moscow, Russia
Hardcover ISBN: 978-0-8218-3667-5
Product Code: GSM/86
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
eBook ISBN: 978-1-4704-2116-8
Product Code: GSM/86.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Hardcover ISBN: 978-0-8218-3667-5
eBook ISBN: 978-1-4704-2116-8
Product Code: GSM/86.B
List Price: $220.00 $177.50
MAA Member Price: $198.00 $159.75
AMS Member Price: $176.00 $142.00
• Graduate Studies in Mathematics
Volume: 86; 2008; 625 pp
MSC: Primary 34; Secondary 14; 32; 13
The book combines the features of a graduate-level textbook with those of a research monograph and survey of the recent results on analysis and geometry of differential equations in the real and
complex domain. As a graduate textbook, it includes self-contained, sometimes considerably simplified demonstrations of several fundamental results, which previously appeared only in journal
publications (desingularization of planar analytic vector fields, existence of analytic separatrices, positive and negative results on the Riemann–Hilbert problem, Ecalle–Voronin and
Martinet–Ramis moduli, solution of the Poincaré problem on the degree of an algebraic separatrix, etc.). As a research monograph, it explores in a systematic way the algebraic decidability of
local classification problems, rigidity of holomorphic foliations, etc. Each section ends with a collection of problems, partly intended to help the reader to gain understanding and experience
with the material, partly drafting demonstrations of the more recent results surveyed in the text.
The exposition of the book is mostly geometric, though the algebraic side of the constructions is also prominently featured. On several occasions the reader is introduced to adjacent areas, such
as intersection theory for divisors on the projective plane or geometric theory of holomorphic vector bundles with meromorphic connections. The book provides the reader with the principal tools
of the modern theory of analytic differential equations and intends to serve as a standard source for references in this area.
Graduate students and research mathematicians interested in analysis and geometry of differential equations in real and complex domain.
□ Chapters
□ Chapter I. Normal forms and desingularization
□ Chapter II. Singular points of planar analytic vector fields
□ Chapter III. Local and global theory of linear systems
□ Chapter IV. Functional moduli of analytic classification of resonant germs and their applications
□ Chapter V. Global properties of complex polynomial foliations
□ First aid
□ The authors provide a crash course on functions of several complex variables and elements of the theory of Riemann spaces.
Mathematical Horizon
□ The book is easy to read. The ideas and directions are clearly indicated before going into details. ... Moreover, it is easy to open the book at any section and start reading.
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Volume: 86; 2008; 625 pp
MSC: Primary 34; Secondary 14; 32; 13
The book combines the features of a graduate-level textbook with those of a research monograph and survey of the recent results on analysis and geometry of differential equations in the real and
complex domain. As a graduate textbook, it includes self-contained, sometimes considerably simplified demonstrations of several fundamental results, which previously appeared only in journal
publications (desingularization of planar analytic vector fields, existence of analytic separatrices, positive and negative results on the Riemann–Hilbert problem, Ecalle–Voronin and Martinet–Ramis
moduli, solution of the Poincaré problem on the degree of an algebraic separatrix, etc.). As a research monograph, it explores in a systematic way the algebraic decidability of local classification
problems, rigidity of holomorphic foliations, etc. Each section ends with a collection of problems, partly intended to help the reader to gain understanding and experience with the material, partly
drafting demonstrations of the more recent results surveyed in the text.
The exposition of the book is mostly geometric, though the algebraic side of the constructions is also prominently featured. On several occasions the reader is introduced to adjacent areas, such as
intersection theory for divisors on the projective plane or geometric theory of holomorphic vector bundles with meromorphic connections. The book provides the reader with the principal tools of the
modern theory of analytic differential equations and intends to serve as a standard source for references in this area.
Graduate students and research mathematicians interested in analysis and geometry of differential equations in real and complex domain.
• Chapters
• Chapter I. Normal forms and desingularization
• Chapter II. Singular points of planar analytic vector fields
• Chapter III. Local and global theory of linear systems
• Chapter IV. Functional moduli of analytic classification of resonant germs and their applications
• Chapter V. Global properties of complex polynomial foliations
• First aid
• The authors provide a crash course on functions of several complex variables and elements of the theory of Riemann spaces.
Mathematical Horizon
• The book is easy to read. The ideas and directions are clearly indicated before going into details. ... Moreover, it is easy to open the book at any section and start reading.
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Multiphase Flow Properties & Pressure Gradient Calculation - Production Technology
Multiphase Flow Properties & Pressure Gradient Calculation
For oil wells, the main component of pressure loss is the gravity or hydrostatic term. Calculation of the hydrostatic pressure loss requires knowledge of the proportion of the pipe occupied by
liquid (holdup) and the densities of the liquid and gas phases. Accurate modeling of fluid PVT properties is essential to obtain in-situ gas/liquid proportions, phase densities, and viscosities.
Calculation of holdup is complicated by the phenomenon of gas/liquid slip. Gas, being less dense than liquid flows with a greater vertical velocity than liquid. The difference in velocity between
the gas and liquid is termed the slip velocity. The effect of slip is to increase the mixture density and hence the gravity pressure gradient.
In the next paragraphs, two-phase flow properties (holdup, densities, velocity, and viscosity) will be detailed. Then the pressure gradient equation which is applicable to any fluid flowing in a pipe
inclined at an angle φ is depicted. As well as, the two-phase flow procedure to calculate the outlet pressure is detailed.
Two-Phase Flow Properties:
With reference to multiphase flow in pipes, the fraction of a particular fluid present in an interval of a pipe. In multiphase flow, each fluid moves at a different speed due to different
gravitational forces and other factors, with the heavier phase moving slower, or being more held up, than the lighter phase. The holdup of a particular fluid is not the same as the proportion of the
total flow rate due to that fluid, also known as its ”cut”. To determine in-situ flow rates, it is necessary to measure the holdup and velocity of each fluid. The sum of the holdups of the fluids
present is unity.
1- Liquid and Gas Holdup (HL & Hg):
HL is defined as the ratio of the volume of a pipe segment occupied by the liquid to the volume of the pipe segment. The remainder of the pipe segment is of course occupied by gas, which is referred
to as Hg.
Hg = 1 – HL
2- No-Slip Liquid and Gas Holdup (λL & λg):
The No-Slip correlation assumes homogeneous flow with no slippage between the phases. Fluid properties are
taken as the average of the gas and liquid phases and friction factors are calculated using the single phase
Moody correlation.
λL is defined as the ratio of the volume of the liquid in a pipe segment divided by the volume of the pipe segment which would exist if the gas and liquid traveled at the same velocity (no-slippage).
It can be calculated directly from the known gas and liquid volumetric flow rates from :
1- Liquid Density (ρL):
ρL may be calculated from the oil and water densities with the assumption of no slippage between the oil and water phases as follows:
2- Two-Phase Density:
Calculation of the two-phase density requires knowledge of the liquid holdup. Three equations for two-phase density are used by various investigators in two-phase flow:
1- Superficial Gas and Liquid Velocities (vsg & vsL):
2- Actual Gas and Liquid Velocities (vg & vL):
3- Two-Phase Velocity (vm):
4- Slip Velocity (vs):
1- Liquid Viscosity (μL):
μL may be calculated from the oil and water viscosities with the assumption of no slippage between the oil and water phases as follows:
2- Two-Phase Viscosity:
Calculation of the two-phase viscosity requires knowledge of the liquid holdup. Two equations for two-phase viscosity are used by various investigators in two-phase flow:
3- Liquid Surface Tension (σL):
General Pressure Gradient Equation:
The pressure gradient equation which is applicable to any fluid flowing in a pipe inclined at an angle φ from horizontal was derived previously. This equation is usually adapted for two-phase flow by
assuming that the two-phase flow regime and two-phase properties can be considered homogeneous over a finite volume of the pipe.
Many correlations have been developed for predicting two-phase flow pressure gradients which differ in the manner used to calculate the three terms of pressure gradients equation (elevation change,
friction, and acceleration terms):
a. No slip, no flow regime considerations: the mixture density is calculated based on the no slip holdup. No distinction is made for different flow regimes.
b. Slip considered, no flow regime consideration: The same correlations for liquid holdup and friction factor are used for all flow regimes.
c. Slip considered, flow regime considered: Usually a different liquid holdup and friction factor prediction methods are required in each flow regimes.
Two-Phase Flow Procedure for Outlet Pressure Calculation:
1. Starting with the known inlet pressure and flow rates.
2. Select a length increment, ΔL, and estimate the pressure drop in this increment, ΔP.
3. Calculate the average pressure and, for non-isothermal cases, the average temperature in the increment.
4. Determine the gas and liquid properties (based on black-oil or compositional model) at average pressure and temperature conditions.
5. Calculate the pressure gradient, dP/dL, in the increment at average conditions of pressure, temperature, and pipe inclination, using the appropriate pressure gradient correlation.
6. Calculate the pressure drop in the selected length increment, ΔP=ΔL(-dP/dL).
7. Compare the estimated and calculated values of ΔP. If they are not sufficiently close, estimate a new value and return to step 3.
8. Repeat the steps 2 to 7 for the next pipe length increment.
Flow Correlation Information
Multi-phase flow correlations are used to predict the liquid holdup and frictional pressure gradient. Correlations in common use consider liquid/gas interactions – the oil and water are lumped
together as one equivalent fluid. They are therefore more correctly termed two-phase flow correlations. Depending on the particular correlation, flow regimes are identified and specialized holdup
and friction gradient calculations are applied for each flow regime.
There is no universal rule for selecting the best flow correlation for a given application. It is recommended that a Correlation Comparison always be carried out. By inspecting the predicted flow
regimes and pressure results, the User can select the correlation that best models the physical situation.
You May Also Like…
Tagged acceleration, elevation change, fluid density, fluid velocity, fluid viscosity, friction, Liquid and Gas Holdup, liquid holdup, no-slip correlation, No-Slip Liquid and Gas Holdup, pressure
gradient equation, slip velocity, Superficial Gas and Liquid Velocities, two-phase density, two-phase flow calculation procedure, Two-Phase Flow Correlations. Bookmark the permalink. | {"url":"https://production-technology.org/multiphase-flow-correlations-properties/","timestamp":"2024-11-13T19:13:12Z","content_type":"text/html","content_length":"87721","record_id":"<urn:uuid:eb4ac803-d52f-432f-b82c-f1ae596fd9aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00637.warc.gz"} |
A New Method for Finding the Thevenin and Norton Equivalent Circuits
Engineering, 2010, 2, 328-336
doi:10.4236/eng.2010.25043 Published Online May 2010 (http://www.SciRP.org/journal/eng)
Copyright © 2010 SciRes. ENG
A New Method for Finding the Thevenin and Norton
Equivalent Circuits
George E. Chatzarakis
Department of Electrical Engineering Educators, School of Pedagogical and Technological Education,
Athens, Greece
E-mail: geaxatz@otenet.gr, geaxatz@mail.ntua.gr, gea.xatz@aspete.gr
Received December 3, 2009; revised February 7, 2010; accepted February 12, 2010
The paper presents a new pedagogical method for finding the Thevenin and Norton equivalent circuits of a
linear electric circuit (LEC) at the n-different pairs of terminals simultaneously, regardless of the circuit to-
pology and complexity. The proposed method is appropriate for undergraduate electrical and electronic en-
gineering students leading to straightforward solutions, mostly arrived at by inspection, so that it can be re-
garded as a simple and innovative calculation tool for Thevenin equivalents. Furthermore, the method is eas-
ily adapted to computer implementation. Examples illustrating the method’s scientific and pedagogical reli-
ability, as well as real test results and statistically-sound data assessing its functionality are provided.
Keywords: Circuit Analysis, Equivalent Circuits, Inspection, Mesh Analysis, Nodal Analysis, Thevenin and
Norton Equivalent Circuits
1. Introduction
One of the principal and fundamental topics taught to
undergraduate Electrical and Electronic Engineering
students within their Electric Circuits course is finding
the Thevenin and Norton equivalent circuits of a linear
electric circuit (LEC) at a specific pair of terminals.
The ability to find these parameters will proves to be
necessary to the students, when the focus is on a particu-
lar part of the circuit (i.e. finding the voltage, current or
power dissipated to a resistance) or when faced with
problems of load matching.
However, the methods developed and recorded in text-
books so far [1-8], address the finding of Thevenin and
Norton equivalent circuits of a LEC at one pair of termi-
nals only. This means that finding the said circuits at a
different pair of terminals, necessitates repetition of the
whole procedure all over again. This is a time-consuming
approach which may lead to wrong calculations and
which has a negative effect upon students who, in order
to find the new parameters, have to go through the same
procedure again.
Many times the question is posed by the students
themselves: is there a relationship between the Thevenin
and Norton equivalent circuits at different pairs of ter-
minals since the circuit topology remains largely the
same? And, consequently, is there a way to calculate
these parameters at different pairs of terminals simulta-
Given that the circuit remains the same, the question
arises whether there could be a way for finding the
equivalent circuits at different pairs of terminals simul-
The question led the author of this paper to seek and
substantiate a method for finding the equivalent circuits
at more than one pair of terminals simultaneously, based
on a new approach for finding the Thevenin equivalent
circuit in combination with the mesh or nodal analysis
developed by Chatzarakis et al. [9].
In addition and as it is shown in the following sections
the proposed method can be adopted for all kinds of
LECs including or not dependent sources, coupled circuit
components and sinusoidal sources of different frequen-
cies (using superposition principle) so that it can prove a
very powerful and innovative tool for the students and
possibly for field engineers that would like to proceed to
fast and accurate calculations under certain circum-
2. Method Description
Consider the dc LEC and the n-different pairs of terminals
,,22,11 , shown in Figure 1(a).
G. E. CHATZARAKIS329
Figure 1. dc LEC.
According to the method proposed by Chatzarakis et al.
[9], resistances are connected to the ter-
minals , respectively, as shown in
Figure 1(b).
RR R
Since a dc LEC may be either planar or nonplanar, the
method to be used for calculating the currents through
and the voltages across the resistances in
each of the two cases may be mesh or nodal analysis for
the former [1-8], and nodal analysis for the latter
RR R
Consider now the following three possible cases:
Case 1) If mesh analysis is used for planar circuits,
defining the currents of the loops made by connecting the
above mentioned resistances at the n-different pairs of
terminals as 12
II, and the currents at the nm
remaining loops, possibly existing into the circuit, as
nn ,
I, leads to the following mesh-current
equations by inspection:
nn nnnmn
mmmn mmm
cR RRRIV
RcRRR IV
RR cRRIV
To calculate the currents, flowing
through the resistances respectively, the determi-
nants and should be found. Hence,
I1, 2,,k
nn nnn
m mmnmm
cR RRR
DRR cRR
k colum
cRVR R
which lead to (2) and (3), respectively. Thus,
iji j
Da RaRdR
where are real constants, and
aa b
()()() ()
kk k
ki i
jk jk
DaRaRdR b
where are real constants. Hence, for any
() ()()
aa bnk
1 Equations (2) and (3) give
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS
()()() ()
kk k
ij ijj
jk jk
iji jj
aRa RdRb
()()() ()
kk k
ij ijj
jk jk
k k
But, since and , (4) and (5) give
()()() ()
11 ()
kk k
ij ijj
ii k
jk jk
Iim a
aRa RdRb
()()() ()
11 ()
kk kk
ij ijj
ii k
jk jk
k k
iji jj
aRa RdRb
Vim R
THN k
RR a
Equations (6), (7) and (8) define the Thevenin and Nor-
ton equivalent circuits at the n-different pairs of termi-
nals simultaneously. Note that the cal-
culation, which is of great importance for the transmis-
sion and distribution lines of electric energy, is made
possible through Equation (8) by finding the determinant
D only (Equation (2)), which is explained by the fact that
the determinant D contains all the necessary information
for finding the said equivalent resistances.
Example 1
For the circuit in Figure 2(a), obtain the Thevenin and
Norton equivalents as seen from terminals A-B, B-C, and
C-E (taken from [2], p. 158).
Solution: In Figure 2(b), mesh analysis gives by in-
Hence, it is a matter of elementary calculations to find
32321 48020056 RRRRD
Thus, it is clear that , ,
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS331
Figure 2. Circuit for Example 1.
345a,, , and, con-
3.857 3.214
4V 15V
45 3.214
266 5.911 A
266 19 V
Case 2) If nodal analysis is used for nonplanar circuits
specifically, following a similar procedure for finding the
conductance matrix determinant D, the voltages , and
the currents ,
I nk
1(as in Case 1), the following
functions are always derived:
iji jj
Dq GqGwGb
()()() ()
kk k
ij ijj
jk jk
iji jj
qGq GwGb
qGq GwGb
()()() ()
kk k
ij ijj
jk jk
k k
iji jj
qGq GwGb
where , are real constants.
qq b0
() ()()
kk k
qq b
But, since and , (10) and (11) give
()()() ()
11 ()
kk k
ij ijj
ii k
jk jk
iji jj
qGq GwGb
Vim b
qGq GwGb
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS
()()() ()
11 ()
0, 11
kk kk
ij ijj
ii k
jk jk
k k
Gjk n
iji jj
GGG b
Iim G
qGq GwGb
RR k
k (14)
Equations (12), (13) and (14) define the Thevenin and
Norton equivalent circuits at the n-different pairs of ter-
minals. It is worth mentioning again that calculating the
is made possible through Equation (14)
by finding determinant D only (Equation 9).
Example 2
For the circuit in Figure 3, obtain the Thevenin and
Norton equivalents as seen from terminals A-B, C-B, and
Solution: Since this circuit is nonplanar, the method to
be used is the nodal analysis. Hence, in view of Figure 4,
nodal analysis gives by inspection
Hence, it is a matter of elementary calculations to find
1.25 1.7
1.651.5 1.8751.115 1
510.2512.15 10.05DGG G G
32.2511.66.875DGGG G
838.7 16.1 11.613.17DGGGGGGG
810.95 16.16.295DD GGGG
Thus, it is clear that , ,
, , ,
and, consequently,
1.5 1.875
1.5 1.875
10.05 6.875
10.05V6.875 V
10.05 6.875
6.7 A3.667A
1.5 1.875
1.115 1.115
6.295 6.295V
6.295 5.646 A
Case 3) By a similar procedure, the proposed method
holds also for circuits containing dependent voltage
and/or current sources (active circuits), as well. The
Figure 3. Circuit for Example 2.
Figure 4. Method application circuit for Example 2.
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS333
difference lies in the fact that in this case the determinant
D is taken to be the final determinant of the linear system
which results after taking into consideration the relations
describing the dependent sources. Hence, the determi-
nant is no longer resistance or conductance determinant
(Case 1 and Case 2, respectively). This fact, however,
does not affect the proposed method as regards the cal-
culation of the Thevenin and Norton equivalent circuits
at more than one pairs of terminals simultaneously.
It must be noted that students often find it difficult to
treat dependent sources in active circuits. The traditional
serial procedure requires the solution of two circuits, one
for finding the and one for finding the , for
each pair of terminals separately. This approach demands
special attention as regards the dependent sources, since
one/some of them may be zero-valued (in the second
circuit), also influencing other circuit components that
must eventually be removed. The proposed method ad-
dresses the above issues by treating the circuit only once
and without having to remove any of its components,
which constitutes a great advantage over the traditional
Example 3
For the circuit in Figure 5(a), obtain the Thevenin and
Norton equivalents as seen from terminals A-B and K-L
(taken from [3], p. 310)
Solution: In Figure 5(b), mesh analysis gives by in-
Taking into account the fact that the
above matrix equation becomes
4( )
4( )
or, equivalently,
Hence, it is a matter of elementary calculations to find
160256 6401024DRRRR
10880 15360DR
2560 10240DD R
Figure 5. Circuit for Example 3.
Thus, it is clear that, ,
, and consequently
17 A10 A
68 V16 V
3. Evaluation of the Suggested Method
The theoretical development of the proposed method
(Section 2), and, particularly, its results enable the author
to support the pedagogical value of its findings.
Evidence of the effectiveness of the proposed method
on the student learning outcomes came from data gath-
ered during an evaluation process encompassing fourth
cycles of testing the students’ reactions to the suggested
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS
Copyright © 2010 SciRes. ENG
Figures 6-8 show the number of students who re-
sponded to each of the two methods within the allotted
time, managing, at the same time, to arrive at an accurate
solution in each of the 4 problem-solving situations. The
outcomes show that there is a definite advantage to the
proposed method as to the degree of its retention and its
easy retrieval from memory, even if too much time has
passed since its last exemplification and/or application
(tests conducted up to 1 year later).
method in comparison with the traditional serial proce-
dure. The sample was taken from ASPETE, a School of
Pedagogical & Technological Education, located in
Athens, GR, and consisted of 4 groups of Electrical and
4 groups of Electronic Engineering students. Each group
was limited to a maximum of 25 (n = 200), and there
were no significant changes between cycles as to the
group’s internal arrangements.
The first cycle comprised six 60-minute tests, corre-
sponding to the three examples examined within Cases1),
2) and 3) above, conducted 2 weeks after each method’s
presentation in class (two 60-minute tests (×) three ex-
amples = six 60-minute tests).
It is worth noticing that the decrease rate in the num-
ber of students who responded successfully in the 4 cy-
cles of tests remains constant between successive tests in
the case of proposed method, i.e.,
175/143143/116116 / 921.2
(Figure 6)
The cycle was repeated three more times (5 weeks, 1
semester and 2 semesters later), with problem-solving
situations of similar complexity, and with no prior notice
given to the cohorts.
132/101101/7777/ 601.3
(Figure 7)
156 /125125/100100 / 801.25
(Figure 8)
2 weeks
la ter
5 weeks
la ter
1 semester
la ter
2 semesters
la ter
No. of Students
Proposed method
Classical method with
serial procedure
Case 1)
Figure 6. Accuracy of answers with the two methods (Case 1).
2 weeks later5 weeks later1 semester
2 semesters
No. of Students
Proposed method
Classical method with
serial procedure
Case 2)
Figure 7. Accuracy of answers with the two methods (Case 2).
G. E. CHATZARAKIS335
2 weeks later5 weeks later1 semester
2 semesters
No. of Students
Proposed method
Classical method with
serial procedure
Case iii)
Figure 8. Accuracy of answers with the two methods (Case 3).
On the contrary, this rate increases significantly in the
case of classical method with serial procedure, i.e.,
28 /1255 /2880/551.451.2 (Figure 6)
20 / 740 / 2068 / 401.71.3 (Figure 7)
4/1 15/444/152.9 1.25 (Figure 8)
Semi-structured interviews with students and teachers,
administered by the author at the end of the observation
cycles, cross-checked the above outcomes. The respon-
dents not only validated the above findings, they also
demonstrated an extremely positive attitude towards the
proposed method in comparison with the traditional se-
rial procedure. More particularly, students across the 8
participating groups and faculty members that were
asked to apply the proposed method in their own classes
appeared to have identical views regarding the fact that
the proposed method “requires much less time to apply”,
and that the students feel “less anxious about” and, hence,
“more confident” and/or “competent” in working with it
than with the traditional serial procedure. When asked to
comment on the method’s effectiveness regarding stu-
dent performance in terms of real test results, student and
teacher comments validated qualitative impressions re-
corded by the author as field notes and/or empirical data.
4. Conclusions
The study presented in this paper, pertaining to the si-
multaneous finding of the Thevenin and Norton equiva-
lent circuits of a dc LEC at the n-different pairs of ter-
minals, has led to the following conclusions:
Its application by the undergraduate students of the
Departments of Electrical and Electronics Engineering
does not require a high level of mathematical knowledge.
Basics of linear algebra pertaining to the solution of lin-
ear systems could be enough. In any case, the specific
subject area is included in the taught courses of first year
study programs in almost all universities.
It is literally applied by simple inspection, regard-
less of circuit complexity and topology. This reduces the
amount of anxiety experienced by the students regarding
the way to the solution of the problem.
Correctness of results is ensured using either calcu-
lators that can treat large matrices with symbolic vari-
ables or inexpensive math software for personal com-
puters. Furthermore, the method is easily adapted to
computer implementation.
Correctness of results is also ensured by the fact
that the method does not allow for an increase in the er-
rors that are likely to be made from one pair of terminals
to the other etc due to the repeated application of almost
the same procedure. In other words the student can arrive
at correct results for a number of terminals within the
time period required for the calculation of these parame-
ters for only one pair of terminals.
Since the calculations are mostly made by inspec-
tion, the students can easily apply the proposed method
at a much later time than the time the specific subject
area of electric circuits was taught to them.
From a purely educational point of view, the pro-
posed method enables the students and their instructors
to calculate the said equivalent circuits at more than one
pairs of terminals without having to repeat calculations
right from the beginning. Hence, the approach may be
said to hold a particular interest for both instructors and
students, since they do not have to think and/or act de-
pending on the dc LEC topology; the method leads them
to an easy, systematic, time-saving and absolutely accu-
rate calculation of the above parameters at more than one
pairs of terminals simultaneously.
We are able to know the behavior of a given circuit
at any pair of terminals, with regards to the open-circuit
voltage, the short-circuit current, and the equivalent re-
Copyright © 2010 SciRes. ENG
G. E. CHATZARAKIS
sistance. This means that in theory we are able to know
the pair of terminals with respect to which we may drive
the maximum power the specific circuit can supply, since
the matching loads are equal to the Thevenin equivalent
resistances, and the voltages equal to the corresponding
Thevenin voltages at every pair of terminals.
In practice, however, connecting loads to a circuit is
limited to specific pairs of terminals which are not many.
This, in combination with the fact that the load is known,
enables us to select from the few pairs of terminals the
one with respect to which we may achieve the best pos-
sible matching. The criterion for determining the pair of
terminals where the specific load L
R will be connected,
is its minimum divergence from the Thevenin equivalent
resistances, i.e. the appropriate pair kk
, where the
load connection will cause the minimum power reflec-
tions, is determined by the
kRR min. In view of
this, a major contribution of this study is the fact that in
real world applications, it is very useful for the engineer
to know the Thevenin equivalent circuit at more than one
pairs of terminals simultaneously, in order to confront
possible faults.
The approach holds equally well for ac LECs, the
sources of which operate at the same frequency, noting
that the coefficients giving the values of the equivalent
circuits parameters are all complex numbers.
Finally, the accuracy and the effectiveness of the
proposed method can also be verified in a laboratory
environment of electric or electronic circuits where the
time students spend on a weekly basis can not be very
long. Following this method, students can obtain easily
and fast Thevenin equivalents of multi-terminal LECs,
either for practice or on purpose, that could validate the
theoretic results and justify the time savings achieved.
The author hopes that the reader will be inspired to
re-examine his or her own text from a more critical
viewpoint and perhaps add additional critical contribu-
tions to the literature. The author also hopes he has ef-
fectively affirmed that research (both technical and
pedagogical) in circuit analysis can be both interesting
and profitable.
. References
[1] J. W. Nilsson and S. A. Riedel, “Electric Circuits,” Read-
ing, Addison Wesley Publishing Company, 5th Edition,
MA, 1996.
[2] C. K. Alexander and M. N. O. Sadiku, “Fundamentals of
Electric Circuits,” McGraw-Hill, New York, 2000.
[3] G. E. Chatzarakis, “Electric Circuits,” Tziolas Publica-
tions, Thessaloniki, Vol. 1, 1998.
[4] G. E. Chatzarakis, “Electric Circuits,” Tziolas Publica-
tions, Thessaloniki, Vol. 2, 2000.
[5] W. H. Hayt and J. E. Kemmerly, “Engineering Circuit
Analysis,” McGraw-Hill, New York, 1993.
[6] C. A. Desoer and E. S. Kuh, “Basic Circuit Theory,”
McGraw-Hill, New York, 1969.
[7] R. A. Decarlo and P.-M. Lin, “Linear Circuit Analysis,”
Oxford University Press, New York, 2001.
[8] D.E. Johnson et al., “Electric Circuit Analysis,” Pren-
tice-Hall, Upper Saddle River, 1997.
[9] G. E. Chatzarakis et al., “Powerful Pedagogical Ap-
proaches for Finding Thevenin and Norton Equivalent
Circuits for Linear Electric Circuits,” International Jour-
nal of Electrical Engineering Education, Vol. 41, No. 4,
October 2005, pp. 350-368.
[10] G. E. Chatzarakis, “Nodal Analysis Optimization Based
on Using Virtual Current Sources: A New Powerful
Pedagogical Method,” IEEE Transactions on Education,
Vol. 52, No. 1, February 2009, pp. 144-150.
Copyright © 2010 SciRes. ENG | {"url":"https://file.scirp.org/Html/1833.html","timestamp":"2024-11-12T17:00:13Z","content_type":"text/html","content_length":"191034","record_id":"<urn:uuid:5090b99f-04b6-478c-ba20-c72f96890659>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00266.warc.gz"} |
Statistics for the Social Sciences
Learning Objectives
• Use mean and median to describe the center of a distribution.
Choosing between Median and Mean
We now have a choice between two measurements of center. We can use the median, or we can use the mean. How do we decide which measurement to use?
In these next examples, we learn that the shape of the distribution and the presence of outliers helps us answer this question.
Homework Scores with an Outlier
Here is a dotplot of the 26 homework scores earned by a student. Notice that the distribution of scores has an outlier. This student typically scores between 80 and 90 on homework, but there is one
score of 0. Which measurement of center gives a better summary of this distribution?
• Median = 84.5
• Mean = 81.8
Both measures of center are in the B grade range, but the median is a better summary of this student’s homework scores. The outlier does not affect the median. This makes sense because the median
depends primarily on the order of the data. Changing the lowest score does not affect the order of the scores, so the median is not affected by the value of this point.
The mean is not a good summary of this student’s homework scores. The outlier decreases the mean so that the mean is a bit too low to be a representative measure of this student’s typical
performance. This makes sense because when we calculate the mean, we first add the scores together, then divide by the number of scores. Every score therefore affects the mean.
Note: In the distribution above, there are 26 homework scores for this student. If the teacher made fewer homework assignments, a zero would have a greater impact on the mean. We can see this in the
distribution below. This distribution has only 10 scores. The one grade of 0 moves the mean into the C grade range.
Skewed Incomes
In this example, we look at how skewness in a data set affects the mean and median. The following histogram shows the personal income of a large sample of individuals drawn from U.S. census data for
the year 2000. Notice that it is strongly skewed to the right. This type of skewness is often present in data sets of variables such as income.
The mean and median for this data set are
• Mean = $24,000
• Median = $16,900
Here again we see that the mean income does not represent the typical income for this sample very well. The small number of people with higher incomes increase the mean. The mean is too high to
represent the large number of people making less than $20,000 a year. A small number of high incomes gives the misleading impression that the typical income in the sample is $24,000. The small number
of people with higher incomes does not impact the median, so the median income of $16,900 better represents the typical income in this sample.
What’s the Main Point?
These examples illustrate some general guidelines for choosing a measure of center:
• Use the mean as a measure of center only for distributions that are reasonably symmetric with a central peak. When outliers are present, the mean is not a good choice.
• Use the median as a measure of center for all other cases.
Both of these examples also highlight another important principle: Always plot the data.
We need to use a graph to determine the shape of the distribution. By looking at the shape, we can determine which measures of center best describe the data.
Instructions for using the simulation:
• To add a point, move the slider to the value you want, then click Add.
• To remove a point, move the slider to the value you want, then click Minus.
• To reset the simulation, click the button in the upper left corner that says Reset.
Click here to open this simulation in its own window.
Let’s Summarize
• We have two different measurements for determining the center of a distribution: mean and median. When we use the term center, we mean a typical value that can represent the distribution of data.
• The mean is the average. We calculate the mean by adding the data values and dividing by the number of individual data points.
• The mean has the following properties:
□ It is the fair-share measure. For example, imagine that you have 10 homework scores. Say that your scores vary, but the mean is 84. Then you have 84(10) = 840 points, which is like having an
84 on each of the 10 assignments.
□ The mean is also referred to as the balancing point of a distribution. If we measure the distance between each data point and the mean, the distances are balanced on each side of the mean.
• The median is the physical center of the data when we make an ordered list. It has the same number of values above it as below it.
• General Guidelines for Choosing a Measure of Center
□ Use the mean as a measure of center only for distributions that are reasonably symmetric with a central peak. When outliers are present, the mean is not a good choice.
□ Use the median as a measure of center for all other cases.
• Always plot the data. We need to use a graph to determine the shape of the distribution. By looking at the shape, we can determine which measures of center best describe the data. | {"url":"https://courses.lumenlearning.com/atd-herkimer-statisticssocsci/chapter/mean-and-median-2-of-2/","timestamp":"2024-11-02T04:37:10Z","content_type":"text/html","content_length":"53979","record_id":"<urn:uuid:c1a6e004-ab57-4910-bc28-003dfc71ac9b>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00666.warc.gz"} |
operator == abstract method
Test whether this value is numerically equal to other.
If both operands are doubles, they are equal if they have the same representation, except that:
• zero and minus zero (0.0 and -0.0) are considered equal. They both have the numerical value zero.
• NaN is not equal to anything, including NaN. If either operand is NaN, the result is always false.
If one operand is a double and the other is an int, they are equal if the double has an integer value (finite with no fractional part) and the numbers have the same numerical value.
If both operands are integers, they are equal if they have the same value.
Returns false if other is not a num.
Notice that the behavior for NaN is non-reflexive. This means that equality of double values is not a proper equality relation, as is otherwise required of operator==. Using NaN in, e.g., a HashSet
will fail to work. The behavior is the standard IEEE-754 equality of doubles.
If you can avoid NaN values, the remaining doubles do have a proper equality relation, and can be used safely.
Use compareTo for a comparison that distinguishes zero and minus zero, and that considers NaN values as equal.
bool operator ==(Object other); | {"url":"https://main-api.flutter-io.cn/flutter/dart-core/num/operator_equals.html","timestamp":"2024-11-11T20:31:20Z","content_type":"text/html","content_length":"9460","record_id":"<urn:uuid:5ae67211-6f67-4a89-a785-23eaf7b80c23>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00506.warc.gz"} |
Linear Regression in Machine Learning
Linear Regression is one of the most popular and frequently used algorithms in Machine Learning. According to a Kaggle survey, more than 80% of people from the machine-learning community prefer this
algorithm over others. We might have got an idea about its popularity. Hence, Linear Regression is essential to become an expert in the Machine Learning and data science domain. In this article, we
will get a detailed overview of this algorithm.
Key takeaways from this blog
After going through this blog, we will be able to understand the following things,
1. What is Linear regression in Machine Learning?
2. Mathematical understanding of Linear Regression.
3. What are the types of Linear regression?
4. The loss function for Linear Regression.
5. What is the Ordinary Least Squares (OLS) method?
6. How to measure the goodness of fit for Linear Regression?
7. What is polynomial regression, and why is it considered linear regression?
8. How to prepare a dataset to fit the Linear Regression model best?
9. A python-based implementation of Linear regression.
10. Possible interview questions on linear regression.
So, let’s start without any further delay.
What is Linear Regression in Machine Learning?
Linear Regression is a supervised machine learning algorithm that learns a linear relationship between one or more input features (X) and the single output variable (Y). As a standard paradigm of
Machine Learning, the output variable is dependent on the input features.
Mathematical Understanding of Linear Regression
In a machine learning problem, linearity is defined as the linearly dependent nature of a set of independent features X and the dependent quantity y. Mathematically, if X = [x1, x2, …, xn] is a set
of independent features, and y is a dependent quantity, we try to find a function that maps y → X as,
y = β0 + β1*x1 + β2*x2 + ..... + βn*xn + ξ
The βi’s are the parameters (also called weights) that our Linear Regression algorithm learns while mapping X to Y using supervised historical data. ξ is the error due to fitting imperfection, as we
can not assume that all the data samples will perfectly follow the expected function.
Let’s understand these mathematical terms via an example. Suppose we want to predict the price of any house. The crucial features that can affect this price are the size of the house, distance from
the railway station/airport, availability of schools, etc. Let’s treat all these records as a separate feature, then X1 = Size of house, X2 = Distance from the airport, X3 = Distance from school, and
so on.
Do all these features contribute equally to determining the house price? The answer would be No. Every feature has a certain weightage, like “size” matters the most and “distance from the airport”
matters the least. Hence we need to multiply them with a real number describing their weightage, and βs in the above equation represent the same. Also, even if we learn the price prediction strategy,
there will be minute differences in the predicted and actual prices of the house. The term ξ shows this imperfection in the above equation.
What are the types of Linear Regression?
A model is linear when it is linear in parameters relating the input to the output variables. The dependency needs to be more linear in terms of inputs for the model to be linear. For example, all
equations below are linear regressions, defining the model representing the linear relationship between the model parameters.
There are mainly two types of Linear Regression:
Simple Linear Regression
Suppose the number of independent features in X is just one. In that case, it becomes a category of simple linear regression where we try to fit a conventional linear line Y = m*X + c, where “m” is
the slope, and “c” is the intercept. For example, suppose we want to predict the house price by knowing the house size.
Multiple Linear Regression
If the number of independent features in X is more than 1, it becomes a category of multiple linear regression.
y = β0 + β1*x1 + β2*x2 + ..... + βn*xn + ξ
For example, considering all essential features for predicting house price becomes a multiple linear regression. Most of the industry problems are based on this.
Loss Function in Linear Regression
As we said, there will be some imperfections in the fitting. In the image above shown above, the imperfection is shown as ei. Suppose the actual value for input X1 is Y, and our linear regression
model predicted Y' for the same input X1. Then the error (also known as residual) can be calculated as,
ei = |y - y'| or ei = (y - y')^2
This is for one sample, so if we go ahead and calculate the cumulative error for all the samples present in our dataset, it will be called the Loss function for our model. In a regression problem,
the Sum of Squared Residuals (SSR) is one of the most common loss functions, where we sum up the squares of all the errors.
What is the Ordinary Least Squares (OLS) method?
When we fit the Linear regression model with this loss function, it varies the parameters (β0, β1, .., β1) and tries to find the best set of these parameters for which the average of the loss
function becomes minimum. The loss function averaged over all the samples is called a Cost function for linear regression. With SSR as our loss function, finding the best parameters is termed the
Ordinary Least Squares (OLS) method.
Finding the best parameters such that the cost would be minimum is an optimization problem and can be solved using several techniques like Gradient Descent, Linear Algebra, and differential calculus.
To know the Gradient Descent method’s working to solve this optimization problem, please look at this article.
How to measure the goodness of fit for Linear Regression?
We are trying to solve a regression problem using the Linear Regression model, so several evaluation metrics, like MSE, RMSE, and R² score, can determine the goodness of fit. Formulas and their
detailed explanation can be found here.
What is Polynomial Regression?
In real-life scenarios, there can be possibilities where a linear line won’t fit the data perfectly. Independent variables do not possess a linear relationship with the dependent variable here;
hence, we need our machines to learn a curvy relationship. When our machine learns this curvilinear trend, we call it a polynomial regression. The black line shows the linear fit on a non-linear
trend in the image below; hence the performance was worst.
Why is polynomial regression a Linear Regression?
In polynomial regression, we try to fit a polynomial function with degrees >1, e.g., X², X³. So the mapping function would look like this,
y = β0 + β1*x1 + β2*x2 + β3*x2^2 + ξ
Please note that we can have multiple features (X1, X2 in the above equation), and the function can contain higher degrees corresponding to any of the features. Let’s take the case of multiple linear
regression, where we had multiple features present in our dataset and treat higher-order terms from the above equation as a new feature, let’s say X3. Then the same equation will look like this,
y = β0 + β1*x1 + β2*x2 + β3*x3 + ξ
This is precisely the same as the multiple linear regression case, so we only consider polynomial regression as linear regression.
How to prepare a dataset to fit the Linear Regression model best?
As this algorithm has existed for more than 200 years, much research and studies have been done on this algorithm. The literature suggests different heuristics that can be kept in mind while
preparing the dataset. As we discussed, Ordinary Least Square is the most common technique for implementing Linear regression models, so one could try the methods below and see if the R² score is
• Gaussian distributed data is more reliable: One can use different transformation techniques to fit the input data into a gaussian distribution as the prediction made by LR in such a case is more
• Outliers should be removed: In OLS (ordinary least square), we sum the residuals of all the data samples. In such a case, when outliers are present, the predictions will be biased, and eventually
model will perform poorly.
• Input should be rescaled: Providing rescaled inputs via standardization or normalization can produce more accurate results. This is more useful when there is a multiple-linear regression problem
• Transformation of data for linearity between input and output: It should be ensured that the input and output are linearly related; otherwise, data transformation can make them linear, like
logarithmic scaling.
• Collinearity should be removed: Model can be overfitted in case of collinearity. So with the help of correlation value, the most correlated can be removed for better generalization.
Too much theory now. Let's get some hands-on and implement the Linear Regression model.
Python Implementation of Linear Regression
Problem Statement
In our blog on coding machine learning from scratch, we discussed a problem statement on finding the relationship between input and output variables and wrote our program from scratch. But here, we
will try to solve a similar but slightly complex problem of finding the below relationship from the recorded data samples of x and y using the Scikit-Learn library.
Y = -(x-1)*(x+2)*(x-3)*(x+4)
We are going to solve this problem statement in 5 easy steps:
Step 1: Importing the necessary libraries
Required libraries to solve the problem statement are:
• numpy for algebraic operations
• matplotlib for plotting scattered data points and fitted curve
• LinearRegression from sklearn.linear_model for performing the regression operations
• Metrics such as r2_score to evaluate the goodness of fit of the performed regression
• Library for preprocessing features, such as PolynomialFeatures, to obtain higher input dimensions to process polynomial regression.
import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.metrices import mean_squared_error, r2_score
from sklearn.preprocessing import PolynomialFeatures
Step 2: Data Formation
Machines learn from historical observations, but we can directly create data samples using a python program for this problem statement. A total of 50 data points are chosen from the same polynomial
equation. We have added a randomized, normally distributed noise value to avoid perfect fitting conditions to increase the complexity.
Y = -(x-1)*(x+2)*(x-3)*(x+4)
The term np.random.normal(mean, variance, number of points) creates a normally distributed noise with the chosen mean and variance. We will add this noise to our original function to introduce the
imperfection in data.
#creating and plotting dataset with curve-linear relationship
x = np.arange(-5,5,0.2) + 0.1*np.random.normal(0,1,50)
y = -1*(x**4) - 2*(x**3) + 13*(x**2) + 14*(x) - 24 + 10*np.random.normal(-1,1,50)
plt.scatter(x,y, color='red', s=25, label='data')
Step 3: Increasing the dimensionality of input variables
We have just the pair of X and Y values in our dataset. We need to increase the dimensionality in the dataset to perform the regression analysis. The function poly_data takes in input: the original
data and the degree of the polynomial data and, based on the degree value, generates more features to perform the regression.
def poly_data(x,y,degree):
x = x[:,np.newaxis]
y = y[:,np.newaxis]
polynomial_features = PolynomialFeatures(degree=degree)
x_poly = polynomial_features.fit_transform(x)
return (x_poly,y)
For example, poly_data(x,y,2) will generate a second order of the polynomial by mapping X → [1,x, x²], increasing the number of features for performing multiple regression.
Step 4: Linear Regression Model Formation and Evaluation
PolyLinearRegression function takes the input data generated by the poly_data function, unpacks it, and fits it in the LinearRegression model imported from Scikit-learn. Once the model is formed, we
can use it for prediction.
The RMSE and R² scores are computed for the predicted output value. The formed model is a data structure similar to lists, hash, or queues, so it can easily store these evaluation metric values.
def PolyLinearRegression(data, degree=1):
x_poly,y = data
model = LinearRegression()
model.fit(x_poly, y) #it will fit the model
y_poly_pred = model.predict(x_poly) #prediction
rmse = np.sqrt(mean_squared_error(y,y_poly_pred))
r2 = r2_score(y,y_poly_pred)
model.degree = degree
model.RMSE = rmse
model.r2 = r2
return model
Step 5: Plotting the Results
Plotting the result for the regression analysis is done using the function Regression_plots, which returns the plot for a regression model of a different order.
def Regression_plots(data,model,axis):
x, y = data
axis.plot(x[:,1],model.predict(x), color=color[degree=1],
label = str("Model Degree: %d"%model.degree)
+ str("; RMSE:%.3f"%model.RMSE
+ str("; R2 Score: %.3f"%model.r2))
Step 6: Fitting various degree polynomials on the same data
We need to find out whether degree 3 or 4 will fit this dataset better by looking at the dataset. So we need to check different degree fitments and the RMSE and R² scores to find the best fitment.
This section compiles the functions and plots to perform regression analysis of varying-order polynomial functions. Polynomial degrees are varied from 1 to 4, and corresponding RMSE and R² values are
present in the image below.
if __name__ == "__main__":
x = np.arange(-5,5,0.2) + 0.1*np.random.normal(0,1,50)
y = -1*(x**4) - 2*(x**3) + 13*(x**2) + 14*(x) - 24 + 10*np.random.normal(-1,1,50)
_,axis = plt.subplots()
color = ['black','green','blue','purple']
axis.scatter(x[:,np.newaxis], y[:,np.newaxis], color='red',
s=25, label='data')
for degree in range(1,5):
data = poly_data(x,y,degree = degree)
model = PolyLinearRegression(data, degree=degree)
It is observed that when we increase the model’s degree, the RMSE is reduced, and the R² score improves, which means the higher-dimension polynomials fit the data better than the lower dimensions.
Hyperparameters for Linear regression
If we keep increasing the polynomial degree, we will suddenly land in the overfitting problem. In our case, we fabricated the data and knew that the polynomial with degree 4 would fit it perfectly.
But, in real scenarios, the fitting degree of the polynomial can not be guessed by looking at the scatter plot of data samples. Here, we keep increasing the degree, and when the model overfits, we
use regularization techniques to cure it. While applying the regularization, a hyper-parameter of lambda needs to be tuned.
Industrial Applications of Linear Regression
Linear Regression is the most loved machine learning algorithm. Some most popular industrial applications of this algorithm are:
1. Life Expectancy Prediction: Based on geographical and economic factors, World Health Organisation tries to predict the average life of any living thing.
2. House price prediction: Based on factors like the size of the house, locality, and distance from the airport, the real-estate business tries to expect the price of any home.
3. Forecasting company sales: Linear Regression can predict the coming month's sales based on the previous month’s sales.
Possible Interview Questions on Linear Regression
Linear regression is the most used algorithm in machine learning and data science. In every interview for machine learning engineer or data scientist positions, interviewers love to ask questions to
check the basic understanding of how this algorithm works. Some of those questions can be,
1. What is a Linear Regression algorithm, and Why do we say it is a Machine Learning algorithm?
2. What is the Ordinary Least Square method?
3. What are residuals?
4. If input variables are not linearly related, can it still be a linear regression algorithm?
5. How to ensure the better performance of Linear regression models?
In this article, we discussed the most famous algorithm in machine learning, i.e., Linear regression. We implemented the linear regression model on our constructed data step-wise to understand all
the verticals involved. We also discussed the methods using which we can increase the performance of the linear regression model. We hope you enjoyed it.
