content stringlengths 86 994k | meta stringlengths 288 619 |
|---|---|
BCIT Astronomy 7000: A Survey of Astronomy
Chapter 3 Orbits and Gravity
3.3 Newton’s Universal Law of Gravitation
Learning Objectives
By the end of this section, you will be able to:
• Explain what determines the strength of gravity
• Describe how Newton’s universal law of gravitation extends our understanding of Kepler’s laws
Newton’s laws of motion show that objects at rest will stay at rest and those in motion will continue moving uniformly in a straight line unless acted upon by a force. Thus, it is the straight line
that defines the most natural state of motion. But the planets move in ellipses, not straight lines; therefore, some force must be bending their paths. That force, Newton proposed, was gravity.
In Newton’s time, gravity was something associated with Earth alone. Everyday experience shows us that Earth exerts a gravitational force upon objects at its surface. If you drop something, it
accelerates toward Earth as it falls. Newton’s insight was that Earth’s gravity might extend as far as the Moon and produce the force required to curve the Moon’s path from a straight line and keep
it in its orbit. He further hypothesized that gravity is not limited to Earth, but that there is a general force of attraction between all material bodies. If so, the attractive force between the Sun
and each of the planets could keep them in their orbits. (This may seem part of our everyday thinking today, but it was a remarkable insight in Newton’s time.)
Once Newton boldly hypothesized that there was a universal attraction among all bodies everywhere in space, he had to determine the exact nature of the attraction. The precise mathematical
description of that gravitational force had to dictate that the planets move exactly as Kepler had described them to (as expressed in Kepler’s three laws). Also, that gravitational force had to
predict the correct behavior of falling bodies on Earth, as observed by Galileo. How must the force of gravity depend on distance in order for these conditions to be met?
The answer to this question required mathematical tools that had not yet been developed, but this did not deter Isaac Newton, who invented what we today call calculus to deal with this problem.
Eventually he was able to conclude that the magnitude of the force of gravity must decrease with increasing distance between the Sun and a planet (or between any two objects) in proportion to the
inverse square of their separation. In other words, if a planet were twice as far from the Sun, the force would be (1/2)^2, or 1/4 as large. Put the planet three times farther away, and the force is
(1/3)^2, or 1/9 as large.
Newton also concluded that the gravitational attraction between two bodies must be proportional to their masses. The more mass an object has, the stronger the pull of its gravitational force. The
gravitational attraction between any two objects is therefore given by one of the most famous equations in all of science:
where F[gravity] is the gravitational force between two objects, M[1] and M[2] are the masses of the two objects, and R is their separation. G is a constant number known as the universal
gravitational constant, and the equation itself symbolically summarizes Newton’s universal law of gravitation. With such a force and the laws of motion, Newton was able to show mathematically that
the only orbits permitted were exactly those described by Kepler’s laws.
Newton’s universal law of gravitation works for the planets, but is it really universal? The gravitational theory should also predict the observed acceleration of the Moon toward Earth as it orbits
Earth, as well as of any object (say, an apple) dropped near Earth’s surface. The falling of an apple is something we can measure quite easily, but can we use it to predict the motions of the Moon?
Recall that according to Newton’s second law, forces cause acceleration. Newton’s universal law of gravitation says that the force acting upon (and therefore the acceleration of) an object toward
Earth should be inversely proportional to the square of its distance from the center of Earth. Objects like apples at the surface of Earth, at a distance of one Earth-radius from the center of Earth,
are observed to accelerate downward at 9.8 meters per second per second (9.8 m/s^2).
It is this force of gravity on the surface of Earth that gives us our sense of weight. Unlike your mass, which would remain the same on any planet or moon, your weight depends on the local force of
gravity. So you would weigh less on Mars and the Moon than on Earth, even though there is no change in your mass. (Which means you would still have to go easy on the desserts in the college cafeteria
when you got back!)
The Moon is 60 Earth radii away from the center of Earth. If gravity (and the acceleration it causes) gets weaker with distance squared, the acceleration the Moon experiences should be a lot less
than for the apple. The acceleration should be (1/60)^2 = 1/3600 (or 3600 times less—about 0.00272 m/s^2. This is precisely the observed acceleration of the Moon in its orbit. (As we shall see, the
Moon does not fall to Earth with this acceleration, but falls around Earth.) Imagine the thrill Newton must have felt to realize he had discovered, and verified, a law that holds for Earth, apples,
the Moon, and, as far as he knew, everything in the universe.