Next Blog: Life Expectancy Prediction using linear regression
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SET UP THE INTEGRALS NEEDED TO SOLVE EACH OF THE... INCLUDE AN ILLUSTRATION OF THE VARIABLE.
1. A tank contains 288 ft3 of water. If the density of water is 62.4 lbs/ft3, how much work is needed to
pump all of the water out of the tank? How would your integral change if the tank is on one of its other
sides? Which position would produce the greatest amount of work?
8 ft
4 ft
12 ft
2. A tank is full of water. How much work is needed to pump all of the water out of the tank to a point
3 feet above the tank? How would your integral change if the tank is on its side?
12 ft
5 ft
3. A conical tank filled with kerosene is buried 4 feet underground. The density of kerosene is 51.2
lbs/ft3. The kerosene is pumped out until the level drops 5 feet. How much work is needed to pump the
kerosene to the surface if the variable is given as
A. The distance between the vertex of the cone and the “slice”?
B. The distance between the top of the cone and the “slice”?
C. The distance between the surface and the “slice”?
D. The distance between the final level of the kerosene and the “slice”?
4 ft
6 ft
8 ft
4. A chain is L feet long and weighs W pounds. How much work is needed to pull the chain to the top
of a bridge that is L 5 feet tall?
5. A block of ice weighing 500 pounds will be lifted to the top of a 200 foot building. In the 20 minutes
it will take to do this, the block will lose 12 pounds. How much work is needed to lift the block of ice
to the top of the building?
6. A banner in the shape of an isosceles triangle is hung over the side of a building. The banner has a
base of 25 feet (at the roof line), a height of 20 feet, and weighs 40 pounds. How much work is needed
to lift the banner onto the roof of the building? | {"url":"https://studylib.net/doc/17685530/set-up-the-integrals-needed-to-solve-each-of-the...-inclu...","timestamp":"2024-11-06T08:22:16Z","content_type":"text/html","content_length":"55054","record_id":"<urn:uuid:338914bc-3dd0-4e76-ad5c-18e60366d0c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00647.warc.gz"} |
Number and nature of solutions to non-linear systems | Secondary Maths | UK Secondary (7-11)
So far, we have looked at solving systems of linear equations, such as the pair of equations $y=x+1$y=x+1 and $y=4-x$y=4−x. These are not the only kinds of systems of equations that we might come
across, however.
The Earth's orbit around the Sun is close to a circle in shape, and we can model this with the equation $x^2+y^2=150^2$x2+y2=1502 (where the units are in millions of km). A stray comet is passing
near to the sun, and follows a path given by the equation $y=\frac{x^2}{240}-60$y=x2240−60. How many times does the comet's path cross the Earth's path around the sun?
Recall that the real solutions to a system of linear equations can be thought of as the points of intersection of their graphs. For a system of two linear equations, there are three possibilities:
• If the two lines are not parallel, such as $y=x+1$y=x+1 and $y=4-x$y=4−x, they intersect at one point. This type of system has one real solution.
• If the two lines are parallel and distinct, such as $y=x+1$y=x+1 and $y=x$y=x, they never intersect. This type of system has no real solutions.
• If the two lines are equivalent, such as $y=x+1$y=x+1 and $2y=2x+2$2y=2x+2, they overlap and intersect at every point. This type of system has infinitely many real solutions.
We can take the same approach for any system of equations. By drawing the curves on the same coordinate plane, we can identify the number of real solutions by looking at the number of points of
Here is a graph of the equations $x^2+y^2=150^2$x2+y2=1502 and $y=\frac{x^2}{240}-60$y=x2240−60:
We can immediately see from this graph that there are two real solutions to the system $x^2+y^2=150^2$x2+y2=1502 and $eq=2$eq=2. That is, there are two places where the comet's path crosses the
Earth's orbit.
Of course, as long as the Earth is not at those specific points at the same time as the comet, there won't be any collisions.
For systems of non-linear equations there are different sets of possible solutions, depending on the types of equations involved. Importantly, systems of non-linear equations can still have:
• no real solutions, when the graphs have no points of intersection.
• a finite number of real solutions (one or more), when the graphs intersect a finite number of times.
• infinitely many real solutions, which most commonly happens when the two equations are identical after rearrangement and intersect at every point.
Here are examples of each case.
No real solutions
The system of equations $y=x^2+1$y=x2+1 and $y=-x^2-1$y=−x2−1 has no real solutions. We can see that the graphs have no points of intersection:
Finitely many real solutions
The system of equations $y=x^2$y=x2 and $x^2+y^2=4$x2+y2=4 has two real solutions. We can see that their graphs have two points of intersection:
Infinitely many real solutions
The system of equations $y=\frac{2}{x-1}$y=2x−1 and $x=\frac{2}{y}+1$x=2y+1 has infinitely many real solutions. We could rearrange one equation to obtain the other, and we can see that their graphs
are identical (and so they intersect at every point):
The real solutions to a system of equations can be thought of as the points of intersection of their graphs. So to determine the number of solutions to a particular system, we can sketch the graphs
and see how many times they intersect!
Depending on the particular equations, the system might have no real solutions, a finite number of real solutions (one or more), or infinitely many real solutions.
Practice questions
Question 1
A graph of the equations $y=x+2$y=x+2 and $y=x^2$y=x2 is shown below.
How many real solutions does this system of equations have?
Question 2
A graph of the equations $y=x^2-3x+1$y=x2−3x+1 and $y=-x^2-3x+1$y=−x2−3x+1 is shown below.
How many real solutions does this system of equations have?
Question 3
Determine the number of solutions to the system of equations $y=x\left(x+3\right)$y=x(x+3) and $y=-3x\left(x+3\right)$y=−3x(x+3). | {"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-453/topics/Topic-8416/subtopics/Subtopic-111519/","timestamp":"2024-11-10T05:11:48Z","content_type":"text/html","content_length":"573317","record_id":"<urn:uuid:58b12e98-8695-4eb8-9606-164ea3cc1c22>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00379.warc.gz"} |
Purpose of the AUC-ROC (Area Under the Receiver Operating Characteristic) Score
The AUC-ROC Score measures how well a model can distinguish between two classes (in our case, lived and died). The score ranges from 0 to 1.
• 0.5: Represents a model that makes random predictions.
• 0.5 - 0.7: Indicates a model with low to moderate predictive ability.
• 0.7 - 0.8: Suggests a model with acceptable performance.
• 0.8 - 0.9: Signifies a model with excellent performance.
• 0.9 - 1.0: Reflects a model with outstanding performance, with 1.0 being a perfect model.
What Our AUC-ROC Score Tells Us
An AUC-ROC Score of 0.831 means that our model is excellent at distinguishing between horses that will survive and those that won't. This score indicates that our model has an 83.1% chance of
correctly predicting a live/die outcome. | {"url":"https://horsecolic.thekenpoist.net/accuracy_report","timestamp":"2024-11-05T15:41:30Z","content_type":"text/html","content_length":"11122","record_id":"<urn:uuid:ce40cfb7-fe92-45b3-a7d7-a343ce2da891>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00205.warc.gz"} |
Spectral aspects of commuting conjugacy class graph of finite groups
[1] N. M. M. Abreu, C. T. M. Vinagre, A. S. Bonif_acioa and I. Gutman, The Laplacian energy of some Laplacian integral graph, MATCH Commun. Math. Comput. Chem., 60 (2008) 447-460.
[2] K. Balińska, D. Cvetković, Z. Radosavljević, S. Simić and D. Stevanović, A survey on integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat., 13 (2003) 42-65.
[3] R. Brauer and K. A. Fowler, On groups of even order, Ann. of Math., 62 No. 2 (1955) 565-583.
[4] D. Cvetković, P. Rowlinson, S. Simić, Signless Laplacian of finite graphs, Linear Algebra Appl., 423 (2007) 155-171.
[5] P. Dutta, B. Bagchi and R. K. Nath, Various energies of commuting graphs of finite nonabelian groups, Khayyam J. Math., 6 No. 1 (2020) 27-45.
[6] J. Dutta and R. K. Nath, Spectrum of commuting graphs of some classes of finite groups, Matematika, 33 No. 1 (2017) 87-95.
[7] J. Dutta and R. K. Nath, Finite groups whose commuting graphs are integral, Mat. Vesnik, 69 No. 3 (2017) 226-230.
[8] J. Dutta and R. K. Nath, Laplacian and signless Laplacian spectrum of commuting graphs of finite groups, Khayyam J. Math., 4 No. 1 (2018) 77-87.
[9] W. N. T. Fasfous, R. K. Nath and R. Sharafdini, Various spectra and energies of commuting graphs of finite rings, Hacet. J. Math. Stat., 49 No. 6 (2020) 1915-1925.
[10] S.C. Gong, X. Li, G.H. Xu, I. Gutman and B. Furtula, Borderenergetic graphs, MATCH Commun. Math. Comput. Chem., 74 (2015) 321-332.
[11] I. Gutman, Hyperenergetic molecular graphs, J. Serb. Chem. Soc., 64 (1999) 199-205.
[12] I. Gutman, N. M. M. Abreu, C. T. M. Vinagre, A. S. Bonifácioa and S. Radosavljević, Relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem., 59 (2008) 343-354.
[13] F. Harary and A. J. Schwenk, Which graphs have integral spectra?, Graphs and Combin., Lect. Notes Math., 406 (1974), Springer-Verlag, Berlin, 45-51.
[14] M. Herzog, M. Longobardi and M. Maj, On a commuting graph on conjugacy classes of groups. Comm. Algebra, 37 No. 10 (2009) 3369-3387.
[15] S. Kirkland, Constructably Laplacian integral graphs, Linear Algebra Appl., 423 (2007) 3-21.
[16] J. Liu, and B. Liu, On the relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem., 61 (2009) 403-406.
[17] R. Merris, Degree maximal graphs are Laplacian integral, Linear Algebra Appl., 199 (1994) 381-389.
[18] A. Mohammadian, A. Erfanian, D. G. M. Farrokhi and B. Wilkens, Triangle-free commuting conjugacy class graphs. J. Group Theory, 19 (2016) 1049-1061.
[19] R. K. Nath, Various spectra of commuting graphs n-centralizer finite groups, International Journal of Engineering Science and Technology, accepted for publication.
[20] M. A. Salahshour and A. R. Ashrafi, Commuting conjugacy class graph of finite CA-groups, Khayyam J. Math., 6 No. 1 (2020) 108-118.
[21] S. K. Simić and Z. Stanić, Q-integral graphs with edge-degrees at most five, Discrete Math., 308 (2008) 4625-4634.
[22] D. Stevanović, I. Stanković, and M. Milošević, More on the relation between energy and Laplacian energy, MATCH Commun. Math. Comput. Chem., 61 (2008) 395-401.
[23] F. Tura, L-borderenergetic graphs, MATCH Commun. Math. Comput. Chem., 77 (2017) 37-44.
[24] H.B. Walikar, H.S. Ramane and P.R. Hampiholi, On the energy of a graph, Graph Connections, Allied Publishers, New Delhi, 1999, pp. 120-123, Eds. R. Balakrishnan, H. M. Mulder, A. Vijayakumar. | {"url":"https://as.yazd.ac.ir/article_1979.html","timestamp":"2024-11-06T15:38:21Z","content_type":"text/html","content_length":"52835","record_id":"<urn:uuid:cbe6cba9-3086-4256-bc70-81ec695282d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00096.warc.gz"} |
Naive Bayes Classifier Tutorial
Source: Medium
I hope you’re excited to learn about another fantastic class of machine learning models: Naive Bayes. Naive Bayes is wonderful because its core assumptions can be described in about a sentence, and
yet it is immensely useful in many different problems.
But before we dive into the specifics of Naive Bayes, we should spend some time discussing the difference between two categories of machine learning models: discriminative and generative models.
Naive Bayes will be the first generative algorithm we look at, though other common examples include hidden markov models, probabilitistic context-free grammars, and the more hip generative
adversarial networks.
Recall that in our running car example of the past few posts, we are given a dataset of cars along with labels indicating whether they are cheap or expensive. From each car, we have extracted a set
of input features such as the size of the trunk, the number of miles driven, and who the car manufacturer is.
We start from the distribution we are trying to learn $P(X_1, X_2, X_3, Y)$. We can expand the distribution using a few rules of probability along with Bayes’ Rule:
$P(X_1, X_2, X_3, Y) = P(Y) \cdot P(X_1|Y) \cdot P(X_2|X_1, Y) \cdot P(X_3|X_1, X_2, Y)$
This formulation was derived from a few applications of the chain rule of probability. Now we get to the big underlying assumption of the Naive Bayes model.
We now assume the input features are conditionally independent given the outputs. In English, what that means is that for a given feature $X_2$, if we know the label $Y$, then knowing the value of an
additional feature $X_1$ doesn’t offer us any more information about $X_2$.
Mathematically, this is written as $P(X_2|X_1, Y) = P(X_2|Y)$. This allows us to simplify the right side of our probability expression substantially:
$P(X_1, X_2, X_3, Y) = P(Y) \cdot P(X_1|Y) \cdot P(X_2|Y) \cdot P(X_3|Y)$
And with that, we have the expression we need to train our model!
Naive Training
So, how do we actually train the model? In practice, to get the most likely label for a given input, we need to compute these values $P(X_1|Y)$, $P(X_2|Y)$, etc. Computing these values can be done
through the very complicated process of counting! 🙂
Let’s take a concrete example to illustrate the procedure. For our car example, let’s say $Y$ represents cheap and $X_1$ represents the feature of a car’s manufacturer.
Let’s say we have a new car manufactured by Honda. In order to compute $P(X_1=\textrm{Honda}|Y=\textrm{cheap})$, we simply count all the times in our dataset we had a car manufactured by Honda that
was cheap.
Assume our dataset had 10 cheap, Honda cars. We then normalize that value by the total number of cheap cars we have in our dataset. Let’s say we had 25 cheap cars in total. We thus get $P(X_1=\textrm
{Honda}|Y=\textrm{cheap}) = 10 / 25 = 2/5$.
We can compute similar expressions (e.g. $P(X_2=\textrm{40000 miles driven}|Y=\textrm{cheap})$) for all the features of our new car. We then compute an aggregated probability that the car is cheap by
multiplying all these individual expressions together.
We can compute a similar expression for the probability that our car is expensive. We then assign the car the label with the higher probability. That outlines how we both train our model by counting
what are called feature-label co-occurrences and then use these values to compute labels for new cars.
Final Thoughts
Naive Bayes is a super useful algorithm because its extremely strong independence assumptions make it a fairly easy model to train. Moreover, in spite of these independence assumptions, it is still
extremely powerful and has been used on problems such as spam filtering in some early version email messaging clients.
In addition, it is a widely used technique in a variety of natural language processing problems such as document classification (determining whether a book was written by Shakespeare or not) and also
in medical analysis (determining if certain patient features are indicative of an illness or not).
However the same reason Naive Bayes is such an easy model to train (namely its strong independence assumptions) also makes it not a clear fit for certain other problems. For example, if we have a
strong suspicion that certain features in a problem are highly correlated, then Naive Bayes may not be a good fit.
One example of this could be if we are using the language in an email message to label whether it has positive or negative sentiment, and we use features for whether or not a message contains certain
The presence of a given swear word would be highly correlated with the appearance of any other swear word, but Naive Bayes would disregard this correlation by making false independence assumptions.
Our model could then severely underperform because it is ignoring information about the data. This is something to be careful about when using this model!
Shameless Pitch Alert: If you’re interested in practicing MLOps, data science, and data engineering concepts, check out Confetti AI the premier educational machine learning platform used by students
at Harvard, Stanford, Berkeley, and more! | {"url":"https://www.mihaileric.com/posts/not-so-naive-bayes/","timestamp":"2024-11-02T11:15:06Z","content_type":"text/html","content_length":"112542","record_id":"<urn:uuid:2c642fcb-1756-4782-b240-4a51c5ee3266>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00784.warc.gz"} |
The following are special types of parallelograms, with specific properties about their sides, angles, and/or diagonals that help identify them:
These are three examples of rectangles:
These are two examples of rhombi:
Explore the applet by dragging the vertices of the polygons.
1. Which polygon(s) are always rectangles? Can you create a rectangle with any of the polygons?
2. How do you know which polygon(s) form a rhombus versus a square?
3. Which polygon(s) are always parallelograms? Can you create a parallelogram with any of the polygons?
The following theorems relate to the special parallelograms:
Squares have the same properties as both a rectangle and rhombus.
Note that these theorems are for parallelograms, so if we are only told that a polygon is a quadrilateral, then they may not meet the conditions stated. | {"url":"https://mathspace.co/textbooks/syllabuses/Syllabus-1190/topics/Topic-22480/subtopics/Subtopic-285877/?ref=blog.mathspace.co","timestamp":"2024-11-05T06:00:08Z","content_type":"text/html","content_length":"405809","record_id":"<urn:uuid:b3b76d54-fbbf-4fc7-a5c6-c67f6306e452>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00143.warc.gz"} |
5.2: Estimation of Original Gas In-Place, OGIP, Using the Volumetric Method
The Volumetric Method for OGIP is essentially the same for gas reservoirs. The method uses static geologic data to determine the volume of the pore space of the reservoir. Once the volume of the pore
space is estimated, then the gas formation volume factor, ${B}_{g}$ can be used to estimate the OGIP.
By simply using the definition of reservoir volumes, the original-gas-in-place of the reservoir be determined by:
$G=\frac{{V}_{grv}\left(\overline{{h}_{n}}/\overline{{h}_{g}}\right)\overline{\varphi }\text{}\overline{{S}_{g\iota }}}{\overline{{B}_{g\iota }}}$
or, equivalently (after applying the Saturation Constant):
$G=\frac{{V}_{grv}\left(\overline{{h}_{n}}/\overline{{h}_{g}}\right)\overline{\varphi }\left(1-\overline{{S}_{w\iota }}\right)}{\overline{{B}_{g\iota }}}$
• $G$ is the original-gas-in-place of the reservoir, SCF
• ${V}_{grv}$ is the gross rock volume, ft^3
• $\left(\overline{{h}_{n}}/\overline{{h}_{g}}\right)$ is the net-to-gross thickness ratio, fraction
□ $\overline{{h}_{n}}$ is the average net reservoir thickness, ft
□ $\overline{{h}_{g}}$ is the average gross reservoir thickness, ft
• $\overline{{S}_{g\iota }}$ is the average initial gas saturation, fraction
• $\overline{{S}_{w\iota }}$ is the average initial water saturation, fraction
• $\overline{\varphi }$ is the average reservoir porosity, fraction
• $\overline{{B}_{g\iota }}$ is the average formation volume factor of the gas at initial conditions (p[ri] and T[r]), ft^3/SCF
As with the oil volumetric equation (Equation 4.02), Equation 5.01 is evaluated at initial pressure and saturation conditions because the original in-place volume is the desired result. Also, as with
the volumetric oil equation, the net-to-gross ratio in the volumetric gas equation is simply the thickness fraction that converts the total reservoir thickness to the thickness that contributes to
hydrocarbon storage and flow in the reservoir. We use thickness to apply the net-to-gross ratio because rock formations are formed as layers during deposition, and we assume that layers of
poor-quality rock may be deposited and intermixed with layers of good quality reservoir rock. We can further define the gross rock volume as:
• ${V}_{grv}$ is the gross rock volume, ft^3
• $43,560$ is a unit conversion constant, ft^2/acre
• $A$ is the mapped area of the reservoir, acres
• $\overline{{h}_{g}}$ is the average gross reservoir thickness, ft
As with oil reservoirs, the volumetric method for estimating the in-place volumes is also considered to be less accurate than the material balance method. The reason again being the use of all of the
averages in Equation 5.01. This can also be improved by the use of the iso-contour technique describe in Lesson 4. | {"url":"https://www.e-education.psu.edu/png301/node/538","timestamp":"2024-11-06T10:59:54Z","content_type":"text/html","content_length":"44274","record_id":"<urn:uuid:52089e8c-0a8e-423b-a056-54bc90aff2a1>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00171.warc.gz"} |
Defining a periodic function.
Defining a periodic function.
I am a new sage user. I'd like to define simple periodic maps over \R which plot and integrate correctly (eg. a square signal (which of course is discontinuous but which I would still like to be able
to plot in a way that makes it clear that the function is not multivalued at discontinuity points)). I tried different approaches none of which gave satisfactory results.
Any hint on how to do that nicely (or what would be the obstacles)?
Thank you
3 Answers
Sort by ยป oldest newest most voted
Thank-you for the quick answer.
frac() gives a function which is periodic on positive reals. I just adapted your solution to get it periodic on \RR.
f = lambda x: 1 if (x - RR(x).floor()) < 1/2 else 0
I would have liked to be able to define a symbolic function though. Is it doable?
I was also trying to get a function whose plot is correct (without asking it to be pointwise) as it is the case for Piecewise().
Below are some of the things (not chronologically ordred though) I had tried (just for completness as I guess they are just full of beginer's classical mistakes).
which does not evaluate
returns ValueError: cannot convert float NaN to integer.
returns ValueError: Value not defined outside of domain.
also returns ValueError: Value not defined outside of domain.
I also tried to redefine unit_step:
def echelon_unite(x):
if x<0:
return hres
problem integral(echelon_unite(x),x,-10,3) returns 13
numerical integral returns a coherent result.
Other tentative with incoherent result (still with integrate and not numerical_integral)
def Periodisation_int(f,a,b):
x = var('x')
h0(x) = (x-b)-(b-a)*floor((x-b)/(b-a))
hres = compose(f,h0)
return hres
sage: g=Periodisation_int(sin,0,1)
sage: integrate(g(x),x,0,2)
-cos(1) + 2
sage: integrate(g(x),x,0,1)
-cos(1) + 1
sage: integrate(g(x),x,1,2)
-1/2*cos(1) + 1
My guess is I was using integrate on inappropriate objects. I would still like to know how to define corresponding symbolic function (if it is possible).
Thanks again,
best regards.
You should tell us about your different approaches, and what was wrong with them. I am not sure whether you ask about defining a function or plotting it. Here would be my direct lazy approach,
without more informations about your needs. To define a periodic function, use the .frac() method of real numbers. In your case:
sage: f = lambda x: 1 if RR(x).frac() < 1/2 else 0
sage: plot(f, 0, 4)
If you do not like the vertical line at the discontinuities, you can look at the options of the plot() function:
sage: plot?
In your case, you can try:
sage: plot(f, 0, 4, plot_points='1000', linestyle='', marker='.')
For the integral, since the finction is defined point by points (not a symbolic function), you can do a numerical integration:
sage: numerical_integral(f, 0, 4)
(2.0, 2.2315482794965646e-14)
The object you get out of Periodisation_int is not a symbolic function, but when you call it with x you get a symbolic expression anyway, so it doesn't really matter (and most symbolic machinery is
for expressions anyway). If you really want a symbolic function you can do:
def per(f,a,b):
x = SR.var('x')
CSRx = CallableSymbolicExpressionRing( (x,))
t = f((x-b)-(b-a)*floor((x-b)/(b-a)))
h = CSRx(t)
return h
where the whole bit with CSRx is to actually get a symbolic function (something you can call). I'm not so sure that integrate knows what to do with this symbolic function, though. | {"url":"https://ask.sagemath.org/question/10539/defining-a-periodic-function/?answer=15453","timestamp":"2024-11-14T17:36:56Z","content_type":"application/xhtml+xml","content_length":"67635","record_id":"<urn:uuid:17f855c6-416a-40ba-a673-4018a50c34d3>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00360.warc.gz"} |
e S
Comparing Symmetries in Models and Simulations
Springer Handbook of Model-Based Science
We distinguish mathematical modeling, computer implementations of these models and purely computational approaches by their symmetries and by randomness.
Computer simulations brought remarkable novelties to knowledge construction. In this chapter, we first distinguish between mathematical modeling, computer implementations of these models and purely
computational approaches. In all three cases, different answers are provided to the questions the observer may have concerning the processes under investigation. These differences will be highlighted
by looking at the different theoretical symmetries of each frame. In the latter case, the peculiarities of agent-based or object oriented languages allow to discuss the role of phase spaces in
mathematical analyses of physical versus biological dynamics. Symmetry breaking and randomness are finally correlated in the various contexts where they may be observed.
Keywords: Phase Space, Symmetry Breaking, Chaotic Dynamic, Object Oriented Programming, Genetically Modify Organism
Comparing Symmetries in Models and Simulations
Computer simulations brought remarkable novelties in knowledge construction. In this paper, we first distinguish between mathematical modeling, computer implementations of these models and purely
computational approaches. In all three cases, different answers are provided to the questions the observer may have concerning the processes under investigation. These differences will be highlighted
by looking at the different theoretical symmetries of each frame. In the latter case, the peculiarities of Agent Based or Object Oriented Languages allow to discuss the role of phase spaces in
mathematical analyses of physical vs. biological dynamics. Symmetry breaking and randomness are finally correlated in the various contexts where they may be observed.
Keywords: computer simulation, symmetries, randomness, theoretical framework, biology, equational modeling
1 Introduction
Mathematical and computational modeling have become crucial in Natural Sciences, as well as in architecture, economics, humanities, ….
Sometimes the two modeling techniques, typically over continuous or discrete structures, are conflated into or, even, identified to natural processes, by considering nature either intrinsically
continuous or discrete, according to the preferences of the modeler.
We analyze here the major differences that exist between continuous (mostly equational) computational (mostly discrete and algorithmic) modeling, often referred to as computer simulations. We claim
that these different approaches propose different insights into the intended processes: they actually organize nature (or the object of study) in deeply different ways. This may be understood by an
analysis of symmetries and symmetry breakings, which are often implicit but strongly enforced by the use of mathematical structures.
We organize the World by symmetries. They constitute a fundamental “principle of (conceptual) construction”, in the sense of (Bailly and Longo 2011), from Greek geometry, to XXth century physics and
mathematics. All axioms by Euclid may be understood as “maximizing the symmetries of the construction” (see (Longo 2010)). Euclid’s definitions and proofs proceed by rotations and translations, which
are symmetries of space.
Symmetries govern the search for invariants and their preserving transformations that shaped mathematics from Descartes spaces to Grothendieck toposes and all XXth century Mathematics (see (Zalamea
2012)). Theoretical physics has been constructed by sharing with mathematics the same principle of (conceptual) construction. Among them, symmetries, which describe invariance, and order, which is
needed for optimality, play a key role from Galileo’s inertia to the geodetic principle and to Noether’s theorems (see (Van Fraassen 1989; Kosmann-Schwarzbach 2004; Longo and Montévil 2014)). The
fundamental passage from Galileo’s symmetry group, which describes the transformation from an inertial frame to another while preserving the theoretical invariants, to Lorentz-Poincaré group
characterizes the move from classical to relativistic physics. The geodetic principle is an extremizing principle and a consequence of conservation principles, that is of symmetries in equations
Well beyond mathematics and modern physics, the choice of symmetries as organizing principle is rooted in our search for invariants of action, in space and time, as moving and adaptive animals. We
adjust to changing environment by trying to detect stabilities or by forcing them into the environment. Our bilateral symmetry is an example of this evolutionary adjustment between our biological
structure and movement: its symmetry plane is given by the vertical axis of gravitation and the horizontal one of movement^[5]. In this perspective, the role we give to symmetries in mathematics and
physics is grounded on pre-human relations to the physical world, well before becoming a fundamental component of our scientific knowledge construction.
By this, we claim that an analysis of the intended symmetries and their breaking, in theorizing and modeling, is an essential part of an investigation of their reasonable effectiveness and at the
core of any comparative analysis.
2 Approximation?
Before getting into our main theme, let’s first clarify an “obvious” issue that is not so obvious to many: the discrete is not an approximation of the continuum. They simply, or more deeply, provide
different insights. Thus, in no way we will stress the superiority of one technique over the other. We will just try to understand continuous vs. discrete frames in terms of different symmetries.
It should be clear that, on one hand, we do not share the view of many, beautifully expressed by Réné Thom, on the intrinsically continuous nature of the World, where the discrete is just given by
singularities in continua. On the other hand, many mythical descriptions of a Computational World or just of the perfection of computational modeling seem to ignore the limits of discrete
approximation as well as some more basic facts, which are well-known, since always, in Numerical Analysis (the first teaching job, for a few years, of the first author). There is no way to
approximate long enough a continuous non-linear dynamics by an algorithm on discrete data types when the mathematical description yields some sensitivity to initial/border conditions. Given any
digital approximation, the discrete and the continuous trajectories quickly diverge by the combination of the round-off and the sensitivity. However, in some cases (some hyperbolic dynamics), the
discrete trajectory may be indefinitely approximated by a continuous one, but not conversely. The result is proved by difficult “shadowing theorems”, see (Pilyugin 1999). Note that this is the
opposite of the “discrete approximating the continuum”, which is given for granted by many.
We are hinting here just to a comparison between mathematical techniques that provably differ, but which, a priori, says nothing about the actual physical processes that are not continuous nor
discrete, as they are what they are. Yet, it is very easy to check an algorithmic description of a double pendulum against the actual physical device (on sell for 50 euros on the web): very soon the
computational imitation has nothing to do with the actual dynamics. The point is that there is no way to have a physical double pendulum to iterate exactly on the “same” initial conditions (i.e. when
started in the same interval of the best possible measurement), as this device is sensitive to minor fluctuations (thermal, for example), well below the unavoidable interval of measurement. By
principle and in practice, instead, discrete data types allow exact iteration of the computational dynamics, on exactly the same initial data. Again, this is a difference in symmetries and their
In conclusion, on one side, a mathematical analysis of the equations allows to display sensitivity properties, from “mixing”, a weak form of chaos, to high dependence on minor variations of the
initial conditions (as well as topological transitivity, a property related to the density of orbits, etc). These are mathematical properties of deterministic chaos. We stress by this that
deterministic chaos and its various degrees are a property of the mathematical model: by a reasonable abuse one may then say that the modeled physical process is chaotic, if one believes that the
mathematical model is a good/faithful/correct representation of the intended process. But this is an abuse: the dice or a double pendulum know very well where they will go: along a unique physical
geodetics, extremizing a Lagrangian action, according to Hamilton principle. If we are not able to predict it, it is our problem due to the non-linearity of the model, which “amplifies fluctuations”,
and due to our approximated measurements.
As it happens, the interval of measurement, the unavoidable approximated interface between us and the World, is better understood by continua than over discrete data types (we will go back to this)
and, thus, physicists usually deal with equations within continuous frames.
On the other side, the power of discrete computations allows to …compute, even forever, and, by this, it gives fantastic images of deterministic chaos. As a matter of fact, this was mathematically
described and perfectly understood by Poincaré in 1892, yet it came to the limelight only after Lorentz computational discovery of “strange attractors” (and Ruelle’s work, (Ruelle and Takens 1971)).
As deterministic chaos is an asymptotic notion, there is no frame where one can better see chaotic dynamics, strange attractor or alike than on a computer. Yet, just push the restart button and the
most chaotic dynamics will iterate exactly, as we observed and further argue below, far away from any actual physical possibility. And this is not a minor point: it is “correctness of programs” a
major scientific issue in Computer Science. Of course, one can artificially break the symmetry, by asking a friend to change the 16th decimal in the initial conditions. Then, the chaotic dynamics
will follow a very different trajectory on the screen, an interesting information, per se. However, our analysis here is centered on symmetry breaking intrinsic to a theory, that is on changes which
have a physical meaning. This control, available in computer simulations, is thus an artifact from a physical perspective.
3 What do equations and computations do?
3.1 Equations
In physics, equations follow symmetries, either in equilibrium systems, where equations are mostly derived from conservation properties (thus from symmetries, see below), or in far from equilibrium
systems, where they describe flows, at least in the stationary cases — very little is known in non stationary cases. This is the physical meaning of most equational descriptions.
Then one “computes” from equations and, in principle, derives knowledge on physical processes, possibly by obtaining and discussing solutions — or the lack of solutions: a proof of non-analyticity,
such as Poincaré’s Three Body Theorem for example, may be very informative. But these derivations are not just formal: they are mostly based on proofs of relevant theorems. The job of mathematical
deductions, in physics in particular, is to develop the consequences of “meaningful” writings. Mathematics is not a formal game of signs, but a construction grounded on meaning and handled both by
formal “principles of proofs” and by semantically rich “principles of constructions” (Bailly and Longo 2011). Typically, one reasons by symmetries, uses order, including well-ordering, the genericity
of the intended mathematical object or generalized forms of induction that logicians analyze by very large cardinals, an extension of the order of integer numbers obtained by alternating limits and
successor operations (Barwise 1978). Once more, theoretical symmetries and meaning step in while proving theorems and solving/discussing equations; also the passage from Laplace’s predictability of
deterministic process, to Poincaré’s proof of deterministic though unpredictable processes is a breaking of the observable symmetries (see below for more).
As a matter of fact, in order to solve equations, or discuss their solvability, we invented very original mathematical structures, from Galois’ groups to differential geometry. The use of enriched
construction principles, often based on or yielding new mathematical meaning, has been constantly stimulated by the analysis of equations. This is part of the common practice of mathematical
reasoning. However, well beyond the extraordinary diagonal trick by Gödel, it is very hard to prove that “meaningful” procedures are unavoidable in actual proofs, that is to show that meaning is
essential to proofs. An analysis of some recent “concrete” incompleteness result is in (Longo 2011): meaning, as well-ordering, a geometric judgment, provably and inevitably steps in proofs even of
combinatorial theorems (of Arithmetic!). Or, very large, infinite cardinals may be shown to be essential to proofs (Friedman 1998). In this precise sense, formal deductions as computations, with
their finitistic principles of proof, are provably incomplete.
In particular, physico-mathematical deductions, used to discuss and solve equations, are not just formal computations, i.e. meaningless manipulations of signs. They transfer symmetries in equations
to further symmetries, or prove symmetry changes or breaking (non-analyticity, typically). In Category Theory, equations are analyzed by drawing diagrams and inspecting their symmetries.
3.2 From Equations to Computations
The mathematical frame of modern computers was proposed within an analysis of formal deductions. Actually, Gödel, Kleene, Church, Turing …invented computable functions, in the 1930’s, in order to
disprove the largely believed completeness hypothesis of formal/axiomatic systems and their formally provable consistency^[6]. Turing, in particular, imagined the logical Computing Machine, imitating
a man in the least action of sign manipulation according to formal instructions (write or erase 0 and 1, move left or right of one square in a “child’s notebook”), and invented by this the modern
split between software and hardware. He then wrote an equation that easily defines an incomputable arithmetic function. Turing’s remarkable work for this negative result produced the modern notion of
program and digital computer, a discrete state machine working on discrete data types. As we said, computing machinery, invented as an implementation of formal proofs, are provably incomplete even in
arithmetic, let alone in proper extension of it, based on principles richer that arithmetic induction (well-ordering, symmetries, infinite ordinals …).
Thus, beyond the limits set by the impossibility of approximation mentioned above, there is also a conceptual gap between proving over equations and computing solutions by algorithms on discrete
data. The first deals with the physical meaning of equations, their symmetries and their breaking, transfers this meaning to consequences, by human reasoning, grounded on “gestures" (such as drawing
a diagram) and common understanding. It is based on the invention, if needed, of new mathematical structures, possibly infinitary ones, from Galois’ groups to Hilbert Spaces to the modern fine
analysis of infinitary proofs (Rathjen 2006). These, in some cases such as for well-ordering or the large infinite cardinals mentioned above, may even be proved to be unavoidable, well beyond
computations and formalisms (see the reference above). Do algorithms transfer “physical meaning” along the computation? Do they preserve symmetries? Are those broken in the same way we understand
they are in the natural process under scrutiny?
Our claim is that algorithmic approaches (with the notable exception of interactive automated formal calculus, within its limits) involve a modification of the theoretical symmetries used to describe
and understand phenomena in physics, in particular by continua. This means that algorithmic approaches usually convey less or a different physical meaning than the original equational approaches. In
other words, the modification of the equations needed for a completely finitary and discrete approach to the determination of a phenomenon leads to losses of meaningful aspects of the mathematization
and to the introduction of arbitrary or new features.
As far as losses are concerned, the most preeminent ones probably stem from the departure from the continuum, an invention resulting from measurement, from Pythagoras’ theorem to the role of
intervals in physical measurement. As we already hinted, in the computing world, deterministic unpredictability does not make sense. A program determines and computes on exact data: when those are
known, exactly (which is always possible), the program iterates exactly, thus allows a perfect prediction, as the program itself yields the prediction. The point is that deterministic
unpredictability is due to the non-linearity, typically, of the “determination” (the equations) and triggered by non-observable fluctuations or perturbation, below the (best) interval of measurement.
Now, approximation is handled, in mathematics, by topologies of open intervals over continua, the so called “natural topology” over the real numbers.
At this regards, note that a key assumption, bridging mathematics of continua and classical physics, is that any sequence of measurements of increasing, arbitrary precision converge to a well defined
state. This is mathematically a Cauchy condition of completeness, which implies that the rational numbers are not sufficient to understand the situation. Cantor’s real numbers have been invented
exactly to handle this kind of problems (among other reasons, such as the need to mathematize rigorously the phenomenal continuum in its broadest sense, the continuum of movement, say).
Also, the fundamental relation between symmetries and conservation properties exhibited by Noether’s theorems depend on the continuum (e.g. continuous time translations), so that these results can no
longer be derived on a discretized background. In short, these theorems rely on the theoretical ability to transform states continuously along continuous symmetries in equations (of movement, for
example) since the intended conserved quantity cannot change during such a transformation. With a discrete transformation the observed quantities can be altered (and it is the case usually in
simulations) because there is no continuity to enforce their conservation.
Reciprocally, the changes due to the discretization introduce features that are arbitrary from a physical perspective. For example a basic discretization of time introduces an arbitrary fundamental
time-scale. In Numerical Analysis, the methodology is to have the (differential) equations as the locus of objectivity and to design algorithms that can be shown to asymptotically converge (in a
pertinent mathematical sense, and hopefully rapidly in practice) towards the mathematical solutions of the physically meaningful equations. In these frames, the theoretical meaning of the numerical
(or algorithmic) approaches is entirely derivative: such numerical approaches are sound only with respect to, and inasmuch as there are mathematical results showing a proximity with the original
equations and the trajectories determined by them. The mathematical results (convergence theorems) define the nature of this proximity, and are usually limited to specific cases, so that entire
research communities develop around the topic of the simulation of a specific family of equations (Navier-Stokes or alike for turbulence, Schrödinger in Quantum Physics, …). As a result, the methods
to approach different (non-linear) equations by computing rely on specific discretizations and their close, often ad hoc, analysis.
3.3 Computations
As we said, we are just singling-out some methodological differences or gaps between different modeling techniques. On the “side of algorithms”, the main issue we want to stress here is that
equational approaches force uniform phase spaces. That is, the list of pertinent observables and parameters, including space and/or time, of course, must be given a priori. Since the work by
Boltzmann and Poincaré, physicists usually consider the phase spaces made out of (position, momentum) or (energy,time) as sufficient for writing the equational determination. By generalizing the
Philosopher’s (Kant) remark on Newton’s work, the (phase) space is the very “condition of possibility” for the mathematical intelligibility of physics. Or, to put it as H. Weyl, the main
epistemological teaching of Relativity Theory is that physical knowledge begins when one fixes the reference system (that is to say, the way to describe the phase space) and the metrics on it. Then
Einstein’s Invariantentheorie allows to inspect the relevant invariants and transformations, on the grounds of Lorentz-Poincaré symmetry groups, typically, within a pre-given list of observables and
Now, there exists a rich practice of computational modeling, which does not need to pass through equations, skips this a priori. Varenne nicely describes the dynamic mixture of different
computational contexts as a “simulat”, a neologism which recalls “agrégat” (an aggregate) (Varenne 2012). This novelty has been introduced, in particular, by the peculiar features of Object Oriented
Programming (OOP), but other “agent oriented systems” exist.
As a matter of fact, procedural languages require all values to share the same representation — this is how computer scientists call names for observables and parameters^[7]. “Objects" instead may
interact even with completely different representations as long as their interfaces are compatible^[8]. Thus, objects behave autonomously and do not require knowledge of the private (encapsulated)
details of those they are interacting with. As a consequence, only the interface is important for external reactions ( (Cook 1991; Bruce, Cardelli, and Pierce 1997)).
In biological modeling, aggregating different techniques, with no common a priori “phase space”, is a major contribution to knowledge construction. Organisms, niches, ecosystems may be better
understood by structuring them in different levels of organization, each with a proper structure of determination, that is phase space and description of the dynamics. For example, networks of cells
are better described by tools from statistical physics, while morphogenesis, e.g. organ formation, are currently and mostly modeled by differential equations in continua. Each of these approaches
requires pregiven phase spaces, which may radically differ (and the communities of researchers in the two fields hardly talk to each other). In a computer, by its high parallelism, one may mix these
different techniques, with some more or less acceptable approximations, in spite of their differences. Even more so, ad hoc algorithms may describe specific interactions, independently of a unified
equational description that may be impossible. Then “objects", in the sense above, may interact only on the grounds of the actual interface, both within a level of organization and between different
levels, without reference to the proper or internal (to the object, to the level), causal structure.
In other words, OOP allows independent objects’ dynamics, reminiscent of individual cell dynamics. Then, proliferation with variation and motility, the default state of life (see (Longo et al. 2015))
may be added to the models of morphogenesis that usually consider cells as inertial bullets, which they are not; that is, their proliferation, changes and motility are not entailed by physical forces
that contribute to shape organs (in particular, when organs function for the exchange of energy and matter). By the computational power of modern computers, agent or object based programming styles
(such as OOP) may implement autonomous agency for each cell, have them simultaneously interact within a morphogenetic field shaping the dynamics or a network ruled by statistical laws.
In summary, in computer simulation, one may “put together” all these techniques, design very complex “simulat” as aggregation of algorithms, including stochastic equations, probabilities
distributions and alike. In particular, OOP allows the simulation of discrete dynamics of individual cells in an organism or of organisms in an ecosystem. And this with no need to write global first
equations: one directly goes to algorithms in their changing environment.
However, let the process, or images on a computer, run …then push the restart button. Since access to discrete data is exact, as we said and keep stressing, the computer will iterate on the same
initial conditions, exactly, with the same discrete algorithms. Thus, it will go exactly along the same computation and produce exactly the same trajectories, images and genesis of forms. This has no
physical meaning as an unstable or chaotic system would never “iterate identically”. It is even less biologically plausible, as biology is, at least, the “never identical iteration of a morphogenetic
process” (see (Longo et al. 2015)). Observe now that exact iteration is a form of (time-shift/process-identity) symmetry; while non identical iteration is a symmetry breaking (see below for more on
randomness vs. symmetry breaking). Noise, of course, may be introduced artificially, but this makes a deep conceptual difference, at the core of our analysis.
Note finally, that stochastic equations, probability values and their formal or algorithmic descriptions, are expressions and measurement of randomness, they do not implement randomness. And this is
a key issue.
4 Randomness in Biology
Theoretical Physics proposes at least two forms of randomness: classical and quantum. They are separated by different probability theories and underying logic: entanglement modifies the probability
correlations between quantum events (Belavkin 2000). Even the outcome of the mesurement of generic states is contextual which means that this outcome depends on the other measurements performed and
cannot be assumed to be predefined (Abbott, Calude, and Svozil 2014; Cabello 2008), and this situation departs from classical ones which are not contextual. A new form of randomness seems to be
emerging from computer networks; or, at least, it is treated, so far, by yet different mathematics (Longo, Palamidessi, and Paul 2010). In particular, some analysis of randomness are carried without
using probabilities.
In the same way that we said that the world is neither intrinsically continuous or discrete, randomness is not in the world: it is in the interface between our theoretical descriptions and “reality”
as accessed by measurement. Randomness is unpredictability with respect to the intended theory and measurement. Both classical and quantum randomness, though different, originate in measurement.
The classical one is present in dynamics sensitive to initial or border conditions: a fluctuation or perturbation below measurement, which cannot be exact by physical principles (it is an interval,
as we said), is amplified by the dynamics, becomes measurable and …“we have a random phenomenon” (Poincaré 1902). This amplification is mathematically described by the non-linearity of the intended
equations or evolution function, with a subtle difference though. If a solution of the non-linear system exists, then the analysis of the Lyapounov exponents, possibly, yields some information on the
speed of divergence of trajectories, initially indistinguishable by measurement: a non measurable fluctuation is amplified and produces an unpredictable and measurable event, yet the amplification is
computable. In the case of non-existence or non-analyticity of solutions of the given differential equations, one may have bifurcations or an unstable homoclinic trajectories (i.e. trajectories at
the intersection of stable and unstable manifolds). The choice at bifurcation or the physical trajectory is then highly unpredictable, thus random, and may be also physically ascribed to fluctuations
or perturbations below measurement. However, in this case, one does not have, in general, a criterium of divergence, such as Lyapounov exponents. The fluctuation or perturbation “causes” the
unpredictable event, thus Curie’s principle is preserved: “a physical effect cannot have a dissymmetry absent from its efficient cause” — a symmetry conservation principle, or “symmetries cannot
decrease”. Yet, at the level of measured observables one witness a symmetry breaking, as the causing dissymmetry cannot be observed.
Quantum randomness is grounded on non-commutativity of the measurement of conjugated variables (position and momentum or energy and time), given by a lower bound — Planck’s h. It is represented by
Schroedinger’s equation that defines the trajectory of a probability amplitude (or law), in a very abstract mathematical space (a Hilbert space). As hinted above, measurement of entangled particles
gives probabilities that are different from the classical contexts (Bell inequalities are not respected, see (Aspect 1999)).
In quantum physics, though, there is another fundamental difference: in classical and relativistic mechanics, from Aristotle, to Galileo and Einstein, it is assumed that “every event has a cause”. As
mentioned above in reference to Curie’s principle, the unpredictable, but measurable, classical event is “caused” by the (initial or border) undetectable fluctuation. Instead, in current
interpretations of QM, random events may be a-causal — the spin up / spin down of an electron, say, is pure contingency, it does not need to have a cause. This radically changes the conceptual frame
— and many still do not accept it and keep looking, in vain, for hidden variables (hidden causes), along the classical paradigm.
Surprisingly enough, a quantum event at the molecular level may have a phenotypic effect, in biology. This is the result of recent empirical evidence, summarized and discussed in (Buiatti and Longo
2013). Thus, a phenotype, that is a structural property of an organism, possibly a new organism, may result from an a-causal event, happening at a completely different level of organization
(molecular vs. organs or organisms). This micro event may be amplified by classical dynamics of molecules, including as their enthalpic oscillations and their Brownian motion. Brownian motion is
omnipresent in cells’ proteome, where macromolecules are very “sticky” and their chemical interactions are largely stochastic — though canalized by strong chemical affinities and cell
compartmentalization. So, quantum and classical randomness may “superpose” in a highly constrained environment. Moreover, it is increasingly recognized that gene expression is mostly stochastic, see
(Elowitz et al. 2002; Arjun and Oudenaarden 2008).
This leads to the fully general fact that:
macromolecular interactions and dynamics are stochastic, they must be described in terms of probabilities and these probabilities depend on the context.
This context includes the global proteomic composition, the torsion and pressure on the chromatin (Lesne and Victor 2006), the cell activity in a tissue (Bizzarri et al. 2011; Barnes 2014), the
hormonal cascades…up to the ecosystem, as containing fundamental constraints to biological dynamics. The up and down interactions between different levels of organization yield a proper form of
biological randomness, a resonance between levels, called bio-resonance in (Buiatti and Longo 2013). Bio-resonance destabilizes and stabilizes organisms; it both yields and follows from variability,
as correlated variations contribute also to the changing structural stability of organisms. Note that variability produces adaptation and diversity, at the core of biological dynamical stability: an
organism, a population, a species is “biologically stable”, while changing and adapting, also because it is diverse. Both stability and diversity are also the result of randomness. “Also”, because,
as we said, randomness is highly canalized in biology, by cellular compartments of molecules, tissues tensegrity, organismal control (hormones, immune and neural systems …) and the ecosystem may
downward influence these constraints (methylation and demethylation, which may regulate gene expression, can be induced by the environment), (Gilbert and Epel 2009). Variability and diversity are
constrained by history as well: phenotypes are the result of an evolutionary history that canalizes, but does not determine (at least in view of quantum events) further evolution. For example, as for
historical “canalization” there are good reasons to believe that we, the vertebrates, we will never get out of the “valley” of tetrapodes — at most we may lose, and some of us have lost, podia and
keep just traces of them.
In conclusion, randomness has a constitutive role in biology, as variability and diversity contribute to structural stability, beginning with gene expression. We developed above a comparative
analysis in terms of symmetries of physical processes with respect to their equational and computational modeling. We now hinted to the different ways randomness is understood in various physical and
biological frames. In biology, this later issue becomes particularly relevant, in view of the organizing role of randomness, including for small numbers (a population of a few thousands individuals
is biologically more stable when diverse). Further on, we will propose a ‘general ‘thesis’’ relating randomness and symmetry breaking.
5 Symmetries and information, in physics, in biology.
5.1 Turing, Discrete State Machines and Continuous Dynamics
We already stressed the key role of invariants and invariant preserving transformations in the construction of mathematical and physical knowledge. The sharing of construction principles in these two
disciplines, first of all, symmetry principles and order principles, are the reason of the reasonable, though limited, effectiveness of mathematics for physics: these disciplines have been actually
co-constituted on the grounds of these common construction principles, see (Bailly and Longo 2011). However, since so few physical processes can be actually predicted — frictions and many-body
interactions, i.e. non-linearity, are everywhere —, the effectiveness of mathematics stays mostly in the reasonable intelligibility we have of a few phenomena, when we can organize them in terms of
invariants and their transformations, thus of symmetries, well beyond predictability.
In the account above, changing fundamental symmetries produced the change from one theoretical frame to another, such as from classical to relativistic physics. Further useful examples may be given
by thermodynamics and hydrodynamics. The irreversibility of time, a symmetry breaking, steps in the first by the proposal of a new observable, entropy; the second assumes incompressibility and
fluidity in continua, two symmetries that are irreducible to the quantum mechanical ones, so far.
There is a common fashion in projecting the sciences of information onto biological and even physical processes. The DNA, the brain, even the Universe would be (possibly huge) programs or Turing
Machines, sometimes set up in networks — note that the reference to networks is newer, it followed actual network computing by a many years delay.
We do not discuss here the Universe nor the brain. It may suffice to quote the inventor of computing by discrete state machines, Turing: “ …given the initial state of the machine and the input signal
it is always possible to predict all future states. This is reminiscent of Laplace’s view that from the complete state of the universe at one moment of time, as described by the positions and
velocities of all particles, it should be possible to predict all future states. The prediction which we are considering is, however, rather nearer to practicability than that considered by Laplace.
The system of the ’universe as a whole’ is such that quite small errors in the initial conditions can have an overwhelming effect at a later time. The displacement of a single electron by a billionth
of a centimeter at one moment might make the difference between a man being killed by an avalanche a year later, or escaping. It is an essential property of the mechanical systems which we have
called ’discrete state machines’ that this phenomenon does not occur. Even when we consider the actual physical machines instead of the idealized machines, reasonably accurate knowledge of the state
at one moment yields reasonably accurate knowledge any number of steps later“ ((Turing 1950), p. 440)^[9].
As for the brain, Turing continues: “The nervous system is certainly not a discrete-state machine. A small error in the information about the size of a nervous impulse impinging on a neuron, may make
a large difference to the size of the outgoing impulse” ((Turing 1950), p. 451). As a matter of fact, the notions of spontaneous symmetry breaking, “catastrophic instability”, random fluctuations…are
at the core of Turing’s analysis of continuous morphogenesis, (Turing 1952), far remote from his own invention of the elaboration of information by the “Discrete State Machine” (DSM, his renaming in
1950 of his Logical Computing Machine of 1936).
It is worth stressing here the breadth and originality of Turing’s work. He first invented the split hardware/software and the DSM, in Logic. Then, when moving to bio-physics, he invented a
continuous model for morphogenesis, viewed just as physical matter (hardware) that undergoes continuous deformations, triggered by (continuous) symmetry breaking of an homogeneous field, in a
chemical reaction-diffusion system. The model is given by non-linear equations: a linear solution is proposed, the non-linear case is discussed at length.
A key property of Turing’s continuous model is that it is “a falsification” (his words on page 37) of the need for a (coded) “design”. This clearly appears from the further comments on the role of
genes, mentioned below. In discussions reported by Hodges (Hodges 1997), Turing turns out to be against Huxley’s “new synthesis”, which focused on chromosomes as fully determining ontogenesis and
phylogenesis (Huxley 1942). He never refers to the already very famous 1944 booklet by Schrödinger (Schrödinger 1944), where Schrödinger proposes to understand the chromosomes as loci of a coding,
thus as a Laplacian determination of embryogenesis, as he says explicitly (“once their structure will be fully decoded, we will be in the position of Laplace’s daemon” says Schrödinger in chapter 2,
The hereditary code-script). As a matter of fact, in his 1952 paper, Turing quotes only Child, D’arcy Thompson and Waddington as biologists, all working on dynamics of forms, at most constrained
(Waddington), but not determined nor “pre-designed” by chromosomes. Indeed, Turing discusses the role of genes, in chromosomes, which differ from his “morphogenes” as generators of forms by a
chemical action/reaction system. He sees the function of chromosomal genes as purely catalytic and, says Turing, “genes may be said to influence the anatomical form of the organism by determining the
rates of those reactions that they catalyze …if a comparison of organisms is not in question, the genes themselves may be eliminated from the discussion”, page 38 (a remarkable proposal, in the very
fuzzy, ever changing notion of “gene”, see (Fox Keller 2002)). No (predefined) design, no coded or programmed Aristotelian homunculus in the chromosomes (the myth of the chromosomes as a program),
for Turing, the man who invented coding and programming. This is science: an explicit proposal of a (possibly new) perspective on nature, not the transfer of familiar tools (the ones he invented, in
this case!) on top of a different phenomenology.
Note finally that, when comparing his DSM to a woman’s brain in (Turing 1950), Turing describes an “imitation game”, while he talks of a “model” as for morphogenesis. This beautiful distinction,
computational imitation vs. continuous model, is closely analyzed in (Longo 2009).
5.2 Classifying information
Let’s further analyze the extensive use of “information” in biology, molecular biology in particular. Information branches in at least two theories:
• elaboration of information (Turing, Church, Kleene and many others, later consistently extended to algorithmic information theory: Martin-Loef, Chaitin, Calude, see (Calude 2002) and
• transmission of information (Shannon, Brillouin, see (SHANNON 1948)).
In (Longo et al. 2012), we stressed the key differences between these two theories that are confusedly identified in molecular biology, with unintelligible consequences in the description of the
relationship of information to entropy and complexity …two relevant notions in biology^[10].
As scientific constructions, both information theories are grounded on fundamental invariants. And this is so since at least Morse practical invention, with no theory, of information transmission.
Information is independent of the specific coding and the material support. We can transmit and encode information as “bip-bip”, by short and long hits, as flashes, shouts, smoke clouds …by bumping
on wood, metal, by electricity in cables or whatever and this in a binary, ternary, or other code …. Information is the invariant with respect to the transformation of these coding and material
supports: this is its fundamental symmetry. Up to Turing’s fundamental invention: distinguish software from hardware. So, a rich Theory of Programming was born, largely based on Logic, Typed and
Typed-free languages, term rewriting systems etc, entirely independent of the specific encoding, implementation and hardware. The computer’s soul is so detached from its physical realization that
Descartes dualism is a pale predecessor of this radical and most fruitful split. And when the hardware of your computer is dying, you may transfer the entire software, including the operating system,
compilers and interpreters, to another computer. This symmetry by transfer is called “metempsychosis”, we think. Now, it does not apply in biology, nowhere.
The DNA is not a code, carrying information. There is no way to detach a soft content from it and transfer it to another material structure: it cannot be replaced by metal bullets, or bumps on a
piece of wood. What gets transferred to RNA and then proteins is a chemical and physical structure, a most relevant one, as the DNA is an extraordinary chemical trace of an history. And it transmits
to other chemicals an entirely contingent physico-chemical conformation. If a stone bumps against other stones in a river and de-forms them (in-forms them, would say Aristotle), there is no meaning
to speak of a transmission of information, in the scientific invariant sense above, unless in reference to the Aristotelian sense. No informational invariant can be extracted, but the ones proper to
the physico-chemical processes relative to stone bumping. Life is radically contingent and material: no software/hardware split. The pre-scientific reference to information, sometimes called
“metaphorical”, has had a major misleading role. First, it did not help to find the right invariants. The physico-chemical structure of cellular receptors, for example, has some sort of generality,
which yields some stereospecificity (Kuiper et al. 1997). Yet, this is still strictly related to a common chemistry that has nothing to do with an impossible abstract information theoretic
description. The proposal of a too abstract and matter independent invariant did not help to find the right scientific level of invariance. Or, more severely so, it forced exact stereospecifity of
macromolecular interaction, as a consequence of the information theoretic bias.
Monod, one of the main theoreticians of molecular biology, claims that the molecular processes are based on the “oriented transmission of information …(in the sense of Brillouin)”. In (Monod 1970),
he derives from this that the “necessarily stereospecific molecular interactions explain the structure of the code …a Boolean algebra, like in computers” and that “genes define completely the
tridimensional folding of proteins, the epigenetic environment only excludes the other possible foldings”. Indeed, bio-molecular activities “are a Cartesian Mechanism, autonomous, exact, independent
from external influences”. Thus, the analysis based on the search for how information could be transmitted, forced an understanding inspired by the Cartesian exactness proper to computers as well as
the Laplacian causal structure, Turing would say, proper to information theories. It induced the invention of exact stereospecificity, which is “necessary” to “explain” the Boolean coding! That is,
stereospecificity was logically, not empirically, derived, while, since 1957 (Novick and Weiner 1957), robust evidence had already shown the stochasticity of gene expression (see (Kupiec 1983; Kupiec
and Sonigo 2003; Arjun and Oudenaarden 2008) and (Heams 2014) for a recent synthesis).
We now know that the protein folding is not determined by the coding (yet, Monod did consider this possibility). Macromolecular interactions, including gene expression, are largely random: they must
at least be given in probabilities, as we said, and these probabilities would then depend on the context. No hardware independent Boolean algebra governs the chemical cascades from DNA to RNA to
proteins, also because these cascades depend, as we already recalled, on the pressure and tensions on the chromatin, the proteome activities, the intracellular spatial organization, the cellular
environment and many other forms of organismal regulations, see for example (Weiss et al. 2004; Lesne and Victor 2006).
In summary, the informational bias introduced a reasoning based on Laplacian symmetries, far away from the largely turbulent structure of the proteome, empowered also by chaotic enthalpic
oscillations of macromolecules. This bias was far from neutral in guiding experiments, research projects and conceptual frames. For example, it passed by the role of endocrine disruptors of the more
than 80,000 molecules we synthesized and used in the XXth century, an increasingly evident cause of major pathologies, including cancer, (Zoeller et al. 2012; Soto and Sonnenschein 2010; Demeneix
2014). These molecules were not supposed to interfere with the exact molecular cascades of key-lock correspondences, a form of stereospecificity. The bias guided the work on GMO, which have been
conceived on the grounds of the “central dogma of molecular biology” and of Monod’s approach above: genetic modifications would completely guide phenotypic changes and their ecosystemic interactions
(see (Buiatti 2003)).
One final point. Information theories are “code independent”, or analyze code in order to develop general results and transmission stability as code insensitive (of course cryptography goes
otherwise: but secrecy and code breaking are different purposes, not exactly relevant for organisms). Information on discrete data is also “dimension independent”: by a polynomial translation one may
encode discrete spaces of any finite dimension into one dimension. This is crucial to computing, since it is needed to define Turing’s Universal Machine, thus operating systems and compilers.
Biology instead is embedded in a physical world where the space dimension is crucial. In physics, heat propagation and many other phenomena, typically field theories, strictly depend on space
dimension. By “mean field theories” one can show that life, as we know it, is only possible in three dimensions (see (Bailly and Longo 2011)). Organisms are highly geometric in the sense that
“geometric” implies sensitivity to coding and dimensions. In this sense, continuous models more consistently propose some intelligibility: in “natural” topologies over continua, that is when the
topology derives from the interval of physical measurement, dimension is a topological invariant, a fundamental invariant in physics, to be preserved in biology, unless the reader believes that he/
she can live encoded in one dimension, just exchanging information, like on the tape of a Turing Machine. A rather flat Universe …yet, with no loss of information. But where one has only information,
not life.
Missing the right level of invariance and, thus, the explanatory symmetries, is a major scientific mistake. Sometimes, it may seem just a “matter of language”, as if language mattered little, or of
informal metaphors, as if metaphors were not carrying meaning, forcing insight and guiding experiments. They actually transfer the conceptual structure or the intended symmetries of the theory they
originate from, in an implicit, thus more dangerous and un-scientific way. Just focusing on language, consider the terminology used when referring to DNA/RNA as the “universal code for life”, since
all forms of life are based on it. This synchronic perspective on life — all organisms yield these molecules and the basic chemical structure of their interactions, thus there is a universal code —
misses the historical contingency of life. There is no universality in the informational sense of an invariant code with respect to an independent hardware. Life is the historical result of
contingent events, the formation somewhere and somehow of DNA or RNA or both, sufficiently isolated in a membrane, which occurred over that hardware only. Then, the resulting cell reproduced with
variation and diversified, up to today’s evolutionary diversity. One contingent material origin, then diversification of that matter, of that specific hardware and no other. Invariance, symmetries
and their breaking are different from those proper to “information”, in this strictly material, evolutionary perspective.
6 Theoretical symmetries and randomness
In this section, we would like to elaborate on a “thesis”, already hinted in (Longo and Montévil 2014). In physical theories, where the specific trajectory of an object is determined by its
theoretical symmetries, we propose that randomness appears when there is a change in some of these symmetries along a trajectory and reciprocally that changes of symmetries are associated to
Intuitively theoretical symmetries enable to understand a wide set of phenomenal situations as equivalent. In the end of the day, the trajectory that a physical object will follow, according to a
theory, is the only trajectory which is compatible with the theoretical symmetries of a given system. Symmetries, in this context, enable to understand conservation properties, the uniqueness of the
entailed trajectory and ultimately the associated prediction, if any.
Now, what happens when, over time or with respect to a pertinent parameter, a symmetry of the system is broken? A symmetry corresponds to a situation where the state or the set of possible states and
the determination of a system does not change according to specific transformations (the symmetries). After the symmetry breaking, the state(s) becomes no longer invariant by these transformations;
typically, the trajectory goes to one of the formerly symmetric states and not to the others (a ball on top of a mathematical hill falls along one of the equivalent sides). Since the initial
situation is exactly symmetric (by hypothesis), all the different “symmetric” states are equivalent and there is no way to single out any of them. Then, in view of the symmetry breaking, the physical
phenomena will nevertheless single out one of them. As a result we are confronted with a non-entailed change: it is a random change.
This explanation provides a physico-mathematical meaning to the philosophical notion of contingency as non-necessity: this description of randomness as symmetry breaking captures contingency as a
lack of entailment or of necessity in an intended theory. Note that usually the equivalent states may not be completely symmetric as they may be associated to different probabilities, nevertheless
they have the same status as “possible” states.
For now, we discussed the situation at the level of the theoretical determination alone, but the same reasoning applies mutadis mutandis to prediction. Indeed, we access to a phenomenon by
measurement, but measurement may be associated to different possible states, not distinguishable individually. These states thus are symmetric with respect to the measurement, but the determination
may be such that these (non-measurably different) states lead to completely different measurable consequences. This reasoning is completely valid only when the situation is such for all allowed
measurements, so that randomness cannot be associated to the possible crudeness of an arbitrary specific measurement.
Reciprocally, when we consider a random event, it means that we are confronted with a change that cannot be entailed from a previous observation (and the associated determination). When the possible
observations can be determined (known phase space), this means that the different possibilities have a symmetric status before the random event (precisely because they are all pre-defined
possibilities) but that one (or several) of them are singled out by the random event in the sense that it becomes the actual state. We recognize in this statement the description of a symmetry that
is broken during the random event.
Let us now review the main physical cases of randomness.
• Spontaneous symmetry breaking in quantum field theories and theories of phase transitions (from a macroscopic viewpoint) are the most straightforward examples of the conjecture we describe. In
these cases, the theoretical determination (Hamiltonian) is symmetric and the change of a parameter leads the systems equilibrium to shift from a symmetric state to an asymmetric one (for example
isotropy of a liquid shifting to a crystal with a specific orientation). Randomness stems then just from the “choice” of a specific orientation, triggered by fluctuations in statistical
• Classical mechanics can, in spite of its deterministic nature, lead to unpredictability as a consequence of the symmetrizing effect of measurement on one side (there are always different states
which are not distinguished by a measurement), and a determination that leads those states to diverge (which breaks the above symmetry). This reasoning applies to chaotic dynamics but also to
phase transitions where, from a strictly classical viewpoint, fluctuations below the observation determine the orientation of the symmetry changes.
• In classical probabilities, applied to “naive” cases such as throwing a dice or to more sophisticated framework such as statistical mechanics, our reasoning also applies. When forgetting about
the underlying classical mechanics, the probabilistic framework is a strict equivalence between different possibilities, except for their expected frequencies which may differ: those are given by
the associated probabilities. In order to define theoretically these probabilities, some underlying theoretical symmetries are required. In our examples, the symmetries are the symmetry between
the sides of a dice and for statistical mechanics, the symmetry between states with the same energy for the microcanonical ensemble. From a strictly classical viewpoint, these symmetries are
assumed to be established on average by the properties of the considered dynamics. In the case of dice, it is the rotation, associated to the dependence on many parameters which leads to a
sufficient mixing, generating the symmetry between the different sides of the dice. In the case of statistical mechanics, it is the property of topological mixing of chaotic dynamics (a property
met by these systems by definition). This property is assumed in order to justify the validity of statistical mechanics from the point of view of classical mechanics. In both cases, a specific
state or outcome corresponds to a breaking of the relevant symmetry.
• In quantum mechanics, the usual determination of the trajectory of a state is deterministic, randomness pops out during measurement. The operator corresponding to the measurement performed
establishes a symmetry between its different eigen vectors, which also correspond to the different outcomes corresponding to the eigen values. This symmetry is partially broken by the state of
the system, which provides different weights (probabilities) to these possibilities. The measurement singles out one of the eigen vectors which becomes the state of the system and this breaks the
former symmetry.
We can conclude from this analysis and these examples that randomness and symmetry breaking are tightly associated. We can put this relationship into one sentence:
A symmetry breaking means that equivalent “directions” become no longer equivalent and precisely because the different directions were initially equivalent (symmetric) the outcome cannot be predicted
As discussed elsewhere (Longo and Montévil 2011, 2014), we assume that theoretical symmetries in biology are unstable. It follows that randomness, understood as associated to symmetry breaking,
should be expected to be ubiquitous; however, this approach leads also to propose a further form of randomness. In order to show that randomness can be seen as a symmetry breaking, we needed to
assume that the set of possibilities was determined before the event. In biology, the instability of the theoretical symmetries does not allow such an assumption in general. On the opposite, a new
form of randomness appears through the changes of phase spaces, and this randomness does not take the form of a symmetry breaking stricto sensu inasmuch as it does not operate on a pre-defined set.
In other words, these changes cannot be entailed but they cannot even be understood as the singling out of one possibility among others — the list of possibilities (the phase space) is not pre-given.
In brief, theoretical symmetries in physics enable to single-out a specific trajectory in a phase space, formed by a combination of observables. Thus, a symmetry breaking corresponds to the need of
one or several supplementary quantities to further specify a system on the basis of already defined quantities (which were formerly symmetric and thus not useful to specify the situation). In
biology, instead, the dynamic introduces new observable quantities which get integrated to the determination of the object as the latter is associated to the intended quantities and symmetries. This
dynamics of the very phase space may be analyzed a posteriori as a symmetry breaking. Thus, randomness moves from within a phase space to the very construction of a phase space, a major mathematical
We would like to thank Kim Bruce for reminding and updating us on the Foundations of Object Oriented Languages, an area which he contributed to trigger, in part by joint work with GL, and by starting
the FOOL series conferences, twenty years ago.
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Statement of Principles from the Endocrine Society.” Endocrinology 153 (9): 4097–4110. doi: 10.1210/en.2012-1422. | {"url":"https://montevil.org/publications/chapters/2017-lm-comparing-symmetries-models/","timestamp":"2024-11-06T14:00:07Z","content_type":"text/html","content_length":"272484","record_id":"<urn:uuid:0ebd328b-ca8f-432f-b723-60a2ceceb173>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00078.warc.gz"} |
Square Root of 41 - How to Find the Square Root of 41? - Cuemath
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
Square Root of 41
Did you know that the sum of the first six prime numbers, i.e., 2, 3, 5, 7, 11, and 13 is 41? 41 is an odd prime number. Prime numbers have been described as the building blocks of mathematics, or
more particularly, the "atoms" of mathematics. Let us determine the square root of 41 in this article.
• Square Root of 41: √41 = 6.40312423743
• Square of 41: 41^2 = 1681
1. What Is the Square Root of 41?
2. Is Square Root of 41 Rational or Irrational?
3. How to Find the Square Root of 41?
4. FAQs on Square Root of 60
5. Important Notes
6. Challenging Questions
7. FAQs on Square Root of 41
What Is the Square Root of 41?
Let us look at an example of perfect squares first. 3² = 3 × 3 = 9 is an example. Here 3 is called the square root of 9, but 9 is a perfect square. Does that mean non-square numbers cannot have a
square root? Non-square numbers also have a square root, but they are not whole numbers.
Square Root of 41
Square root of 41 in the radical form is expressed as √41 and in the exponent form, it is expressed as 41^½. The square root of 41, rounded to 5 decimal places, is ± 6.40312.
Is the Square Root of 41 Rational or Irrational?
A number that cannot be expressed as a ratio of two integers is an irrational number. The decimal form of the irrational number will be non-terminating (i.e. it never ends) and non-recurring (i.e.
the decimal part of the number never repeats a pattern). Now let us look at the square root of 41.
Do you think the decimal part stops after 6.40312423743? No, it is never-ending and you cannot observe any pattern in the decimal part.
Thus, √41 is an irrational number.
How to Find the Square Root of 41?
Square roots can be calculated using various methods, such as:
• By simplifying the radical of the numbers that are perfect squares.
• By long division method for perfect and non-perfect squares
41 is a prime number and hence, it is not a perfect square. Therefore, the square root of 41 can only be found by the long division method.
Simplified Radical Form of Square Root of 41
To simplify the square root of 41, let us first express 41 as the product of its prime factors. Prime factorization of 41 = 1 × 41. Therefore, √41 is in the lowest form and cannot be simplified
further. Thus, we have expressed the square root of 41 in the radical form. Can you try and express the square root of 29 in a similar way?
Square Root of 41 By Long Division Method
Let us follow these steps to find the square root of 41 by long division.
• Step 1: Group the digits into pairs (for digits to the left of the decimal point, pair them from right to left) by placing a bar over it. Since our number is 41, let us represent it inside the
division symbol.
• Step 2: Find the largest number such that when you multiply it with itself, the product is less than or equal to 41. We know that 6 × 6 = 36 and is less than 41. Now, let us divide 41 by 6.
• Step 3: Let us place a decimal point and pairs of zeros and continue our division. Now, multiply the quotient by 2 and the product becomes the starting digits of our next divisor.
• Step 4: Choose a number in the unit's place for the new divisor such that its product with a number is less than or equal to 500. We know that 2 is in the ten's place and our product has to be
500 and the closest multiplication is 124 × 4 = 496.
• Step 5: Bring down the next pair of zeros and multiply the quotient 64 (ignore the decimal) by 2, which is 128. This number forms the starting digits of the new divisor.
• Step 6: Choose the largest digit in the unit's place for the new divisor such that the product of the new divisor with the digit at one's place is less than or equal to 400. We see that 1281,
when multiplied by 1, gives 1281 which is greater than 400. Therefore, we will take 1280 × 0 = 0 which is less than 400.
• Step 7: Add more pairs of zeros and repeat the process of finding the new divisor and product as in step 2.
Note that the square root of 41 is an irrational number, i.e. it is never-ending. So, you can stop the process after 4 or 5 iterations.
Explore square roots using illustrations and interactive examples
• The square root of 41 in the radical form is expressed as √41
• In exponent form, the square root of 41 is expressed as 41^½.
• The real roots of √41 are ± 6.40312.
• What is the value of √√√√41?
• Simplify ((41)^½)^¾
• Determine the square root of 4141.
Square Root of 41 Solved Examples
1. Example 1: Fredrick told his friends that the value of -√41 is the same as √-41. What do you think?
Negative square root cannot be real numbers.
-√41 is a real number.
But √-41 is an imaginary number.
Hence, they are not the same. -√41 is not the same as √-41.
2. Example 2
Joel had a doubt. He knew that 6.403 is the square root of 41. He wanted to know if -6.403 is also a square root of 41? Can you clarify his doubt?
Let us take an example of a perfect square number and extend the same logic to clarify Joel's doubt.
We know that 3 is a square root of 9 because when 3 is multiplied to itself it gives 9. But, what about -3? Let us multiply and check.
-3 × -3 = 9 (- × - = +)
Therefore, -3 is also a square root of 9. Thus, -6.403 is also the square root of 41.
3. Example 3
Help Tim simplify √√41.
By long division, we learnt that √41 = 6.403.
We need to find the value of √6.403. Going by the same long division steps as discussed above, we get √6.403 = 2.530.
Great learning in high school using simple cues
Indulging in rote learning, you are likely to forget concepts. With Cuemath, you will learn visually and be surprised by the outcomes.
Interactive Questions
Check Answer >
FAQs on Square Root of 41
What is the square root of 41?
Square root of 41: √41 = 6.40312423743.
What is the square of 41?
Square of 41 is 1681.
How do you work out the square root of 41?
We can find the square root of 41 using various methods.
1. Repeated Subtraction
2. Prime Factorization
3. Estimation and Approximation
4. Long Division
If you want to learn more about each of these methods, click here.
How do you find √41 times √41?
The value of √41 × √41 = 41
What is the square root of 41 in the simplest radical form?
√41 is the simplest radical form.
Is the square root of 41 an irrational number?
√41 = 6.40312423743
√41 is an irrational number.
Math worksheets and
visual curriculum | {"url":"https://www.cuemath.com/algebra/square-root-of-41/","timestamp":"2024-11-02T18:15:35Z","content_type":"text/html","content_length":"225718","record_id":"<urn:uuid:349a3d73-7f76-4d1f-8192-0657df53e187>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00214.warc.gz"} |
Estimates SARIMA models using a state space representation, the Kalman filter, and maximum likelihood.
vOut = sarimaSS(y, p, d, q, P_s, D_s, Q_s, s, trend, const)#
☆ y (Nx1 vector) – data.
☆ p (Scalar) – the autoregressive order.
☆ d (Scalar) – the order of differencing.
☆ q (Scalar) – the moving average order.
☆ P_s (Scalar) – the seasonal autoregressive order.
☆ D_S (Scalar) – the seasonal order of differencing.
☆ Q_s (Scalar) – the seasonal moving average order.
☆ s (Scalar) – the seasonal frequency term.
☆ trend (Scalar) – an indicator variable to include a trend in the model. Set to 1 to include trend, 0 otherwise.
☆ const (Scalar) – an indicator variable to include a constant in the model. Set to 1 to include trend, 0 otherwise.
amo (struct) –
An instance of an arimamtOut structure containing the following members:
amo.aic Scalar, value of the Akaike information criterion.
amo.b Kx1 vector, estimated model coefficients.
amo.e Nx1 vector, residual from fitted model.
amo.ll Scalar, the value of the log likelihood function.
amo.sbc Scalar, value of the Schwartz Bayesian criterion.
amo.lrs Lx1 vector, the Likelihood Ratio Statistic.
amo.vcb KxK matrix, the covariance matrix of estimated model coefficients.
amo.mse Scalar, mean sum of squares for errors.
amo.sse Scalar, the sum of squares for errors.
amo.ssy Scalar, the sum of squares for Y data.
amo.rstl an instance of the kalmanResult structure.
library tsmt;
airline = loadd( getGAUSSHome() $+ "pkgs/tsmt/examples/airline.dat");
// Transform data
y = ln(airline);
p = 0;
d = 1;
q = 1;
P_s = 0;
D_s = 1;
Q_s = 1;
trend = 0;
const = 0;
struct arimamtOut amo;
amo = sarimaSS( y, p, d, q, P_s, D_s, Q_s, s, trend, const ); | {"url":"https://docs.aptech.com/gauss/tsmt/sarimass.html","timestamp":"2024-11-01T20:43:08Z","content_type":"text/html","content_length":"21198","record_id":"<urn:uuid:22259875-03ac-4f54-963f-6f8589ab8e58>","cc-path":"CC-MAIN-2024-46/segments/1730477027552.27/warc/CC-MAIN-20241101184224-20241101214224-00671.warc.gz"} |
Numpy - Get the Sum of Diagonal Elements - Data Science Parichay
The Numpy library in Python comes with a number of useful functions to work with and manipulate the data in arrays. In this tutorial, we will look at how to get the sum of the diagonal elements of a
2d array in Numpy.
How to get the sum of diagonal elements in Numpy?
To get the sum of diagonal elements of a 2d Numpy array, first, use the numpy.diag() function to get the diagonal elements and then calculate their sum using numpy.ndarray.sum().
The following is the syntax –
numpy.diag(v, k).sum()
The numpy.diag() function takes the following parameters –
1. v – The 2d array to extract the diagonal elements from.
2. k – The diagonal to extract the elements from. It is 0 (the main diagonal) by default. Diagonals below the main diagonal have k < 0 and the ones above the main diagonal have k > 0.
It returns the extracted elements from the diagonal as a numpy array.
You can then apply the numpy.ndarray.sum() function on the returned array of diagonal elements to get its sum.
Let’s now look at examples of using the above syntax to get the sum of diagonal elements of a 2d array.
First, we will create a Numpy array that we will use throughout this tutorial.
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import numpy as np
# create a 2D numpy array
arr = np.array([
[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
[10, 11, 12]
# display the matrix
[[ 1 2 3]
[ 4 5 6]
[ 7 8 9]
[10 11 12]]
Here, we used the numpy.array() function to create a 2d array of shape 4×3 (having 4 rows and 3 columns).
Example 1 – Sum of elements on the default diagonal
Let’s now use the numpy.diag() function to get the diagonal elements for the 2d array created above. We will use the default diagonal (k = 0).
# get the diagonal elements
res = np.diag(arr)
# display the diagonal elements
[1 5 9]
We get the diagonal elements of the passed array as a 1d numpy array. You can see that the returned array has the same values as the main diagonal.
To get the sum of the diagonal elements, use the numpy.ndarray.sum() function.
# get the sum of diagonal elements
We get the sum of the main diagonal elements as 15.
Example 2 – Sum of elements on a custom diagonal
In the above example, we extracted the elements of the main diagonal.
The numpy.diag() function comes with an optional parameter, k that you can use to specify the diagonal you want to extract the elements from.
The below image better illustrates the different values of k (representing different diagonals) for our input array.
k is 0 by default. The diagonals below the main diagonal have k < 0 and the diagonals above it have k > 0.
Let’s now use the numpy.diag() function to get the elements on diagonal, k = -1.
# get the elements of the diagonal -1
res = np.diag(arr, k=-1)
# display the diagonal elements
[ 4 8 12]
We get the elements for the k=-1 diagonal.
Now, we’ll use the numpy.ndarray.sum() function to get its sum.
# get the sum of diagonal elements
We get the sum of values in the k=-1 diagonal as 24.
In this tutorial, we looked at how to get the sum of diagonal elements of a 2d array in Numpy. The following are the steps mentioned in this tutorial.
1. Use the numpy.diag() function to get the diagonal elements of a 2d array. You can specify the diagonal for which you want the extract the elements using the optional parameter k. By default, it
represents the main diagonal, k = 0.
2. Once you have an array of the diagonal elements, use the numpy.ndarray.sum() function to get its sum.
You might also be interested in –
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Coulomb's Law: Definition, Formula, Vector Form
Coulomb’s Law
Coulomb’s Law: According to this law, the electrical force between two charged objects is equal to the product of such charged particles and inversely proportional to the square of their distance.
Attractive or repulsive force acts. Electrostatic force describes it. It measures the force between two charged particles. French physicist Charles-Augustin de Coulomb introduced this idea in 1785.
Straight-line force. The force is repulsive if the charged object has the same sign and attractive if it has the opposite sign.
Coulomb’s Law Definition
The force of attraction or repulsion between two electrically charged particles is defined by Coulomb’s law, a physical principle. According to the rule, the strength of the force is inversely
proportional to the square of the distance between the charges and directly proportional to the product of the charges.
Coulomb’s law can be formulated mathematically as follows:
Where: F is the force between the two charges and F = k * q1 * q2 / d2
The Coulomb constant, or k, has a value of 9 * 109 N * m2 / C2 and is a constant.
The two particles’ charges are q1 and q2, respectively.
d represents the separation of the two particles.
If the two charges have opposing signs, the force between them is attracting; if they have the same sign, it is repulsive. If the charges are greater, the force is stronger; if the charges are
farther apart, the force is weaker.The fundamental physical principle known as Coulomb’s law is crucial to the study of electromagnetic, chemistry, and biology, among other branches of science.
Coulomb’s Law Formula
As we are all aware, coulomb’s law states that the product of two charged objects and the distance between them are directly proportional to each other. In mathematics, it can be expressed as:
F ∝ q1q2/d²
F = 1/4π€° x q1q2/d²
After the proportion is subtracted, we have a proportionality constant of 1/4°, which equals the value of 9109 N-m2/C2.
These two charged bodies are q1 and q2.
The distance between the two bodies is squared as d2.
Units, Dimensions, and Value of Coulomb’s Law
€° = 1/4πF x q1q2/d²
As SI Unit of charge is coulomb(C)
Unit of €° = 1CC/ Nm² = C²N^-1m^-2
Dimensions of €° = [M^-1L^-3T^4A^2]
Value of €°= 8.85 x 10^-12 C²N^-1m^-2
Coulomb’s Law Vector form
Scalar and vector quantities are the two different forms of quantities. While a vector quantity determines both the direction and the magnitude, a scalar variable just decides the magnitude. Force is
a vector quantity that has both magnitude and direction, as is common knowledge. Thus, Coulomb’s law is expressed as a vector.
Suppose there are two charges, q1 and q2, respectively. Since both charges have the same sign, the force operating between them will be repulsive. Let F12 be the force on the q1 charge caused by Q2
and F21 be the force on the q2 charge caused by Q1. The force acting on q2 as a result of q1 is opposite to and equal to the force on q2 as a result of q1. The third law of motion of Newton is thus
observed by Coulomb’s law.
F12 = -F21
Therefore, F12 = 1/4° x q1q2/d2 = -F21 represents the force acting in vector form
Coulomb’s law Example and Applications
Here are some examples of how Coulomb’s law is used in daily life:
• Magnets are drawn to one another because their magnetic poles are in opposition to one another. Coulomb’s law governs the force of attraction between two magnets.
• When a balloon is rubbed against your hair, it gets negatively charged due to the attraction between charged particles. If you bring the balloon close to another negatively charged object, the
other thing will be repelled if it is likewise negatively charged. This is due to Coulomb’s law, which states that because the negatively charged balloon and the negatively charged item have the
same sign of charge, they repel one another.
• Atomic stability: Protons and electrons, which are both positively and negatively charged, make up an atom. The electrostatic force, which holds the atom together, pulls the electrons towards the
protons. Coulomb’s law establishes the magnitude of the electric force between electrons and protons.
• Electrostatic precipitation: This method for removing dust and other airborne particles from the atmosphere. By putting the particles through an electric field, the particles are charged. After
that, a grounded plate attracts the charged particles, where they are subsequently gathered.
• Xerography: Xerography is a photocopying technique that imprints an image on paper using electrostatics. A paper is passed over a negatively charged drum, which allows part of the positive charge
from the document to transfer to the drum. Then a powder that is negatively charged is sprinkled onto the drum. The parts of the drum that have been charged by the document are where the powder
adheres. The powder is then melted and fused to the paper by heating the drum.
• Electrostatics are used by ink-jet printers to spray ink onto paper. Above the paper is a plate that is negatively charged. Ink droplets are positively charged after being passed via an electric
field. The negatively charged plate attracts the charged droplets, which are then sprayed onto the paper there.
• In order to apply a coating to a surface, electrostatic painting is a technique. A charged mist of the coating is sprayed onto the surface. The mist’s charged particles are drawn to the surface
and deposition occurs there.
Limitations of Coulomb’s law
The following are some of Coulomb’s law’s drawbacks:
• Coulomb’s law only applies to point charges, meaning that it assumes there is no spatial dimension to the charges. In fact, every charge has a limited size. This indicates that Coulomb’s law does
not perfectly anticipate the force between two charges.
• The consequences of polarization are not taken into consideration: A neutral object will become electrically charged through the process of polarisation when an electric field is present.
Polarization’s effects are not taken into consideration by Coulomb’s law. This means that if the items are polarised, the force between two charges may change slightly from what Coulomb’s law
• Coulomb’s law only applies to static charges and does not apply to moving charges. Magnetic fields are created by moving charges, and the force between moving charges is not just the sum of their
electrostatic forces.
• Coulomb’s law is invalid in extremely powerful electric fields, thus those don’t apply. Since quantum mechanics controls how charges behave in these fields, Coulomb’s law is insufficient to
adequately describe the force that exists between charges.
Exercise of Coulomb’s law Problems
Question 1: Two-point charges, q1 = +12 μC and q2 = 8μC are separated by a distance r = 20 cm. Calculate the magnitude of the electric force.
Answer: Here, q1 = +12 μC = 12 x 10^-6 C
q2 = 8μC = 8 x 10^-6C
r = 20 cm = 0.20m
According to Coulomb’s law, F = 1/4π€° x q1q2/d²
F = (9×10^9) (12 x 10^-6) (8 x 10^-6)/ (0.20)²
F = 25.92 N | {"url":"https://www.adda247.com/school/coulombs-law/","timestamp":"2024-11-12T03:11:11Z","content_type":"text/html","content_length":"650996","record_id":"<urn:uuid:ec336830-bf59-40d6-9d4d-669e190f28f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00203.warc.gz"} |
Countif not working for duplicate check in column
Has anyone been having issues with the countif formula to check for duplicates in a column? My output keeps showing zero even though there are 2 of the same numbers in one column.
Best Answer
• I have seen this issue when there are leading zeros. Try using an @cell reference.
=COUNTIF([Column Name]:[Column Name], @cell = [Column Name]@row)
• I have seen this issue when there are leading zeros. Try using an @cell reference.
=COUNTIF([Column Name]:[Column Name], @cell = [Column Name]@row)
• That is really weird! Any idea of the underlying cause of that? A bug?
There is nothing in the documentation for @cell to suggest it's use here, yet I just replicated Warren's issue and your fix. It's not just numbers with leading zeros, it's any number stored as
text - but text/number combinations work fine.
Ex. Without the @cell, the formula fails with '0420 and '5555, but not with 00Jeff.
All the red cells contain: =COUNTIF(TypeB:TypeB, TypeB@row). The yellow cells use @cell = TypeB@row.
Here's a weirder part: The failed formula doesn't always show a value of 0! I added the two numeric 8888 cells at the bottom, got the correct count of 2. Then I added the formula to the bottom
row and entered '8888, and it gave the cell a count of 2 (even though there are 3 cells with '8888 in the column.)
Then I entered the formula on the next line down, and entered '5555, and I got 4, just like I did for the numeric 5555 above. I entered another '5555 and another '8888, but neither count updated.
BUT THEN... I added another numeric 5555 and another numeric 8888, and guess what? The counts updated for '5555 and '8888! < What??? >
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• Now you are mixing data types. You have text and numerical. You would need to convert it all to text (so you can keep leading zeros and whatnot) and then your COUNTIFS should be working.
• I know why they shouldn't match. The weird thing is that it IS matching across data types - but only in one direction! The formula trying to match a number stored as text doesn't count the
matching number stored as text, but it does match the number stored as number. But vice versa, matching a number stored as number it ignores the same number stored as text.
(I don't have a particular issue to solve, I'm just trying to map out the rules here and see if I can understand why adding "@cell =" makes the formula able to match number stored as text to
number stored as text.)
To recap, the formula =COUNTIF(TypeB:TypeB, TypeB@row)
When TypeB@row is '8888, the formula ignores '8888, and counts 8888, which is backwards.
When TypeB@row is 8888, the formula counts 8888 and ignores '8888, as it should.
When TypeB@row is 001ABC, the formula counts 001ABC, as it should.
Jeff Reisman
Link: Smartsheet Functions Help Pages Link: Smartsheet Formula Error Messages
If my answer helped solve your issue, please mark it as accepted so that other users can find it later. Thanks!
• @Jeff Reisman I haven't really tested it much personally as I always just go with converting everything to text unless I know for sure everything will be numbers.
Have you checked to see which order the data is in? If text is the first value in the column vs if number is the first value in the column?
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Erratum to `Towards a homotopy theory of higher dimensional transition
Erratum to `Towards a homotopy theory of higher dimensional transition systems'
Philippe Gaucher
Counterexamples for Proposition~8.1 and Proposition~8.2 in the article Theor. Appl. Categ. 25(2011), pp 295-341 are given. They are used in the paper only to prove Corollary~8.3. A proof of this
corollary is given without them. The proof of the fibrancy of some cubical transition systems is fixed.
Keywords: higher dimensional transition system, locally presentable category, topological category, combinatorial model category, left determined model category, Bousfield localization, bisimulation
2010 MSC: 18C35,18G55,55U35,68Q85
Theory and Applications of Categories, Vol. 29, 2014, No. 2, pp 17-20.
Published 2014-03-05.
TAC Home | {"url":"http://www.tac.mta.ca/tac/volumes/29/2/29-02abs.html","timestamp":"2024-11-07T04:29:03Z","content_type":"text/html","content_length":"1669","record_id":"<urn:uuid:6866b752-4de3-4659-8712-9e213a9e997e>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00666.warc.gz"} |
Enable trac explicit sync for metacrs subprojects
Reported by: strk Owned by: strk
Priority: normal Milestone:
Component: SysAdmin/Trac Keywords:
Cc: rouault
post-commit SVN hook for the metacrs project only signals changes to the metacrs trac instance, no matter which subproject is being committed to.
This ticket is to make the post-commit hook smarter to distinguish which project has to be notified.
This is a spin-off of ticket #1910
Change History (8)
Metacrs subprojects are:
• geotiff
• csmap
• proj4j
• proj4js
I've tweaked the post-commit hook with this code (now under a local git repo):
MOD=`svnlook changed "${REPOS}" -r "${REV}" | head -1 | sed 's@.* \([^/]*\)/.*@\1@'`
if test "${MOD}" = "csmap" -o "${MOD} = "geotiff" -o "${MOD}" = "proj4j" -o "${MOD}" = "proj4js"; then
And disabled the per-request sync again.
Even, could you try again sending a dumb commit to geotiff please (and if it works to any of the others) ?
NOTE: if things go as I expect, the MetaCRS trac instance should also get *less* revisions notified too (not sure if this is something we want though)
There's a syntax issue in the hook :
$ svn commit -m "Dummy commit"
Envoi README
Transmission des données .done
Transaction de propagation...
Révision 2764 propagée.
Attention : post-commit hook failed (exit code 2) with output:
/var/www/svn/repos/metacrs/hooks/post-commit: 22: Syntax error: Unterminated quoted string
Can you please try again ?
Thanks, since you're at it could you also ask what they want to do with the "metacrs" trac instance itself ? Do they want to see *all* commits there (as it has been so far) or is it ok to only show
commits that are NOT part of any of the known submodules ?
Nobody cares about metacrs itself. It is just ... too meta ;-)
Resolution: → wontfix
Status: new → closed
I think this ticket is moot as I think most of the projects listed here have moved off trac and to github. So closing. | {"url":"https://trac.osgeo.org/osgeo/ticket/1911","timestamp":"2024-11-06T20:24:15Z","content_type":"text/html","content_length":"35563","record_id":"<urn:uuid:8b7d3cf2-7f54-40ff-b23c-44f8a22855b2>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00155.warc.gz"} |
How to write a fraction in Word ~ Tech Aisa
How to write a fraction in Word
How to write a fraction in Word: While editing text, there are times when you need to type fractions in Word to enter fraction formulas for mathematical, physical, or chemical expressions. But not
all users know how to make a fraction in Word.
In Word documents, not only plain text and numbers are used, formulas and symbols are also written there. A fraction in the Word is often used in materials devoted to mathematics, medicine, sports,
science, photography, technology, or even cooking, for example, when it is necessary to designate some value as part of a whole.
We were all taught in school how to write fractions by hand and how to apply them in real life. But how to write them in digital form, how to write a fraction in Word?
Microsoft Word is the most popular word processor used in various fields. Therefore, it makes sense for many users to learn how to enter fraction formulas in Word.
Using fractions in Word
As a rule, fractions are parts of a whole. We use fractions to represent data in more readable forms.