Calculating Weight
By what factor would a person’s weight at the surface of Earth change if Earth had its present mass but eight times its present volume?
With eight times the volume, Earth’s radius would double. This means the gravitational force at the surface would reduce by a factor of (1/2)^2 = 1/4, so a person would weigh only one-fourth as much.
Check Your Learning
By what factor would a person’s weight at the surface of Earth change if Earth had its present size but only one-third its present mass?
With one-third its present mass, the gravitational force at the surface would reduce by a factor of 1/3, so a person would weight only one-third as much.
Gravity is a “built-in” property of mass. Whenever there are masses in the universe, they will interact via the force of gravitational attraction. The more mass there is, the greater the force of
attraction. Here on Earth, the largest concentration of mass is, of course, the planet we stand on, and its pull dominates the gravitational interactions we experience. But everything with mass
attracts everything else with mass anywhere in the universe.
Newton’s law also implies that gravity never becomes zero. It quickly gets weaker with distance, but it continues to act to some degree no matter how far away you get. The pull of the Sun is stronger
at Mercury than at Pluto, but it can be felt far beyond Pluto, where astronomers have good evidence that it continuously makes enormous numbers of smaller icy bodies move around huge orbits. And the
Sun’s gravitational pull joins with the pull of billions of others stars to create the gravitational pull of our Milky Way Galaxy. That force, in turn, can make other smaller galaxies orbit around
the Milky Way, and so on.
Why is it then, you may ask, that the astronauts aboard the Space Shuttle appear to have no gravitational forces acting on them when we see images on television of the astronauts and objects floating
in the spacecraft? After all, the astronauts in the shuttle are only a few hundred kilometers above the surface of Earth, which is not a significant distance compared to the size of Earth, so gravity
is certainly not a great deal weaker that much farther away. The astronauts feel “weightless” (meaning that they don’t feel the gravitational force acting on them) for the same reason that passengers
in an elevator whose cable has broken or in an airplane whose engines no longer work feel weightless: they are falling ([link]).^1
Astronauts in Free Fall.
Figure 1. While in space, astronauts are falling freely, so they experience “weightlessness.” Clockwise from top left: Tracy Caldwell Dyson (NASA), Naoko Yamzaki (JAXA), Dorothy Metcalf-Lindenburger
(NASA), and Stephanie Wilson (NASA). (credit: NASA)
When falling, they are in free fall and accelerate at the same rate as everything around them, including their spacecraft or a camera with which they are taking photographs of Earth. When doing so,
astronauts experience no additional forces and therefore feel “weightless.” Unlike the falling elevator passengers, however, the astronauts are falling around Earth, not to Earth; as a result they
will continue to fall and are said to be “in orbit” around Earth (see the next section for more about orbits).
Orbital Motion and Mass
Kepler’s laws describe the orbits of the objects whose motions are described by Newton’s laws of motion and the law of gravity. Knowing that gravity is the force that attracts planets toward the Sun,
however, allowed Newton to rethink Kepler’s third law. Recall that Kepler had found a relationship between the orbital period of a planet’s revolution and its distance from the Sun. But Newton’s
formulation introduces the additional factor of the masses of the Sun (M[1]) and the planet (M[2]), both expressed in units of the Sun’s mass. Newton’s universal law of gravitation can be used to
show mathematically that this relationship is actually
where a is the semimajor axis and P is the orbital period.
How did Kepler miss this factor? In units of the Sun’s mass, the mass of the Sun is 1, and in units of the Sun’s mass, the mass of a typical planet is a negligibly small factor. This means that the
sum of the Sun’s mass and a planet’s mass, (M[1] + M[2]), is very, very close to 1. This makes Newton’s formula appear almost the same as Kepler’s; the tiny mass of the planets compared to the Sun is
the reason that Kepler did not realize that both masses had to be included in the calculation. There are many situations in astronomy, however, in which we do need to include the two mass terms—for
example, when two stars or two galaxies orbit each other.