Use a few rules to help you decide when to use fractions:
• If you need to represent a fraction instead of the decimal form of some numbers, like ½ instead of 0.5.
• You are working on a document that includes dimension details. An example of this is text with a recipe where the quantities of the ingredients are in proportions.
These are some of the most common cases where fractions work best. Before using them in a particular document, you must determine if this format is really suitable for your content.
Usually, two types of display of fractions are used in documents: ordinary fractions in Word with a horizontal line between numbers (vertical division), and fractions with a slash (slash) that
separates numbers (horizontal division). These are simple vertical and simple diagonal fractions.
In this guide, we will look at several methods on how to write a fraction in Word:
• Enter fractions as plain text on one line.
• Using preformatted fraction symbols.
• Create a fraction using the fraction division slash symbol.
• Using the Equation tool to create a custom fraction.
We’ll look at how each of these approaches works below. From the instructions in this article, you will learn how to insert a fraction in Word in several ways.
How to write a fraction horizontally in Word
The easiest way to write fractions in Microsoft Word is to simply use a slash between the numerator and denominator, that is, between the two numbers that make up the fraction. Thus, you simply use
the syntactic “numerator, fraction bar in Word (/), denominator”, for example, 1/3, 2/3 as a representation of the fraction in the text content of the document.
Although this is an effective method that most users understand, such expressions are not strictly fractions because they are spelled incorrectly.
In this image, the fraction on the left is mistyped and the one on the right is mistyped.
Depending on the program settings, Word often automatically corrects the spelling of fractions, so if you type 1/2, 1/4, 3/4, the application will automatically replace them with ½, ¼, ¾ from the
insert character table.
If it doesn’t, you can enable this setting in the following way:
1. Go to the “File” menu by clicking “Options”.
2. In the Options window, open the Spelling tab.
3. Click the “AutoCorrect Options…” button.
4. In the AutoCorrect window, on the AutoFormat As You Type tab, check the box next to Fractions (1/2) with appropriate characters, and then click OK.
Note that this function only works for fractions that are in the caret symbol table.
After entering the expression, press the “Space” key, and the autocorrect for the correct type of fraction immediately takes place.
How to print a fraction in Word using symbols
Not all characters in MS Word are used for the auto-replace feature, for example, 1/3, 2/3, 1/5 will not turn into proper fractions. Therefore, you will need to convert such expressions into
fractions manually.
Do the following:
1. Open the “Insert” tab, and in the “Symbols” group, click on the “Symbol” button.
2. Select “More Symbols” from the drop-down menu.
3. In the “Symbol” window, in the “Symbols” tab in the “Set:” field, select the “numeric characters” option.
4. Here you will find some simple fractions ready for insertion.
5. You need to click on the appropriate symbol and then close the symbol table.
6. After performing this operation, the selected fraction sign will appear in the Word.
How to put a fraction in Word using code
You can insert fractions into a Word document using a special character code from Unicode. The standard includes 19 characters that can be used to write fractions in a Word document.
Unicode Fraction Integral Character Table.
¼ 00BC
½ 00BD
¾ 00BE
⅚ 215A
⅛ 215B
⅜ 215C
⅝ 215D
⅞ 215E
You will need to first enter the corresponding character code, consisting of numbers, and then, without indenting, press the key combination “Alt” + “X”. After that, the inserted characters will turn
into the corresponding fraction.
How to insert a fraction in Word horizontally using the equation box
Now we will look at a universal way to create a simple fraction using special fields. We will use the EQ field (Equation – equation) using the \f switch. This method creates a fraction with a
horizontal bar between the numerator and denominator.
Do the following:
1. Place the mouse cursor where you want to insert the fraction.
2. Press the “Ctrl” + “F9” keys to insert curly braces. On some laptops, you need to press “Ctrl” + “Fn” + “F9”.
3. In curly brackets, enter the following formula:
eq \f(numerator; denominator)
The EQ field creates a mathematical equation, followed by a space, then a function is output with the numerator and denominator separated by a semicolon.
You should get the following expression:
{ eq \f(numerator; denominator) }
4. Press the “F9” key (“Fn” + “F9”) to update the field.
5. As a result, you will get a fraction with the numerator and denominator separated by a horizontal division line.
Note that you are not limited to just the numbers in the numerator and denominator, if you want to create fractions using words you can do that too. Type words instead of numbers, or use both in
How to make fractions in Word using superscript and subscript characters
Now we will look at one of the ways to write custom fractions using superscript and subscript characters. With this method, by properly formatting the numbers before and after the slash, you get an
almost real fraction, like ^5 ∕ [6] , ^8 ∕ [9] .
Do the following:
1. Type the numbers as a fraction separated by a slash at the selected location in the document.
2. Select the first number (numerator).
3. On the Home tab, in the Font group, click the Superscript icon to convert this number to a superscript.
4. Select the second number (denominator) after the slash.
5. Click the Subscript icon to convert a number to a subscript.
How to make a fraction in Word using the “Equation” function
Now you will learn how to write a formula with a fraction in Word. Microsoft Word has an Equation tool that includes the ability to create a custom fraction that is separate from the rest of the text
in the document.
Go through the steps:
1. Go to the “Insert” tab.
2. In the Symbols group, click the Equation icon.
3. The Equation tab opens.
4. In the “Structures” group, click on “Fraction”.
5. Choose the right shot design.
In the field that appears, add the numerator and denominator, and then left-click to remove the borders.
This option is suitable if you are using a fraction as part of an equation or if you place it separately from the main text in your document. However, in other cases, the equation boxes don’t always
blend well with the surrounding text.
Inserting a fraction using a text box
In some cases, you need to insert a fraction at a specific location in a Word document, and not just in the center, right, or left. To do this, use the text field function.
Do the following:
1. Open the “Insert” tab.
2. In the “Text” group, click on the “Text Box” icon (In Word 2010, Word 2007 – “Inscription”).
3. Select “Simple Lettering”.
4. A text box frame appears on the document page from which you want to remove the selected content.
5. In the “Symbols” group, click on the “Equation” button to insert a suitable fraction into the text box.
6. Drag the frame to the desired location in the Word document.
Article Conclusions
In the process of typing in MS Word, the user may need to type fractions that can be used in documents of various kinds. You can use several methods to insert fractions into a Word document: using
the AutoCorrect feature, inserting symbols, entering character codes, using special fields, or using the equation function.
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Area to Ton Calculator - Online Calculators
Use our Powerful Area to Tons Calculator for your current construction projects! Enter the required data to calculate with our basic and advanced calculator. Furthermore, read our solved examples and
formulas for better understanding that how it works.
Area to Tons Calculator
Enter any 3 values to calculate the missing variable
The formula is:
$\text{TotalTons} = \frac{\text{Area} \times \left(\frac{\text{Depth}}{12}\right) \times \text{MaterialDensity}}{2000}$
Variable Meaning
Total Tons Total weight in tons
Area Area to be covered (in square feet)
Depth Depth or thickness of the material layer (in inches)
Material Density Density of the material (in pounds per cubic foot)
2000 Conversion factor from pounds to tons
Solved Examples:
Example 1:
• Area = 500 square feet
• Depth = 4 inches
• Material Density = 100 pounds per cubic foot
Calculation Instructions
Step 1: Total Tons = $\frac{\text{Area} \times \left(\frac{\text{Depth}}{12}\right) \times \text{MaterialDensity}}{2000}$ Start with the formula.
Step 2: Total Tons = $\frac{500 \times \left(\frac{4}{12}\right) \times 100}{2000}$ Replace the variables with the given values.
Step 3: Total Tons = $\frac{500 \times 0.3333 \times 100}{2000}$ Convert depth from inches to feet (4/12 = 0.3333).
Step 4: Total Tons = $\frac{16666.67}{2000}$ Multiply the area by the depth in feet and then by the material density.
Step 5: Total Tons = 8.33 tons Divide by 2000 to convert pounds to tons.
Answer: You need 8.33 tons of material.
Example 2:
• Area = 1000 square feet
• Depth = 6 inches
• Material Density = 120 pounds per cubic foot
Calculation Instructions
Step 1: Total Tons = $\frac{\text{Area} \times \left(\frac{\text{Depth}}{12}\right) \times \text{MaterialDensity}}{2000}$ Start with the formula.
Step 2: Total Tons = $\frac{1000 \times \left(\frac{6}{12}\right) \times 120}{2000}$ Replace the variables with the given values.
Step 3: Total Tons = $\frac{1000 \times 0.5 \times 120}{2000}$ Convert depth from inches to feet (6/12 = 0.5).
Step 4: Total Tons = $\frac{60000}{2000}$ Multiply the area by the depth in feet and then by the material density.
Step 5: Total Tons = 30 tons Divide by 2000 to convert pounds to tons.
Answer: You need 30 tons of material.
What is Area to Tons Calculator
The Area to Tons Calculator is a powerful tool that helps to calculate the total weight of a material required to cover a specific area at a given depth. This calculation is particularly useful in
construction and landscaping projects where materials like gravel, sand, or soil are involved.
The formula $\text{TotalTons} = \frac{\text{Area} \times \left(\frac{\text{Depth}}{12}\right) \times \text{MaterialDensity}}{2000}$ converts the area and depth into cubic feet, multiplies by the
material’s density to get the total weight in pounds, and then converts the weight into tons.
The Tons Calculator is very useful in construction, landscaping, farming, and digging jobs. By accurately measuring the weight of things needed for a certain area, you can make good choices, use
resources well, and finish projects successfully. | {"url":"https://areacalculators.com/area-to-ton-calculator/","timestamp":"2024-11-03T04:13:07Z","content_type":"text/html","content_length":"112748","record_id":"<urn:uuid:5c91162f-3bd4-4ccc-87d1-06e762d60981>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00189.warc.gz"} |
Elements of Geometry and Plane Trigonometry
Elements of Geometry and Plane Trigonometry: With an Appendix, and Copious Notes and Illustrations
Section 18 221
Section 19 225
Section 20 226
Section 21 243
Section 22 244
Section 23 256
Section 24 260
Section 25 262
Section 9 101
Section 10 125
Section 11 128
Section 12 155
Section 13 181
Section 14 183
Section 15 199
Section 16 200
Section 17 204
Section 26 268
Section 27 283
Section 28 311
Section 29 318
Section 30 338
Section 31 365
Section 32 371
Section 33 392
Popular passages
... if a straight line, &c. QED PROPOSITION 29. — Theorem. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal
to the interior and opposite upon the same side ; and likewise the two interior angles upon the same side together equal to two right angles.
The first of four magnitudes is said to have the same ratio to the second which the third has to the fourth, when any...
If a straight line meets two straight lines, so as to " make the two interior angles on the same side of it taken " together less than two right angles...
A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Componendo, by composition ; when there are four proportionals, and it is inferred that the first together with the second, is to the second, as the third together with the fourth, is to the fourth.
The angle at the centre of a circle is double of the angle at the circumference upon the same base, that is, upon the same part of the circumference.
Thus, for" example, he to whom the geometrical proposition, that the angles of a triangle are together equal to two right angles...
UPON a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle.
All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
The rectangle contained by the sum and difference of two straight lines is equivalent to the difference of the squares of these lines.
Bibliographic information | {"url":"https://books.google.com.jm/books?id=stU3AAAAMAAJ&dq=editions:UOM39015064332870","timestamp":"2024-11-12T18:35:43Z","content_type":"text/html","content_length":"65573","record_id":"<urn:uuid:d97cc1ee-dc15-4acf-b47c-dae0ce415474>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00190.warc.gz"} |
Hierarchical Gaussian Filter · RxInfer.jl
This example has been auto-generated from the examples/ folder at GitHub repository.
# Activate local environment, see `Project.toml`
import Pkg; Pkg.activate(".."); Pkg.instantiate();
In this demo the goal is to perform approximate variational Bayesian Inference for Univariate Hierarchical Gaussian Filter (HGF).
Simple HGF model can be defined as:
\[\begin{aligned} x^{(j)}_k & \sim \, \mathcal{N}(x^{(j)}_{k - 1}, f_k(x^{(j - 1)}_k)) \\ y_k & \sim \, \mathcal{N}(x^{(j)}_k, \tau_k) \end{aligned}\]
where $j$ is an index of layer in hierarchy, $k$ is a time step and $f_k$ is a variance activation function. RxInfer.jl export Gaussian Controlled Variance (GCV) node with $f_k = \exp(\kappa x + \
omega)$ variance activation function. By default the node uses Gauss-Hermite cubature with a prespecified number of approximation points in the cubature. In this demo we also show how we can change
the hyperparameters in different approximation methods (iin this case Gauss-Hermite cubature) with the help of metadata structures. Here how our model will look like with the GCV node:
\[\begin{aligned} z_k & \sim \, \mathcal{N}(z_{k - 1}, \mathcal{\tau_z}) \\ x_k & \sim \, \mathcal{N}(x_{k - 1}, \exp(\kappa z_k + \omega)) \\ y_k & \sim \, \mathcal{N}(x_k, \mathcal{\tau_y}) \end
In this experiment we will create a single time step of the graph and perform variational message passing filtering alrogithm to estimate hidden states of the system. For a more rigorous introduction
to Hierarchical Gaussian Filter we refer to Ismail Senoz, Online Message Passing-based Inference in the Hierarchical Gaussian Filter paper.
For simplicity we will consider $\tau_z$, $\tau_y$, $\kappa$ and $\omega$ known and fixed, but there are no principled limitations to make them random variables too.
To model this process in RxInfer, first, we start with importing all needed packages:
using RxInfer, BenchmarkTools, Random, Plots, StableRNGs
Next step, is to generate some synthetic data:
function generate_data(rng, n, k, w, zv, yv)
z_prev = 0.0
x_prev = 0.0
z = Vector{Float64}(undef, n)
v = Vector{Float64}(undef, n)
x = Vector{Float64}(undef, n)
y = Vector{Float64}(undef, n)
for i in 1:n
z[i] = rand(rng, Normal(z_prev, sqrt(zv)))
v[i] = exp(k * z[i] + w)
x[i] = rand(rng, Normal(x_prev, sqrt(v[i])))
y[i] = rand(rng, Normal(x[i], sqrt(yv)))
z_prev = z[i]
x_prev = x[i]
return z, x, y
generate_data (generic function with 1 method)
# Seed for reproducibility
seed = 42
rng = StableRNG(seed)
# Parameters of HGF process
real_k = 1.0
real_w = 0.0
z_variance = abs2(0.2)
y_variance = abs2(0.1)
# Number of observations
n = 300
z, x, y = generate_data(rng, n, real_k, real_w, z_variance, y_variance);
Let's plot our synthetic dataset. Lines represent our hidden states we want to estimate using noisy observations.
pz = plot(title = "Hidden States Z")
px = plot(title = "Hidden States X")
plot!(pz, 1:n, z, label = "z_i", color = :orange)
plot!(px, 1:n, x, label = "x_i", color = :green)
scatter!(px, 1:n, y, label = "y_i", color = :red, ms = 2, alpha = 0.2)
plot(pz, px, layout = @layout([ a; b ]))
To create a model we use the @model macro:
# We create a single-time step of corresponding state-space process to
# perform online learning (filtering)
@model function hgf(y, κ, ω, z_variance, y_variance, z_prev_mean, z_prev_var, x_prev_mean, x_prev_var)
z_prev ~ Normal(mean = z_prev_mean, variance = z_prev_var)
x_prev ~ Normal(mean = x_prev_mean, variance = x_prev_var)
# Higher layer is modelled as a random walk
z_next ~ Normal(mean = z_prev, variance = z_variance)
# Lower layer is modelled with `GCV` node
x_next ~ GCV(x_prev, z_next, κ, ω)
# Noisy observations
y ~ Normal(mean = x_next, variance = y_variance)
@constraints function hgfconstraints()
# Structuted factorization constraints
q(x_next, x_prev, z_next) = q(x_next)q(x_prev)q(z_next)
@meta function hgfmeta()
# Lets use 31 approximation points in the Gauss Hermite cubature approximation method
GCV() -> GCVMetadata(GaussHermiteCubature(31))
hgfmeta (generic function with 1 method)
The code below uses the infer function from RxInfer to generate the message passing algorithm given the model and constraints specification. We also specify the @autoupdates in order to set new
priors for the next observation based on posteriors.
function run_inference(data, real_k, real_w, z_variance, y_variance)
autoupdates = @autoupdates begin
# The posterior becomes the prior for the next time step
z_prev_mean, z_prev_var = mean_var(q(z_next))
x_prev_mean, x_prev_var = mean_var(q(x_next))
init = @initialization begin
q(x_next) = NormalMeanVariance(0.0, 5.0)
q(z_next) = NormalMeanVariance(0.0, 5.0)
return infer(
model = hgf(κ = real_k, ω = real_w, z_variance = z_variance, y_variance = y_variance),
constraints = hgfconstraints(),
meta = hgfmeta(),
data = (y = data, ),
autoupdates = autoupdates,
keephistory = length(data),
historyvars = (
x_next = KeepLast(),
z_next = KeepLast()
initialization = init,
iterations = 5,
free_energy = true,
run_inference (generic function with 1 method)
Everything is ready to run the algorithm. We used the online version of the algorithm, thus we need to fetch the history of the posterior estimation instead of the actual posteriors.
result = run_inference(y, real_k, real_w, z_variance, y_variance);
mz = result.history[:z_next];
mx = result.history[:x_next];
pz = plot(title = "Hidden States Z")
px = plot(title = "Hidden States X")
plot!(pz, 1:n, z, label = "z_i", color = :orange)
plot!(pz, 1:n, mean.(mz), ribbon = std.(mz), label = "estimated z_i", color = :teal)
plot!(px, 1:n, x, label = "x_i", color = :green)
plot!(px, 1:n, mean.(mx), ribbon = std.(mx), label = "estimated x_i", color = :violet)
plot(pz, px, layout = @layout([ a; b ]))
As we can see from our plot, estimated signal resembles closely to the real hidden states with small variance. We maybe also interested in the values for Bethe Free Energy functional:
plot(result.free_energy_history, label = "Bethe Free Energy")
As we can see BetheFreeEnergy converges nicely to a stable point. | {"url":"https://reactivebayes.github.io/RxInfer.jl/stable/examples/problem_specific/Hierarchical%20Gaussian%20Filter/","timestamp":"2024-11-14T07:59:16Z","content_type":"text/html","content_length":"23500","record_id":"<urn:uuid:66540017-51bd-4761-9155-222ae169604d>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00644.warc.gz"} |
Marilyn Burns on Conducting Math Interviews with K–5 Students - MARILYN BURNS MATH
Marilyn: I’ve been thinking about that question for a long time. In some ways there are similarities in that in both of the areas, in reading and math, we’re trying to help children learn to bring
meaning to black marks on paper. And so I’m astonished that when you read things, that literally black marks on paper will evoke huge emotional responses in me and hopefully in kids too. I’d like to
think the same in math without it being just negative responses.
But I think there are differences as well. Read is a verb. We want kids to read, but we don’t want kids to math. Reading requires the ability to decode, after which you have access to all the books
and the library in the world. I could even read Spanish. I don’t always understand, so the comprehension part is so important. So in reading, it’s accuracy and fluency and comprehension. And in math
it’s similar. We want kids to be accurate. We want them to be fluent, and we want them to understand. And in math world we don’t use the word comprehension. We use the word understanding. But
basically I think of them as the same thing.
But I think there’s a difference, and I’m not sure about this. In reading, the ability to decode depends on learning the system. Learning what the sounds are, having some skills to attack words,
having ability with sight words. And in a way that’s kind of social convention learning. You have to be taught that from an outside source. We’re not born with the ability to decode words on pages.
But in math it’s all based on logical structures. Kids come to school knowing that one and one is two. They learn the counting part. Those are words, but young kids count in kind of weird ways, but
they have an understanding of some of the mathematical relationships that they’ve encountered. They sometimes they have wrong ideas like my half is bigger than your half. But the ideas in math are
rooted in logic. So the logic and the learning about the symbols and the equations, the black marks, are kind of meshed together.
And I think in reading, I think kids who can read even accurately and fluently have no idea what they’re reading. So they’re similar and they’re different. I think the people who are devoted to
thinking hard about math and people who are devoted to thinking hard about reading, we can learn from each other. I’m not sure we always do.
Brett: Oh absolutely. Well with comprehension in mind, if a student in math has memorized math facts, how can that be misleading to a student’s proficiency?
Marilyn: We all memorized our times tables, and you really need to know them from memory if you want access to them. It’s really laborious to say, okay, 7×6, let’s figure out, 6×6, whatever you have
to go. You need to know 7×6=42 and it helps you reason mentally with numbers. It helps you make estimates. It’s a part of what you need to know. But without understanding, that kind of memorizing
isn’t going to serve a child very well.
So I suppose the same is true in reading. When you memorize the sight words and you’ll have had skills to get them, and you want that to be as fast as possible so you have more fluency, but the
difference in math, it’s different somehow.
It’s kind of hard to make sense of it, but I think that the understanding is always … The logical underpinnings are always there in all aspects of math. So for kids learning how to write numbers and
count without reasoning, it’s not going to work. It doesn’t work for them.
Brett: You had a couple of tweets go viral recently, and I’m going to paraphrase the two of them together here, but essentially one of them said, “Correct answers can hide confusion, while incorrect
answers can mask understanding.” What is happening there?
Marilyn: What I have seen on paper and pencil assignments and quizzes are correct answers that when I talk to a student about what they’re doing, it doesn’t seem to be any understanding. It’s kind of
like the dumb luck approach. I Interviewed a boy about comparing fractions and he said, “One-fourth and five-sixths,” he said, “Without question, five-sixths is greater.” That was my question, which
is greater. Now if that was on a worksheet where it would have said circle the greater fraction, he would have circled five-sixths. I wouldn’t have learned anything. Then I said to him, “Well, how’d
you decide?” He says, “Well, one is far away from four, but five is real close to six.” And I’m thinking like, “Whoa. That was interesting.” What I didn’t ask him as a followup, which I kind of wish
I did, what if I had asked him about five-sixths and six-sevenths? They’re each one away. How would he figure out which was the greater one there? There’s always a way to keep probing and pushing,
but I would have assumed from the worksheet that Michael understood comparing fractions, and I would have been woefully misinformed.
Brett: And that really gets to the power of interviewing. Why is interviewing so important?
Marilyn: Interviewing is important because I really get a window into how a child is thinking. I sometimes say to kids, when I’m interviewing them, “I’m going to ask you to explain your reasoning. If
I could, I would like to open up your head and look inside and say, ‘Oh, that’s how you’re thinking,’ but I can’t. So you’ll have to tell me. So I’m really interested because how you explain your
thinking will help me understand you better and that’ll make me a better teacher. So let’s go.”
So the interviewing is always about how did you figure that out? Even when a kid is right or especially when the student is right. I think in classroom, my early teaching you would ask a question and
a child would give an answer and then you’d say, “Are you sure about that?” Or, “Would you say that again?” It’s the hint like, “You’ve got it wrong kid. Go back to work.” So I’ve shifted that in
interviewing where we ask the kids how they think, whether they’re right or wrong. We don’t ever give any feedback about whether they’re right and wrong. We could talk about why that’s important.
But it’s a question of hearing how the student is making sense. Math is all about making sense. And I think that too often students think that math is about following the rules.
Brett: Well and you take in your interviewing process, you take the emphasis off of getting the right answer by focusing on thinking.
Marilyn: Well, yes, but I’m careful about that because the right answers are important. So sometimes people say, “Well, the answers aren’t important. It’s more important how they think.” No, the
answer is important. Well, what the answers are, are important because sometimes we ask kids for an estimate or sometimes you have to be accurate. I think knowing when you have to be accurate and
when an estimate would suffice is part of learning. But whatever the question I’m asking, the answer that I get is my first indication into what a student knows, but I don’t stop there. I’ve got to
find out what the student really knows. So the answers are important, but they’re not the Holy Grail. I’m not looking just for the answer. I’m looking for answers and explanations. Like in reading,
I’m looking at can a child read a passage accurately and fluently and can they understand what they read?
Brett: When we think about interviewing, someone might compare it to a running record, they might compare it to conferring. Interviewing is different.
Marilyn: I think it’s similar and different in both of those situations. Now I don’t have a lot of experience with running records, but I did go and spend some time with teachers to learn more about
running records when we were working on the math interviews. So in running records you are checking for accuracy. So are we in our interviews. We are checking for fluency. So are we in our
interviews. And you’re checking for comprehension, but the comprehension is related to the specific passage that the student has read. So you have the questions you’re going to ask.
In the interviews with math, I think it’s a little more open. When I say like I did to Michael, “Which is greater, one-fourth or five-sixths,” then I have to say, “How did you figure that out?” Or,
“How did you decide,” or, “How did you know?” And I don’t know what the answer is. With the running records, you kind of know what you’re listening for. I want to not pay attention to what I hope he
would say and listen to what he might say. So it’s similar but I think it may be subtle, but I think it’s a profound difference.
Brett: One of the things that’s very special about your interview style, you stay in the silence. Why is that so important?
Marilyn: I learned about wait time years ago, and it affected my teaching amazingly. Ask a question and then shut up. And I think as teachers we talk, we talk, we talk. And so if I stop and give kids
time to collect their thoughts, then they have time to collect their thoughts. Otherwise, I’m the one who’s doing all the learning because I’m doing all the talking. Whoever is doing the talking is
probably doing more learning than the person who’s listening because you’re working it out in your head.
So I’ve learned in my classroom teaching about wait time. I’ve learned about ways in the classroom to make that wait time productive time when kids are thinking. And so it seems natural in an
interview situation, especially when you watch the kid … You’re sitting face to face. You can watch them thinking. Their faces are scrunched up or they’re kind of subvocalizing or they’re tapping.
Whatever they’re doing, as long as the kid’s involved, why would I interrupt the child?
It’s fascinating. In the moment it’s not hard to do because you’re sort of curious about what’s this kid going to come up with. But sometimes I rewatched interviews on video tape and then we have
plenty of opportunities for teachers to see them. When a child sits for 20, 22 seconds, that is a long time. That’s the amount of time we’re supposed to be washing our hands that we’re worried about
the coronavirus. That’s the time that I learned yesterday is how long it takes to sing the Happy Birthday song twice. Let me tell you, that’s a long time to sit because I was interested in showing
teachers some of these examples of wait time and I’m thinking, “Well that would be the most boring professional learning session. It’s like watching paint dry.” But you’re not. You’re watching wheels
turn. I’ve become comfortable with silence.
Brett: And through Listening to Learn, you’re showing us just how vastly different every student processes, how every student thinks. Talk a little bit about how different each student can be.
Marilyn: So here’s a question that’s our first question in our interview three which focuses on adding and subtracting. And interview three is numbers within 20. Interview two is numbers within 10,
and then four and five, and the numbers get greater. So the first question is how much is 4+9? Well, the reason we selected that question is because when we started asking that question, we were
amazed at how many different ways kids came up to solving that problem. Ways that I had never even thought were possible. From the kid who starts with the four and has to count on, five, six and gets
all the way up maybe to 13 and that’s a risky strategy. So the kid who says, “Well look, I’m starting with a bigger number nine,” or a kid who says, “Well, you know, 4+9, 9+1=10.” Every time I think
I’ve heard it all, a kid comes up with it. There’s this girl says, “Okay, I take 4+4=8.” I’m thinking, “Well, all right, kids know their doubles.
Where’s she going with that?” So, “Eight and then I took that four out of the nine so there’s five left over. So eight and two more makes 10, and then I had three leftover and it was 13,” and it’s
like, “Go girl.” So I showed that clip to a lot of people and they say, “Well what do you do, because it’s not particularly efficient.” I said, “She’s eight years old. Every child has a right to be
eight for a whole year before they have to be nine.” That’s her thinking. She made sense of mathematics. I’m ecstatic.
Brett: So what is our takeaway from that in terms of what it means for classroom instruction?
Marilyn: One the things that I think many teachers are comfortable and excited about doing on number talks kind of mini-lesson discussions, 15 minutes at beginning of the class. One possible way for
number talks is you put out a problem on 4+9 and you’d say, “What are all the ways we can think about figuring out that answer in our heads?” And in our heads, it wouldn’t really, there’s nothing to
do on paper and pencil. You’re just going to have to count. So one of the things on number talk, put up the problem, 4+9, you might say, “How many different ways can we think about this?” And then I
want kids to hear and learn from one another. I want a community of math learners. So when the child does the counting on, I want to honor that. I want to put it up there in the board and say that’s
where Abel did it. But when the little girl Hanasis took out the 4+4, I want to say, “Oh, this is how Hanasis did it.”
Something happens in those number talks. I had a kid do something. It’s one in those areas of like, “Huh, I never thought about this.” He took the nine, he took one off the nine and he put it on the
four and he changed the problem from 4+9 to 5+8 and said, “That’s 13.” I’m thinking like, “All right. Yes, that’s right,” but I still to this day, don’t know. Did he know … Was 5+8 his friend and 4+9
wasn’t? It’s sort of like the fact that he was compensating in addition, seeing if I take one off one end and and put it on the other, the sum stays the same. He didn’t verbalize that generalization,
but that’s pretty sophisticated mathematically.
Brett: Why do we want students to develop number sense?
Marilyn: It’s part of life. I mean, I think about all the decisions that we make where we use arithmetic in our lives. What we have to add, subtract, multiply or divide to solve some problem that we
need. It could be keeping score when you’re playing a game. It could be buying wallpaper when I actually suggest that you overestimate, not underestimate because as soon as you run short, that dye
lot’s going to be gone. Understanding our finances, figuring out the tip in a restaurant.
There’s so many times when we’re in the supermarket, “I got $20, I don’t want to go over, I don’t have my credit card with me.” Keep track of where things are. We think about numbers all the time.
It’s just part of our world. So if we’ve done this with teachers where we ask them to list all the ways and we get long, long lists about it, about what they do and then you say, okay, if we sort
those and whether you figure it out with paper and pencil or you use a calculator or you do it in your head, how does that come out? And you know, most of the things in our daily life where we use
arithmetic, cooking, gardening, playing games, we do it in our head. You figure out, “Okay, what time do I have to leave to get to the movies? The Showtime is 7:15. Will the commute traffic been
done? What time do I leave?” Look at all the math you’re doing.
The biggest math problem of the year? What time do you put the turkey in the oven to have it ready for Thanksgiving? Because now they say so much a pound. So much a pound with the stuffing. Turkey
seems to cook more quickly these days, but what are you going to do? That gives people anxiety, but basically we think about math all the time. So people say, well, “Oh yeah, that’s why we should
learn fractions because of cooking.” I say, “Well if I’m doubling a recipe and it calls for two-thirds of a cup, then I just take two-thirds and two-thirds. If I’m halving a recipe and it calls for
three-fourths of a cup.” But then again it’s not really important. You kind of can measure it.
So I want kids to be playful with numbers. Think about numbers, find numbers useful to them. Not be afraid of numbers, be curious about numbers. There’s so much that I want math to be playful and
exploring and interesting and enjoyable.
Brett: And along with that, to you it’s very important that students see the meaning in all the math. You redefined your teaching to specifically get there. Why?
Marilyn: Yeah, well without meaning, what is it? I mean, math is such a weird thing. I majored in math for an embarrassing reason. I didn’t like writing. I figured you’d just had to do the problems.
I was kind of goody two-shoes in school, and I was good in math. I memorized my way through a lot of college math courses.
It didn’t work. Made me a better teacher. I know what it feels like to feel really dumb as the professor’s filling up the board and all of a sudden, “Uh-oh, I missed a step.” And then when you miss a
step, you’re out. And then I had to keep writing everything down that was going up on the board so I could try and figure it out later. So making sense is something that the kid does. That the source
of the understanding is in the kid’s head. My job is to stimulate the child with ways that they can create that understanding so they can bring meaning to math. What else is it about?
Brett: I mean selfishly as a student I would be that student that would say to the teacher, “Why are we doing this?”
Marilyn: Yes, and for many teachers it’s hard because they learn math by “Yours is not to question why, just invert and multiply,” and get threatened or annoyed at the kid who says that whereas you
get thrown. I mean, I get thrown by questions.
A kid, third grader was convinced that 90 was an odd number because it had a nine. And so I said, “Brayden,” I can still see his face, I said, “I’ll get back to you tomorrow. I got to think about
this.” I really wanted to think about why did he think that? Well, nine is an odd number and the 90 starts with nine, must be odd. So I could have said, “No, no, no. If the nine comes first, it
doesn’t count. The zero is an even number.” Then I realized many teachers I’ve asked, “Is zero even or odd?” And there are people who aren’t sure or maybe it’s neither because it’s kind of that pivot
number. So we all know what that flutter is. “Uh-oh. I don’t get it.”
Brett: You can see us thinking there, even in the question. I love that.
Marilyn: Kids are my laboratory. They never, never fail to amaze me and surprise me.
Brett: How did the role of mental math come in?
Marilyn: Well, that was part of the common sense thing when you realize that everything you do in life is mental math. Okay. Reading, writing, arithmetic. That bothered me so much as a child because
arithmetic didn’t start with R. Reading. Writing didn’t start with R, but it sounded like it. But it bothers me now for a different reason. It should be reading, writing and reasoning because
arithmetic is … I don’t want it to think of it as what we do with paper and pencil with following algorithmic procedures. Algorithms are important. They have an important place in mathematics, but I
want the reasoning to be the leader of it. So it’s the reasoning and that requires thinking mentally. But I think that as teachers, we’re under a lot of pressure to send home work.
And I know there are many teachers who give their kids their work packet at the beginning of the week, and they have all week to work on. And there’s a lot of practice, and it’s a good way for kids
to practice their skills. But what do you send home if you’re doing mental math all day? The parents are going to think you didn’t do any math. Especially if they say, “What’d you do in math today?”
And the kid says, “Nothing,” because all they did was talk and reason and exchange ideas. So I think that workbooks and textbook pages have really not encouraged teachers to give mental math the due
it deserves and requires.
Brett: How do students react to your interviewing? To the interviews themselves?
Marilyn: When we developed these, I often go and borrow a class and ask the teacher if I can interview some kids. I’ll go and introduce myself to the kids and explain my usual thing. “I can’t open up
your head. I can’t look inside,” the kids are a little nervous. And I’ve always said to the teacher, “Don’t give me your top kid. Don’t give me your kid who everybody knows is really having a
difficult time with it. Just give me a kid, a regular kid.” I take the kid out and they come back and everybody’s looking, and the kid comes back and we have fun. When do kids get all this attention?
This is a person they don’t even know and I’m friendly, and we’re going to talk about this and I’m interested and they come back. Sometimes I have kids come back, they say, “That was good.” Then
they’re all raising their hand, “Take me, take me.” I have never met a kid that didn’t want to be interviewed.
I remember one time a kid did not want to be videotaped and he whispered the whole thing, which I thought was a very creative solution to this avoiding my videotaping him. But I don’t know. We’ve
been teachers for a long time. You walk into a classroom and we know how to relate to children. We know how to deal with mobs of children, and we know how to deal with individual kids. So, “Come,
we’re going to have a conversation. Help me out here.”
Brett: Marilyn, you’re developing a new project called Listening to Learn. You call it a new digital interview tool. What can you share with us about it?
Marilyn: Well, I’ve been interviewing kids for years, and I want teachers to have the experience of really finding out what their students understand and encouraging them to do interviews. And the
reason that it’s a digital tool is important because what I’ve figured out is it’s hard to listen and learn. You have to learn to listen in order to listen to learn.
And that means really hearing what a student says and capturing that. But what am I going to do with that information? As I’ve thought about it, and it’s kind of embarrassing to think that it was
later in my career that I thought about, there are certain reasoning strategies that became evident to us as we interviewed kids that we want kids to be able to have access to. So if I think about
addition and subtraction as a content focus, which is separate from multiplication and division or then fractions and decimals, or even before that, some foundation. So if I focus on addition and
subtraction, I want kids to be able to decompose numbers. I want them to understand the inverse relationship between addition and subtraction. I want them to be able to break numbers apart and to
place value parts. I want them to use benchmark numbers.
I’m rattling these off, but we have the 10 reasoning strategies that we want all kids to have access to in order to be able to add and subtract mentally with numbers within 10, 20, 100 and 1000. The
same ten. This is not a lot. And once I figured those out I said, “Wow, this is pretty slick.” But when I ask a kid, “How much is 4+9,” the child’s answer that I want to help teachers is listen to
what the child said. And we have captured all the different ways, over 1000 kids that we’ve interviewed, of how they answered them and we give them, so this one is the most close to that. Then we
take, in the digital tool, which I couldn’t do otherwise, is the heavy lifting of mapping the kid’s explanation onto one of those 10 strategies I just said.
So I want all teachers to know the 10 strategies. I don’t want you to teach the strategies as if they’re separate isolated skills because something will always emerge as connected to more than one
strategy. If a kid says, “Well, 4+9 okay, I’m going to start with the nine and then I’m going to take one off the four, make it 10 so it’s 10+3,” the kid is reversing the two addends. So it’s
demonstrating applying the commutative property. The kid is using the benchmark number of 10. The kid is decomposing numbers within five. They’re doing so many things. We capture all that. So what
the digital tool can do, which has got me so excited, is it can tell you which of those strategies the student has demonstrated. And I can look across my class and say which of the strategies that
kids seem to have access to and which do I need to spend more time with? That is truly information that could inform my instructional decisions.
Brett: Well that was what I wanted to ask you next. What can a teacher do with this tool in their teaching?
Marilyn: So we were interviewing a third grade class at interviewing and we found out that none of the kids use the inverse operation. For example, 11-9. That’s within 20. A kid may count back 11,
10, nine, eight. It’s a risky countback strategy, but 11-9 like, “9+2=11.” That’s the inverse of, “I can use addition to solve a subtraction problem.” I have had kids, older kids, you ask 1000-998,
it should be slam dunk two. For some kids it isn’t. So imagine if you had that information about your entire class of who had chosen to use the inverse relationship to solve problems because it was
the most efficient.
Now, if a kid says to me, “1000-998. Okay, 1000-900=100. 100-90=10. 10-8=2.” A little long winded, but I’m impressed. But I want to know if they could have access, so we have several questions in
each interview, which should give kids the chance to show that they could use that. But suppose I have no kids demonstrating that or suppose in my class I’ve got all but a third of the kids aren’t.
So that tells me, “Okay, in my number talks, let’s talk about problems where it’s more efficient to subtract, to solve it or more efficient to add, to subtract the problem.” It tells me what I need
to have so that kids who are either developing understanding of that important strategy or cementing it or really can extend it to even more complex numbers, greater numbers.
I have to have that information, but I want a community of learners so that I can engage it. It helps me make intentional decisions about my instruction, and that has been the most stunning thing to
me. My instructional choices have become so much more intentional once I have the information from interviews.
Brett: And you’ve seen it strengthen the classroom community as well.
Marilyn: Oh gosh. Yes. I understand the need for differentiation and it’s hard. Yes, I understand the need for classroom community. I know how hard teaching is. Some years you get a class where
everybody’s sort of within a range. Some years you get a class where you have kids who are so disparate in two different groups. I’ve seen it all. They’re all tough, but at all times we are a
community of learners. And that’s when I need all the tools I have pedagogically.
I need to know how to do number talks. I need to know how to engage the kids in problem solving investigations that we can all learn from. I need to know how to give them independent work and truly
differentiate. I need time to confer with kids and help them. I need to do it all. They do that in reading and writing instruction. I need to do it in math, but it needs to be a community even when
kids are coming to it with their own way.
Brett: I’ve heard you say, “Math can be a way in for a teacher to really see their students.”
Marilyn: Yeah. I remember a principal in New York City told me that math was easier because you can make it concrete and it’s most astonishing thing when you’re sitting with a young child and we have
10 counters in front of us. I agree, the child agrees we have 10. And I say, “Watch what I’m going to do,” and I take one of the counters away and I say, “How many are there now?”
Some kids have to count them. Other kids will just say, “Well nine.” The difference between those two kids, I would not know that one kid understands that nine comes before ten, nine is nested in 10,
10 … There’s so much that happens. We think of counting as being simple, but it’s really pretty complex and without the ability to see it, I could actually see it there. That gives me understanding
about what that child does and doesn’t know.
So the child who isn’t sure probably needs some work with smaller quantities. I tweeted out recently a question that came to me one morning and it was like, if you look at the pips on a di e, if you
look at the arrangement of dots, you don’t have to count one, two, three, four, five. Well you don’t, but some kids do. It’s a brain function. You can see five so it’s subitizing. Is that the same as
sight words? And I tweeted that out. A bunch of people sent me some really interesting information to think about with this idea of subitizing. What do I see? What do I know? And I want teachers to
be curious about that with their kids.
So one of the things we suggest in Listening to Learn, we help teachers with the protocol of giving interviews. We interpret the students’ explanations and map them onto the strategies. And we also
provide some “what next” instructional suggestions. Teachers have their programs and they’re basically, those are their default roadmaps for instruction, but we give some suggestions. It’s common for
many teachers to do dot patterns where you show kids random dot patterns. Sometimes few enough to see if they will … You just sort of show-hide, take a quick look. But if you have a bunch of them
arranged in such a way, a quick look where you can’t subitize that many, but you could see perceptually a group of four and a group of two and you’d say, “Oh, that’s six.”
So it’s kind of like a conceptual subitizing rather than just perceptual subitizing. I mean, I’m amazed how complicated counting is, and I taught secondary math. I encourage secondary math teachers
to go down the grades and talk to the kids about what they understand. Things that we took so much for granted. When the professor in the math class said, “It is now obvious that,” I was the kid to
whom it was not obvious. | {"url":"https://marilynburnsmath.com/math-podcasts/marilyn-burns-on-conducting-math-interviews-with-k-5-students/","timestamp":"2024-11-14T12:18:29Z","content_type":"text/html","content_length":"130975","record_id":"<urn:uuid:d3d35809-1ad9-4cfe-b7a5-2aedd7ea211c>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00610.warc.gz"} |
CCD Continuous Counter-Current Decantation Cyanidation Flowsheet - 911Metallurgist
PROCESS FLOWSHEET DESCRIPTION: Continuous counter-current decantation cyanidation flowsheet.
ORE TREATED: Gold and silver ores amenable to the cyanidation process and where economics justify plant outlay.
ADVANTAGES: By producing the precious metals in bullion form, the highest net return is realized on many gold and silver ores. Where the tonnage available for treatment justifies the capital outlay,
and where this process gives high recovery with satisfactory chemical consumption, this flowsheet is recommended.
PROCESSING COMMENTS: The counter-current decantation washing circuit for removing the solutions carrying the dissolved precious metals is rapidly supplanting other methods. In this circuit, wash
water and barren solution are added in the last thickener units and flow toward the head of the plant, becoming enriched and are finally passed to the clarification and precipitation units where the
precious metals are precipitated and recovered. The ore pulp (carrying the dissolved gold and silver) flows in the opposite direction, or counter-current, and becomes depleted in soluble value until
finally discharged as a tailing with little or no contained soluble values.
Compared with the use of filters, lower capital cost, lower power cost, and lower labor cost more than offset the slightly higher loss in dissolved values, and improve net operating profits.
The Positive Type Washing Tray Thickener, requiring a small fraction of former floor and building space, is now replacing individual thickeners and allows the use of this attractive circuit in colder
sections where previously building and heating costs prevented this circuit from being considered.
Continuous Counter Current Decantation Calculations
The Problem with Continuous counter current decantation
Continuous counter current decantation calculations have always been a problem to the cyanide or chemical engineer because of the necessity of using simultaneous equations. These are tedious to
solve, involve considerable time, and there is always plenty of opportunity for mathematical error. There is really no fool proof short cut which can be applied to the average typical cyanide
circuit, so for reference we present a typical problem which may be used as a guide.
Note for the solution balance in the above flowsheet the barren solution from the precipitation plant is returned to next to the last washing thickener. This keeps loss of dissolved precious metal to
a minimum.
To scale up these 1955 numbers, just multiply by 9X and that will approximate 2016
Conditions assumed:
1. 100 tons ore per 24 hour ground in cyanide solution.
2. All thickener underflows discharged at 50% solids.
3. $20.00 value dissolved per ton of ore.
4. 75% dissolved in grinding circuit; balance in agitators.
5. 400 tons of solution precipitated to $0.015/ton.
6. Pulp agitated at 40% solids (1½ to 1).
7. Let V, W, X, Y and Z represent dollar value of solution discharged from each thickener.
Equating out and into each thickener we have:
• Thickener V—100V+400V=500W+(0.75X$20X 100 tons)
• Thickener W—100W+550W=500X+50W+(0.25X$20X100 tons)+100V
• Thickener X—100X+500X= 100W+500Y
• Thickener Y—100Y+500Y=100Z+100X +(400X $0.015)
• Thickener Z—100Z+100Z=100Y+100 tons water, value $0.00
1. V=W+3 V=$5.00444
2. W=X+1.6 W=$2.00444
3. X=Y+0.32 X=$0.40444
4. Y=0.2Z+0.076 Y=$0.08444
5. 2Z=Y Z=$0.04222
To check these figures:
From the foregoing the following results are deduced:
Assay value of pregnant solution. Value of V =$5.00444
Assay value of discharged solution. Value of Z =$0.04222
Loss of dissolved value per ton of ore =$0.04222
Dissolved value saved =99.8%
With an ore of the value as used in these calculations it would be desirable to increase the volume of solution to precipitation and thereby cut down on the dissolved loss. This extra solution from
thickener W would be added to the classifier overflow feeding into the primary thickener V. Only sufficient solution is added to the grinding circuit to meet the requirements of the grind necessary.
It is highly desirable to cut the solution loss to less than 2 cents per ton.
Calculations from the mechanical loss of cyanide:
Assumed conditions:
1. Neglect the cyanide consumption throughout the system
2. Strength of cyanide (NaCN) solution 1.00 lb. per ton
3. Let V, W, X, Y and Z represent the strength in lbs. of NaCN solution discharged from the respective thickeners.
Equating out and into each thickener:
1. V=1.0
2. 100W+550W=50W+100V+500X
3. 100X+500X=100W+500Y
4. 100Y+500Y=100Z+400V+100X
5. 100Z+100Z=100Y+100 tons H2O at no value.
Mechanical loss of cyanide per ton of ore=Z=0.4449 lbs.
While there have been certain short cut methods developed for calculation of solution values and losses, it is very important for the cyanide plant metallurgist to fully understand the application of
the simultaneous equation method as presented. One such short cut method is detailed in the AIME Paper entitled “Continuous Counter Current Decantation Calculations,” by T. B. Counselman, Volume 187,
February, 1950. Other sources of good information are in the “Handbook of Cyanidation,” by Hamilton and “Cyanidation Concentration of Gold and Silver Ores,” by Dorr and Bosqui.
In actual practice, the best procedure is sample the solution at various points in the system and assay for contained values. This also applies to determining the cyanide and lime strength which is
done by titration methods. The assumed conditions for the algebraic calculations are never quite met in practice, for the values due to a certain extent continue to dissolve in the primary, thicken
and may also continue on through to washing stages. This is particularly so with ores containing both gold and silver, for the latter usually dissolves at a slower rate.
Design of Counter Current Decantation in Copper Metallurgy by joseph kafumbila on Scribd | {"url":"https://www.911metallurgist.com/blog/ccd-continuous-counter-current-decantation-cyanidation/","timestamp":"2024-11-04T23:21:28Z","content_type":"text/html","content_length":"131935","record_id":"<urn:uuid:2512a747-90fa-4b84-8c9c-750d0770f6e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00895.warc.gz"} |
Travel optimization
Consider a two-dimensional world with no obstacles. You must go from point A to pointB, and then return to A. When traveling, you can either walk or take the train. There are only two train
stations, C and D, with trains going in both directions. On the outside, stations have a public timer displaying the time until the next train. When looking at the timer, you can decide to either
wait for the next train, or to walk to your destination. Assume that the time displayed when arriving at a station follows a uniform distribution from 0 to U, and that there is no correlation between
the schedule of the trains in the two directions. Suppose that it takes no time to enter or leave a train.
Given the positions of the four differents points A, B, C and D, the walking speed W, the train speed T, and the constant U, can you compute the minimum expected time to go fromA to B, and from B
to A, assuming a perfect strategy?
Input consists of several cases with real numbers, each with the coordinates of A, B, C andD, followed by W, T, and U. Assume T > W > 0 and U > 0.
For each case, print the expected time to go from A to B, and to go from B to A, both with four digits after the decimal point. The input cases have no precision issues. | {"url":"https://jutge.org/problems/P62842_en","timestamp":"2024-11-13T02:26:21Z","content_type":"text/html","content_length":"24638","record_id":"<urn:uuid:b648930b-2393-450d-9aad-3fc0b4d6bd44>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00265.warc.gz"} |
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Multi-normed spaces
Dissertationes Mathematicae 488 (2012), 1-165 MSC: Primary 43A10, 43A20; Secondary 46J10. DOI: 10.4064/dm488-0-1
We modify the very well known theory of normed spaces $\newcommand{\norm}{\Vert\cdot\Vert}(E, \norm)$ within functional analysis by considering a sequence $\newcommand{\norm}{\Vert\cdot\Vert}(\norm_n
: n\in\mathbb N)$ of norms, where $\newcommand{\norm}{\Vert\cdot\Vert}\norm_n$ is defined on the product space $E^n$ for each $n\in\mathbb N$.
Our theory is analogous to, but distinct from, an existing theory of `operator spaces'; it is designed to relate to general spaces $L^p$ for $p\in [1,\infty]$, and in particular to $L^1$-spaces,
rather than to $L^2$-spaces.
After recalling in Chapter 1 some results in functional analysis, especially in Banach space, Hilbert space, Banach algebra, and Banach lattice theory, that we shall use, we shall present in Chapter
2 our axiomatic definition of a `multi-normed space' $\newcommand{\norm}{\Vert\cdot\Vert}((E^n, \norm_n) : n\in \mathbb N)$, where $\newcommand{\norm}{\Vert\cdot\Vert}(E, \norm)$ is a normed space.
Several different, equivalent, characterizations of multi-normed spaces are given, some involving the theory of tensor products; key examples of multi-norms are the minimum, maximum, and $(p,q)
$-multi-norms based on a given space. Multi-norms measure `geometrical features' of normed spaces, in particular by considering their `rate of growth'. There is a strong connection between
multi-normed spaces and the theory of absolutely summing operators.
A substantial number of examples of multi-norms will be presented.
Following the pattern of standard presentations of the foundations of functional analysis, we consider generalizations to `multi-topological linear spaces' through `multi-null sequences', and to
`multi-bounded' linear operators, which are exactly the `multi-continuous' operators. We define a new Banach space ${\mathcal M}(E,F)$ of multi-bounded operators, and show that it generalizes
well-known spaces, especially in the theory of Banach lattices.
We conclude with a theory of `orthogonal decompositions' of a normed space with respect to a multi-norm, and apply this to construct a `multi-dual' space.
Applications of this theory will be presented elsewhere. | {"url":"https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/all/488/0/88110/multi-normed-spaces","timestamp":"2024-11-04T20:20:05Z","content_type":"text/html","content_length":"48101","record_id":"<urn:uuid:6a7b0426-3bde-4c8e-87f9-a1b443939297>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00667.warc.gz"} |
The Quantum Bit
The best place to start our journey through quantum computing is to recall how classical computing works and try to extend it. Since our final quantum computing model will be a circuit model, we
should informally discuss circuits first.
A circuit has three parts: the “inputs,” which are bits (either zero or one); the “gates,” which represent the lowest-level computations we perform on bits; and the “wires,” which connect the outputs
of gates to the inputs of other gates. Typically the gates have one or two input bits and one output bit, and they correspond to some logical operation like AND, NOT, or XOR.
A simple example of a circuit.
If we want to come up with a different model of computing, we could start regular circuits and generalize some or all of these pieces. Indeed, in our motivational post we saw a glimpse of a
probabilistic model of computation, where instead of the inputs being bits they were probabilities in a probability distribution, and instead of the gates being simple boolean functions they were
linear maps that preserved probability distributions (we called such a matrix “stochastic”).
Rather than go through that whole train of thought again let’s just jump into the definitions for the quantum setting. In case you missed last time, our goal is to avoid as much physics as possible
and frame everything purely in terms of linear algebra.
Qubits are Unit Vectors
The generalization of a bit is simple: it’s a unit vector in $ \mathbb{C}^2$. That is, our most atomic unit of data is a vector $ (a,b)$ with the constraints that $ a,b$ are complex numbers and $ |a|
^2 + |b|^2 = 1$. We call such a vector a qubit.
A qubit can assume “binary” values much like a regular bit, because you could pick two distinguished unit vectors, like $ (1,0)$ and $ (0,1)$, and call one “zero” and the other “one.” Obviously there
are many more possible unit vectors, such as $ \frac{1}{\sqrt{2}}(1, 1)$ and $ (-i,0)$. But before we go romping about with what qubits can do, we need to understand how we can extract information
from a qubit. The definitions we make here will motivate a lot of the rest of what we do, and is in my opinion one of the major hurdles to becoming comfortable with quantum computing.
A bittersweet fact of life is that bits are comforting. They can be zero or one, you can create them and change them and read them whenever you want without an existential crisis. The same is not
true of qubits. This is a large part of what makes quantum computing so weird: you can’t just read the information in a qubit! Before we say why, notice that the coefficients in a qubit are complex
numbers, so being able to read them exactly would potentially encode an infinite amount of information (in the infinite binary expansion)! Not only would this be an undesirably powerful property of a
circuit, but physicists’ experiments tell us it’s not possible either.
So as we’ll see when we get to some algorithms, the main difficulty in getting useful quantum algorithms is not necessarily figuring out how to compute what you want to compute, it’s figuring out how
to tease useful information out of the qubits that otherwise directly contain what you want. And the reason it’s so hard is that when you read a qubit, most of the information in the qubit is
destroyed. And what you get to see is only a small piece of the information available. Here is the simplest example of that phenomenon, which is called the measurement in the computational basis.
Definition: Let $ v = (a,b) \in \mathbb{C}^2$ be a qubit. Call the standard basis vectors $ e_0 = (1,0), e_1 = (0,1)$ the computational basis of $ \mathbb{C}^2$. The process of measuring $ v$ in the
computational basis consists of two parts.
1. You observe (get as output) a random choice of $ e_0$ or $ e_1$. The probability of getting $ e_0$ is $ |a|^2$, and the probability of getting $ e_1$ is $ |b|^2$.
2. As a side effect, the qubit $ v$ instantaneously becomes whatever state was observed in 1. This is often called a collapse of the waveform by physicists.
There are more sophisticated ways to measure, and more sophisticated ways to express the process of measurement, but we’ll cover those when we need them. For now this is it.
Why is this so painful? Because if you wanted to try to estimate the probabilities $ |a|^2$ or $ |b|^2$, not only would you get an estimate at best, but you’d have to repeat whatever computation
prepared $ v$ for measurement over and over again until you get an estimate you’re satisfied with. In fact, we’ll see situations like this, where we actually have a perfect representation of the data
we need to solve our problem, but we just can’t get at it because the measurement process destroys it once we measure.
Before we can talk about those algorithms we need to see how we’re allowed to manipulate qubits. As we said before, we use unitary matrices to preserve unit vectors, so let’s recall those and make
everything more precise.
Qubit Mappings are Unitary Matrices
Suppose $ v = (a,b) \in \mathbb{C}^2$ is a qubit. If we are to have any mapping between vector spaces, it had better be a linear map, and the linear maps that send unit vectors to unit vectors are
called unitary matrices. An equivalent definition that seems a bit stronger is:
Definition: A linear map $ \mathbb{C}^2 \to \mathbb{C}^2$ is called unitary if it preserves the inner product on $ \mathbb{C}^2$.
Let’s remember the inner product on $ \mathbb{C}^n$ is defined by $ \left \langle v,w \right \rangle = \sum_{i=1}^n v_i \overline{w_i}$ and has some useful properties.
• The square norm of a vector is $ \left \| v \right \|^2 = \left \langle v,v \right \rangle$.
• Swapping the coordinates of the complex inner product conjugates the result: $ \left \langle v,w \right \rangle = \overline{\left \langle w,v \right \rangle}$
• The complex inner product is a linear map if you fix the second coordinate, and a conjugate-linear map if you fix the first. That is, $ \left \langle au+v, w \right \rangle = a \left \langle u, w
\right \rangle + \left \langle v, w \right \rangle$ and $ \left \langle u, aw + v \right \rangle = \overline{a} \left \langle u, w \right \rangle + \left \langle u,v \right \rangle$
By the first bullet, it makes sense to require unitary matrices to preserve the inner product instead of just the norm, though the two are equivalent (see the derivation on page 2 of these notes). We
can obviously generalize unitary matrices to any complex vector space, and unitary matrices have some nice properties. In particular, if $ U$ is a unitary matrix then the important property is that
the columns (and rows) of $ U$ form an orthonormal basis. As an immediate result, if we take the product $ U\overline{U}^\text{T}$, which is just the matrix of all possible inner products of columns
of $ U$, we get the identity matrix. This means that unitary matrices are invertible and their inverse is $ \overline{U}^\text{T}$.
Already we have one interesting philosophical tidbit. Any unitary transformation of a qubit is reversible because all unitary matrices are invertible. Apparently the only non-reversible thing we’ve
seen so far is measurement.
Recall that $ \overline{U}^\text{T}$ is the conjugate transpose of the matrix, which I’ll often write as $ U^*$. Note that there is a way to define $ U^*$ without appealing to matrices: it is a
notion called the adjoint, which is that linear map $ U^*$ such that $ \left \langle Uv, w \right \rangle = \left \langle v, U^*w \right \rangle$ for all $ v,w$. Also recall that “unitary matrix” for
complex vector spaces means precisely the same thing as “orthogonal matrix” does for real numbers. The only difference is the inner product being used (indeed, if the complex matrix happens to have
real entries, then orthogonal matrix and unitary matrix mean the same thing).
Definition: A single qubit gate is a unitary matrix $ \mathbb{C}^2 \to \mathbb{C}^2$.
So enough with the properties and definitions, let’s see some examples. For all of these examples we’ll fix the basis to the computational basis $ e_0, e_1$. One very important, but still very simple
example of a single qubit gate is the Hadamard gate. This is the unitary map given by the matrix
$ \displaystyle \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\\ 1 & -1 \end{pmatrix}$
It’s so important because if you apply it to a basis vector, say, $ e_0 = (1,0)$, you get a uniform linear combination $ \frac{1}{\sqrt{2}}(e_1 + e_2)$. One simple use of this is to allow for
unbiased coin flips, and as readers of this blog know unbiased coins can efficiently simulate biased coins. But it has many other uses we’ll touch on as they come.
Just to give another example, the quantum NOT gate, often called a Pauli X gate, is the following matrix
$ \displaystyle \begin{pmatrix} 0 & 1 \\\ 1 & 0 \end{pmatrix}$
It’s called this because, if we consider $ e_0$ to be the “zero” bit and $ e_1$ to be “one,” then this mapping swaps the two. In general, it takes $ (a,b)$ to $ (b,a)$.
As the reader can probably imagine by the suggestive comparison with classical operations, quantum circuits can do everything that classical circuits can do. We’ll save the proof for a future post,
but if we want to do some kind of “quantum AND” operation, we get an obvious question. How do you perform an operation that involves multiple qubits? The short answer is: you represent a collection
of bits by their tensor product, and apply a unitary matrix to that tensor.
We’ll go into more detail on this next time, and in the mean time we suggest checking out this blog’s primer on the tensor product. Until then! | {"url":"https://www.jeremykun.com/2014/12/15/the-quantum-bit/","timestamp":"2024-11-02T03:12:42Z","content_type":"text/html","content_length":"19704","record_id":"<urn:uuid:5c9b208d-d124-49c9-8400-1f17ceb86ebf>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00298.warc.gz"} |
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Verifiable Random Functions (VRFs)
Internet-Draft VRF November 2021
Goldberg, et al. Expires 21 May 2022 [Page]
Intended Status:
Standards Track
Verifiable Random Functions (VRFs)
A Verifiable Random Function (VRF) is the public-key version of a keyed cryptographic hash. Only the holder of the private key can compute the hash, but anyone with public key can verify the
correctness of the hash. VRFs are useful for preventing enumeration of hash-based data structures. This document specifies several VRF constructions that are secure in the cryptographic random oracle
model. One VRF uses RSA and the other VRF uses Elliptic Curves (EC).¶
This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶
Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current
Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.¶
Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as
reference material or to cite them other than as "work in progress."¶
This Internet-Draft will expire on 21 May 2022.¶
Copyright (c) 2021 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document.
Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Simplified BSD License
text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License.¶
A Verifiable Random Function (VRF) [MRV99] is the public-key version of a keyed cryptographic hash. Only the holder of the private VRF key can compute the hash, but anyone with corresponding public
key can verify the correctness of the hash.¶
A key application of the VRF is to provide privacy against offline enumeration (e.g. dictionary attacks) on data stored in a hash-based data structure. In this application, a Prover holds the VRF
private key and uses the VRF hashing to construct a hash-based data structure on the input data. Due to the nature of the VRF, only the Prover can answer queries about whether or not some data is
stored in the data structure. Anyone who knows the public VRF key can verify that the Prover has answered the queries correctly. However, no offline inferences (i.e. inferences without querying the
Prover) can be made about the data stored in the data structure.¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119].¶
The following terminology is used through this document:¶
alpha or alpha_string:
beta or beta_string:
pi or pi_string:
A VRF comes with a key generation algorithm that generates a public VRF key PK and private VRF key SK.¶
The prover hashes an input alpha using the private VRF key SK to obtain a VRF hash output beta¶
The VRF_hash algorithm is deterministic, in the sense that it always produces the same output beta given the same pair of inputs (SK, alpha). The prover also uses the private key SK to construct a
proof pi that beta is the correct hash output¶
The VRFs defined in this document allow anyone to deterministically obtain the VRF hash output beta directly from the proof value pi by using the function VRF_proof_to_hash:¶
Thus, for VRFs defined in this document, VRF_hash is defined as¶
and therefore this document will specify VRF_prove and VRF_proof_to_hash rather than VRF_hash.¶
The proof pi allows a Verifier holding the public key PK to verify that beta is the correct VRF hash of input alpha under key PK. Thus, the VRF also comes with an algorithm¶
that outputs (VALID, beta = VRF_proof_to_hash(pi)) if pi is valid, and INVALID otherwise.¶
VRFs are designed to ensure the following security properties.¶
Uniqueness means that, for any fixed public VRF key and for any input alpha, there is a unique VRF output beta that can be proved to be valid. Uniqueness must hold even for an adversarial Prover that
knows the VRF private key SK.¶
More precisely, "full uniqueness" states that a computationally-bounded adversary cannot choose a VRF public key PK, a VRF input alpha, and two proofs pi1 and pi2 such that VRF_verify(PK, alpha, pi1)
outputs (VALID, beta1), VRF_verify(PK, alpha, pi2) outputs (VALID, beta2), and beta1 is not equal to beta2.¶
For many applications, a slightly weaker security property called "trusted uniqueness" suffices. Trusted uniqueness is the same as full uniqueness, but it is guaranteed to hold only if the VRF keys
PK and SK were generated in a trustworthy manner.¶
As further discussed in Section 7.1.1, some VRFs specified in this document satisfy only trusted uniqueness, while others satisfy full uniqueness. VRFs in this document that satisfy only trusted
uniqueness but not full uniqueness MUST NOT be used if the key generation process cannot be trusted.¶
Like any cryptographic hash function, VRFs need to be collision resistant. Collison resistance must hold even for an adversarial Prover that knows the VRF private key SK.¶
More precisely, "full collision resistance" states that it should be computationally infeasible for an adversary to find two distinct VRF inputs alpha1 and alpha2 that have the same VRF hash beta,
even if that adversary knows the private VRF key SK.¶
For many applications, a slightly weaker security property called "trusted collision resistance" suffices. Trusted collision resistance is the same as collision resistance, but it is guaranteed to
hold only if the VRF keys PK and SK were generated in a trustworthy manner.¶
As further discussed in Section 7.1.1, some VRFs specified in this document satisfy only trusted collision resistance, while others satisfy full collision resistance. VRFs in this document that
satisfy only trusted collision resistance but not full collision resistance MUST NOT be used if the key generation process cannot be trusted.¶
Pseudorandomness ensures that when an adversarial Verifier sees a VRF hash output beta without its corresponding VRF proof pi, then beta is indistinguishable from a random value.¶
More precisely, suppose the public and private VRF keys (PK, SK) were generated in a trustworthy manner. Pseudorandomness ensures that the VRF hash output beta (without its corresponding VRF proof
pi) on any adversarially-chosen "target" VRF input alpha looks indistinguishable from random for any computationally bounded adversary who does not know the private VRF key SK. This holds even if the
adversary also gets to choose other VRF inputs alpha' and observe their corresponding VRF hash outputs beta' and proofs pi'.¶
With "full pseudorandomness", the adversary is allowed to choose the "target" VRF input alpha at any time, even after it observes VRF outputs beta' and proofs pi' on a variety of chosen inputs
"Selective pseudorandomness" is a weaker security property which suffices in many applications. Here, the adversary must choose the target VRF input alpha independently of the public VRF key PK, and
before it observes VRF outputs beta' and proofs pi' on inputs alpha' of its choice.¶
As further discussed in Section 7.2, VRFs specified in this document satisfy both full pseudorandomness and selective pseudorandomness, but their quantitative security against the selective
pseudorandomness attack is stronger.¶
It is important to remember that the VRF output beta does not look random to the Prover, or to any other party that knows the private VRF key SK! Such a party can easily distinguish beta from a
random value by comparing beta to the result of VRF_hash(SK, alpha).¶
Also, the VRF output beta does not look random to any party that knows the valid VRF proof pi corresponding to the VRF input alpha, even if this party does not know the private VRF key SK. Such a
party can easily distinguish beta from a random value by checking whether VRF_verify(PK, alpha, pi) returns (VALID, beta).¶
Also, the VRF output beta may not look random if VRF key generation was not done in a trustworthy fashion. (For example, if VRF keys were generated with bad randomness.)¶
As explained in Section 3.3, pseudorandomness is guaranteed only if the VRF keys were generated in a trustworthy fashion. For instance, if an adversary outputs VRF keys that are deterministically
generated (or hard-coded and publicly known), then the outputs are easily derived by anyone and are therefore not pseudorandom.¶
There is, however, a different type of unpredictability that is desirable in certain VRF applications (such as [GHMVZ17] and [DGKR18]). This property is similar to the unpredictability achieved by an
(ordinary, unkeyed) cryptographic hash function: if the input has enough entropy (i.e., cannot be predicted), then the correct output is indistinguishable from uniform.¶
A formal definition of this property appears in Section 3.2 of [DGKR18]. The VRF schemes presented in this specification are believed to satisfy this property if the public key was generated in a
trustworthy manner. Additionally, the ECVRF is believed to also satisfy this property even if the public key was not generated in a trustworthy manner, as long as the public key satisfies the key
validation procedure in Section 5.6.¶
The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in Section 3. Its security
follows from the standard RSA assumption in the random oracle model. Formal security proofs are in [PWHVNRG17].¶
The VRF computes the proof pi as a deterministic RSA signature on input alpha using the RSA Full Domain Hash Algorithm [RFC8017] parametrized with the selected hash algorithm. RSA signature
verification is used to verify the correctness of the proof. The VRF hash output beta is simply obtained by hashing the proof pi with the selected hash algorithm.¶
The key pair for RSA-FDH-VRF MUST be generated in a way that it satisfies the conditions specified in Section 3 of [RFC8017].¶
In this section, the notation from [RFC8017] is used.¶
Parameters used:¶
• (n, e) - RSA public key¶
• K - RSA private key¶
• k - length in octets of the RSA modulus n (k must be less than 2^32)¶
Fixed options:¶
• Hash - cryptographic hash function¶
• hLen - output length in octets of hash function Hash¶
Primitives used:¶
RSAFDHVRF_prove(K, alpha_string)¶
• K - RSA private key¶
• alpha_string - VRF hash input, an octet string¶
• pi_string - proof, an octet string of length k¶
• pi_string - proof, an octet string of length k¶
• beta_string - VRF hash output, an octet string of length hLen¶
Important note:¶
• RSAFDHVRF_proof_to_hash should be run only on pi_string that is known to have been produced by RSAFDHVRF_prove, or from within RSAFDHVRF_verify as specified in Section 4.3.¶
1. two_string = 0x02¶
2. beta_string = Hash(two_string || pi_string)¶
3. Output beta_string¶
RSAFDHVRF_verify((n, e), alpha_string, pi_string)¶
• (n, e) - RSA public key¶
• alpha_string - VRF hash input, an octet string¶
• pi_string - proof to be verified, an octet string of length n¶
• ("VALID", beta_string), where beta_string is the VRF hash output, an octet string of length hLen; or¶
The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that satisfies the trusted uniqueness, trusted collision resistance, and full pseudorandomness properties defined in Section 3. The
security of this VRF follows from the decisional Diffie-Hellman (DDH) assumption in the random oracle model. Formal security proofs are in [PWHVNRG17].¶
To additionally satisfy "full uniqueness" and "full collision resistance", the Verifier MUST additionally perform the validation procedure specified in Section 5.6 upon receipt of the public VRF key.
Notation used:¶
Fixed options (specified in Section 5.5):¶
Type conversions (specified in Section 5.5):¶
Parameters used (the generation of these parameters is specified in Section 5.5):¶
ECVRF_prove(SK, alpha_string)¶
• SK - VRF private key¶
• alpha_string = input alpha, an octet string¶
• pi_string - VRF proof, octet string of length ptLen+n+qLen¶
1. Use SK to derive the VRF secret scalar x and the VRF public key Y = x*B¶
(this derivation depends on the ciphersuite, as per Section 5.5;¶
these values can be cached, for example, after key generation, and need not be rederived each time)¶
2. H = ECVRF_hash_to_curve(Y, alpha_string)¶
3. h_string = point_to_string(H)¶
4. Gamma = x*H¶
5. k = ECVRF_nonce_generation(SK, h_string)¶
6. c = ECVRF_hash_points(H, Gamma, k*B, k*H) (see Section 5.4.3)¶
7. s = (k + c*x) mod q¶
8. pi_string = point_to_string(Gamma) || int_to_string(c, n) || int_to_string(s, qLen)¶
9. Output pi_string¶
• pi_string - VRF proof, octet string of length ptLen+n+qLen¶
• "INVALID", or¶
• beta_string - VRF hash output, octet string of length hLen¶
Important note:¶
• ECVRF_proof_to_hash should be run only on pi_string that is known to have been produced by ECVRF_prove, or from within ECVRF_verify as specified in Section 5.3.¶
ECVRF_verify(Y, pi_string, alpha_string)¶
• Y - public key, an EC point¶
• pi_string - VRF proof, octet string of length ptLen+n+qLen¶
• alpha_string - VRF input, octet string¶
• ("VALID", beta_string), where beta_string is the VRF hash output, octet string of length hLen; or¶
The ECVRF_hash_to_curve algorithm takes in the VRF input alpha and converts it to H, an EC point in G. This algorithm is the only place the VRF input alpha is used for proving and verifying. See
Section 7.6 for further discussion.¶
This section specifies a number of such algorithms, which are not compatible with each other. The choice of a particular algorithm from the options specified in this section is made in Section 5.5.¶
The following ECVRF_hash_to_curve_try_and_increment(Y, alpha_string) algorithm implements ECVRF_hash_to_curve in a simple and generic way that works for any elliptic curve.¶
The running time of this algorithm depends on alpha_string. For the ciphersuites specified in Section 5.5, this algorithm is expected to find a valid curve point after approximately two attempts
(i.e., when ctr=1) on average.¶
However, because the running time of algorithm depends on alpha_string, this algorithm SHOULD be avoided in applications where it is important that the VRF input alpha remain secret.¶
ECVRF_hash_to_try_and_increment(Y, alpha_string)¶
• Y - public key, an EC point¶
• alpha_string - value to be hashed, an octet string¶
• H - hashed value, a finite EC point in G¶
Fixed option (specified in Section 5.5):¶
• arbitrary_string_to_point - conversion of an arbitrary octet string to an EC point.¶
1. ctr = 0¶
2. PK_string = point_to_string(Y)¶
3. one_string = 0x01¶
4. zero_string = 0x00¶
5. H = "INVALID"¶
6. While H is "INVALID" or H is the identity element of the elliptic curve group:¶
7. Output H¶
The ECVRF_hash_to_curve_h2c_suite(Y, alpha_string) algorithm implements ECVRF_hash_to_curve using one of the several hash-to-curve options defined in [I-D.irtf-cfrg-hash-to-curve]. The specific
choice of the hash-to-curve option (called Suite ID in [I-D.irtf-cfrg-hash-to-curve]) is given by the h2c_suite_ID_string parameter.¶
ECVRF_hash_to_curve_h2c_suite(Y, alpha_string)¶
Fixed option (specified in Section 5.5):¶
The encode function is provided by the hash-to-curve suite whose ID is h2c_suite_ID_string, as specified in [I-D.irtf-cfrg-hash-to-curve], Section 8. The domain separation tag DST, a parameter to the
hash-to-curve suite, SHALL be set to¶
where "ECVRF_" is represented as a 6-byte ASCII encoding (in hexadecimal, octets 45 43 56 52 46 5F).¶
The following algorithms generate the nonce value k in a deterministic pseudorandom fashion. This section specifies a number of such algorithms, which are not compatible with each other. The choice
of a particular algorithm from the options specified in this section is made in Section 5.5.¶
ECVRF_nonce_generation_RFC6979(SK, h_string)¶
• SK - an ECVRF secret key¶
• h_string - an octet string¶
• k - an integer between 1 and q-1¶
The ECVRF_nonce_generation function is as specified in [RFC6979] Section 3.2 where¶
The following is from Steps 2-3 of Section 5.1.6 in [RFC8032].¶
ECVRF_nonce_generation_RFC8032(SK, h_string)¶
• SK - an ECVRF secret key¶
• h_string - an octet string¶
• k - an integer between 0 and q-1¶
ECVRF_hash_points(P1, P2, ..., PM)¶
• P1...PM - EC points in G¶
• c - hash value, integer between 0 and 2^(8n)-1¶
• pi_string - VRF proof, octet string (ptLen+n+qLen octets)¶
This document defines ECVRF-P256-SHA256-TAI as follows:¶
This document defines ECVRF-P256-SHA256-SSWU as identical to ECVRF-P256-SHA256-TAI, except that:¶
This document defines ECVRF-EDWARDS25519-SHA512-TAI as follows:¶
This document defines ECVRF-EDWARDS25519-SHA512-ELL2 as identical to ECVRF-EDWARDS25519-SHA512-TAI, except:¶
The ECVRF as specified above is a VRF that satisfies the "trusted uniqueness", "trusted collision resistance", and "full pseudorandomness" properties defined in Section 3. In order to obtain "full
uniqueness" and "full collision resistance" (which provide protection against a malicious VRF public key), the Verifier MUST perform the following additional validation procedure upon receipt of the
public VRF key. The public VRF key MUST NOT be used if this procedure returns "INVALID".¶
Note that this procedure is not sufficient if the elliptic curve E or the point B, the generator of group G, is untrusted. If the prover is untrusted, the Verifier MUST obtain E and B from a trusted
source, such as a ciphersuite specification, rather than from the prover. Ciphersuites in this document specify E and B.¶
This procedure supposes that the public key provided to the Verifier is an octet string. The procedure returns "INVALID" if the public key in invalid. Otherwise, it returns Y, the public key as an EC
• PK_string - public key, an octet string¶
• "INVALID", or¶
• Y - public key, an EC point¶
Note that if the cofactor = 1, then Step 3 need not multiply Y by the cofactor; instead, it suffices to output "INVALID" if Y is the identity element of the elliptic curve group. Moreover, when
cofactor>1, it is not necessary to verify that Y is in the subgroup G; Step 3 suffices. Therefore, if the cofactor is small, the total number of points that could cause Step 3 to output "INVALID" may
be small, and it may be more efficient to simply check Y against a fixed list of such points. For example, the following algorithm can be used for the edwards25519 curve:¶
(bad_pk[0], bad_pk[2], bad_pk[3] each match two bad public keys, depending on the sign of the x-coordinate, which was cleared in step 5, in order to make sure that it does not affect the comparison.
bad_pk[1] and bad_pk[4] each match one bad public key, because x-coordinate is 0 for these two public keys. bad_pk[5] and bad_pk[6] are simply bad_pk[0] and bad_pk[1] shifted by p, in case the
y-coordinate had not been modular reduced by p. There is no need to shift the other bad_pk values by p, because they will exceed 2^255. These bad keys, which represent all points of order 1, 2, 4,
and 8, have been obtained by converting the points specified in [X25519] to Edwards coordinates.)¶
Note to RFC editor: Remove before publication¶
A reference C++ implementation of ECVRF-P256-SHA256-TAI, ECVRF-P256-SHA256-SSWU, ECVRF-EDWARDS25519-SHA512-TAI, and ECVRF-EDWARDS25519-SHA512-ELL2 is available at https://github.com/reyzin/ecvrf.
This implementation is neither secure nor especially efficient, but can be used to generate test vectors.¶
A Python implementation of an older version of ECVRF-EDWARDS25519-SHA512-ELL2 from the -05 version of this draft is available at https://github.com/integritychain/draft-irtf-cfrg-vrf-05.¶
A C implementation of an older version of ECVRF-EDWARDS25519-SHA512-ELL2 from the -03 version of this draft is available at https://github.com/algorand/libsodium/tree/draft-irtf-cfrg-vrf-03/src/
A Rust implementation of an older version of ECVRF-P256-SHA256-TAI from the -05 version of this draft, as well as variants for the sect163k1 and secp256k1 curves, is available at https://crates.io/
A C implementation of a variant of ECVRF-P256-SHA256-TAI from the -05 version of this draft adapted for the secp256k1 curve is available at https://github.com/aergoio/secp256k1-vrf.¶
An implementation of an earlier version of RSA-FDH-VRF (SHA-256) and ECVRF-P256-SHA256-TAI was first developed as a part of the NSEC5 project [I-D.vcelak-nsec5] and is available at http://github.com/
The Key Transparency project at Google uses a VRF implementation that is similar to the ECVRF-P256-SHA256-TAI, with a few changes including the use of SHA-512 instead of SHA-256. Its implementation
is available at https://github.com/google/keytransparency/blob/master/core/crypto/vrf/¶
An implementation by Ryuji Ishiguro following an older version of ECVRF-EDWARDS25519-SHA512-TAI from the -00 version of this draft is available at https://github.com/r2ishiguro/vrf.¶
An implementation similar to ECVRF-EDWARDS25519-SHA512-ELL2 (with some changes, including the use of SHA-3) is available as part of the CONIKS implementation in Golang at https://github.com/
Open Whisper Systems also uses a VRF similar to ECVRF-EDWARDS25519-SHA512-ELL2, called VXEdDSA, and specified here https://whispersystems.org/docs/specifications/xeddsa/ and here https://
moderncrypto.org/mail-archive/curves/2017/000925.html. Implementations in C and Java are available at https://github.com/signalapp/curve25519-java and https://github.com/wavesplatform/curve25519-java
Applications that use the VRFs defined in this document MUST ensure that the VRF key is generated correctly, using good randomness.¶
The ECVRF as specified in Section 5.1-Section 5.5 satisfies the "trusted uniqueness" (see Section 3.1) and "trusted collision resistance" (see Section 3.2) properties as long as the VRF keys are
generated correctly, with good randomness. If the Verifier trusts the VRF keys are generated correctly, it MAY use the public key Y as is.¶
However, if the ECVRF uses keys that could be generated adversarially, then the Verifier MUST first perform the validation procedure ECVRF_validate_key(PK_string) (specified in Section 5.6) upon
receipt of the public key as an octet string. If the validation procedure outputs "INVALID", then the public key MUST not be used. Otherwise, the procedure will output a valid public key Y, and the
ECVRF with public key Y satisfies the "full uniqueness" and "full collision resistance" properties.¶
The RSA-FDH-VRF satisfies the "trusted uniqueness" and "trusted collision resistance" properties as long as the VRF keys are generated correctly, with good randomness. These properties may not hold
if the keys are generated adversarially (e.g., if the RSA function specified in the public key is not bijective). Meanwhile, the "full uniqueness" and "full collision resistance" are properties that
hold even if VRF keys are generated by an adversary. The RSA-FDH-VRF defined in this document does not have these properties. However, if adversarial key generation is a concern, the RSA-FDH-VRF may
be modified to have these properties by adding additional cryptographic checks that its public key has the right form. These modifications are left for future specification.¶
Without good randomness, the "pseudorandomness" properties of the VRF may not hold. Note that it is not possible to guarantee pseudorandomness in the face of adversarially generated VRF keys. This is
because an adversary can always use bad randomness to generate the VRF keys, and thus, the VRF output may not be pseudorandom.¶
[PWHVNRG17] presents cryptographic reductions to an underlying hard problem (e.g. Decisional Diffie-Hellman for the ECVRF, or the standard RSA assumption for RSA-FDH-VRF) that prove the VRFs
specified in this document possess full pseudorandomness as well as selective pseudorandomness (see Section 3.3 for an explanation of these notions). However, the cryptographic reductions are tighter
for selective pseudorandomness than for full pseudorandomness. This means that the VRFs have quantitively stronger security guarantees for selective pseudorandomness.¶
Applications that are concerned about tightness of cryptographic reductions therefore have two options.¶
• They may choose to ensure that selective pseudorandomness is sufficient for the application. That is, that pseudorandomness of outputs matters only for inputs that are chosen independently of the
VRF key.¶
• If full pseudorandomness is required for the application, the application may increase security parameters to make up for the loose security reduction. For RSA-FDH-VRF, this means increasing the
RSA key length. For ECVRF, this means increasing the cryptographic strength of the EC group G. For both RSA-FDH-VRF and ECVRF, the cryptographic strength of the hash function Hash may also
potentially need to be increased.¶
The security of the ECVRF defined in this document relies on the fact that the nonce k used in the ECVRF_prove algorithm is chosen uniformly and pseudorandomly modulo q, and is unknown to the
adversary. Otherwise, an adversary may be able to recover the private VRF key x (and thus break pseudorandomness of the VRF) after observing several valid VRF proofs pi. The nonce generation methods
specified in the ECVRF ciphersuites of Section 5.5 are designed with this requirement in mind.¶
Side channel attacks on cryptographic primitives are an important issue. Implementers should take care to avoid side-channel attacks that leak information about the VRF private key SK (and the nonce
k used in the ECVRF), which is used in VRF_prove. In most applications, VRF_proof_to_hash and VRF_verify algorithms take only inputs that are public, and thus side channel attacks are typically not a
concern for these algorithms.¶
The VRF input alpha may be also a sensitive input to VRF_prove and may need to be protected against side channel attacks. Below we discuss one particular class of such attacks: timing attacks that
can be used to leak information about the VRF input alpha.¶
The ECVRF_hash_to_curve_try_and_increment algorithm defined in Section 5.4.1.1 SHOULD NOT be used in applications where the VRF input alpha is secret and is hashed by the VRF on-the-fly. This is
because the algorithm's running time depends on the VRF input alpha, and thus creates a timing channel that can be used to learn information about alpha. That said, for most inputs the amount of
information obtained from such a timing attack is likely to be small (1 bit, on average), since the algorithm is expected to find a valid curve point after only two attempts. However, there might be
inputs which cause the algorithm to make many attempts before it finds a valid curve point; for such inputs, the information leaked in a timing attack will be more than 1 bit.¶
ECVRF-P256-SHA256-SSWU and ECVRF-EDWARDS25519-SHA512-ELL2 can be made to run in time independent of alpha, following recommendations in [I-D.irtf-cfrg-hash-to-curve].¶
The VRF proof pi is not designed to provide secrecy and, in general, may reveal the VRF input alpha. Anyone who knows PK and pi is able to perform an offline dictionary attack to search for alpha, by
verifying guesses for alpha using VRF_verify. This is in contrast to the VRF hash output beta which, without the proof, is pseudorandom and thus is designed to reveal no information about alpha.¶
The VRFs specified in this document allow for read-once access to the input alpha for both signing and verifying. Thus, additional prehashing of alpha (as specified, for example, in [RFC8032] for
EdDSA signatures) is not needed, even for applications that need to handle long alpha or to support the Initialized-Update-Finalize (IUF) interface (in such an interface, alpha is not supplied all at
once, but rather in pieces by a sequence of calls to Update). The ECVRF, in particular, uses alpha only in ECVRF_hash_to_curve. The curve point H becomes the representative of alpha thereafter. Note
that the suite_string octet and the public key are hashed together with alpha in ECVRF_hash_to_curve, which ensures that the curve (including the generator B) and the public key are included
indirectly into subsequent hashes.¶
Hashing is used for different purposes in the two VRFs (namely, in the RSA-FDH-VRF, in MGF1 and in proof_to_hash; in the ECVRF, in hash_to_curve, nonce_generation, hash_points, and proof_to_hash).
The theoretical analysis assumes each of these functions is a separate random oracle. This analysis still holds even if the same hash function is used, as long as the four queries made to the hash
function for a given SK and alpha are overwhelmingly unlikely to equal each other or to any queries made to the hash function for the same SK and different alpha. This is indeed the case for the
RSA-FDH-VRF defined in this document, because the first octets of the input to the hash function used in MGF1 and in proof_to_hash are different.¶
This is also the case for the ECVRF ciphersuites defined in this document, because:¶
• inputs to the hash function used during nonce_generation are unlikely to equal inputs used in hash_to_curve, proof_to_hash, and hash_points. This follows since nonce_generation inputs a secret to
the hash function that is not used by honest parties as input to any other hash function, and is not available to the adversary.¶
• the second octets of the inputs to the hash function used in proof_to_hash, hash_points, and ECVRF_hash_to_curve_try_and_increment are all different.¶
• the last octet of the input to the hash function used in proof_to_hash, hash_points, and ECVRF_hash_to_curve_try_and_increment is always zero, and therefore different from the last octet of the
input to the hash function used in ECVRF_hash_to_curve_h2c_suite, which is set equal to the nonzero length of the domain separation tag by [I-D.irtf-cfrg-hash-to-curve].¶
For the RSA VRF, if future designs need to specify variants of the design in this document, such variants should use different first octets in inputs to MGF1 and to the hash function used in
proof_to_hash, in order to avoid the possibility that an adversary can obtain a VRF output under one variant, and then claim it was obtained under another variant¶
For the elliptic curve VRF, if future designs need to specify variants (e.g., additional ciphersuites) of the design in this document, then, to avoid the possibility that an adversary can obtain a
VRF output under one variant, and then claim it was obtained under another variant, they should specify a different suite_string constant. This way, the inputs to the hash_to_curve hash function used
in producing H are guaranteed to be different; since all the other hashing done by the prover depends on H, inputs all the hash functions used by the prover will also be different as long as
hash_to_curve is collision resistant.¶
Note to RFC Editor: if this document does not obsolete an existing RFC, please remove this appendix before publication as an RFC.¶
• 00 - Forked this document from draft-goldbe-vrf-01.¶
• 01 - Minor updates, mostly highlighting TODO items.¶
• 02 - Added specification of elligator2 for Curve25519, along with ciphersuites for ECVRF-ED25519-SHA512-Elligator. Changed ECVRF-ED25519-SHA256 suite_string to ECVRF-ED25519-SHA512. (This change
made because Ed25519 in [RFC8032] signatures use SHA512 and not SHA256.) Made ECVRF nonce generation a separate component, so that nonces are deterministic. In ECVRF proving, changed + to - (and
made corresponding verification changes) in order to be consistent with EdDSA and ECDSA. Highlighted that ECVRF_hash_to_curve acts like a prehash. Added "suites" variable to ECVRF for
futureproofing. Ensured domain separation for hash functions by modifying hash_points and added discussion about domain separation. Updated todos in the "additional pseudorandomness property"
section. Added a discussion of secrecy into security considerations. Removed B and PK=Y from ECVRF_hash_points because they are already present via H, which is computed via hash_to_curve using
the suite_string (which identifies B) and Y.¶
• 03 - Changed Ed25519 conversions to little-endian, to match RFC 8032; added simple key validation for Ed25519; added Simple SWU cipher suite; clarified Elligator and removed the extra x0 bit, to
make Montgomery and Edwards Elligator the same; added domain separation for RSA VRF; improved notation throughout; added nonce generation as a section; changed counter in try-and-increment from
four bytes to one, to avoid endian issues; renamed try-and-increment ciphersuites to -TAI; added qLen as a separate parameter; changed output length to hLen for ECVRF, to match RSAVRF; made
Verify return beta so unverified proofs don't end up in proof_to_hash; added test vectors.¶
• 04 - Clarified handling of optional arguments x and PK in ECVRF_prove. Edited implementation status to bring it up to date.¶
• 05 - Renamed ed25519 into the more commonly used edwards25519. Corrected ECVRF_nonce_generation_RFC6979 (thanks to Gorka Irazoqui Apecechea and Mario Cao Cueto for finding the problem) and
corresponding test vectors for the P256 suites. Added a reference to the Rust implementation.¶
• 06 - Made some variable names more descriptive. Added a few implementation references.¶
• 07 - Incorporated hash-to-curve draft by reference to replace our own Elligator2 and Simple SWU. Clarified discussion of EC parameters and functions. Added a 0 octet to all hashing to enforce
domain separation from hashing done inside hash-to-curve.¶
• 08 - Incorporated suggestions from crypto panel review by Chloe Martindale. Changed Reyzin's affiliation. Updated references.¶
• 09 - Added a note to remove the implementation page before publication.¶
• 10 - Added a check in ECVRF_decode_proof to ensure that s is reduced mod q. Connected security properties (Section 3) and security considerations (Section 7) with more cross-references.¶
This document also would not be possible without the work of Moni Naor (Weizmann Institute), Sachin Vasant (Cisco Systems), and Asaf Ziv (Facebook). Shumon Huque, David C. Lawerence, Trevor Perrin,
Annie Yousar, Stanislav Smyshlyaev, Liliya Akhmetzyanova, Tony Arcieri, Sergey Gorbunov, Sam Scott, Nick Sullivan, Christopher Wood, Marek Jankowski, Derek Ting-Haye Leung, Adam Suhl, Gary Belvinm,
Piotr Nojszewski, Gorka Irazoqui Apecechea, and Mario Cao Cueto provided valuable input to this draft. Riad Wahby was very helpful with the integration of the hash-to-curve draft.¶
Papadopoulos, D., Wessels, D., Huque, S., Vcelak, J., Naor, M., Reyzin, L., and S. Goldberg, "Making NSEC5 Practical for DNSSEC", in ePrint Cryptology Archive 2017/099, , <https://eprint.iacr.org
Micali, S., Rabin, M., and S. Vadhan, "Verifiable Random Functions", in FOCS, , <https://dash.harvard.edu/handle/1/5028196>.
Bernstein, D.J., "How do I validate Curve25519 public keys?", , <https://cr.yp.to/ecdh.html#validate>.
Gilad, Y., Hemo, R., Micali, Y., Vlachos, Y., and Y. Zeldovich, "Algorand: Scaling Byzantine Agreements for Cryptocurrencies", in Proceedings of the 26th Symposium on Operating Systems Principles
(SOSP), , <https://eprint.iacr.org/2017/454>.
David, B., Gazi, P., Kiayias, A., and A. Russell, "Ouroboros Praos: An adaptively-secure, semi-synchronous proof-of-stake protocol", in Advances in Cryptology - EUROCRYPT, , <https://
Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and D. C. Lawrence, "NSEC5, DNSSEC Authenticated Denial of Existence", Work in Progress, Internet-Draft, draft-vcelak-nsec5-08, , <https://
"Public Key Cryptography for the Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA)", ANSI X9.62, .
The test vectors in this section were generated using the reference implementation at https://github.com/reyzin/ecvrf.¶
The example secret keys and messages in Examples 1 and 2 are taken from Appendix A.2.5 of [RFC6979].¶
Example 1:¶
Example 2:¶
The example secret key in Example 3 is taken from Appendix L.4.2 of [ANSI.X9-62-2005].¶
Example 3:¶
The example secret keys and messages in Examples 4 and 5 are taken from Appendix A.2.5 of [RFC6979].¶
Example 4:¶
Example 5:¶
The example secret key in Example 6 is taken from Appendix L.4.2 of [ANSI.X9-62-2005].¶
Example 6:¶
The example secret keys and messages in Examples 7, 8, and 9 are taken from Section 7.1 of [RFC8032].¶
Example 7:¶
Example 8:¶
Example 9:¶
The example secret keys and messages in Examples 10, 11, and 12 are taken from Section 7.1 of [RFC8032].¶
Example 10:¶
Example 11:¶
Example 12:¶ | {"url":"https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-vrf-10","timestamp":"2024-11-09T09:52:31Z","content_type":"text/html","content_length":"231936","record_id":"<urn:uuid:67e9d7ce-ef93-4040-83d0-3180acb39f17>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00003.warc.gz"} |
BioArt WS15/Physarum polycephalum and unconventional computing
Physarum polycephalum is one more protist which doesn't fit kingdoms of plantae, animals and fungi. Neither bacteria. So it is already interesting. Another interesting thing is that our chair is
involved in the European project Phychip, which is focusing on physarum polycephalum and unconventional computing.