Including the mass term allows us to use this formula in a new way. If we can measure the motions (distances and orbital periods) of objects acting under their mutual gravity, then the formula will
permit us to deduce their masses. For example, we can calculate the mass of the Sun by using the distances and orbital periods of the planets, or the mass of Jupiter by noting the motions of its
Indeed, Newton’s reformulation of Kepler’s third law is one of the most powerful concepts in astronomy. Our ability to deduce the masses of objects from their motions is key to understanding the
nature and evolution of many astronomical bodies. We will use this law repeatedly throughout this text in calculations that range from the orbits of comets to the interactions of galaxies.
Calculating the Effects of Gravity
A planet like Earth is found orbiting its star at a distance of 1 AU in 0.71 Earth-year. Can you use Newton’s version of Kepler’s third law to find the mass of the star? (Remember that compared to
the mass of a star, the mass of an earthlike planet can be considered negligible.)
In the formula a^3 = (M[1] + M[2]) × P^2, the factor M[1] + M[2] would now be approximately equal to M[1] (the mass of the star), since the planet’s mass is so small by comparison. Then the formula
becomes a^3 = M[1] × P^2, and we can solve for M[1]:
Since a = 1, a^3 = 1, so
So the mass of the star is twice the mass of our Sun. (Remember that this way of expressing the law has units in terms of Earth and the Sun, so masses are expressed in units of the mass of our Sun.)
Check Your Learning
Suppose a star with twice the mass of our Sun had an earthlike planet that took 4 years to orbit the star. At what distance (semimajor axis) would this planet orbit its star?
Again, we can neglect the mass of the planet. So M[1] = 2 and P = 4 years. The formula is a^3 = M[1] × P^2, so a^3 = 2 × 4^2 = 2 × 16 = 32. So a is the cube root of 32. To find this, you can just ask
Google, “What is the cube root of 32?” and get the answer 3.2 AU.
You might like to try a
that lets you move the Sun, Earth, Moon, and space station to see the effects of changing their distances on their gravitational forces and orbital paths. You can even turn off gravity and see what
Key Concepts and Summary
Gravity, the attractive force between all masses, is what keeps the planets in orbit. Newton’s universal law of gravitation relates the gravitational force to mass and distance:
The force of gravity is what gives us our sense of weight. Unlike mass, which is constant, weight can vary depending on the force of gravity (or acceleration) you feel. When Kepler’s laws are
reexamined in the light of Newton’s gravitational law, it becomes clear that the masses of both objects are important for the third law, which becomes a^3 = (M[1]+ M[2]) × P^2. Mutual gravitational
effects permit us to calculate the masses of astronomical objects, from comets to galaxies.
1. 1 In the film Apollo 13, the scenes in which the astronauts were “weightless” were actually filmed in a falling airplane. As you might imagine, the plane fell for only short periods before the
engines engaged again.
the mutual attraction of material bodies or particles | {"url":"https://pressbooks.bccampus.ca/a7000y2018/chapter/3-3-newtons-universal-law-of-gravitation/","timestamp":"2024-11-05T20:16:35Z","content_type":"text/html","content_length":"121190","record_id":"<urn:uuid:cbfbd545-d3b7-49d7-a7d1-af64af414090>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00038.warc.gz"} |
Multiplication Charts Up To 100 | Multiplication Chart Printable
Multiplication Charts Up To 100
Printable Multiplication Chart Up To 100 PrintableMultiplication
Multiplication Charts Up to 100
Multiplication Charts Up to 100 – A Multiplication Chart is a practical tool for kids to discover exactly how to multiply, separate, and locate the smallest number. There are several uses for a
Multiplication Chart.
What is Multiplication Chart Printable?
A multiplication chart can be made use of to assist children discover their multiplication realities. Multiplication charts been available in many types, from complete page times tables to solitary
page ones. While individual tables are useful for providing pieces of details, a complete web page chart makes it simpler to evaluate truths that have actually already been mastered.
The multiplication chart will typically include a left column and also a leading row. The leading row will certainly have a listing of items. When you wish to discover the item of two numbers, choose
the first number from the left column as well as the 2nd number from the top row. When you have these numbers, move them along the row or down the column up until you reach the square where the two
numbers meet. You will after that have your item.
Multiplication charts are practical knowing tools for both children and also grownups. Multiplication Charts Up to 100 are offered on the Internet as well as can be published out and also laminated
flooring for longevity.
Why Do We Use a Multiplication Chart?
A multiplication chart is a diagram that demonstrates how to multiply 2 numbers. It normally consists of a left column and a top row. Each row has a number representing the item of the two numbers.
You pick the very first number in the left column, relocate down the column, and afterwards select the 2nd number from the top row. The product will certainly be the square where the numbers meet.
Multiplication charts are practical for several reasons, including helping youngsters learn how to separate and streamline fractions. Multiplication charts can also be handy as workdesk sources due
to the fact that they offer as a consistent tip of the pupil’s progression.
Multiplication charts are also valuable for assisting students remember their times tables. As with any ability, memorizing multiplication tables takes time and technique.
Multiplication Charts Up to 100
Printable 100 Multiplication Chart PrintableMultiplication
Multiplication Chart To 100
Multiplication Table Up To 100 Multiplication Table Multiplication
Multiplication Charts Up to 100
If you’re looking for Multiplication Charts Up to 100, you’ve come to the appropriate place. Multiplication charts are offered in different formats, including complete dimension, half dimension, as
well as a selection of cute layouts.
Multiplication charts and also tables are indispensable tools for children’s education. You can download and print them to make use of as a mentor help in your child’s homeschool or classroom. You
can additionally laminate them for sturdiness. These charts are wonderful for usage in homeschool mathematics binders or as class posters. They’re specifically beneficial for youngsters in the 2nd,
3rd, and also 4th qualities.
A Multiplication Charts Up to 100 is an useful tool to enhance mathematics realities and also can assist a kid find out multiplication promptly. It’s likewise a fantastic tool for avoid checking and
also discovering the moments tables.
Related For Multiplication Charts Up to 100 | {"url":"https://multiplicationchart-printable.com/multiplication-charts-up-to-100/","timestamp":"2024-11-07T16:59:59Z","content_type":"text/html","content_length":"42982","record_id":"<urn:uuid:a00b8001-1823-4c0d-9b38-46cc84f7f95c>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00705.warc.gz"} |
C Program to Calculate the Distance Between Two Points Using Structure - Ccodelearner
• C Program to Calculate the Distance Between Two Points Using Structure
In mathematics, finding the distance between two points is a common problem that arises in various applications. Whether you’re designing a game, simulating physical interactions, or solving
geometric problems, knowing how to calculate distances accurately is essential. In this blog post, we will explore a C program that uses structures to calculate the distance between two points in
a two-dimensional space. By the end of this post, you will have a good understanding of how to implement this program and apply it to your own projects.
Understanding Structures in C
Before we dive into the program, let’s briefly understand what structures are in the C programming language. Structures provide a way to group related data into a single unit. They allow us to
create custom data types that can store different types of variables together. In our case, we will use structures to represent points in a two-dimensional space.
To define a structure in C, we use the struct keyword followed by the structure name and a list of variables inside curly braces. Here’s an example of defining a structure for a point:
struct Point {
int x;
int y;
In the above code, we have defined a structure named Point that contains two integer variables x and y. These variables represent the respective coordinates of a point.
Calculating the Distance between Two Points
Now that we have a good grasp of structures, let’s move on to calculating the distance between two points. To calculate the distance, we can use the well-known Euclidean distance formula. The
Euclidean distance between two points (x1, y1) and (x2, y2) is given by:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
To implement this in C, we will create a function that takes two points as input and returns the calculated distance.
#include <stdio.h>
#include <math.h>
struct Point {
int x;
int y;
double calculateDistance(struct Point p1, struct Point p2) {
int x_diff = p2.x - p1.x;
int y_diff = p2.y - p1.y;
double distance = sqrt((x_diff * x_diff) + (y_diff * y_diff));
return distance;
int main() {
struct Point p1, p2;
printf("Enter coordinates for Point 1: ");
scanf("%d %d", &p1.x, &p1.y);
printf("Enter coordinates for Point 2: ");
scanf("%d %d", &p2.x, &p2.y);
double dist = calculateDistance(p1, p2);
printf("Distance between the two points: %.2lf\n", dist);
return 0;
In the above code, we start by including the necessary header files stdio.h and math.h for input/output operations and mathematical calculations, respectively. We then define our Point structure.