Physarum Polycephalum and its life cycle
"Physarum polycephalum, literally the "many-headed slime", is a slime mold that inhabits shady, cool, moist areas, such as decaying leaves and logs. Like slime molds in general, it is sensitive to
light; in particular, light can repel the slime mold and be a factor in triggering spore growth."(wikipedia A) It feeds on bacteria, spores and other microbial creatures.
• Vegetative phase: plasmodium (consists of networks of protoplasmic veins, and many nuclei)
• sclerotium (hardened multinucleated tissue)
• sporangia
• "Physarum Polycephalum exhibit a form of intelligence
• When separated they will pull themselves back together
• They also exhibit self-sacrifice
• They will gather and form a stalk and then a fruiting body
• Those self making up the stalk will die. Those at the top will clump into a ball made of life spores" (Bonner)
Physarum Experiments
HOWTO @ Adamatzky (2010). Chapter 2
Physarum Machine
"Unconventional computing is an interdisciplinary of science where computer scientists, physicists, mathematicians, apply principles of information processing in natural systems to design novel
computer devices and architectures" (Adamatzky 2007)
“The plasmodium functions as a parallel amorphous computer with parallel inputs and parallel outputs. Data are represented by spatial configurations of sources of nutrients. A program of computation
is coded via configurations of repellents and attractants. Results of computation are presented by the configuration of the protoplasmic network and the localisation of the plasmodium.”(Adamatzky
“.. plasmodium is unique biological substrate that mimics universal storage modification machines, namely the Kolmogorov-Uspensky machine. In the plasmodium implementation of the storage modification
machine data are represented by sources of nutrients and memory structure by protoplasmic tubes connecting the sources. In laboratory experiments and simulation we demonstrate how the
plasmodium-based storage modification machine can be programmed.”(Adamatzky & Jones 2009)
Computable Discrete Elements in the Turing Machine
In a 1936 paper by Turing, the concept of the machine is proposed as the simple idea of an apparatus which is able to compute discrete values – zeros and ones. In the same paper, Turing introduces a
computing machine with an infinite length of tape and a tape head acting upon seven commands: a) read the tape, b) move the tape left, c) move tape right, d) write “zero” on the tape, e) write “one”
on the tape, f) jump to another command, and g) halt. The idea of these commands is to show that output B could be processed having an initial state and some input A. The position of the tape head on
the proposed apparatus processing the information is dependent on the information stored on the tape: If the input information is defined, so is the output. The problem in such a computational model
is any numerically undefined variable which would cause the machine to stop processing information, or to "halt." The halting state or, according to Turing, the “decision problem"
(Enscheidungsproblem) is the problem of digital computation being defined by numerical variables. Thus, the Turing machine is limited to computing all input information and to solving all given
problems (Turing 1936).
Turing Machines: https://www.youtube.com/watch?v=gJQTFhkhwPA
Markov chain
"A Markov chain (discrete-time Markov chain or DTMC[1]), named after Andrey Markov, is a random process that undergoes transitions from one state to another on a state space. It must possess a
property that is usually characterized as "memorylessness": the probability distribution of the next state depends only on the current state and not on the sequence of events that preceded it. This
specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes."(wikipedia (b))
Kolmogorov Machine
"Kolmogorov, or Kolmogorov-Uspensky, machines [Ko1, KU, US] are similar to Turing machines except that the tape can change its topology."(Gurevich) Also, as far as I understand, Kolmogorov Machine
isn't described by discrete 0 and 1 values. Also its functions could be updated in real time over the recursive method. On the other hand both Turing Machine and Kolmogorov machine, could emulate
each other, so at the end the difference is just in the way how the machines compute their functions.
"Мы остановимся на следующих вариантах математического опреде ления вычислимой функции или алгоритма: A) Определение вычислимой функции как функции, значения которой выводимы в некотором логическом
исчислении (Гёдель [4], Чёрч [5]1)). Б) Определение вычислимой функции как функции, значения кото рой получаются при помощи исчисления Х-коиверсии Чёрча [5], [7]. B) Определение вычислимой функции
как функции частично-рекур сивной (см. работу Клини [8])2) или —для случая всюду определенной функции —как общерекурсивной (Клини [10]). (Термины «частично-рекур сивная» и «общерекурсивная»
понимаются здесь в смысле приложения I). Г) Вычислительная машина Тьюринга [ И ] 3 ) . Д) Финитный комбинаторный процесс Поста [13]. Е) Нормальный алгорифм А. А. Маркова [1], [2]." (Колмогоров &
Успенский 1958)
"Kolmogorov machines tape similarly to Schönhage’s tape is a finite connected graph with a distinguished (active) node. They work upon partly recursive function, changing instructions in real time."
"Instructions: 1. add a new node together with a pair of edges of some colors between the active node and the new one, 2. remove a node and the edges incident to it, 3. add a pair of edges of some
colors between two existing nodes, 4. remove the two edges between two existing nodes, 5. halt. "(Gurevich)
"Grigoriev [Gr] exhibited a function real-time computable by some KU machine but not real-time computable by any Turing machine."(Gurevich) | {"url":"https://www.uni-weimar.de/kunst-und-gestaltung/wiki/GMU:BioArt_WS15/Physarum_polycephalum_and_unconventional_computing","timestamp":"2024-11-13T22:36:06Z","content_type":"text/html","content_length":"47282","record_id":"<urn:uuid:0e4cae90-6c3e-4f1a-b765-bb05919ebd66>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00163.warc.gz"} |
Free Printable Law of Cosines Worksheets for Students
The law of cosines, commonly referred to as the cosine rule or the cosine formula in trigonometry, basically connects the length of the triangle to the cosines of one of its angles. It claims that we
can determine the length of the third side of a triangle if we know the length of the first two sides and the angle between them. It may be calculated using the equation c2 = a2 + b2 – 2ab cosγ,
where a, b, and c are the triangle's sides and γ is the angle between a and b. If you are struggling with problems concerning the law of cosines, please don’t be worried. On this website, we have
prepared various law of cosines worksheets for extra practice. These Math worksheets will give students a chance to practice a variety of problems and activities to help students dive deeper into the
topic. Try our law of cosines worksheets if you're seeking a way to reteach and offer further help when it comes to this concept. We’re sure that it would be a great reinforcement resource. | {"url":"https://worksheetzone.org/math/law-of-cosines-worksheet","timestamp":"2024-11-10T02:58:33Z","content_type":"text/html","content_length":"609778","record_id":"<urn:uuid:af789b77-1d34-4246-a0af-df8f807174ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00880.warc.gz"} |
What is the SHA-2 224 bit digest?
SHA-224 is a one-way hash function that provides 112 bits of security, which is the generally accepted strength of Triple-DES [3DES]. This document makes the SHA-224 one-way hash function
specification available to the Internet community, and it publishes the object identifiers for use in ASN.
What is message digest in SHA?
In cryptography, SHA-1 (Secure Hash Algorithm 1) is a cryptographically broken but still widely used hash function which takes an input and produces a 160-bit (20-byte) hash value known as a message
digest – typically rendered as a hexadecimal number, 40 digits long.
Is message digest and hash same?
A message digest algorithm or a hash function, is a procedure that maps input data of an arbitrary length to an output of fixed length. Output is often known as hash values, hash codes, hash sums,
checksums, message digest, digital fingerprint or simply hashes.
What is the basic difference between SHA-224 and SHA 512 224?
SHA-512 is roughly 50% faster than SHA-224 and SHA-256 on 64-bit machines, even if its digest is longer. The speed-up is due to the internal computation being performed with 64-bit words, whereas the
other two hash functions employ 32-bit words.
What is SHA-2 used for?
SHA-1 and SHA-2 are the Secure Hash Algorithms required by law for use in certain U.S. Government applications, including use within other cryptographic algorithms and protocols, for the protection
of sensitive unclassified information.
How does SHA work?
Secure Hash Algorithms, also known as SHA, are a family of cryptographic functions designed to keep data secured. It works by transforming the data using a hash function: an algorithm that consists
of bitwise operations, modular additions, and compression functions.
How does message digest work?
Message Digest is used to ensure the integrity of a message transmitted over an insecure channel (where the content of the message can be changed). The message is passed through a Cryptographic hash
function. This function creates a compressed image of the message called Digest.
What is message digest explain?
A message digest is a fixed size numeric representation of the contents of a message, computed by a hash function. A message digest can be encrypted, forming a digital signature. Messages are
inherently variable in size. A message digest is a fixed size numeric representation of the contents of a message.
What are message digests?
What is message digest used for?
A message digest is a cryptographic hash function containing a string of digits created by a one-way hashing formula. Message digests are designed to protect the integrity of a piece of data or media
to detect changes and alterations to any part of a message.
What is the digest length of the SHA-512 224?
SHA-256 and SHA-512 are novel hash functions computed with eight 32-bit and 64-bit words, respectively….SHA-2.
Digest sizes 224, 256, 384, or 512 bits
Structure Merkle–Damgård construction with Davies–Meyer compression function
Rounds 64 or 80
Best public cryptanalysis
What sha512 224?
What is sha512/224? SHA-512/224 or Secure Hash Algorithm 2 is one of several cryptographic hash functions that takes input and produces a 224-bit (28-byte) hash value. This message digest is usually
then rendered as a hexadecimal number which is 56 digits long. | {"url":"https://www.rwmansiononpeachtree.com/what-is-the-sha-2-224-bit-digest/","timestamp":"2024-11-13T11:51:55Z","content_type":"text/html","content_length":"76096","record_id":"<urn:uuid:8fbc29b0-1764-4913-9139-0ccf11320a5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00821.warc.gz"} |
Radiation Models
Updated 23-JAN-2003 to include additional discussion and references.
Updated 7-JUN-2000 to correct errors in curve fits for CO[2] and H[2]O.
Updated 5-FEB-1998 to include information on CH[4] and CO, as well as CO[2] and H[2]O.
Most of the current TNF target flames could not be described as strongly radiating flames. However, thermal radiation from all but the most highly diluted hydrogen jet flames reduces the local
temperatures sufficiently to affect the production rate of NO. Detailed treatment of radiative transfer within a turbulent flame is computationally very expensive. In order to include the effect of
radiation in turbulent combustion models, without significantly increasing computational expense, a highly simplified treatment of radiative heat loss is needed. Because the primary focus of the TNF
Workshop has been on the modeling of turbulence-chemistry interaction, we have agreed adopted an optically thin radiation model, with radiative properties based on the RADCAL model by Grosshandler of
NIST [1].
The characteristics of some flames selected for the workshop are such that a model based upon the assumption of optically thin radiative heat loss should yield reasonable accuracy. This has been
demonstrated for the simple hydrogen jet flames [2]. However, there is some evidence that the optically thin model significantly over predicts radiative losses from the CH4 flames in the TNF library,
due to strong absorption by the 4.3-micron band of CO[2] [3-5].
Radiation mainly affects the NO predictions in the TNF target flames. In general, one can expect an adiabatic flame calculation to over predict NO levels (if all other submodels are correct), while
an optically thin model is expected to under predict NO levels. The answer corresponding to a detailed radiation model should be somewhere between these limits. At present, the majority of modelers
in the TNF Workshop are satisfied with this limitation of the present radiation model. Further discussion of radiation in the TNF hydrocarbon flames is included in the TNF5 and TNF6 Proceedings.
The model described here is also documented in [6], which may be used as reference.
Under the assumptions that these flames are optically thin, such that each radiating point source has an unimpeded isotropic view of the cold surroundings, the radiative loss rate per unit volume may
be calculated as:
Q(T,species) = 4 SUM{p[i]*a[p,i]} *(T^4-Tb^4)
• sigma=5.669e-08 W/m^2K^4 is the Steffan-Boltzmann constant,
• SUM{ } represents a summation over the species included in the radiation calculation,
• p[i] is the partial pressure of species i in atmospheres (mole fraction times local pressure),
• a[p,i] is the Planck mean absorption coefficient of species i,
• T is the local flame temperature (K), and
• Tb is the background temperature (300K unless otherwise specified in the experimental data).
Note that the T[b] term is not consistent with an emission-only model. It is included here to eliminate the unphysical possibility of calculated temperatures in the coflowing air dropping below the
ambient temperature. In practice, this term has a negligible effect on results.
CO[2] and H[2]O are the most important radiating species for hydrocarbon flames. As an example, Jay Gore reports that the peak temperature in a strained laminar flame calculation decreased by 50K
when radiation by CO[2] and H[2]O was included. Inclusion of CH[4] and CO dropped the peak temperature by another 5K. On the rich side of the same flames the maximum effect of adding the CH[4] and CO
radiation was an 8K reduction in temperature (from 1280 K to 1272 K for a particular location). These were OPPDIF calculations (modified to include radiation) of methane/air flames with 1.5 cm
separation between nozzles and 8 cm/s fuel and air velocities.
In the context of the expected accuracy of the optically thin model, it would appear that the inclusion of CH[4] and CO radiation is not essential for calculations of methane flames. Radiation from
CO may be important for CO/H[2] flames and methanol flames. We do not see a need to repeat expensive calculations to include this detail. However, we suggest that all four species be included in
future calculations.
Curve fits for the Planck mean absorption coefficients for H[2]O, CO[2], CH[4], and CO are given below as functions of temperature. These are fits to results from the RADCAL program [1].
Curve Fits: The following expression should be used to calculate a[p] for H[2]O, and CO[2] in units of (m-1atm-1). These curve fits were generated for temperatures between 300K and 2500K and may be
very inaccurate outside this range.
a[p] = c0 + c1*(1000/T) + c2*(1000/T)^2 + c3*(1000/T)^3 + c4*(1000/T)^4 + c5*(1000/T)^5
The coefficients are:
H[2]O CO[2]
c0 -0.23093 18.741
c1 -1.12390 -121.310
c2 9.41530 273.500
c3 -2.99880 -194.050
c4 0.51382 56.310
c5 -1.86840E-05 -5.8169
A fourth-order polynomial in temperature is used for CH[4]:
a[p,ch4] = 6.6334 – 0.0035686*T + 1.6682e-08*T^2 + 2.5611e-10*T^3 – 2.6558e-14*T^4
A fit for CO is given in two temperature ranges:
a[p,co] = c0+T*(c1 + T*(c2 + T*(c3 + T*c4)))
if( T .le. 750 ) then
c0 = 4.7869
c1 = -0.06953
c2 = 2.95775e-4
c3 = -4.25732e-7
c4 = 2.02894e-10
c0 = 10.09
c1 = -0.01183
c2 = 4.7753e-6
c3 = -5.87209e-10
c4 = -2.5334e-14
Acknowledgments: Nigel Smith (AMRL), Jay Gore (Purdue University), JongMook Kim (Purdue University), and Qing Tang (Cornell) provided information for this radiation model.
Grosshandler, W. L., RADCAL: A Narrow-Band Model for Radiation Calculations in a Combustion Environment, NIST technical note 1402, 1993.
Barlow, R. S., Smith, N. S. A., Chen, J.-Y., and Bilger, R. W., Combust. Flame 117:4-31 (1999).
Frank, J. H., Barlow, R. S., and Lundquist, C., Proc. Combust. Inst. 28:447-454 (2000).
Zhu, X. L., Gore, J. P., Karpetis, A. N., and Barlow, R. S., Combust. Flame 129:342-345 (2002).
Coelho, P. J., Teerling, O. J., Roekaerts, D., “Spectral Radiative Effects and Turbulence-Radiation-Interaction in Sandia Flame D,” in TNF6_Proceedings.pdf (2002).
Barlow, R. S., Karpetis, A. N., Frank, J. H., and Chen, J.-Y., Combust. Flame 127:2102-2118 (2001). | {"url":"https://tnfworkshop.org/radiation/","timestamp":"2024-11-03T04:05:16Z","content_type":"text/html","content_length":"36576","record_id":"<urn:uuid:b74024c0-7ddd-45c4-a40b-b0b7f021f921>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00116.warc.gz"} |
Predicting Junction Temperature
The junction temperature of a silicon device can be an important design consideration for linear regulators. Let's say that we want to verify the effects of a given heat sink and prove whether or not
it is effective at keeping the regulator safe at ambient temperatures and in the rated conditions. This article explores how we can create a lumped element electrical model to represent the physical
thermal system and then use this to help us choose/test an appropriate heat sink.
The Linear Regulator Circuit
For our example let's say we have designed a simple 5Vdc linear regulator as shown below and that we need a continuous output current of about 1A to the output load.
Simple Linear Regulator Circuit.
The regulator will have to reliably provide the required output current at the ambient temperature of 50°C maximum and, above all, the regulator will have to work safely for hours without getting
Power Dissipation
Our first step is to figure out the power dissipated by the regulator U1. To do this we will measure the voltage across the regulator itself between the input and output pins and the output current
in the Load. Proteus can help with this by using a DC Voltmeter and a DC Ammeter connected as shown in the picture above. This gives us 6.32Vdc and 0.97Adc in the load which, for an output of 5Vdc
and 1A nominal, implies that U1 will dissipate a power of
From the datasheet we can get the maximum power dissipation at the needed ambient temperature. The curve is shown below:
Power Dissipation Characteristics for LM7805.
At the maximum ambient temperature of 50°C we can see that average power that regulator can dissipate without a heat-sink is well below the expected projected value of 6.13W. This also implies that
the junction temperature of the regulator will dangerously rise up and the regulator will likely be damaged. So, we need to provide a means to dissipate the temperature from the regulator by using a
suitable heat-sink.
From the regulator datasheet (LM7805) we get the value of the junction to case thermal resistance (metal lead) for TO220 case and the maximum junction temperature as follows:
Rjc = 1.7°C/W Jtmax=150°C
The Physical Thermal Model
Let's now look at a physical thermal model of a generic device with heat-sink and identify all the thermal parameters.
Generic Physical Thermal Model.
The formula we need is:
Tj = Pd (Rj-c + Rc-s + Rs-a) + Ta
• Rj-c is the junction-case
• Rc-s is the case-sink
• Rs-a is the sink-ambient resistances
• Pd is the dissipated power
• Ta is the maximum ambient temperature
• Tj is the junction temperature
Tj will be the junction temperature established from the thermal equilibrium due to heat transfers elements, ambient temperature value and dissipated power.
'Lumped Element' Thermal Model
What we now want to do is transpose this thermal model into an electrical model using Proteus. We will use a thermal RC model, also known as a 'lumped element' model
Lumped Element Model.
All thermal elements representing heat transfer resistances are simulated with electrical resistors. Also, the dissipated power is simulated with a current generator the which current is equal to Pd
value and represents the heat flow. The ambient temperature is represented by a voltage generator and, finally, the capacitors represent the junction and heat-sink thermal capacities, CTHJ and CTH.
A useful technique in Proteus is to represent the values for each element of the model with generic variable names on the device. You'll see this in the screenshot above. We can then apply part
specific values in a *DEFINE block when we calculate them for a particular heat sink.
Calculating Element Values
So, we know that our *DEFINE block in the thermal RC model defines all numerical values of the thermal elements . Now we set the values as applicable to our circuit. RJC we get from the datasheet and
is set to 1.7 °C/W. RCS is defined as the resistance introduced by the thermal compound between metal case and heat-sink. For a generic compound a value of 0.5°C/W is reasonable. RSA is the thermal
resistance of the heat-sink. The selected heat-sink is the one shown below.
This is an aluminium vertical mounting heat-sink from OHMITE, model RA-T2X-38E. The weight is 38g with an heat transfer coefficient (natural convection) of 3.9 °C/W.
We will use the weight value to calculate the thermal capacity value, CTH, of the heat-sink. Knowing that the Specific Heat of Aluminium is 0.9 J/g°C we calculate the thermal capacity CTH using:
CTH = m * c
• m is the mass in grams
• c is the specific heat
This gives us:
CTH = 38g * 0.9 J/g°C = 34.2 J/°C.
Temperature Simulation
We have now got all the element values to run Proteus with a graph simulation of the Junction (green), Case (red) and Heat-sink (light blue)temperatures. We can see that junction temperature will be
kept below 90°C @ 50°C ambient temperature. This is well below the maximum junction temperature allowed of 150°C with a reasonable safe margin and tells us that our regulator will work safe providing
1A continuously.
Junction vs Case vs Heat-sink Temperatures.
In a lot of situations we can use these techniques in Proteus to make a prediction analysis of the maximum temperatures that a generic power silicon devices will reach at the rated conditions for
safe operations.
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Joe Lee
PyTorch - Classification with Nerual Networks
This my solution and a brief summary to classification section of the PyTorch course that I'm going through right now.
The first exercise was to classify a moon data(interleaving half-circles) from the scikit learn dataset. As always, the first step is to visualize the data of the given problem.
It is possible to observe that a linear activation function might not be able to accurately describe the dataset as the data itself appears to be non-linear.
# Linear Model
class MoonModelV1(nn.Module):
def __init__(self, input_features, output_features, hidden_units=10):
self.linear_layer_stack = nn.Sequential(
nn.Linear(in_features=input_features, out_features=hidden_units),
nn.Linear(in_features=hidden_units, out_features=hidden_units),
nn.Linear(in_features=hidden_units, out_features=output_features)
def forward(self, x):
return self.linear_layer_stack(x)
model_moon = MoonModelV1(input_features=2, output_features=1)
# Bincary cross entropy with Sigmoid layer loss function with SGD optimizer
loss_fn = nn.BCEWithLogitsLoss()()
optimizer = torch.optim.SGD(model_moon.parameters(), lr=0.1)
In fact, without a non-linear layer within the model, it is difficult for the model to accurately capture the pattens of the data.
However, if we utilize non-linear activation functions between our layers, it is possible to capture the non-linear patterns from the moon data set. For this model, the ReLU function was used.
class MoonModelV2(nn.Module):
def __init__(self, input_features, output_features, hidden_units):
self.linear_layer_stack = nn.Sequential(
nn.Linear(in_features=input_features, out_features=hidden_units),
nn.Linear(in_features=hidden_units, out_features=hidden_units),
nn.Linear(in_features=hidden_units, out_features=output_features)
def forward(self, x):
return self.linear_layer_stack(x)
# return self.layer_3(self.relu(self.layer_2(self.relu(self.layer_1(x))))) <-- equivalent
model_moon2 = MoonModelV2(input_features=2, output_features=1, hidden_unites=10)
# Binary Cross Entropy Loss Fn with SGD optimizer
loss_fn = nn.BCEWithLogitsLoss()
optimizer = torch.optim.SGD(model_moon.parameters(), lr=0.1)
Now, it is possible to observe a significant improvement in accuracy compared to the previous non-linear model.
The other exercise was to generate a model that could classify a spiral shaped data set with 3 classes as shown below. The visualized data appears to be non-linear, and the ReLU activation function
was used here to adress that as well.
This model is very similar to the previous classifications; however, achieving the predicted data during the training and testing process, as well as the loss function is a bit different from binary
For binary classifictions:
# Binary cross entropy loss function is used
loss = torch.nn.BCELossWithLogits or torch.nn.BCELoss
# logits --> prediction probabilities --> prediciton data
y_pred_probs = torch.sigmoid(y_logits) # convert the logits into probabilities using sigmoid function
y_pred = torch.round(y_pred_probs) # classify 1 or 0 at the break point of 0.5
For multi-class classifictions:
# Cross entropy loss function is used
loss = torch.nn.CrossEntropyLoss
# logits --> prediction probabilities --> prediciton data
y_pred_probs = torch.softmax(y_logits) # convert the logits into probabilities using softmax function
y_pred = torch.argmax(y_pred_probs) # return the class with the highest probability
To summarize, classification requires some type of activation functions that could normalize the data into some probability distributions. Binary classification utilizes the sigmoid function as it
maps/squeezes the values into some values between 0 and 1(proabilities!). Multi-class classification uses the softmax function to squeeze the values into a probability distribution. Then, these
probabilities can be retrieved to identify a class that is more likely.
class SpiralModel(nn.Module):
def __init__(self, input_features, output_features, hidden_units):
self.linear_layer_stack = nn.Sequential(
nn.Linear(in_features=input_features, out_features=hidden_units),
nn.Linear(in_features=hidden_units, out_features=hidden_units),
nn.Linear(in_features=hidden_units, out_features=output_features)
def forward(self, x):
return self.linear_layer_stack(x)
model_spiral = SpiralModel(input_features=2, output_features=3, hidden_units=10)
loss_fn = nn.CrossEntropyLoss()
optimizer = torch.optim.Adam(model_spiral.parameters(), lr=0.1)
Using this non-linear multi-class classification model, it was possible to accurately classify the spiral dataset and its features as shown below. Thanks for reading!
© Joe Lee.RSS | {"url":"https://www.shyun.dev/posts/week21","timestamp":"2024-11-08T15:07:33Z","content_type":"text/html","content_length":"40974","record_id":"<urn:uuid:d110c099-7fd4-447a-9093-dca7e8ad393c>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00689.warc.gz"} |
What is the new way to do multiplication?
What is the new way to do multiplication?
First you divide the larger number into its separate parts. Here, 23 becomes 20 and 3. Next, you multiply each separate part — 20 x 7 and 3 x 7. Finally, you add all the products together.
What is the 12 trick for multiplication?
To multiply by 12, we add a zero to the number we’re going to multiply and then we add double the original number to the result.
What are some of the methods in multiplying numbers?
Four multiplication methods are: addition method, long multiplication, grid method, and drawing lines. Each of these methods will result in the same correct product.
How can I memorize my multiplication fast?
5. Break memorization down into easy steps
1. Encourage students and set time for them to practice verbally or in writing.
2. Introduce new multiplication facts one by one, gradually and incrementally opening the concept to the more advanced steps of multiplying by 2, 3, 4 and so on.
How to teach multiplication in 6 Easy Steps?
You write a number 1 through 10 on the board (preferably the number you are working on in the classroom.)
Any child in the room tosses the ball to another child.
Both children compete in trying to be the first to say the answer,multiplying the number on the board by the number called out by the child who caught the
How do you multiply with fingers?
Hold your hands out in front of you with your palms facing up. Each of your ten fingers represent a number.
Point the finger you want to multiply by nine down towards your body.
Solve the problem by counting fingers to the left and right.
Try this with other multiples of nine. How would you multiply 9 and 2 with your fingers? What about 9 and 7?
How do you count on fingers?
Finger-counting, also known as dactylonomy, is the act of counting using one’s fingers. There are multiple different systems used across time and between cultures, though many of these have seen a
decline in use because of the spread of Arabic numerals.. Finger-counting can serve as a form of manual communication, particularly in marketplace trading – including hand signaling during open
How to learn your multiplication?
Select the times tables you want to try
Use the drop down boxes and select the one you think is the correct answer
Once you have completed all the questions press the OK Done button
You then see your results!
Then do more ( the test is always different! ),or come back here and choose another table | {"url":"https://www.resurrectionofgavinstonemovie.com/what-is-the-new-way-to-do-multiplication/","timestamp":"2024-11-03T10:17:04Z","content_type":"text/html","content_length":"34307","record_id":"<urn:uuid:a28cbe72-c1e7-457f-8bac-87f7ef36028d>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00099.warc.gz"} |
Computing a Minimum Weight k-Link Path in Graphs with the Concave Monge Property for Journal of Algorithms
Journal of Algorithms
Computing a Minimum Weight k-Link Path in Graphs with the Concave Monge Property
View publication
Let G be a weighted, complete, directed acyclic graph whose edge weights obey the concave Monge condition. We give an efficient algorithm for finding the minimum weight k-link path between a given
pair of vertices for any given k. The algorithm runs in n2o(√log k log n) time, for k = Ω(log n). Our algorithm can be applied to get efficient solutions for the following problems, improving on
previous results: (1) computing length-limited Huffman codes, (2) computing optimal discrete quantization, (3) computing maximum k-cliques of an interval graph, (4) finding the largest k-gon
contained in a given convex polygon, (5) finding the smallest k-gon that is the intersection of k half-planes out of n half-planes defining a convex n-gon. © 1998 Academic Press. | {"url":"https://research.ibm.com/publications/computing-a-minimum-weight-k-link-path-in-graphs-with-the-concave-monge-property","timestamp":"2024-11-02T09:12:44Z","content_type":"text/html","content_length":"73685","record_id":"<urn:uuid:d00e7dbe-4121-425c-89f1-d64ee20b14cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00886.warc.gz"} |
P Q R Bands in IR SpectraP Q R Branches in IR Spectra
P Q R Branches in IR Spectra
P Q R Branches in IR Spectra
We know that each vibrational energy level of a molecule has a series of closely spaced rotational energy levels. Vibrational - rotational transitions are caused by the absorption of infra red light
for which the selection rule is ΔJ = -1, 0 +1, where J and v are the rotational and vibrational quantum numbers respectively.
If ΔJ = -1, we get P Branch (Lines with frequency lower than the fundamental frequency)
If ΔJ = 0, we get Q Branch (Vibration transition occurs without being accompanied by rotational transition)
If ΔJ = +1, we get R Branch (Lines with frequency higher than the fundamental frequency)
For a diatomic molecule like HCl, we have only one fundamental vibrational mode and hence, we expect only one infrared absorptional band according to 3n − 5 rule-
3 x 2 − 5 = 1
Diatomic molecules are always linear. However, even under low resolution, it gives two bands designated as R and P bands.
The Q band is not at all expected because it is possible only when the moment of inertia I = 0 as we know that-
E[J] = [h^2/8π^2I] J(J + 1)
ΔE[J], J = 0 only when I = 0.
Hence, it shows only ΔJ = +1 (R Branch) and ΔJ = -1 (P Branch) transition under 𝜈[o] → 𝜈[1]. These R and P bands split under high resolution due to the so many associated rotational transitions. | {"url":"https://www.maxbrainchemistry.com/p/p-q-r-bands-in-ir-spectra.html","timestamp":"2024-11-05T07:09:55Z","content_type":"application/xhtml+xml","content_length":"196951","record_id":"<urn:uuid:4ff20ad9-c836-4f51-aee3-829f76093918>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00412.warc.gz"} |
Optimization of Machine Process Parameters in EDM for EN 31 Using Evolutionary Optimization Techniques
Department of Mechanical Engineering, BIT, Mesra, Ranchi 835215, India
Author to whom correspondence should be addressed.
Submission received: 7 May 2018 / Revised: 30 May 2018 / Accepted: 4 June 2018 / Published: 5 June 2018
Electrical discharge machining (EDM) is a non-conventional machining process that is used for machining of hard-to-machine materials, components in which length to diameter ratio is very high or
products with a very complicated shape. The process is commonly used in automobile, chemical, aerospace, biomedical, and tool and die industries. It is very important to select optimum values of
input process parameters to maximize the machining performance. In this paper, an attempt has been made to carry out multi-objective optimization of the material removal rate (MRR) and roughness
parameter (Ra) for the EDM process of EN31 on a CNC EDM machine using copper electrode through evolutionary optimization techniques like particle swarm optimization (PSO) technique and biogeography
based optimization (BBO) technique. The input parameter considered for the optimization are Pulse Current (A), Pulse on time (µs), Pulse off time (µs), and Gap Voltage (V). PSO and BBO techniques
were used to obtain maximum MRR and minimize the Ra. It was found that MRR and SR increased linearly when discharge current was in mid-range however non-linear increment of MRR and Ra was found when
current was too small or too large. Scanning Electron Microscope (SEM) images also indicated a decreased Ra. In addition, obtained optimized values were validated for testing the significance of the
PSO and BBO technique and a very small error value of MRR and Ra was found. BBO outperformed PSO in every aspect like computational time, less percentage error, and better optimized values.
1. Introduction
The definition of the word “machining” has evolved significantly over the last couple of decades. In a broader term, machining can be defined as the process to remove material from a work piece or a
process where a piece of material is given desired shape and size through a controlled material removal process. The traditional processes such as milling, turning, grinding, drilling, etc., to
remove material through mechanical abrasion, micro chipping, or chip formation. There are many reasons due to which the traditional processes are not economical and even possible when hardness of the
material to be machined is quite high i.e., above 400 HB, or material is too brittle to be machined, the machining forces are too high for the delicate and slender work piece specimen, the complexity
of the part to be machined, and the residual stresses in the machined component which are not at all acceptable [
The above disadvantages have led to the development of other material removal mechanisms like mechanical, chemical, thermal, electrochemical, and different hybrid mechanism. These material removal
mechanisms have therefore resulted in machining processes referred to as non-traditional machining processes. Owing to advantages offered by the non-traditional machining processes, these are
available for an extensive range of industrial applications. The source of energy used differ from process to process and therefore can be categorised accordingly: thermal & electro-thermal processes
like laser beam machining, ion beam machining, electric discharge machining etc.; chemical and electrochemical processes such as electrochemical machining, electro-chemical honing, etc.; mechanical
processes for instance ultrasonic machining water jet machining, etc.; and hybrid processes such as abrasive EDM, etc.
Electrical discharge machining (EDM), also termed as die sinking, spark machining, wire burning, spark eroding, or wire erosion, is a non-traditional (non-conventional) manufacturing process where a
desired shape is obtained with the help of electrical discharges (sparks) [
]. Now-a-days, EDM is a commonly used non-conventional machining process because of its of applications to produce complex shapes with ease, and components which require excellent surface finish
which can be used in biomedical, automobile, chemical, aerospace, and tool and die industries to name a few. EDM also offers other advantages such as production of tapered holes, fines holes,
delicate components, weak materials machining, and also machining of small work pieces which might get damaged on machining with conventional tool cutting because of their high cutting tool pressure.
In EDM process, both work piece and the tool electrode are submerged inside a dielectric fluid. During machining, a spark channel is generating in between the electrode and work piece gap with
generation of very high temperature of about 10,000 °C. This high temperature is sufficient to melt and vaporizes small amount of material from the work piece surface that leads to the material
removal from the work piece surface.
2. Need for Optimization
In this modern era of digitalization of technologies and global competitiveness, enterprises and industries need to reduce time and save money and the only way they see it is through optimization of
process parameters. EDM process suffers from a number of limitations such as higher power consumption, high cost on initial investment and large floor space. Further, it can machine only conductive
materials and is more expensive when compared to a traditional process like milling and turning. Therefore, a careful selection of the different parameters and careful planning is recommended before
commencing the machining process. Selection of suitable machining parameters is critical to achieve optimum machining results. Many researchers and academicians have tried performing optimization for
various manufacturing processes (both conventional and non-conventional) through traditional and non-traditional optimization methods considering various process parameters to obtain optimum results.
For EDM, MRR (Material Removal Rate), and Ra (Roughness Parameter, average roughness) are considered as the most important parameters and the need to maximize MRR and minimize Ra has always been the
goal for many researchers and industries.
3. Literature Survey
To find an optimized set of inputs for achieving appropriate outputs has always been a challenge and a tedious task for researchers for many years. Effects of EDM process parameters on machining
responses or performances has been studied by many researchers. However, machining performances and responses considered, relationship between different process parameters and machining responses and
also optimization techniques used by different researchers are quite diverse. For modelling the mathematical relationship between machining responses and process parameters, there have been different
techniques that have been used by different researchers but the one that is mostly used and preferred for the EDM process is second order polynomial model or response surface model (RSM) [
]. Though response surface model is simple for implementing, there is a very big error of estimation value in this technique when responses are nonlinear. Many other techniques like Artificial Neural
Network (ANN) [
], which supports vector machine regression [
], and adaptive neuro-fuzzy interference have also been used by researchers in the past [
]. Though when compared with SRM models or conventional polynomials these regression model has found to be better than that. Kriging model developed in geostatic in South Africa is also being used to
define the relationship between different process parameters and machining responses in EDM, and it has been found to be better than both SRM and ANN when highly nonlinear characteristic and
efficiency in cost [
] of experimentation is considered though Kringing model may require more points to capture non-linear behaviour [
]. When optimizing the process parameters, it helps an EDM operator to get the most optimum machining outputs for a required purpose. There are other methods that have been also been used for
optimization of EDM machining such as Taguchi [
], genetic algorithm [
], and TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) [
]. Taguchi uses signal to noise ratio for reaching optimal level of process parameters though the flaw in this method is that it does not promises ‘real optimal’ solution since it only addresses the
discrete control factors or design variables [
]. TOPSIS, is simply a multi-criteria decision making method, and is not a standard mathematical optimization technique and does not promise real optimal solution. Genetic Algorithm is a quite
popular optimization technique having many engineering applications [
].Khalid proposed the idea of EDiMfESO (Electrical Discharge Machine using Fuzzy Fitness Evolutionary Strategies Optimization) and he successfully optimized MRR, Ra, and Tool Wear Rate (TWR) [
]. This technique uses evolutionary strategy (ES) combined with dynamic fuzzy to predict the most appropriate multi-objective optimization setting for carrying out operation in EDM. Lin et al., used
Grey-Taguchi technique to obtain multiple performance characteristics like high MRR, low working gap and low electrode wear when machining with Inconel 718 alloy [
]. Similarly, Tang & Du performed operation on EDM in tap water on Ti–6Al–4Vusing Taguchi technique and Grey Relational Analysis technique and tried to optimize the process parameters and presented a
comparative analysis on using tap water with dielectric fluids [
]. Aliakbari and Baseri optimized machining parameters in the rotary EDM process with the help of the Taguchi technique [
4. Research Gap
It can be observed from the above literature survey that traditional methods such as graphical solution technique, Taguchi, TOPSIS, etc. have mostly been used for the parametric optimization of the
EDM process. However, the traditional methods are restricted over a broad range of spectrum of problem domains. The complexity of the optimization problem further limits the application of the
traditional optimization methods. The use of non-traditional methods such as GA (Genetic Algorithm) provides a near optimal solution. Therefore, to overcome the drawbacks of the traditional
optimization techniques and non-traditional methods such as GA, PSO, ABC (Ant Bee Colony), and other heuristic and evolutionary optimization techniques are being used by the scientific community. The
evolutionary optimization algorithm is based on biological genetics. These optimization techniques are more robust in comparison to the traditional techniques because instead of making use of
functional derivatives, fitness information is used by the evolutionary optimization techniques. Therefore, efforts are being made to use such optimization techniques to achieve a more accurate
Therefore, in this research paper, PSO was adopted for the optimization of Ra and MRR for the EDM of EN31. Pulse Current (A), Gap Voltage (V), Pulse on time (µs), and Pulse off time (µs) are
considered as the process parameters. Further, the multiple-objective optimization is performed for the Ra and MRR through PSO. The PSO technique is compared with BBO technique. A comparative
analysis is presented between both the optimization techniques to find out the best optimization technique in terms of computational time, best parameters obtained, less percentage errors, etc.
5. Experimental Setup
Die-Sinking EDM machine (Electronica EMT-43 Machine) was used to conduct the experiments. EDM process was carried out on the work piece EN 31 material having cylindrical cross-section having
dimension of 25 mm × 15 mm. The tool for EDM operation was rectangular positive (+) polarity of Copper (Cu) having dimension 25 mm × 25 mm. The dielectric medium chosen was paraffin oil. Before the
start of the EDM operation, initial weight of the work piece was measured and it was compared with weight of the work piece after the EDM operation and henceforth the difference in the weight gave us
the MRR of EN 31 material. Ra on the machined surfaces were obtained with the Talysurf (Make—Taylor Hobson, Leicester, UK) and were measured on the centre of the workpiece.
Table 1
Table 2
below represents the chemical composition and mechanical properties of the EN 31 tool steel.
Table 3
represents the mechanical properties of the electrode.
Table 4
shows specification of die sinking EDM machine.
5.1. Input& Output Parameters
5.1.1. Output Parameters
The four input parameters considered for the EDM experimentation are: Pulse Current (A); Pulse on time (µs); Pulse off time (µs); and Gap Voltage (V).The above parameters were chosen on the basis of
extensive literature surveys that have reported these parameters to be most influential for the MRR and the Ra.
5.1.2. Output Parameters
The two output parameters considered were: MRR and Ra.
Material Removal Rate (MRR)
Equation (1) could be used for the determination of the MRR (cubic centimetre/min) in the EDM process:
is the initial weight of the work piece before machining,
is the final weight of the work piece after machining, and
is the time period of trials.
MRR is directly proportional on the amount of current passed and the machining time (Pulse on Time, Pulse off Time). Besides these critical factors the MRR is also dependent on the type of voltage
Average Roughness (Ra)
The deviation of a surface from its ideal level is defined in terms of surface roughness. The surface roughness is defined according to ISO 4287:1997 international standard. The term average
roughness is often referred to as roughness and determines the surface texture. The average roughness is calculated by the deviations, i.e., deviation of surface from a theoretical centre line. If
the deviations are large, the surface roughness is high, whereas the surface is considered to be smooth for small deviations. This is known as arithmetic mean surface roughness Ra.
5.2. Experimentation
The experimentation carried out on EDM considering input parameters mentioned above and the output parameters obtained are shown as below in
Table 5
5.3. Regression Model
The relationship between the input parameters and the output parameters can be obtained using the statistical tool of regression analysis. Minitab 16 was used for obtaining the relationship between
MRR and input variables and also for obtaining the relationship between Ra and the input variables. Equations (2) and (3) are the relations for MRR and Ra respectively.
MRR = 0.000253A + 0.011500B + 0.008014C − 0.250000D + 0.058007
Ra = 0.0254A + 1.3904B + 0.3726C + 0.5101D + 2.0411.
Here, A has been substituted for are Pulse Current (A), B for Pulse on time (µs), C for Pulse off time (µs), and D for Gap Voltage (V).
5.4. PSO Technique
Kennedy and Eberhart [
] developed the PSO technique which is one of the evolutionary (heuristic) optimization techniques. The common evolutionary computational attributes exhibited by the PSO technique simulates
graphically the choreography of the bird flock. The PSO technique incepts with the initialization with population of random solution and then updating the generations to achieve the optimal solution.
The potential solutions which are referred to as particles are then flown through the space of the problem which follows the current optimum solution. The particles in the problem space keeps track
of its position. The coordinate of each particle is associated with the best solution achieved so far. The best position achieved by the particle is referred to as ‘pBest’. The best position achieved
by any particle in the problem space is known as the ‘gBest’. Thus the PSO technique is based on accelerating the particles towards their ‘pBest’ and ‘gBest’ locations. The random terms weight the
acceleration and therefore separate set of random numbers are generated for acceleration towards ‘gBest’ and ‘pBest’. The updated velocity i.e., the new velocity of the particle is obtained using the
Equation (4). Equation (5) on the other hand gives the updated position of the particle in the problem space.
$V i + 1 = w V i + c 1 r 1 ( p B e s t i − X i ) + c 2 r 2 ( g B e s t i − X i )$
$X i + 1 = X i + V i + 1$
The random numbers
are generated randomly and lie in the range [0, 1].
are the acceleration constants that weights the acceleration terms. The confidence of the particle in itself is represented by
represents the confidence a particle has in a swarm.
are referred to as cognitive and social parameters, respectively. The values of these two parameters determines the change in amount of tension in the system. The low values of the parameters result
in the particles roaming far away from the target regions whereas a higher value results in an abrupt movement towards the target solution [
]. The exploration abilities of the swarm particles are controlled by the inertia weight w and is therefore very critical in determining the convergence behaviour of PSO. The lower values of w
restrict the velocity updates to nearby region in the search space, whereas higher values result in velocity updates for a wider space in the problem space. Berg and Engelbrecht [
] have investigated the effect on convergence if benchmark functions of w,
, and
Heuristics have been developed to obtain the values of
, and w but these are applicable for single-objective optimization problems only. Determination of the parameters and the inertia weights for multi-objective optimization problems is relatively
difficult and therefore a time variant of PSO has been described by Tripathi et al. [
]. In this variant of PSO, the parameters and the inertia weights are adaptive and changes with the iterations. The search space can be explored more efficiently with the adaptive nature of the time
variant PSO technique. The premature convergence was taken care off by the mutation operator.
The PSO technique doesn’t require complex encoding and decoding process as required by genetic algorithm. The real number represents a particle which update their internal velocity in search of the
best solution. In PSO technique the particles tend to converge towards the best solution by iteratively looking for the best position which comprises of the evolution phase. It is very important to
handle the non-linear equation constraints and evaluate the infeasible particles. This is mainly due to the fact that the particles generated during the process may violate the constraints of the
system and thereby resulting in infeasible particles. Various approaches such as repair of infeasible particles, rejection of the infeasible particles, penalty function methods etc. can be adopted to
cope with the evolutionary problems with constraints. The recent developments however suggest the use of penalty method for addressing the same [
The flowchart for PSO implementation is shown below in
Figure 1
and algorithm for PSO is shown in
Figure 2
5.5. BBO Technique
The BBO technique was initially introduced by Dan Simon [
] in the year 2008, who is a professor at Cleveland State University in the department of Electrical and Computer Engineering. BBO falls into the category of evolutionary optimization techniques in
heuristic based optimization techniques and it tries to optimize a function stochastically and through iteration trying to improve a candidate’s solution through fitness function or given measure of
quality quite similar to PSO technique. Like many other evolutionary optimization techniques BBO drew its inspiration from nature. It is based on the study of distribution of species or speciation
(the evolution of new species), the migration of species between islands, and the extinction of species.
This algorithm is based on a mathematical model, describing the migration of species between habitats, in the form of emigration from non-suitable habitats and immigration to suitable habitats. The
suitability of habitats, is computed and stored as Habitat Suitability Index (HSI), and its definition is completely related to the objective function of the optimization problem, being solved. The
HSI (Habitat Suitability Index) here is the fitness value similar to the one in PSO.
The concept of BBO can be understood as follows:
An island which has higher suitability will have large or more number of species living in them compared to that of an island which has low suitability and henceforth the number of species living in
them will be low. The suitable island for species will therefore be having High Suitability Index (HSI) and variables characterizing HSI are called as Suitability Index Variable (SIV). Hence in a
habitat the SIVs are the independent variables whereas HSIs are the dependent variables.
An island having a higher number of HSI will be having more number of species and henceforth will be having high number of emigration rate and low immigration rate. HSI would tend to be static.
Species will move to the nearest island since they have high emigration level and vice-versa. Now, species which have immigrated to another island would not disappear from their origins completely
and such species will appear to be present at both islands at same time. Hence, migration process would make the bad solution accept some features from the better solutions. Apart from the migration
process, mutation and elitism also takes place in BBO. Mutation rate is the probability of mutation on few habitats. Islands which will be having high HSI will have a low mutation rate and similarly
islands with low HSI will have a high mutation rate compared to the previous one. Henceforth, a good solution is barely chosen to be mutated because of the fact that it can last until the next
generation. This mutation will replace the old habitat having low HSI rate with new habitat. Solutions having low HSI would be more dominant if there is no mutation so that they can easily be trapped
to the local optima. Rate of mutation can be calculated as follows from Equation(6) below:
$m k = m m a x ( 1 − P k ) P m a x$
• m[k] = mutation rate,
• m[max] = maximum mutation rate,
• P[k] = probability of number of species,
• P[max] = maximum probability,
New habitat from mutation will replace the old habitat and with elitism best solution found earlier would continue to remain. Mutations might take place on all the solutions except the one having
best solution with the highest probability (P[k]).