The calculateDistance function takes two Point structures p1 and p2 as input. Inside the function, we calculate the differences in the x and y coordinates of the two points using subtraction. We
then apply the Euclidean distance formula by squaring the differences, adding them together, and taking the square root. The calculated distance is returned as a double value.
In the main function, we declare two Point variables p1 and p2. We prompt the user to enter the coordinates for both points and store them in the respective x and y variables of the Point
structures using scanf. Next, we call the calculateDistance function, passing p1 and p2 as arguments. The calculated distance is stored in the dist variable, which is then printed using printf.
Example Usage
Let’s run a sample execution of our program to see how it works:
Enter coordinates for Point 1: 2 3
Enter coordinates for Point 2: 5 7
Distance between the two points: 5.00
In this example, we entered the coordinates (2, 3) for Point 1 and (5, 7) for Point 2. The program calculated and displayed the distance between these two points as 5.00.
Further Enhancements
Our current program calculates the distance between two points accurately. However, there are numerous ways we can enhance it further based on specific requirements. Here are a few ideas:
1. Error Handling: Add input validation to ensure the user enters valid coordinates. For example, you can check if the input values are within a specific range, such as x and y being
non-negative integers.
2. Multiple Calculations: Modify the program to calculate distances between more than two points. You can prompt the user to enter the number of points they want to calculate distances for and
then iterate over the input process accordingly.
3. Using Floating-Point Coordinates: Extend the program to handle floating-point coordinates. You can modify the x and y variables in the Point structure to be of type double instead of int.
Additionally, you need to update the input and output formatting accordingly.
4. Applying to Three-Dimensional Space: If you want to calculate distances between points in a three-dimensional space, you can expand the Point structure to include a third coordinate z.
Similarly, you would need to update the formula to account for the additional dimension.
In this blog post, we learned how to calculate the distance between two points using a C program that utilizes structures. Overall, we explored the concept of structures and saw how they provide
a convenient way to group related data together. By implementing the Euclidean distance formula, we were able to accurately calculate distances and obtain the desired results.
Remember, this program serves as a starting point, and you can build upon it to create more advanced and customized solutions. Experiment with different enhancements, explore additional features,
and integrate it into your own projects to make the most of this powerful distance calculation technique!
Happy coding! | {"url":"https://ccodelearner.com/c-examples/c-program-to-calculate-distance-between-two-points/","timestamp":"2024-11-09T09:08:20Z","content_type":"text/html","content_length":"271594","record_id":"<urn:uuid:6f897eca-912b-49fb-8eef-8f8b5dc40fbb>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00871.warc.gz"} |
CARLO AMBROGIO FAVERO - Personal Page - Universita' Bocconi
Didattica > Materiali didattici
20630 Introduction to Sport Analytics
This course provides the analytics requirements of a Sports Management program. It is also an opportunity for applied work for all students interested in Data Science. All applications in the course
will be based on the statistical software R. The course is taught through a combination of lectures, class discussion, group presentations. Students are required to read assignments from the texts as
well as additional sources provided by the instructor. Students must attend class prepared to engage in discussions; have, articulate and defend a point of view; and ask questions and provide
comments based on their reading and on their own R applications.
Projects will be allocated to groups of attending students. Project reports and their presentation will be part of the evaluation for attending students.
Presentations on the use of analytics in the Sport Business:
Using Analytics for a Euroleague Basketball Team, presentation by Mario Fioretti, Assistant Coach, Olimpia Milano
Using Analytics in the European Soccer Industry, presentation by Mark Nervegna, Head of Strategy and Analytics, Raiola Global
Pre-Requisites: Students are expected to have attended a core course in statistics and to be familiar with basic calculus and linear algebra.