Let us consider at time
, the recipient island has
species with probability
), and
are consecutively the immigration and emigration rates at the presence of
species on that particular island. Then the variation from
) to
t +
) can be described in Equation(7) below:
$P S ( t + Δ t ) = P S ( t ) ( 1 − λ S Δ t − μ S Δ t ) + P S − 1 ( t ) λ S − 1 Δ t + P S + 1 ( t ) μ S + 1 Δ t .$
Thus, using the known values of
) and
$P ˙$[S]
), the value of
t +
) given in Equation (7) can be approximated as:
$P S ( t + Δ t ) = P S ( t ) + P ˙ S ( t ) Δ t .$
The above equation (Equation (8)) will be used in the programming of BBO for calculating
t +
The algorithm for the BBO is shown below in
Figure 3
6. Optimization of Mrr and Ra Using PSO and BBO
The PSO technique incepts with the setting of PSO parameters which determines the performance of the PSO algorithm. The values of parameters were taken from the work of Hu and Eberhart [
]. Therefore,
= 0.5 + (rand/2),
= 1.49445, number of iterations =100 and population size = 50 particles.
As a second step to PSO technique, initialization of the random position and velocity vectors is followed. The fitness values are derived from the following function given by Equation (9):
$M i n F ( x ) = − w 1 f 1 + w 2 f 2 .$
are normalized values of MRR and Ra. The values of inertia weights considered are 0.5 for MRR and 0.5 for Ra as we have given equal importance to both the parameters. The initial values have been
tabulated in the
Table 6
Finding pBest and gBest is a third step to the PSO technique. The lowest fitness value is selected as the gBest and for the first iteration the pBest will be same as gBest. The change in position is
then obtained using Equations (4) and (5) for each of the parameters. Table 8 shows the new position of particles after first iteration and the change in position of each particle has been tabulated
in the
Table 7
As can be observed from
Table 8
, particle 2 has the minimum Fitness Value. Therefore, the pBest for each parameter will be the values corresponding to particle 2. However, it is higher than the initial gBest. Hence the gBest
achieve is not updated. The process is continued for 100 iterations and the final optimised value of gBest gives the final solution.
A similar process was adopted for the multiple objective optimization using BBO. The HSI or fitness value was calculated and similar to PSO the best values were updated after each iteration and the
iteration is continued until the final optimised value is obtained.
7. Results and Discussion
Repeating the above steps of PSO until the termination condition takes place (maximum number of iterations) we get the optimised value of MRR and Ra. Final values in the corresponding dimensions of
the particle, i.e., Gbest[A] = 40, Gbest[B] = 110, Gbest[C] = 230, Gbest[D] = 50 (GBEST archive), are selected as optimal values. The values corresponding to the optimised value of MRR & Ra for the
various input parameters are: A = 40; B = 110; C = 230; D = 50. MRR obtained through these parameters 235 mm^3/min and Ra is 18.4 µm.
Similarly, running the algorithm for BBO technique until the termination condition takes place (maximum number of iterations) we get the optimised value of MRR and Ra. Final values in the
corresponding dimensions of the particle, are selected as optimal values. The values corresponding to the optimized MRR & Ra for the various input parameters are: A = 40; B = 100; C = 210; and D =
50. MRR 242 mm^3/min and Ra 17.98 µm were obtained using these parameters.
Figure 4
below shows the 3D contour graphs of the relation between different input and output process parameters. It can be clearly inferred from the graph that when current is increased MRR and Ra increases
linearly with the current, however, when the value of current is very large or low MRR and Ra shows non-linearity and increases slowly. It can be explained with the phenomenon that when current is
low, amount of heat generating from the spark entering work piece is also very small and is easily affected by cooling effect of dielectric environment.
When the impacts of various input process parameters and their response or their effects on machining parameters are considered, the global effects of these responses can be understood from
Figure 5
given below. It can be clearly observed that Pulse Current has the major impact or effect on MRR as well as Ra. Other parameters, like Pulse on time and Pulse off time, have low effects compared to
Pulse Current whereas Gap Voltage has the least effect of all the parameters considered. Because of the duty cycle in EDM (ratio of pulse on time and the total cycle time) it is fairly understood
that Pulse on time and pulse off time have considerable impact.
Error in experimental and optimized data for MRR is 1.26% and for Ra it is 1.36 through PSO as shown in
Table 9
, whereas a relatively low percentage error was obtained using BBO technique gave 0.83% in MRR and 0.166% for Ra which are shown in
Table 10
. A relatively low percentage error of 1.26% − 0.83% = 0.43% for MRR and 1.63% − 0.66% = 0.97% for Ra was obtained with BBO when compared with PSO. A lower surface roughness and a greater MRR was
obtained using the BBO technique and that too more accurately.
The surface morphology of the work-piece before machining and after machining is studied using SEM images shown in
Figure 6
a,b, respectively. It is evident from the
Figure 6
a that before machining, the work surface is almost smooth and after machining using initial parameters is rough as compare to previous figure.
Figure 6
c shows the SEM image of the workpiece made by using optimal parameters using BBO where as
Figure 6
d depicts the same for PSO. It is observed that the surface roughness of optimum condition is better than the initial condition in both cases but between two BBO provides a smoother surface.
In the current work, 100 runs were carried out in the PSO and BBO algorithms. The PSO reaches a good fitness function as can be observed in
Figure 7
; in iteration 40 and after that there is no significant improvement. The same can be said for the BBO technique but at an even lower iteration of around 30. Moreover, it can also be concluded, from
Figure 8
, that the processing time for BBO is less when compared with that of PSO because of the lesser complexity in the BBO algorithm and simple formula to compute fitness or HSI function. Hence BBO is a
better choice compared with that of PSO since it requires less computation time which means less production time and henceforth production cost is reduced.
8. Conclusions
In the present paper, the multiple-objective optimization of the MRR and Ra for the EDM process of EN31 was carried out using the PSO and BBO algorithm. The optimum value of MRR and Ra were
established. It can beconcluded that the BBO has outperformed PSO in every aspect like having better optimized values with less relative percentage error. Even the computation time was less when in
BBO compared with that of PSO. Better optimized values were obtained like greater MRR and less Ra was obtained with BBO which was the main goal of the whole experiment. The solution obtained with PSO
and BBO were verified and validated with experimental results. The multiple objective optimization can be used to fulfil the specific requirement of the manufacturing unit. Therefore, the BBO
algorithm proved to be a cost effective solution for establishing the optimum values of input process parameters for enhancing the machining performance of the EDM process.
9. Scope of Future Research
PSO and BBO can be further used for multi-objective optimization for MRR and Ra combined with additional responses like EWR (Electrode Wear Ratio), overcut, etc. Further, the optimization can be
performed by taking into account more input parameters like flushing pressure, electrode material, etc. The machining optimization of EDM for composite materials can be investigated with PSO and BBO.
It would be quite interesting to note how performance of BBO could be compared with other evolutionary optimization techniques like GA, DE, etc. Moreover PSO and BBO algorithm can be used for the
optimization of process parameters for other non-conventional machining processes such as ultrasonic machining, Electro Chemical Machining, etc.
Author Contributions
Both authors have contributed equally towards the presented research work.
The authors sincerely acknowledge the comments and suggestions of the reviewers that have been instrumental for improving and upgrading the chapter in its final form.
Conflicts of Interest
The authors declare no conflict of interest.
Figure 4. 3D contour plots or surfaces showing relationship between different input and output process parameters (a) MRR with Gap Voltage and Pulse Current; (b) MRR with Pulse-on-time and
Pulse-off-time; (c) Ra with Gap Voltage and Pulse Current;and (d) Ra with Pulse-on-time and Pulse-off-time.
Figure 6. SEM images: (a) before machining; (b) after machining with initial parameters; (c) after machining with optimal parameters computed using BBO; and (d) after machining with optimal
parameters computed using PSO.
Elements C Mn Si P S Cr Fe
Chemical Composition (wt. %) 1.07 0.58 0.32 0.04 0.03 1.12 96.84
Thermal Conductivity (w/mk) 46.6
Density (gm/cc) 7.81
Electrical Resistivity (ohm-cm) 0.0000218
Specific heat capacity (j/gm-°C) 0.475
Copper (99% Pure)
Thermal Conductivity (w/mk) 391
Density (gm/cc) 1083
Electrical Resistivity (ohm-cm) 1.69
Specific heat capacity (j/gm-°C) 0.385
Machining Conditions
Machine Used CNC EDM (EMT 43) (Electronica)
Electrode Polarity Positive
Dielectric EDM Oil
Work piece Oil Hardened Non Shrinking Steel (48–50 HRC)
Electrode Electrolytic Copper (99.9% Purity)
Flushing Condition Pressure Flushing through 6 mm hole through work piece
S. No. IP (A) [A] T[on] (µs) [B] T[off] (µs) [C] V (V) [D] MRR mm^3/min (Ra) (µm) S. No. IP (A) [A] T[on] (µs) [B] T[off] (µs) [C] V (V) [D] MRR mm^3/min (Ra) (µm)
1 1 5 11 60 1.2 1.97 26 6 200 425 50 25.0 10
2 1 10 22 55 2.5 2.4 27 10 50 107 50 50.0 10
3 1 20 43 55 2.1 3.1 28 10 75 160 50 48.0 10
4 1 30 64 55 2.3 2.7 29 10 100 213 50 60.0 11.6
5 1 50 107 55 3.1 2.9 30 10 150 319 50 58.0 13.1
6 1.5 5 11 60 2.0 2.4 31 10 200 425 50 56.0 14.4
7 1.5 10 22 55 4.9 3.0 32 20 100 213 50 133.0 15
8 1.5 20 43 55 4.7 3.1 33 20 150 319 50 121.0 16.8
9 1.5 50 107 55 6.2 3.1 34 20 200 425 50 132.0 18.0
10 1.5 100 213 55 4.0 3.3 35 20 500 1063 50 124.0 23.7
11 2 5 11 60 2.3 2.6 36 30 150 319 50 182.0 18.8
12 2 10 22 55 7 2.6 37 30 200 425 50 174.0 20.5
13 2 20 43 55 8 3 38 30 500 1063 50 187.0 27.4
14 2 50 107 55 9 3.6 39 30 1000 2125 50 158.0 34.2
15 3 5 11 50 7.8 2.2 40 40 100 213 50 244.0 18
16 3 10 22 50 11.5 2.7 41 40 150 319 50 240.0 20.4
17 3 20 43 50 12.6 3.7 42 40 200 425 50 218.0 22.3
18 3 50 107 50 13.7 4.9 43 40 500 1063 50 270.0 29.6
19 3 100 213 50 10.3 6 44 40 1000 2125 50 216.0 36.5
20 6 5 11 50 17.0 2.8 45 40 2000 4250 50 240.0 45
21 6 10 22 50 26.0 3.6 46 50 200 425 50 330.0 25
22 6 20 43 50 29.0 4.5 47 50 400 850 50 350.0 32
23 6 50 107 50 31.0 6.3 48 50 500 1063 50 310.0 32
24 6 100 213 50 32.0 7.7 49 50 1000 2125 50 310.0 41
25 6 150 319 50 28.0 8.7 50 50 2000 4250 50 300.0 50
Particle A B C D MRR Ra Fitness Value
1 30 500 1063 50 187 27.4 0.6814
2 40 1000 2125 55 216 36.5 0.5038
3 50 2000 4250 60 300 50 0.5672
Particle A B C D
1 2.92 60 75 5.5
2 2.4 30 35 −4.5
3 0.5 0.5 0.5 0.05
Particle A B C D MRR Ra Fitness Value
1 32.92 560 1138 55.50 184.25 26.50 0.768
2 42.40 1030 2160 50.50 216.35 34.50 0.659
3 50.50 2000.50 4250.5 60.05 302.30 48.75 0.724
I T[on] T[off] V MRR Ra
40 110 230 50 Experimental Optimized %Error Experimental Optimized %Error
238 235 1.26 18.4 18.1 1.63
I T[on] T[off] V MRR Ra
40 100 210 50 Experimental Optimized %Error Experimental Optimized %Error
240 242 0.83 18.01 17.98 0.166
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http:/
Share and Cite
MDPI and ACS Style
Faisal, N.; Kumar, K. Optimization of Machine Process Parameters in EDM for EN 31 Using Evolutionary Optimization Techniques. Technologies 2018, 6, 54. https://doi.org/10.3390/technologies6020054
AMA Style
Faisal N, Kumar K. Optimization of Machine Process Parameters in EDM for EN 31 Using Evolutionary Optimization Techniques. Technologies. 2018; 6(2):54. https://doi.org/10.3390/technologies6020054
Chicago/Turabian Style
Faisal, Nadeem, and Kaushik Kumar. 2018. "Optimization of Machine Process Parameters in EDM for EN 31 Using Evolutionary Optimization Techniques" Technologies 6, no. 2: 54. https://doi.org/10.3390/
Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details
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Changes of State as You May Know, Matter Occurs in Three Familiar States-solid Liquid, and Gas . Suppose You Leave a Small Puddle of | Question AI
Changes of State As you may know, matter occurs in three familiar states-solid liquid, and gas . Suppose you leave a small puddle of liquid water on the kitchen counter.When you come back two hours
later, the puddle is gone. Has the liquid water disappeared? No, a physical change happened. The liquid water changed into water vapor (a gas) and mixed with the air.A change in state, such as from a
solid to a liquid or from a liquid to a gas, is an example of a physical change.
Not your question?Search it
Kerry PittmanElite · Tutor for 8 years
### The change is a physical change.
## Step 1: Understanding the Concept of Physical Change<br />### The problem describes a scenario where liquid water evaporates into water vapor, which is a gas. This is an example of a physical
change, specifically a change of state. In a physical change, the substance's physical properties are altered, but its chemical identity remains the same.<br /><br />## Step 2: Identifying the Change
of State<br />### In this case, the liquid water (a liquid state) has changed to water vapor (a gas state). This change from liquid to gas is due to the process of evaporation.<br /><br />## Step 3:
Confirming the Type of Change<br />### The water molecules gain enough energy to overcome the forces holding them together in the liquid state, allowing them to move freely and become a gas. This is
a physical change because the water molecules themselves do not change, only their state does.
Step-by-step video
Changes of State As you may know, matter occurs in three familiar states-solid liquid, and gas . Suppose you leave a small puddle of liquid water on the kitchen counter.When you come back two hours
later, the puddle is gone. Has the liquid water disappeared? No, a physical change happened. The liquid water changed into water vapor (a gas) and mixed with the air.A change in state, such as from a
solid to a liquid or from a liquid to a gas, is an example of a physical change.
All Subjects Homework Helper | {"url":"https://www.questionai.com/questions-tGQiFL4uKD/changes-state-know-matter-occurs-inthree-familiar","timestamp":"2024-11-15T04:00:07Z","content_type":"text/html","content_length":"86050","record_id":"<urn:uuid:ccbfbcd0-f097-4f1b-8cec-fd64e08999ff>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00618.warc.gz"} |
Algebra Homework
Algebra, statistics, and math are three of the most common areas where students struggle when it comes to assignments. These three subjects are pretty much intertwined. It might be impossible to
study and get through one without having to learn about the others. You might also find yourself in a position where you need algebra homework help. This, you should not have to worry about,
especially when you have useful sources of support.
There are some good algebra homework helpers that you can reach out to whenever you are struggling, and be sure they will offer you the best assistance. You might not have a lot of time remaining
before your assignment is due for submission, and this is why you need to get assignment assistance in algebra as soon as possible. Here are some brilliant ideas to work with:
One of the best places where you can get algebra homework help free is the school library. In fact, you can check any library other than your school’s library. You can access online libraries and
search for the algebra unit you are struggling with. You will find examples that are similar to the question you need to answer. Study these carefully and then use the same insight to tackle your
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You can also benefit by forming a study group to get pre algebra homework help. With your classmates, there is so much that you can do together. You will be brainstorming based on the information you
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Chess and Mathematics
This page, which is currently under construction, will highlight a few ways in which mathematics relates to the realms of chess:
• There is a legend that, when the inventor of chess showed his invention to the king of Persia, the king asked him what he would like as a reward. The inventor said that he "just" wanted one grain
of wheat for the first square on the chessboard, two grains of wheat for the second square, four for the third square, and so on, doubling the amount on each square. The king agreed, thinking
that to be a small reward for inventing such an interesting game, but before he had calculated the amount required for the first 32 squares of the 64-square chessboard, he changed his mind and
had the inventor executed instead. How much grain would he have had to pay the inventor (see the exponential growth page for answer)? This legend illustrates how quickly a quantity can grow
• Combinatorics: There are 400 different ways for both sides to play the first move in a game of chess. There are 197,281 ways for both sides to play the first two moves. There are an estimated
318,979,564,000 ways of playing the first four moves. There are about 169,518,829,100,544,000,000,000,000,000 ways of playing the first ten moves.
• Recreational mathematics: See, for example, Eight Queens Puzzle, Knight's Tour, and Knight's Tour Magic Square.
You can view a large number of chess games on allthatstuff.info. | {"url":"https://mathlair.allfunandgames.ca/chess.php","timestamp":"2024-11-12T16:06:19Z","content_type":"text/html","content_length":"3797","record_id":"<urn:uuid:88c3885f-5d06-4780-a099-c9edad87a5af>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.63/warc/CC-MAIN-20241112145015-20241112175015-00119.warc.gz"} |
000 mean
SAP är per definition också namnet på ERP-programvaran (Enterprise Från och med 2010 har SAP mer än 140 000 installationer över hela världen, över 25
Angel number 000 means there is much spiritual guidance for new possibilities. 000 reveals the potential of uncertainties and choices.
2020-09-12 2020-03-10 2018-08-02 I have a device which outputs the time in the format hh:mm:ss.000, e.g., 00:04:58.727 and I need to convert these to milliseconds. I can't change the way the device
outputs the times so I have to do it in Excel, but I don't know VB so am looking for a cell-by-cell solution. 2020-06-05 To perform especially well at something. When we were in school, my brother
excelled at art, while I was always better at subjects like science and math. Public speaking is something that Kristy really excels at. See also: excel.
The Microsoft Excel ROUND function returns a number rounded to a specified number of digits. The ROUND function is a built-in function in Excel that is categorized as a Math/Trig Function.It can be
used as a worksheet function (WS) in Excel. Note: Excel uses a default order in which calculations occur. If a part of the formula is in parentheses, that part will be calculated first. 3. To
decrease a number by a percentage, simply change the plus sign to a minus sign.
Remember, when you format a number, you're not changing the numeric value.
19 Sep 2018 This tutorial teaches you how to format numbers in Excel using the VBA Remember that # means “display if there, but do not display if not there”. Thus Formatted and Unformatted using 000
as the NumberFormat code.
The multiple to use for rounding is provided as the significance argument. If the number is already an exact multiple, no rounding occurs and the original number is returned. The only thing you
should keep in mind is that the Equal to logical operator in Excel is case-insensitive, meaning that case differences are ignored when comparing text values.
Total land area, 1 000 sq km Arable land, per cent of land area Forest Number Marriages Divorces Per 1 000, mean population Marriages Divorces To download data from the database you can choose Excel
or PC-Axis.
Then the following formula tells Excel to return a count of students who scored more than 85 marks in both the quarters. So we can see in the above screenshot that the COUNTIFS formula counts the
number of students who have scored more than 85 marks in both the quarters. To perform especially well at something. When we were in school, my brother excelled at art, while I was always better at
subjects like science and math.
I can't change the way the device outputs the times so I have to do it in Excel, but I don't know VB so am looking for a cell-by-cell solution. 2020-06-05 To perform especially well at something.
When we were in school, my brother excelled at art, while I was always better at subjects like science and math. Public speaking is something that Kristy really excels at. See also: excel. Farlex
Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.
Neo fundamentalism
You can use a custom number format: Hope that clarifies things. =LEFT (REPT ("0",9-LEN (B2))&B2,3)&"-"&MID (B2,LEN (B2)-5,2)&"-"&RIGHT (B2,4) Thank you for your responses. The SSN is the primary key
between two files I am joining through Power Query.
In your example, the actual How to format numbers in thousands, million or billions in Excel?
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Angel number 000 means there is much spiritual guidance for new possibilities. 000 reveals the potential of uncertainties and choices.
SGD to SEK - Convert Singapore 000 011 011 101 100 010 010 111 011 001 100 001 101 101 011 000 010 011 The result of step 2 is encoded using tail-biting, meaning we begin the shift apps excel in
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När träder lagar i krafttelefonnummer privatpersoner danmark
18 Dec 2018 Number Formatting feature in Excel allows modifying the appearance of cell values, without changing their actual values. Currency formatting
I have written before about other repeating numbers like 46, 147, 1010, 111, 11:11, 12:12, 222, 333, 444, 555, 666, 777, 888, and 999 if you see those numbers as well. When Excel displays a number in
scientific (exponential) notation, it means that it takes part of a number before E and multiplies it by 10 to the power of a number after E. Here’s a simple example to illustrate it. 123,456,789 =
1.23 * 10^8. The number will stay the same, but it will show it in the shortened, less precise form. The number 000 is a message of love from our Creator, carried to us by our guardian angels. | {"url":"https://enklapengarjdny.web.app/22433/43780.html","timestamp":"2024-11-06T02:36:31Z","content_type":"text/html","content_length":"10304","record_id":"<urn:uuid:70fd7e51-d782-40b8-a419-045f1e9faabe>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00887.warc.gz"} |
luacas – A computer algebra system for users of LuaLaTeX
This package provides a portable computer algebra system capable of symbolic computation, written entirely in Lua, designed for use in LuaLaTeX.
Features include: arbitrary-precision integer and rational arithmetic, factoring of univariate polynomials over the rationals and finite fields, number theoretic algorithms, symbolic differentiation
and integration, and more. The target audience for this package are mathematics students, instructors, and professionals who would like some ability to perform basic symbolic computations within LaTe
X without the need for laborious and technical setup.
Sources /macros/luatex/latex/luacas
Bug tracker https://github.com/cochraef/LuaLaTeX-CAS/issues
Repository https://github.com/cochraef/LuaLaTeX-CAS
Version 1.0.2 2023-05-26
Licenses The LaTeX Project Public License 1.3c
Maintainer Timothy All
Evan Cochrane
TDS archive luacas.tds.zip
Contained in TeXLive as luacas
MiKTeX as luacas
Topics LuaTeX
Download the contents of this package in one zip archive (778.5k).
Community Comments
Maybe you are interested in the following packages as well.
Package Links | {"url":"https://ctan.org/pkg/luacas","timestamp":"2024-11-08T22:45:45Z","content_type":"text/html","content_length":"17902","record_id":"<urn:uuid:58f02c3b-d2ed-49e2-bd5b-846b5d3e920f>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00387.warc.gz"} |
Transmission Line Properties That Affect Impedance—Hidden Features
Here and in several other articles published on the Altium Resource section of the company’s website, the topic of transmission line impedance has been addressed from a number of different
perspectives. I have addressed transmission line impedance previously in my article, The Evolution Of Simulation Technology and Impedance and, it might seem that we may have exhausted the field of
potential information that can be provided on impedance, however, in truth, some features were only addressed in passing. This article will elaborate on those features and their effects along with
basic equations that are used in controlling transmission line impedance.
As discussed in previous articles, the four main variables that determine the impedance of a transmission line on a surface layer include:
• Height of the trace above the plane over which it travels.
• The width of the trace.
• The thickness of the trace.
• The insulating material used to support the trace.
Once the above four variables are known, it is possible to determine which features in a PCB will have a relevant effect on impedance. These features include:
• Changes in trace width in the same layer. This is generally referred to as trace necking.
□ Trace necking refers to the reduction of the trace width when it approaches a narrower pad such as that found on an SMD (surface mount device) or a through-hole that has a diameter that is
less than the width of the trace.
• Changes in trace thickness.
• Changes in height above the plane.
• Stubs along the transmission line.
• Loads along the transmission line.
• Connector transitions.
• Poorly-matched terminations.
• No terminations.
• Larger power plane discontinuities.
• Changes in the relative dielectric constant.
As noted in previous articles, right-angle bends and vias are not on the above list because neither one of these features is a significant source of impedance mismatch.
There are a few equations that are helpful in calculating impedance. They are presented below. As noted previously, the impedance of a transmission line is determined by the capacitance and
inductance that is distributed along the length of the transmission line. And, the equation used for calculating impedance is repeated here in Equation 1.
Equation 1. The Impedance Equation
In the above, Z[0] is the impedance in ohms; jωL[0] is the parasitic inductance in henrys per unit length, jωC[0] is the parasitic capacitance in farads per unit length and R[0] is skin effect loss
(which can be ignored until you get to very high frequencies). G[0] is the loss in the dielectric. As noted above, changing either the parasitic inductance or the parasitic capacitance will change
the impedance of the transmission line. It’s also been demonstrated that changes in impedance cause signal reflections. For convenience, the reflection equation is repeated in Equation 2.
Equation 2. The Reflection Equation
This equation predicts the percentage of the incident EM field that will be reflected back to the source based on the two impedances on each side of a change where Z[l] is the downstream impedance
and Z[0] is the upstream impedance. The equation reflects the voltage amplitude of the reflection.
Based on Equation 1, it is not obvious which variables will have an effect on impedance. Equation 3 is the classic surface microstrip equation. It illustrates the variables in a PCB that determine
Equation 3. The Classic Surface Microstrip Impedance Equation
This equation is included for illustration purposes only so that the variables can be shown. In a separate article following this one, it will be shown that this equation as well as other equations
used to calculate impedance have a limited range over which they are valid. More accurate methods are available and some have been discussed in previous articles. The article following this one will
also contain other methods for determining impedance.
The common characteristics of the features noted above is that they can have a measurable effect on one or both of the variables in Equation 1, parasitic inductance or parasitic capacitance. We can
take those features and show the variables that they affect.
• Change in trace width in the same layer—C[0]
• Change in trace thickness—C[0]
• Change in trace height above the plane—C[0]
• Stubs along the transmission line—C[0]
• Loads along the transmission line—C[0]
• Connector transitions—C[0]
• Large power plane discontinuities—C[0]
• Changes in relative dielectric constant—C[0]
• Poorly matched terminations
• No terminations
As can be seen, with the exception of poorly matched terminations and no terminations, all of the sources of impedance mismatch are caused by something that changed the parasitic capacitance. Within
the limits of trace dimensions in PCBs, compared to C[0], L[0] is relatively constant. This helps when it comes time to design controlled impedance signal paths or troubleshoot impedance problems.
Once it is understood that virtually all impedance changes along the length of a transmission line are due to changes in parasitic capacitance, it becomes easier to manage those changes and create
good impedance control.
Table 1 shows the relative dielectric constant of the laminate that is commonly known as FR-4.
Table 1. Laminate Information for Laminate Commonly Called FR-4
Not only does the relative dielectric constant change with frequency, it also varies with the amount of glass and resin used to make the laminate. As can be seen, there are four ways to make a 4-mil
thick piece of laminate; three ways to make a 5-mil thick piece of laminate and four ways to make a 6-mil thick piece of laminate. Also, note that the ratio of glass to resin is different in each of
these formulations as is the relative dielectric constant. If a PCB stackup is designed to use one of these formulations and the fabricator uses one of the others, the impedance will not come out as
expected. This is the most common reason that changing fabricators results in PCBs with different characteristics. To avoid this problem, it is necessary to specify, on the fabrication drawing, which
laminate formulation is required in each opening in the stackup.
Understanding the variables and features within a PCB that can affect transmission line impedance makes it easier to design for impedance control right the first time, and easier to troubleshoot any
impedance issues that may occur during the design or during the fabrication processes.
Have more questions? Call an expert at Altium or read on to learn more about incorporating impedance calculations into your design rules with Altium Designer^®.
1. Ritchey, Lee W. and Zasio, John J., “Right The First Time, A Practical Handbook on High-Speed PCB and System Design, Volume 1.” | {"url":"https://resources.altium.com/p/transmission-line-properties-affect-impedance-hidden-features","timestamp":"2024-11-03T13:13:50Z","content_type":"text/html","content_length":"113965","record_id":"<urn:uuid:4db7121b-ebbf-40d5-8d52-b2732a8579d3>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00553.warc.gz"} |
Understanding the True Cost of a Car Loan - Powerful and Useful Online Calculators
Understanding the True Cost of a Car Loan
Have you ever wondered what the true cost of a car loan is? Understanding the nuances behind auto loans can help you make informed decisions, fitting your new vehicle neatly into your budget without
unexpected surprises.
Understanding the True Cost of a Car Loan
For many families, purchasing a vehicle represents the second largest expense after housing. Navigating the financial landscape of buying a new or used car wisely can significantly impact your
monthly expenses. Most financial advisors suggest that auto loan payments should stay between 10% and 15% of your monthly income. However, this is just a guideline and varies based on individual
circumstances. Calculating your monthly payment and the total cost of an auto loan is essential, and tools like car loan calculators can be incredibly helpful.
Consider Tony, who just landed a new job in another city. While his current role allows him to commute by train, his new position will require a car. He figures he can manage a monthly car payment of
$500. Upon finding a vehicle listed at $25,000 at a local dealership, Tony needs to estimate the monthly payments beforehand to ensure they are within his budget. Here is where a car loan calculator
comes into play.
Key Factors: Loan Term and Interest Rate
Understanding two primary factors — loan term and interest rate — is crucial as they significantly impact the monthly payment on a car loan.
Interest Rate
The interest rate is what lenders charge for letting you borrow money. The goal is to secure the lowest interest rate possible. This might require speaking with multiple lenders to find the best
Loan Term
The loan term is the duration over which you plan to repay the loan. Most loans for automobiles range from 36 to 84 months (3 to 7 years). While a longer loan term reduces the monthly payment amount,
it increases the total cost due to additional interest charges over time.
Understanding the Car Loan Payment Formula
The formula for calculating a car loan payment can be complex, but knowing it can give you a better understanding of how your monthly payments are determined.
[ \text = \frac{(R \times A)}{(1 – (1+R)^{-n})} ]
• PMT = Monthly Loan Payment
• A = Amount Borrowed
• R = Interest Rate
• n = Loan Term (in months or years)
Note: For monthly payments, use the monthly interest rate. You can calculate this by dividing the annual interest rate by 12.
Example Calculation
For instance, if your bank charges a 5% annual interest rate on your car loan, the monthly interest rate would be:
[ \frac = 0.416% ]
How to Use the Car Loan Calculator
Our car loan calculator simplifies the process by taking a few key inputs and quickly showing the impact on your personal finances.
1. Enter the Loan Amount: This includes the car’s sale price minus any down payment or trade-in value.
2. Enter the Loan Term: Specify the duration in months or years.
3. Enter the Interest Rate: Use the annual interest rate for calculation.
4. Calculate and Review: Upon clicking the “Calculate” button, the calculator will display the monthly payment amount, total cost of the loan (principal plus interest), and total interest paid over
the life of the loan. You can also generate an amortization schedule.
Real Example Calculation
Imagine you and your spouse welcome a new baby, making your current car too small. You decide to trade in your car for a new minivan priced at $26,000. The dealership offers $4,000 for your old car,
reducing the loan amount to $22,000. Your bank offers a 5-year loan at a 6% interest rate. Using the loan calculator:
• Loan Amount: $22,000
• Loan Term: 5 years (60 months)
• Interest Rate: 6%
After calculation, your monthly payment would be $425.32. The total loan cost, including $3,519.30 in interest, would be $25,519.30.
Key Benefits and Helpful Tips
Key Benefits:
1. Ease of Use: No need to memorize formulas. The calculator handles the complexities for you.
2. Comparative Analysis: Quickly compare multiple loan options to find the best fit for your family.
3. Total Cost Awareness: Understand not just the monthly payment, but the overall cost of the loan, including interest.
Helpful Tips:
1. Interest Considerations: Extending your loan term lowers monthly payments but increases total interest paid. To minimize this, negotiate for a lower interest rate, opt for a shorter loan term, or
borrow less.
2. Down Payment: A larger down payment can significantly reduce your monthly payments and overall loan cost.
3. Additional Costs: Dealerships often add extra costs such as taxes and fees. To cover these, factor in an additional 5-10% of the sale price. For example, a car costing $10,000 might actually
require borrowing $10,500 to $11,000.
Summary Table
Expense Amount
Car Price $26,000
Trade-in Value -$4,000
Loan Amount $22,000
Loan Term 5 years (60 months)
Interest Rate 6%
Monthly Payment $425.32
Total Cost (Principal+Interest) $25,519.30
Total Interest Paid $3,519.30
Understanding the true cost of a car loan plays a pivotal role in managing your finances efficiently. By leveraging tools like car loan calculators and being aware of crucial factors such as interest
rates and loan terms, you can make informed decisions that align with your financial situation. Remember to consider additional costs and think strategically about down payments and loan durations.
Making these smart choices ensures you avoid unmanageable debt and enjoy your new vehicle without financial strain. | {"url":"https://calculatorbeast.com/understanding-the-true-cost-of-a-car-loan/","timestamp":"2024-11-13T11:30:04Z","content_type":"text/html","content_length":"130191","record_id":"<urn:uuid:76d079ef-bab5-44ad-8319-15d3e5c980e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00733.warc.gz"} |
Fiona paid $1.50 for 3/4 pounds of almonds. Supposed Fiona wanted a pound. How much would that cost?
The cost of one pound would be $2Step-by-step explanation:The given:Fiona paid $1.50 for 3/4 pounds of almondsFiona wanted a poundWe need to find the cost of 1 poundThe cost of [tex]\frac{3}{4}[/tex]
pound is $1.50To find the cost of one pound use the ratio method→ The cost : Pound→ 1.50 : [tex]\frac{3}{4}[/tex]→ x : 1By using cross multiplication∵ [tex]\frac{3}{4}[/tex] × x = 1.50 × 1∴ [tex]\
frac{3}{4}[/tex] x = 1.50- Divide both sides by [tex]\frac{3}{4}[/tex]∴ x = 2The cost of one pound would be $2Learn more:You can learn more about the ratio in brainly.com/question/2707032# | {"url":"http://math4finance.com/general/fiona-paid-1-50-for-3-4-pounds-of-almonds-supposed-fiona-wanted-a-pound-how-much-would-that-cost","timestamp":"2024-11-06T22:14:42Z","content_type":"text/html","content_length":"29977","record_id":"<urn:uuid:a0d9f478-723e-4aad-96e1-fc374982b0d7>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00784.warc.gz"} |
Area of an Equilateral Triangle- Formula, Definition, Derivation, Examples
The area of an equilateral triangle is the amount of space that it occupies in a 2-dimensional plane. To recall, an equilateral triangle is a triangle in which all the sides are equal and the measure
of all the internal angles is 60°. So, an equilateral triangle’s area can be calculated if the length of its side is known.
Area of an Equilateral Triangle Formula
The formula for the area of an equilateral triangle is given as:
Area of Equilateral Triangle (A) = (√3/4)a^2
Where a = length of sides
Learn more about isosceles triangles, equilateral triangles and scalene triangles here.
Derivation for Area of Equilateral Triangle
There are three methods to derive the formula for the area of equilateral triangles. They are:
• Using basic triangle formula
• Using rectangle construction
• Using trigonometry
Deriving Area of Equilateral Triangle Using Basic Triangle Formula
Take an equilateral triangle of the side “a” units. Then draw a perpendicular bisector to the base of height “h”.
Area of Triangle = ½ × base × height
Here, base = a, and height = h
Now, apply Pythagoras Theorem in the triangle.
a^2 = h^2 + (a/2)^2
⇒ h^2 = a^2 – (a^2/4)
⇒ h^2 = (3a^2)/4
Or, h = ½(√3a)
Now, put the value of “h” in the area of the triangle equation.
Area of Triangle = ½ × base × height
⇒ A = ½ × a × ½(√3a)
Or, Area of Equilateral Triangle = ¼(√3a^2)
Deriving Area of Equilateral Triangle Using Rectangle Construction
Consider an equilateral triangle having sides equal to “a”.
• Now, draw a straight line from the top vertex of the triangle to the midpoint of the base of the triangle, thus, dividing the base into two equal halves.
• Now cut along the straight line and move the other half of the triangle to form the rectangle.
Here, the length of the equilateral triangle is considered to be ‘a’ and the height as ‘h’
So the area of an equilateral triangle = Area of a rectangle = ½×a×h …………. (i)
Half of the rectangle is a right-angled triangle as it can be seen from the figure above.
Thereby, applying the Pythagoras Theorem:
⇒ a^2 = h^2 + (a/2)^2
⇒ h^2 = (3/4)a^2
⇒ h = (√3/2)a ……………(ii)
Substituting the value of (ii) in (i), we have:
Area of an Equilateral Triangle
Deriving Area of Equilateral Triangle Using Trigonometry
If two sides of a triangle are given, then the height can be calculated using trigonometric functions. Now, the height of a triangle ABC will be-
h = b. Sin C = c. Sin A = a. Sin B
Now, area of ABC = ½ × a × (b . sin C) = ½ × b × (c . sin A) = ½ × c (a . sin B)
Now, since it is an equilateral triangle, A = B = C = 60°
And a = b = c
Area = ½ × a × (a . Sin 60°) = ½ × a^2 × Sin 60° = ½ × a^2 × √3/2
So, Area of Equilateral Triangle = (√3/4)a^2
Below is a brief recall about equilateral triangles:
What is an Equilateral Triangle?
There are mainly three types of triangles which are scalene triangles, equilateral triangles and isosceles triangles. An equilateral triangle has all the three sides equal and all angles equal to
60°. All the angles in an equilateral triangle are congruent.
Properties of Equilateral Triangle
An equilateral triangle is the one in which all three sides are equal. It is a special case of the isosceles triangle where the third side is also equal. In an equilateral triangle ABC, AB = BC = CA.
Some important properties of an equilateral triangle are:
• An equilateral triangle is a triangle in which all three sides are equal.
• Equilateral triangles also called equiangular. That means, all three internal angles are equal to each other and the only value possible is 60° each.
• It is a regular polygon with 3 sides.
• A triangle is equilateral if and only if the circumcenters of any three of the smaller triangles have the same distance from the centroid.
• A triangle is equilateral if and only if any three of the smaller triangles have either the same perimeter or the same inradius.
• The area of an equilateral triangle is basically the amount of space occupied by an equilateral triangle.
• In an equilateral triangle, the median, angle bisector and perpendicular are all the same and can be simply termed as the perpendicular bisector due to congruence conditions.
• The ortho-centre and centroid of the triangle is the same point.
• In an equilateral triangle, median, angle bisector and altitude for all sides are all the same and are the lines of symmetry of the equilateral triangle.
• The area of an equilateral triangle is √3 a^2/ 4
• The perimeter of an equilateral triangle is 3a.
Example Questions Using the Equilateral Triangle Area Formula
Question 1: Find the area of an equilateral triangle whose perimeter is 12 cm.
Given: Perimeter of an equilateral triangle = 12 cm
As per formula: Perimeter of the equilateral triangle = 3a, where “a” is the side of the equilateral triangle.
Step 1: Find the side of an equilateral triangle using perimeter.
3a = 12
a = 4
Thus, the length of side is 4 cm.
Step 2: Find the area of an equilateral triangle using formula.
Area, A = √3 a^2/ 4 sq units
= √3 (4)^2/ 4 cm^2
= 4√3 cm^2
Therefore, the area of the given equilateral triangle is 4√3 cm^2
Question 2: What is the area of an equilateral triangle whose side is 8 cm?
The area of the equilateral triangle = √3 a^2/ 4
= √3 × (8^2)/ 4 cm^2
= √3 × 16 cm^2
= 16 √3 cm^2
Question 3: Find the area of an equilateral triangle whose side is 7 cm.
Side of the equilateral triangle = a = 7 cm
Area of an equilateral triangle = √3 a^2/ 4
= (√3/4) × 7^2 cm^2
= (√3/4) × 49 cm^2
= 21.21762 cm^2
Question 4: Find the area of an equilateral triangle whose side is 28 cm.
Side of the equilateral triangle (a) = 28 cm
We know,
Area of an equilateral triangle = √3 a^2/ 4
= (√3/4) × 28^2 cm^2
= (√3/4) × 784 cm^2
= 339.48196 cm^2
Frequently Asked Questions
What is an Equilateral Triangle?
An equilateral triangle can be defined as a special type of triangle whose all the sides and internal angles are equal. In an equilateral triangle, the measure of internal angles is 60 degrees.
What does the Area of an Equilateral Triangle Mean?
The area of an equilateral triangle is defined as the amount of space occupied by the equilateral triangle in the two-dimensional area.
What is the Formula for Area of Equilateral Triangle?
To calculate the area of an equilateral triangle, the following formula is used:
A = ¼(√3a^2)
What is the Formula for Perimeter of Equilateral Triangle?
The formula to calculate the perimeter of an equilateral triangle is:
P = 3a
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1. WELL EXPLAINED
2. This website is so helpful
3. my son always comes here to get his answer . very good work !! | {"url":"https://byjus.com/maths/area-of-equilateral-triangle/","timestamp":"2024-11-11T23:40:42Z","content_type":"text/html","content_length":"613696","record_id":"<urn:uuid:1b6ac7f0-5d0c-4ae4-956f-c8f83c5729e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00728.warc.gz"} |
Consider Kite KlLMN 1)if M 2=63 What Is M 3? 7)if M Angle 3=31,what Is M Angle LMN… - Anti Vuvuzela
Consider Kite KlLMN 1)if M 2=63 What Is M 3? 7)if M Angle 3=31,what Is M Angle LMN…
June 27, 2023 Math No Comments
consider kite KlLMN
1)if m 2=63 what is m 3?
7)if m angle 3=31,what is m angle LMN?
If m 2=63, what is m 3?
Since Triangle LMN has two sides that are congruent, it is
an Isosceles triangle; it also follows that the base angles are congruent. Given
that the measure of one of the base angles, angle 2 = 63, the other base angle
which is MNL is also 63. Notice also that angle 3 is half the vertex angle LMN.
The sum of the measures of the angles in a triangle is 180.
[tex]63+63+2(Angle3) = 180[/tex]
[tex]2(Angle 3)=180-126=54[/tex]
[tex]Angle 3= \frac{54}{2} =27^{0} [/tex]
Angle 3 = 27 degrees
7. If m angle 3 = 31, what is m angle LMN?
The measure of LMN is twice the measure of angle 3. So,
Measure of angle LMN = 2(angle 3) = 2(31) = 62 degrees | {"url":"https://antivuvuzela.org/consider-kite-kllmn-1if-m-331/","timestamp":"2024-11-11T17:01:31Z","content_type":"text/html","content_length":"139842","record_id":"<urn:uuid:94fc7fc5-50f5-422a-b6cc-d6856125325d>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00165.warc.gz"} |
Selective School, OC, HSC Tutoring & Preparation | Wisdom Education
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HSC Mathematics Extension 2 Tutoring
HSC Mathematics Extension 2 course at Wisdom Education will assist students to gain critical thinking skills to solve complex problems in your exams.
The Mathematics 4 Unit course is defined in the same terms as the 3 Unit Course in other subjects. Thus it offers a suitable preparation for study of the subject at tertiary level, as well as a
deeper and more extensive treatment of certain topics than is offered in other Mathematics courses.
This syllabus is designed for students with a special interest in mathematics who have shown that they possess special aptitude for the subject. It represents a distinctly high level in school
mathematics involving the development of considerable manipulative skill and a high degree of understanding of the fundamental ideas of algebra and calculus. These topics are treated in some depth.
Thus the course provides a sufficient basis for a wide range of useful applications of mathematics as well as an adequate foundation for the further study of the subject.
The objectives of this syllabus are addressed through eight topics:
• Graphs
□ Basic Curves
□ Drawing graphs by addition and subtraction of ordinates
□ Drawing graphs by reflecting functions in coordinate axes
□ Sketching functions by multiplication of ordinates
□ Sketching functions by division of ordinates
□ Drawing graphs of various forms
□ General approach to curve sketching
□ Using graphs
• Complex Numbers
□ Arithmetic of complex numbers and solving quadratic equations
□ Geometric representation of a complex number as a point
□ Geometrical representations of a complex number as a vector
□ Powers and roots of complex numbers
□ Curves and Regions
• Conics
□ The Ellipse
□ The Hyperbola
□ The Rectangular Hyperbola
□ General descriptive properties of conics
• Integration
□ Decompostion practice
□ Derivation of formula for integration by parts
□ Further integration techniques
□ Integration by partial fractions
□ Integration by parts
□ Integration by substitution
□ Integration by trigonometric substitution
□ Partial fractions
□ Partial fractions and their integration
□ Reduction formula
□ Summary of Mathematics course integral calculus
□ The integrator (integrates any function + other applications)
• Volumes
□ Disk method for volume
□ Cylindrical Shell Method for volume
□ Computing the volume of water in a tipped glass
□ Volume by shells
• Mechanics
□ Calculating Orbital Motion
□ Forces for Mathematics Extension 2
□ Projectile and orbital motion
□ Resisted Motion and Circular Motion
• Polynomials
□ Integer roots of polynomials with integer coefficients
□ Multiple Roots
□ Fundamental Theorem of Algebra
□ Factoring Polynomials
□ Roots and Coefficients of a Polynomial Equation
□ Partial Fractions
• Harder 3 Unit topics.
□ Geometry of the Circle
□ Induction
□ Inequalities
TERM 4 / 2015 - HSC
bottom of page | {"url":"https://www.wisdomeducation.com.au/hsc-mathematics-ext-2","timestamp":"2024-11-12T05:15:08Z","content_type":"text/html","content_length":"509735","record_id":"<urn:uuid:a0ca0cc0-75cc-4a23-bc6b-a8ce4e1b7e01>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00074.warc.gz"} |
HP Forums
I found this site about the HP 35 history. I know that there are a lot of pages about the same matter and sorry if it was already posted here. I did not know how HP priced its products: the cost
multiplied by a factor of "PI" or "e", that was curious to me. For those who do not know the page, it´s very cool. | {"url":"https://www.hpmuseum.org/forum/archive/index.php?thread-10927.html","timestamp":"2024-11-03T00:52:45Z","content_type":"application/xhtml+xml","content_length":"23809","record_id":"<urn:uuid:ba13847a-2965-48fc-9ee0-55d4abaf8522>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00863.warc.gz"} |
Forecasting electricity prices with Amazon Chronos | fg-research
Forecasting electricity prices with Amazon Chronos¶
Chronos is a foundational model for zero-shot probabilistic forecasting of univariate time series [1]. The model converts a time series into a sequence of tokens through scaling and quantization. The
scaling procedure divides the time series by its mean absolute value, while the quantization process maps the scaled time series values to a discrete set of tokens using uniform binning.
The tokenized time series is then used by a large language model (LLM). The LLM takes as input a sequence of tokens and returns the predicted next token. Subsequent future tokens are generated in an
autoregressive manner by extending the initial input sequence with the previously generated tokens and feeding it back to the model. The generated tokens are then converted back to time series values
by inverting the quantization and scaling transformations.
Chronos was trained using the T5 model architecture [2], even though it is compatible with any LLM. The training was performed in a self-supervised manner by minimizing the cross-entropy loss between
the actual and predicted distributions of the next token, as it is standard when training LLMs. The data used for training included both real time series from publicly available datasets, as well as
synthetic time series generated using different methods.
In this post, we demonstrate how to use Chronos for one-step-ahead forecasting. We will use the US average electricity price monthly time series from November 1978 to July 2024, which we will
download from the FRED database, and generate one-month-ahead forecasts from August 2014 to July 2024. We will use expanding context windows, that is on each month we will provide Chronos all the
data up to that month, and generate the forecast for the next month.
We will compare Chronos' forecasts to the rolling forecasts of a SARIMA model which is re-trained each month on the same data that was provided to Chronos as context. We will find that Chronos and
the SARIMA model have comparable performance.
We start by installing and importing all the dependencies.
pip install git+https://github.com/amazon-science/chronos-forecasting.git fredapi pmdarima
import warnings
import transformers
import torch
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from datetime import datetime
from chronos import ChronosPipeline
from pmdarima.arima import auto_arima
from statsmodels.tsa.statespace.sarimax import SARIMAX
from tqdm import tqdm
from fredapi import Fred
from sklearn.metrics import mean_absolute_percentage_error, mean_absolute_error, root_mean_squared_error
Next, we download the time series from the FRED database. We use the Python API for FRED for downloading the data.
If you don’t have a FRED API key, you can request one for free at this link.
# set up the FRED API
fred = Fred(api_key_file="api_key.txt")
# define the time series ID
series = "APU000072610"
# download the time series
data = fred.get_series(series).rename(series).ffill()
The time series includes 549 monthly observations from November 1978 to July 2024. The time series had one missing value in September 1985, which we forward filled with the previous value.
We generate the forecasts over a 10-year period (120 months) from August 2014 to July 2024.
# date of first forecast
start_date = "2014-08-01"
# date of last forecast
end_date = "2024-07-01"
We use the pmdarima library for finding the best order of the SARIMA model using the data up to July 2014.
# find the best order of the SARIMA model
best_sarima_model = auto_arima(
y=data[data.index < start_date],
For each month in the considered time window, we train the SARIMA model with the identified best order on all the data up to that month, and generate the forecast for the next month.
# create a list for storing the forecasts
sarima_forecasts = []
# loop across the dates
for t in tqdm(range(data.index.get_loc(start_date), data.index.get_loc(end_date) + 1)):
# extract the training data
context = data.iloc[:t]
# train the model
with warnings.catch_warnings():
sarima_model = SARIMAX(
trend="c" if best_sarima_model.with_intercept else None,
# generate the one-step-ahead forecast
sarima_forecast = sarima_model.get_forecast(steps=1)
# save the forecast
"date": data.index[t],
"actual": data.values[t],
"mean": sarima_forecast.predicted_mean.item(),
"std": sarima_forecast.var_pred_mean.item() ** 0.5,
# cast the forecasts to data frame
sarima_forecasts = pd.DataFrame(sarima_forecasts)
We find that the SARIMA model achieves an RMSE of 0.001364 and a MAE of 0.001067.
# calculate the error metrics
sarima_metrics = pd.DataFrame(
columns=["Metric", "Value"],
{"Metric": "RMSE", "Value": root_mean_squared_error(y_true=sarima_forecasts["actual"], y_pred=sarima_forecasts["mean"])},
{"Metric": "MAE", "Value": mean_absolute_error(y_true=sarima_forecasts["actual"], y_pred=sarima_forecasts["mean"])},
We use the t5-large version of Chronos, which includes approximately 710 million parameters.
# instantiate the model
chronos_model = ChronosPipeline.from_pretrained(
For each month in the considered time window, we use as context window all the data up to that month, and generate 100 samples from the predicted distribution for the next month. We use the mean of
the distribution as point forecast, as in the SARIMA model.
Note that, as Chronos is a generative model, different random seeds and different numbers of samples result in slightly different forecasts.
# create a list for storing the forecasts
chronos_forecasts = []
# loop across the dates
for t in tqdm(range(data.index.get_loc(start_date), data.index.get_loc(end_date) + 1)):
# extract the context window
context = data.iloc[:t]
# generate the one-step-ahead forecast
chronos_forecast = chronos_model.predict(
# save the forecast
"date": data.index[t],
"actual": data.values[t],
"mean": np.mean(chronos_forecast),
"std": np.std(chronos_forecast, ddof=1),
# cast the forecasts to data frame
chronos_forecasts = pd.DataFrame(chronos_forecasts)
We find that Chronos achieves an RMSE of 0.001443 and a MAE of 0.001105.
# calculate the error metrics
chronos_metrics = pd.DataFrame(
columns=["Metric", "Value"],
{"Metric": "RMSE", "Value": root_mean_squared_error(y_true=chronos_forecasts["actual"], y_pred=chronos_forecasts["mean"])},
{"Metric": "MAE", "Value": mean_absolute_error(y_true=chronos_forecasts["actual"], y_pred=chronos_forecasts["mean"])},
A Python notebook with the full code is available in our GitHub repository.
[1] Ansari, A.F., Stella, L., Turkmen, C., Zhang, X., Mercado, P., Shen, H., Shchur, O., Rangapuram, S.S., Arango, S.P., Kapoor, S. and Zschiegner, J., (2024). Chronos: Learning the language of time
series. arXiv preprint, doi: 10.48550/arXiv.2403.07815.
[2] Raffel, C., Shazeer, N., Roberts, A., Lee, K., Narang, S., Matena, M., Zhou, Y., Li, W. and Liu, P.J., (2020). Exploring the limits of transfer learning with a unified text-to-text transformer.
Journal of machine learning research, 21(140), pp.1-67. | {"url":"https://fg-research.com/blog/general/posts/electricity-forecasting-chronos.html","timestamp":"2024-11-09T12:21:39Z","content_type":"text/html","content_length":"49476","record_id":"<urn:uuid:3db3609b-16f9-4c25-b150-6ea6ef6898bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00726.warc.gz"} |
Gross Pay vs Net Pay On
Gross Pay vs Net Pay On A Pay Stub - What's The Difference?
Gross Pay vs Net Pay On Pay Stub: Discover the Difference
• Gross Pay vs Net Pay On A Pay Stub - What's The Difference?
Updated: July 9, 2023
6 min read
Both employers and employees have to make important calculations when it comes to payroll. After all, everybody wants to know how much they’re spending on salaries or how much they’re getting paid.
And while these financials are very, very important, being accurate in calculating payments is far from easy.
Whether you’re an employer or an employee, you need to acquaint yourself with two important terms – gross pay and net pay. They signify different things and having a good understanding about both
concepts will lead to much better financial accountability.
What’s the Difference Between Gross and Net Pay?
The simple definition of gross income is the earnings of an individual before taxes and other kinds of deductions are applied. Hence, this one is a larger sum than net pay – the amount of earnings
that a person will receive after the payment of taxes and deductions.
Very often, gross pay is the amount that will be listed in a job ad. And because of that fact, employees can easily mistake it for the amount they’ll end up receiving each month (or however often
they’re getting paid).
Net pay is often called “take home pay” because it is the actual sum employees receive. The difference between the two will depend on location, the field that the respective company operates in and
the nature of additional deductions (more on those in the coming sections of the guide).
As a general rule of thumb, information about both types of payments is featured in an employee’s paystub. Having such a salary breakdown gives workers more clarity on how much they’re actually
making and what percentage of their earnings goes to cover mandatory federal expenses and other deductions.
How Is Gross Pay Calculated?
If you are an employer who is dealing with salary calculations for the very first time, you may feel a little bit lost.
So, how exactly do you calculate gross pay? Is there a formula to simplify the process? The answer depends on how your employees are being paid. In order to make an accurate calculation, you’ll need
to approach salaried and hourly payments in a different way.
Formula for Salaried Workers
The calculation can be made if you know the number of pay periods and the annual salary of an individual.
Pay periods can be as follows:
• Weekly (52 pay periods per year)
• Bi-weekly (26 pay periods per year)
• Twice monthly (24 pay periods per year)
• Monthly (12 pay periods per year)
For most companies, a monthly payment will be the most common scenario.
So, a worker who makes 50,000 dollars per year and is paid every month will have a gross income as follows:
50,000 / 12 = 4,166.67 dollars
Note: If workers earn commissions and bonuses, these amounts are added to the gross pay for the respective pay period. The sum of the salary and the bonus is the one to consider as gross income.
Formula for Those Receiving Hourly Payment
The formula for people who are getting paid on an hourly basis is also a fairly simple one.
In order to calculate the gross pay, you will need to multiply the number of hours completed during the respective pay period by the hourly rate.
Let’s say Michael earns 30 dollars per hour and has worked 160 hours over the course of a month. The gross income for that month is going to be:
30 x 160 = 4,800 dollars
If people work overtime over the course of the period, you will need to multiply the number of overtime hours by the overtime rate. At the time of writing, this rate is 1.5 times the regular hourly
rate that the worker earns.
Once again, commissions and bonuses are added to the base pay in order to calculate the gross amount for the respective worker and for the specific period.
Factors That Impact Net Pay
Now that you know the difference between gross and net pay, it’s easy to conclude that the second sum is going to be noticeably smaller than the first one,
The net pay is heavily dependent on location and taxation specifics. Multiple factors will affect the take-home amount a worker is going to receive:
• Federal income taxes: the federal income tax is dependent on the amount that a person earns. Luckily, the IRS has a tax estimator you can use to determine the withholdings to anticipate over the
course of a pay period.
• State income tax withholdings: apart from federal taxes, there are local deductions to account for when calculating net income. Check out this 2023 guide for more information on the current state
tax percentages.
• FICA contributions: these consist of the Social Security tax and the Medicare tax. The Social Security tax rate is 6.2 percent and for Medicare, the deduction is 1.45 percent. The employee’s
contribution has to be matched by the employer.
• Health insurance premiums
• Retirement savings contributions: including 401(k) and other plans
• Wage garnishments
• Voluntary deductions: like life insurance and job-related expenses
How Is Net Pay Calculated?
In order to make an accurate net pay calculation, you will have to go through the following steps:
• Step 1 – Calculate the employee’s gross pay amount: in order to apply the deductions, you have to start with the right number for the respective pay period.
• Step 2 – Calculate voluntary pre-tax deductions: these are the ones you will need to apply first and they include health benefits, retirement contributions (like company-sponsored 401(k) plans),
insurance premiums and others. These are always going to be pre-tax deductions so the order in which you make the calculations is very important.
• Step 3 – Calculate tax deductions: it’s now time to determine how much will go to federal and state taxes. Depending on where you live, there will be some differences in those numbers. Luckily,
many kinds of payroll and pay stub software products can make such calculations automatically after you enter your location.
• Step 4 – Calculate other mandatory deductions: while voluntary contributions are pre-tax deductions, there are some withholdings you need to apply after calculating and deducting taxes from the
gross pay. Wage garnishments, for example, are considered a post-tax deduction. Whenever an employee has a mandatory garnishment, you as the employer will receive an official notification. It
will include information about the amount you will have to withhold before handing the respective employee their wage.
Where and How Can I Find Information about My Gross and Net Salary
As an employee, you should have a specific answer to the question what’s the difference between net pay and gross pay. The company’s HR department should be capable of answering any questions you may
have on the topic.
We’ve already mentioned the fact that both gross income and net income information should appear on your paycheck stub for the respective period.
A well-constructed paystub should also give you the details about the deductions and their specific breakdown.
Important Information for Employers: How Do Gross and Net Pay Impact Employer Tax Contributions
If you live in a state where employers aren’t obliged to hand out paystubs, you can formally request information about your gross and net income. An employer is legally obliged to provide you with
those details, as well as deduction information.
Information for Employers: Taxes and Net / Gross Payments
Those who run a business are probably wondering about the way in which gross and net salaries affect taxation.
Every employer in the US has a responsibility towards their workers when it comes to covering some of their taxes.
We have already discussed the fact that employers have to match employee contributions in terms of FICA deductions.
Employers also have to make an additional taxation contribution under the Federal Unemployment Tax Act (FUTA). The funds from this program are used to finance unemployment benefits programs. The
calculation for FUTA is simple:
• Six percent of the first 7,000 dollars you pay out as gross wages per employee per fiscal year
In most states, an employer who pays unemployment taxes on time will receive a FUTA tax credit of 5.4 percent. This will significantly diminish the financial burden on the company.
The Importance of Understanding the Difference
Anyone who employs people in their business will need to understand the difference between gross and net pay.
Labor costs can be significant – in the service-based business field these expenses can reach as much as 50 percent of a company’s revenue. Because you are spending so much money on paying workers,
you need to have a good understanding of the total amount, the deductions that are withheld and the sums that actually end up in a worker’s pocket.
Being financially-savvy can also help you build your trustworthiness and maximize employee engagement. Workers will often have questions about their income and what is being deducted from it. The
ability to answer correctly and thoroughly will build trust and a good rapport with skilled laborers.
As an employee, you also need to be educated on financial matters pertaining to income and taxes. There are still people who mistakenly believe that their gross pay is the same as the money they take
home. Such misguided beliefs will result in a ton of disappointment (at best). Lacking the knowledge about income and deduction calculations makes you vulnerable and even likely to accept work
conditions that aren’t optimal for you.
And here’s one final important point that pertains to taxes. Every employee has to pay taxes on the income they earned over the course of the fiscal year. These calculations start with gross rather
than net pay.
Keep in mind, however, that gross income and taxable income are not the same thing. Some income sources are not included in taxable income and this is one more thing you will need to keep in account
when handling financials.
Gross and Net Pay in a Nutshell
The gross income is usually higher than net wages because it doesn’t account for taxes and other deductions. And while this sum looks attractive for marketing purposes (like attracting new employees
to the company), it doesn’t provide an accurate idea in terms of take-home money.
While understanding the difference between the two wages is fairly easy, making the calculations to accurately estimate net pay from gross pay is not a simple task.
Many people will turn to accountants and financial advisors to make sense of the percentages and the order in which the calculations have to be made.
The good news is that software often comes to the rescue. High quality payroll solutions and pay stub generators can automate lots of the complicated mathematical steps that lead from accurate gross
pay to accurate net pay.
When in doubt, look for a reliable software-based platform. If you are an employee, you can request such details from your company’s HR department. Having the raw numbers in your hands will make it
much easier to double-check the details, consult another professional or run the digits through a calculator in order to make sense of your current income.
Kristen Larson
Payroll Specialist
Kristen Larson is a payroll specialist with over 10 years of experience in the field. She received her Bachelor's degree in Business Administration from the University of Minnesota. Kristen has
dedicated her career to helping organizations effectively manage their payroll processes with Real Check Stubs.
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Download VTU BE 2020 Jan ME Question Paper 15 Scheme 4th Sem 15ME42 Kinematics of Machines
Download Visvesvaraya Technological University (VTU) BE ( Bachelor of Engineering) ME (Mechanical Engineering) 2015 Scheme 2020 January Previous Question Paper 4th Sem 15ME42 Kinematics of Machines
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Fourth Semester B.E. Degree Examination, Dec.101-94gn.2020
Kinematics of Machines
Time: 3 hrs.
Max. Marks: 80
Note: Answer FIVE full questions, choosing ONE fUll question from each module.
.1 . )
c --
0 -0
Define the following terms with examples: (i) Kinematic chain (ii) Mechanism
(iii) Lower pair and Higher pair
(06 Marks)
Sketch and explain the following mechanisms :
(i) Drag link mechanism. (ii) Geneva wheel. (10 Marks)
What are quick return motion mechanisms? Where are they used? Sketch and explain the
functioning of Whitworth mechanism.
(08 Marks)
Derive an expression for necessary condition of correct steering and explain Ackerman
steering gear with neat sketch.
(08 Marks)
A four bar mechanism ABCD is made up of four links, pin jointed at the ends. AD is a fixed
link which is 180 mm long. The links AB. BC and CD are 90 mm, 120 mm and 120 mm
long respectively. At certain instant, the link AB (crank) makes an angle of 60? with the link
AD. If the link AB rotates at uniform speed of 100 rpm clockwise determine,
(i) Angular velocity of the links BC and CD
(ii) Angular acceleration of the links CD and CB by using Graphical method. (16 Marks)
State and prove Kennedy's theorem.
(06 Marks)
In a reciprocating engine, the length of crank is 250 mm and length of connecting rod is
1000 mm. The crank rotates at an uniform speed of 300 rpm in clockwise direction and the
crank is inclined at 30? with inner dead centre. The Cg (centre of gravity) of the connecting
rod is 400 mm from the crank end. By Klein's construction determine (i) Velocity and
acceleration of piston (ii) Angular velocity and angular acceleration of connecting rod
(iii) Velocity and acceleration at the centre of gravity of the connecting rod. (10 Marks)
.:thing Complex algebra derive expressions for velocity and acceleration of the piston,
angular acceleration of connecting rod of a reciprocating engine mechanism. With these
expression determine the above quantities, if the crank length is 50 ram, connecting rod is
200 mm, crank speed is constant at 3000 rpm and crank angle is 30?. (16 Marks)
6 a. Derive Freudenstein's equation for slider crank mechanism. (08 Marks)
b. Design a four-link mechanism to coordinate three positions of the input and the output links
as follows:
0, = 20 4), =35
= 35' 4), =45`
0, = 50" 4, = 60
Using Freudenstein's equation for four bar mechanism. (08 Marks)
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t''' ? : 0 ! c:
Fourth Semester B.E. Degree Examination, Dec.101-94gn.2020
Kinematics of Machines
Time: 3 hrs.
Max. Marks: 80
Note: Answer FIVE full questions, choosing ONE fUll question from each module.
.1 . )
c --
0 -0
Define the following terms with examples: (i) Kinematic chain (ii) Mechanism
(iii) Lower pair and Higher pair
(06 Marks)
Sketch and explain the following mechanisms :
(i) Drag link mechanism. (ii) Geneva wheel. (10 Marks)
What are quick return motion mechanisms? Where are they used? Sketch and explain the
functioning of Whitworth mechanism.
(08 Marks)
Derive an expression for necessary condition of correct steering and explain Ackerman
steering gear with neat sketch.
(08 Marks)
A four bar mechanism ABCD is made up of four links, pin jointed at the ends. AD is a fixed
link which is 180 mm long. The links AB. BC and CD are 90 mm, 120 mm and 120 mm
long respectively. At certain instant, the link AB (crank) makes an angle of 60? with the link
AD. If the link AB rotates at uniform speed of 100 rpm clockwise determine,
(i) Angular velocity of the links BC and CD
(ii) Angular acceleration of the links CD and CB by using Graphical method. (16 Marks)
State and prove Kennedy's theorem.
(06 Marks)
In a reciprocating engine, the length of crank is 250 mm and length of connecting rod is
1000 mm. The crank rotates at an uniform speed of 300 rpm in clockwise direction and the
crank is inclined at 30? with inner dead centre. The Cg (centre of gravity) of the connecting
rod is 400 mm from the crank end. By Klein's construction determine (i) Velocity and
acceleration of piston (ii) Angular velocity and angular acceleration of connecting rod
(iii) Velocity and acceleration at the centre of gravity of the connecting rod. (10 Marks)
.:thing Complex algebra derive expressions for velocity and acceleration of the piston,
angular acceleration of connecting rod of a reciprocating engine mechanism. With these
expression determine the above quantities, if the crank length is 50 ram, connecting rod is
200 mm, crank speed is constant at 3000 rpm and crank angle is 30?. (16 Marks)
6 a. Derive Freudenstein's equation for slider crank mechanism. (08 Marks)
b. Design a four-link mechanism to coordinate three positions of the input and the output links
as follows:
0, = 20 4), =35
= 35' 4), =45`
0, = 50" 4, = 60
Using Freudenstein's equation for four bar mechanism. (08 Marks)
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CC r
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a g
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7 a. Derive an expression for minimum number of teeth necessary for a gear to avoid
(08 Marks)
b. A pair of gears 40 and 30 teeth respectively are of 25? involute form. Addendum
= 5 mm.
Module = 2.5 mm. I f the smaller wheel is the driver and rotate at 1500 rpm, find the velocity
of sliding at the point of engagement at pitch and at the point of dis-engagement, length of
path of contact and length of arc of contact. (08 Marks)
8 a. Explain with neat sketch of an epicyclic gear train. (04 Marks)
b. In an epicyclic gear train, the internal wheels 'A', '13' and the compound wheel 'C' and 'D'
rotate independently about the axis '0'. The wheels 'E' and rotates on a pin fixed to the
arm "G', `E' gears with 'A' and. 'C', and ` F' gears with 'B' and 'D'. All the wheels have
same pitch and the number of teeth on `E' and 'F' are 18, C = 28, D = 26.
(i) Sketch the arrangement.
(ii) Find the number of teeth on "A' and 'B'.
(iii) If the arm `G' makes 150 rpm CW and 'A' fixed, find speed of `B'.
(iv) If the arm 'G' makes 150 rpm CW and wheel 'A' makes 15 rpm CCW, find the speed
(12 Markk..1
9 A cam with a base circle radius of 35 mm is rotating at a uniform speed of 100 rpm in
anticlockwise direction. Draw the profile for the disc
Oam with reciprocating knife edge
follower on the centre line of the cam shaft for the folloWing follower motion:
(i) Follower to move upward 30 mm with Simple Harmonic Motion (SHM) in 0.1 sec.
(ii) Follower to dwell in next 0.15 sec.
(iii) Follower to move upward to another 30 mm with Simple Harmonic Motion (SHM) in
0.15 sec.
(iv) Follower to return to its starting position with Uniform Acceleration and Retardation
(UARM) in the remaining period. of one complete revolution of the cam shaft.
However, the acceleration period is twice the retardation period.
Determine the maximum velocity and acceleration of the follower during its return stroke.
(16 Marks)
10 a. Define the terms: --
(i) Base circle
(ii) Lift or Stroke
(iii) Pitch point.
(iv) Cam profile. (04 Marks)
b. Derive an expression, for displacement velocity and acceleration when the flat faced
follower is in contact with any point on the nose. (12 Marks)
2 of 2
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(HP15C)(HP67)(HP41C) Bernoulli Polynomials
08-30-2023, 09:58 AM
Post: #9
Namir Posts: 1,108
Senior Member Joined: Dec 2013
RE: (HP15C)(HP67)(HP41C) Bernoulli Polynomials
(08-29-2023 09:16 PM)John Keith Wrote: The problem here is that all useful methods of calculating Bernoulli numbers involve fairly large numbers, and the classic hp's can only represent numbers
< 10^10 exactly. Albert's method using Stirling numbers may be better than nested summation but will still see cancellation errors for n > 16.
One of my favorite methods uses Euler zigzag numbers. The numbers involved are smaller than factorials, and computing the zigzag numbers requires only addition. Also, this method requires only
one division at the end, which prevents rounding errors from accumulating. However, i have not compared the two methods directly and there may be little or no improvement in practice.
Thanks John for the hint and the link. I am off today for a few weeks vacation in Europe (which kicks off with attending the wedding of a nephew in Greece!), so I may be slow in digesting the Euler
Sizgzag numbers. They do look interesting!
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Properties of number 25642
25642 has 4 divisors (see below), whose sum is σ = 38466. Its totient is φ = 12820.
The previous prime is 25639. The next prime is 25643. The reversal of 25642 is 24652.
It is a semiprime because it is the product of two primes.
It can be written as a sum of positive squares in only one way, i.e., 25281 + 361 = 159^2 + 19^2 .
It is not an unprimeable number, because it can be changed into a prime (25643) by changing a digit.
It is a polite number, since it can be written as a sum of consecutive naturals, namely, 6409 + ... + 6412.
2^25642 is an apocalyptic number.
25642 is a deficient number, since it is larger than the sum of its proper divisors (12824).
25642 is a wasteful number, since it uses less digits than its factorization.
25642 is an evil number, because the sum of its binary digits is even.
The sum of its prime factors is 12823.
The product of its digits is 480, while the sum is 19.
The square root of 25642 is about 160.1311962111. The cubic root of 25642 is about 29.4883608206.
Subtracting from 25642 its reverse (24652), we obtain a triangular number (990 = T[44]).
The spelling of 25642 in words is "twenty-five thousand, six hundred forty-two". | {"url":"https://www.numbersaplenty.com/25642","timestamp":"2024-11-14T05:39:29Z","content_type":"text/html","content_length":"7448","record_id":"<urn:uuid:c6cdd5d1-5222-44f3-bc67-d98f35c71992>","cc-path":"CC-MAIN-2024-46/segments/1730477028526.56/warc/CC-MAIN-20241114031054-20241114061054-00873.warc.gz"} |
Go to the source code of this file.
subroutine zlags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
Function/Subroutine Documentation
subroutine zlags2 ( logical UPPER,
double precision A1,
complex*16 A2,
double precision A3,
double precision B1,
complex*16 B2,
double precision B3,
double precision CSU,
complex*16 SNU,
double precision CSV,
complex*16 SNV,
double precision CSQ,
complex*16 SNQ
Download ZLAGS2 + dependencies
[TGZ] [ZIP] [TXT]
ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
that if ( UPPER ) then
U**H *A*Q = U**H *( A1 A2 )*Q = ( x 0 )
( 0 A3 ) ( x x )
V**H*B*Q = V**H *( B1 B2 )*Q = ( x 0 )
( 0 B3 ) ( x x )
or if ( .NOT.UPPER ) then
U**H *A*Q = U**H *( A1 0 )*Q = ( x x )
( A2 A3 ) ( 0 x )
V**H *B*Q = V**H *( B1 0 )*Q = ( x x )
( B2 B3 ) ( 0 x )
U = ( CSU SNU ), V = ( CSV SNV ),
( -SNU**H CSU ) ( -SNV**H CSV )
Q = ( CSQ SNQ )
( -SNQ**H CSQ )
The rows of the transformed A and B are parallel. Moreover, if the
input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
of A is not zero. If the input matrices A and B are both not zero,
then the transformed (2,2) element of B is not zero, except when the
first rows of input A and B are parallel and the second rows are
UPPER is LOGICAL
[in] UPPER = .TRUE.: the input matrices A and B are upper triangular.
= .FALSE.: the input matrices A and B are lower triangular.
[in] A1 A1 is DOUBLE PRECISION
[in] A2 A2 is COMPLEX*16
A3 is DOUBLE PRECISION
[in] A3 On entry, A1, A2 and A3 are elements of the input 2-by-2
upper (lower) triangular matrix A.
[in] B1 B1 is DOUBLE PRECISION
[in] B2 B2 is COMPLEX*16
B3 is DOUBLE PRECISION
[in] B3 On entry, B1, B2 and B3 are elements of the input 2-by-2
upper (lower) triangular matrix B.
[out] CSU CSU is DOUBLE PRECISION
SNU is COMPLEX*16
[out] SNU The desired unitary matrix U.
[out] CSV CSV is DOUBLE PRECISION
SNV is COMPLEX*16
[out] SNV The desired unitary matrix V.
[out] CSQ CSQ is DOUBLE PRECISION
[out] SNQ SNQ is COMPLEX*16
The desired unitary matrix Q.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
November 2011
Definition at line 158 of file zlags2.f. | {"url":"https://netlib.org/lapack/explore-html-3.4.2/d7/dd9/zlags2_8f.html","timestamp":"2024-11-03T12:29:56Z","content_type":"application/xhtml+xml","content_length":"14439","record_id":"<urn:uuid:ebe3cc71-4920-45c5-a029-80dfc90a0e34>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00219.warc.gz"} |
Random Experiment: Types of Events and Sample Space
Let’s say that you toss a fair coin (not a prank coin ). There are only two possible outcomes – a head or a tail. Also, it is impossible to accurately predict the outcome (a head or a tail). In
mathematical theory, we consider only those experiments or observations, for which we know the set of possible outcomes. Also, it is important that predicting a particular outcome is impossible. Such
an experiment, where we know the set of all possible results but find it impossible to predict one at any particular execution, is a random experiment.
Suggested Videos
Even if a random experiment is repeated under identical conditions, the outcomes or results may fluctuate or vary randomly. Let’s look at another example – you take a fair dice and roll it using a
When the dice lands there are only six possible outcomes – 1, 2, 3, 4, 5, or 6. However, predicting which one will occur at any roll of the dice is completely unpredictable.
Source: Maxpixel
Further, in any experiment, there are certain terms that you need to know:
• Trial – A trial is the performance of an experiment.
• Outcomes – Whenever you perform an experiment, you get an outcome. For example, when you flip a coin, the outcome is either heads or tails. Similarly, when you roll a dice, the outcome is 1, 2,
3, 4, 5, or 6.
• Event – An event is a collection of basic outcomes with specific properties. For example, ‘E’ is the event where our roll of a six-sided dice has an outcome of less than or equal to 3. Therefore,
E is the collection of basic outcomes where the result is 3 or less. Symbolically, E = {O[1], O[2], O[3]}. It is important to note that depending on the event, the outcomes can be of any number
(even zero).
Random Experiment – Types of Events
In a random experiment, the following types of events are possible:
Simple and Compound Events
Simple or Elementary events are those which we cannot decompose further. For example, when you toss a coin, there are only two possible outcomes (heads or tails).
The event that the toss turns up a ‘head’ is a simple event and so is the event of it turning up a ‘tail’. Similarly, when you roll a six-sided dice, then the event that number 3 comes up is a simple
The Compound or Composite events are those which we can decompose into elementary or simple events. In simpler words, an elementary event corresponds to a single possible outcome of an experiment.
On the other hand, a compound event is an aggregate of some elementary events and we can decompose it into simple events.
To give you some examples, when you toss a fair coin, the event ‘turning up of a head or a tail’ is a compound event. This is because we can decompose this event into two simple events – (i) turning
up of the head and (ii) turning up of the tail.
Similarly, when you roll a six-sided dice, the event that an odd number comes up is a compound event. This is because we can break it down into three simple events – (i) Number 1 comes up, (ii)
Number 3 comes up, and (iii) The third odd number 5 comes up.
Learn more about Random Variables here in detail.
Equally Likely Events
If among all possible events, you cannot expect either one to occur in preference in the same experiment, after taking all conditions into account, then the events are Equally Likely Events.
Examples of Equally Likely Events
Back to our favorite coin. Tossing a fair coin has two simple events associated with it. The coin will turn up a ‘head’ or a ‘tail’. Now, there is an equal chance of either turning up and you cannot
expect one to turn up more frequently than the other. Also, in the case of rolling a six-sided dice, there are six equally likely events.
Mutually Exclusive Events
In a random experiment, if the occurrence of one event prevents the occurrence of any other event at the same time, then these events are Mutually Exclusive Events.
Examples of Mutually Exclusive Events
Let’s call the coin back into action, shall we?
When you toss a fair coin, the turning up of heads and turning up of tails are two mutually exclusive events. This is because if one turns up, then the other cannot turn up in the same experiment.
Similarly, when you roll a six-sided dice, there are six mutually exclusive events.
Remember, mutually exclusive events cannot occur simultaneously in the same experiment. Also, they may or may not be equally likely.
Independent Events
Two or more events are Independent Events if the outcome of one does not affect the outcome of the other. For example, (coin again!) if you toss a coin twice, then the result of the second throw is
not affected by the result of the first throw.
Dependent Events
Two or more events are Dependent Events if the occurrence or non-occurrence of one in any trial affects the probability of the other events in other trials.
Examples of Dependent Events
No coin this time.
Let’s say that the event is drawing a Queen from a pack of 52 cards. When you start with a new deck of cards, the probability of drawing a Queen is 4/52. However, if you manage to draw a Queen in one
trial and do not replace the card in the pack, then the probability of drawing a Queen in the remaining trials becomes 3/51.
Exhaustive Events
In order to understand Exhaustive Events, let’s take a quick look at the concept of Sample Space.
Sample Space
The Sample Space (S) of an experiment is the set of all possible outcomes of the experiment.
Going back to our coin, the sample space is S = {H, T} … where H-heads and T-tails. Similarly, when you roll a six-sided dice, the sample space is S = {1, 2, 3, 4, 5, 6}. Every possibility is a
sample point or element of the sample space.
Further, an event is a subset of the sample space and can contain one or more sample points. For example, when you roll a dice, the event that an odd number appears has three sample points.
Coming back to exhaustive events, the total number of possible outcomes of a random experiment form an exhaustive set of events. In other words, events are exhaustive if we consider all possible
Solved Question
Q1. What is a Random Experiment?
Answer: An experiment, where we know the set of all possible results but find it impossible to predict one at any particular execution, is a random experiment. | {"url":"https://www.toppr.com/guides/business-economics-cs/mathematics-of-finance-and-elementary-probability/random-experiment/","timestamp":"2024-11-11T00:25:56Z","content_type":"text/html","content_length":"225787","record_id":"<urn:uuid:e6f5429b-0e1b-487d-9f93-86d20e3c06a2>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00370.warc.gz"} |
Prime Number Calculator
Prime Number Calculator
Prime Number Calculator
Understanding the Prime Number Calculator
The Prime Number Calculator is a tool designed to determine whether a given integer is a prime number. A prime number is a positive integer greater than one that has no positive integer divisors
other than one and itself. This means the number cannot be divided evenly by any other number except for one and itself.
Applications of Prime Numbers
Prime numbers play a crucial role in various branches of science and technology, especially in cryptography. Cryptography relies on large prime numbers to create ciphers and secure information. In
computer algorithms, prime numbers help in hash functions, pseudo-random number generation, and encryption schemes. Prime numbers also find applications in advanced mathematics, such as number theory
and prime factorization.
Benefits of Using the Prime Number Calculator
This calculator provides a quick and accurate way to check for the primality of a number. Rather than manually performing divisibility tests, which can be tedious and error-prone, the calculator
simplifies the process with the click of a button. This can be beneficial for students, educators, and professionals working in fields that require frequent identification of prime numbers.
How the Prime Number Calculator Works
When you input a number in the calculator and press "Check Prime," the calculator performs a series of checks:
1. If the number is less than 2, it is not a prime number.
2. If the number is greater than 1, it checks for divisibility by integers from 2 up to the square root of the number. This is efficient because if a number has a divisor greater than its square
root, it must also have a divisor smaller than its square root.
If no divisors are found other than one and the number itself, it's a prime number. The calculator displays the result accordingly.
Why Prime Numbers Matter
Prime numbers are fundamental building blocks in mathematics. Their unique properties and behavior help in understanding and solving complex problems. They're used in creating algorithms and securing
digital communication. Recognizing prime numbers quickly and accurately with a calculator can simplify many mathematical and computational tasks.
1. What is a prime number?
A prime number is a positive integer greater than one that has no divisors other than one and itself. This means it cannot be divided evenly by any other number except for one and itself.
2. How does the Prime Number Calculator work?
The calculator operates by first checking if the input number is less than 2. If so, it is not a prime number. If the number is 2 or greater, the calculator checks for divisibility by integers from 2
up to the square root of the number. If no divisors are found other than one and the number itself, it is a prime number.
3. Why does the calculator only check up to the square root of a number?
Checking up to the square root of a number is efficient. If a number has a divisor greater than its square root, it must also have a divisor smaller than its square root. This reduces the number of
checks needed to determine if a number is prime.
4. Are there any limitations to the Prime Number Calculator?
The calculator may have performance limitations when working with very large numbers due to the time complexity of the algorithm used. For extremely large numbers, specialized software or algorithms
may be required.
5. Can the calculator handle negative numbers or decimals?
No, the calculator only works with positive integers. Negative numbers, zero, and decimal values are not considered for primality in this context.
6. What is the significance of prime numbers in cryptography?
Prime numbers are essential in cryptography for creating ciphers and securing information. Algorithms like RSA encryption rely on the difficulty of factoring large prime numbers to encrypt and
decrypt data securely.
7. Why is it important to identify prime numbers?
Prime numbers are fundamental in various mathematical theories and practical applications. They are used in cryptography, number theory, and for creating efficient algorithms in computer science.
Identifying prime numbers can simplify solving complex problems.
8. How accurate is the Prime Number Calculator?
The calculator provides accurate results for the primality of numbers within its operational range. For typical use cases involving numbers of reasonable size, it is reliable and efficient.
9. Can this calculator help with understanding other mathematical concepts?
Yes, using the calculator can provide insights into the nature of prime numbers and their properties. This understanding is fundamental in areas like factorization, number theory, and various
algorithms in computer science.
10. Is the Prime Number Calculator useful for educational purposes?
Absolutely. The calculator is a helpful tool for students and educators to quickly verify the primality of numbers, saving time and reducing error in manual calculations. | {"url":"https://www.onlycalculators.com/math/arithmetic/prime-number-calculator/","timestamp":"2024-11-09T22:31:24Z","content_type":"text/html","content_length":"230586","record_id":"<urn:uuid:c7285d73-9e47-401a-b6bc-cef79d880042>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00583.warc.gz"} |
If f is a real function such that f(x)=4xsin−13x,x=0 is conti... | Filo
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If is a real function such that is continuous, , then
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Question Text If is a real function such that is continuous, , then
Updated On Nov 14, 2022
Topic Trigonometry
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 136
Avg. Video Duration 20 min | {"url":"https://askfilo.com/user-question-answers-mathematics/if-is-a-real-function-such-that-is-continuous-then-32373531373235","timestamp":"2024-11-09T00:11:07Z","content_type":"text/html","content_length":"329038","record_id":"<urn:uuid:2948b9ea-921a-4f74-b0f2-446eaeb1569a>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00073.warc.gz"} |
Harvey Greenberg
Many different tolerances are used in mathematical programming systems. They are not used in the same way, and tolerances are related to each other. This Mathematical Programming Glossary Supplement
presents the main concepts with specifics for some MPS’s and examples to illustrate caution. Citation Available as Mathematical Programming Glossary Supplement, 2003, at http://
glossary.computing.society.informs.org/ Article Download … Read more | {"url":"https://optimization-online.org/author/hjgreenberg/","timestamp":"2024-11-10T09:51:45Z","content_type":"text/html","content_length":"82606","record_id":"<urn:uuid:4a1d828a-cfe5-4748-9c08-420b2cf6d66e>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00460.warc.gz"} |
Excel Formula Python: Show Grades Based on Range
In this tutorial, you will learn how to write an Excel formula in Python that shows grades based on a given range. The formula uses the VLOOKUP function to determine the grade based on the range
provided. This tutorial will provide a step-by-step explanation of the formula and provide examples to help you understand how it works.
The VLOOKUP function is a powerful tool in Excel that allows you to search for a value in a specified range and return a corresponding value from another column. In this case, we are using the
VLOOKUP function to search for the value in cell A1 and return the corresponding grade from the range $A$1:$B$9.
To use the formula, you need to have the grade ranges in column A and the corresponding grades in column B. The formula will search for the value in cell A1 in the first column of the range $A$1:$B$9
and return the value from the second column, which is the grade.
The formula also uses the TRUE parameter to perform an approximate match. This means that if the value in cell A1 falls within a range, it will return the corresponding grade. The formula can be
copied and applied to other cells to determine the grades for different values.
Now that you have an understanding of the formula, let's look at some examples to see how it works in practice.
An Excel formula
=VLOOKUP(A1, $A$1:$B$9, 2, TRUE)
Formula Explanation
This formula uses the VLOOKUP function to determine the grade based on the given range.
Step-by-step explanation
1. The VLOOKUP function searches for the value in cell A1 in the first column of the range $A$1:$B$9.
2. The range $A$1:$B$9 contains the grade ranges in column A and the corresponding grades in column B.
3. The number 2 in the formula indicates that the VLOOKUP function should return the value from the second column of the range $A$1:$B$9, which is the grade.
4. The TRUE parameter in the formula specifies that the VLOOKUP function should perform an approximate match. This means that if the value in cell A1 falls within a range, it will return the
corresponding grade.
5. The formula can be copied and applied to other cells to determine the grades for different values.
For example, if we have the following data in the range $A$1:$B$9:
| A | B |
| 75 | A |
| 70 | A |
| 65 | B |
| 60 | B |
| 55 | C |
| 50 | C |
| 45 | D |
| 40 | E |
| 0 | F |
If cell A1 contains the value 80, the formula =VLOOKUP(A1, $A$1:$B$9, 2, TRUE) would return the value "A", because 80 falls within the range 75-100.
Similarly, if cell A1 contains the value 55, the formula would return the value "C", because 55 falls within the range 50-54.
The formula can be applied to other cells to determine the grades for different values. | {"url":"https://codepal.ai/excel-formula-generator/query/oPoPdSje/excel-formula-python-show-grades","timestamp":"2024-11-03T10:59:32Z","content_type":"text/html","content_length":"90994","record_id":"<urn:uuid:1fe2c0b4-feca-44db-9f29-648773e5c9be>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00004.warc.gz"} |
Transactions Online
Chien-Ching CHIU, Ching-Lieh LI, Wei CHAN, "Image Reconstruction of a Buried Conductor by the Genetic Algorithm" in IEICE TRANSACTIONS on Electronics, vol. E84-C, no. 12, pp. 1946-1951, December
2001, doi: .
Abstract: In this paper, genetic algorithms is employed to determine the shape of a conducting cylinder buried in a half-space. Assume that a conducting cylinder of unknown shape is buried in one
half-space and scatters the field incident from another half-space where the scattered filed is measured. Based on the boundary condition and the measured scattered field, a set of nonlinear integral
equations is derived and the imaging problem is reformulated into an optimization problem. The genetic algorithm is then employed to find out the nearly global extreme solution of the object function
such that the shape of the conducting scatterer can be suitably reconstructed. In our study, even when the initial guess is far away from the exact one, the genetic algorithm can avoid the local
extremes and converge to a reasonably good solution. In such cases, the gradient-based methods often get stuck in local extremes. Numerical results are presented and good reconstruction is obtained
both with and without the additive Gaussian noise.
URL: https://global.ieice.org/en_transactions/electronics/10.1587/e84-c_12_1946/_p
author={Chien-Ching CHIU, Ching-Lieh LI, Wei CHAN, },
journal={IEICE TRANSACTIONS on Electronics},
title={Image Reconstruction of a Buried Conductor by the Genetic Algorithm},
abstract={In this paper, genetic algorithms is employed to determine the shape of a conducting cylinder buried in a half-space. Assume that a conducting cylinder of unknown shape is buried in one
half-space and scatters the field incident from another half-space where the scattered filed is measured. Based on the boundary condition and the measured scattered field, a set of nonlinear integral
equations is derived and the imaging problem is reformulated into an optimization problem. The genetic algorithm is then employed to find out the nearly global extreme solution of the object function
such that the shape of the conducting scatterer can be suitably reconstructed. In our study, even when the initial guess is far away from the exact one, the genetic algorithm can avoid the local
extremes and converge to a reasonably good solution. In such cases, the gradient-based methods often get stuck in local extremes. Numerical results are presented and good reconstruction is obtained
both with and without the additive Gaussian noise.},
TY - JOUR
TI - Image Reconstruction of a Buried Conductor by the Genetic Algorithm
T2 - IEICE TRANSACTIONS on Electronics
SP - 1946
EP - 1951
AU - Chien-Ching CHIU
AU - Ching-Lieh LI
AU - Wei CHAN
PY - 2001
DO -
JO - IEICE TRANSACTIONS on Electronics
SN -
VL - E84-C
IS - 12
JA - IEICE TRANSACTIONS on Electronics
Y1 - December 2001
AB - In this paper, genetic algorithms is employed to determine the shape of a conducting cylinder buried in a half-space. Assume that a conducting cylinder of unknown shape is buried in one
half-space and scatters the field incident from another half-space where the scattered filed is measured. Based on the boundary condition and the measured scattered field, a set of nonlinear integral
equations is derived and the imaging problem is reformulated into an optimization problem. The genetic algorithm is then employed to find out the nearly global extreme solution of the object function
such that the shape of the conducting scatterer can be suitably reconstructed. In our study, even when the initial guess is far away from the exact one, the genetic algorithm can avoid the local
extremes and converge to a reasonably good solution. In such cases, the gradient-based methods often get stuck in local extremes. Numerical results are presented and good reconstruction is obtained
both with and without the additive Gaussian noise.
ER - | {"url":"https://global.ieice.org/en_transactions/electronics/10.1587/e84-c_12_1946/_p","timestamp":"2024-11-03T13:03:49Z","content_type":"text/html","content_length":"61604","record_id":"<urn:uuid:23770c68-472a-4de3-82a3-bc092c3c45ac>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00102.warc.gz"} |
Area of Isosceles Triangle - Math Steps, Examples & Questions
Write the answer, including the units.
A=21,465.6 \mathrm{~m}^2
Now to calculate how many cattle will fit into the field, take 21,465.6\mathrm{~m}^2 and divide it by 2,000\mathrm{~m}^2 , because each cattle needs that much space to graze on the field.
21,465.6 \div 2,000=10.7328
The quotient has 10 wholes and 7,328 ten-thousandths. There cannot be part of a cow and 10.7328 is not enough for 11 cows. So, 10 cows will fit in the field.
Where is the base of the isosceles triangle?
The base is considered the side adjacent to the two congruent sides. However, to find the area any side of the triangle can be used as the base. In order to be used as the base measurement, the
height measurement should go directly up from the base, forming a 90 degree angle.
How do you find the perimeter of an isosceles triangle?
To find the perimeter, add up all three sides of an isosceles triangle.
What is the pythagorean theorem?
The pythagorean theorem explains a relationship between the sides of all right triangles. It is used in trigonometry to find a missing third side measurement.
What is the hypotenuse?
The hypotenuse is the longest side of a right triangle. It is opposite of the vertex angle measuring 90 degrees.
90 {~cm}^{2}
45 {~cm}^{2}
180 {~cm}^{2}
21 {~cm}^{2}
225 \mathrm{~cm}^2
2,250 \mathrm{~cm}^2
22,500 \mathrm{~cm}^2
45,000 \mathrm{~cm}^2
8 chickens
11 chickens
66 chickens
9 chickens
18.00 {~cm}^{2}
17.42 {~cm}^{2}
9.00 {~cm}^{2}
8.72 {~cm}^{2}
24.6 {~m}^{2}
18.3 {~m}^{2}
12.3 {~m}^{2}
19.4 {~m}^{2}
80 {~cm}
5 {~cm}
2.5 {~cm}
10 {~cm} | {"url":"https://thirdspacelearning.com/us/math-resources/topic-guides/geometry/area-of-isosceles-triangle/","timestamp":"2024-11-07T03:00:07Z","content_type":"text/html","content_length":"291546","record_id":"<urn:uuid:19ed0a59-b342-4fdf-9bf3-a96965849f70>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00202.warc.gz"} |