Teaching Assistant: Office Hours will be held online viaTeams, the Teaching Assistant will follow students both on projects and on exercises
Gabriele Carta, gabriele.carta@unibocconi.it, office hours
Past Exams: 2019_1, 2019_2
Mock Exam May 2023: exam, data, R code with solutions
Exam 23rd May 2023: exam, data, R code with solutions
Dynamic Documents with R Markdown
build a report with all results and comments
An introduction to R Markdown
an illustrative R Markdown code
Github and Github Desktop
A tutorial online
Project 1: Getting sport data from the web with R
The objective of this project is to illustrate how data on sports could be efficiently retrieved from the Web (via API and/or webscraping). Students should feel free to choose their preferred field
and application.
Accessing APIs from R a tutorial , an R code for the tutorial, Accessing data from Github using an R code
Project 2: Creating Web Applications with Rshiny
The objective of this project is to create a Sport related web application with RShiny. An Illustration based on NBA data is provided together with projects produced in 2020. Students should feel
free to choose their preferred field and application.
Slides of Andrea Maver's presentation
Online tutorials on mastering RShiny
Learning Shiny with NBA DATA (by Julia Wrobel),
http://juliawrobel.com/tutorials/shiny_tutorial_nba.html, https://andreamaver.shinyapps.io/EuroleagueApp/
Programmes for NBA Shiny short version , Programmes for NBA shiny long version
Rshiny example
Instructions for those who have opted for the Shiny Project in 2020 are available HERE.
Project 3: An Application of Unsupervised Machine Learning to Sport Analytics
The objective of this project is to apply unsupervised machine learning, and in particular cluster analysis, to finding groups in Sport Analytics data.
P. Zuccolotto and M. Manisera (2020) Basketball Data Science – With Applications in R, Chapman and Hall/CRC. (Chapter 4)
link to basketball analyzeR: https://bdsports.unibs.it/basketballanalyzer/
James, Witten, Habstie and Tibshirani (2011) An Introduction to Statistical Learning- With Applications in R
LINK to the recorded Presentation of the Cluster Analysis project 2020: https://eu-lti.bbcollab.com/recording/8570729e9532435b951e9b40de8470a5
SLIDES and Rmd codes
Project 4: An Application of Supervised Machine Learning to Sport Analytics
The objective of this project is to apply supervised machine learning techniques , and in particular techniques to solve the many predictor problem to predict top athletes compensations.
Students should use as a benchmark the model presented in the lectures and evaluate it against alternatives generated by modern machine learning techniques.
A further possibility for a group undertaking this project is the costruction of a data challenge related to the topic of the project using the data challenge website of Bocconi University.
James, Witten, Habstie and Tibshirani (2011) An Introduction to Statistical Learning- With Applications in R,
Stock J. and M.Watson (2020) Introduction to Econometrics, 4th edition, Chapter 14
Project 5: Evaluating the Home Advantage Effect from quasi-Natural Experiments
Following the COVID shock many games in many sport were played without attendance within "bubbles" in which no team had the "home advantage effect". The objective of this project is to use sport data
to construct a quasi-natural experiment for the evaluation of the Home Advantage Effect.
Stock J. and M.Watson (2020) Introduction to Econometrics, 4th edition, Chapter 13
Presentation of N.Sita(2020) thesis on Evaluating the Home Advantage in NBA
Project 6: Measuring Competitive Advantage and its effects
The objective of this project is to introduce, discuss the concept of Competitive Balance in the Sport Industry. Both a discussion of the theory and applications are possible.
Berri D.J.,M.B.Schmidt and S. Brook(2006), The Wages of Wins, Stanford University Press, Ch 3,4
Brandes L. and E.Franck(2007) "Who made who? An Empirical Analysis of Competitive Balance in European Soccer Leagues" Eastern Economic Journal
Haddock D. and L.P.Cain(2006) "Measuring Parity:Tying into the Idealized Standard Deviation", Journal of Sport and Economics
Koning R.H.(2000) Balance in competition in Dutch soccer, The Statistician, 49, Part 3, pp.419-431
Szimansky S.(2001) "Income inequality, competitive balance and the attractiveness of team sports:some evidence and a natural experiment from English Soccer" the Economic Journal,111, F69-F84
Project 7: Load Management and Injury Risk
A recent report denied the existence of a significant statistical relationship between load management and injury risk in the NBA. The objective of this project is a critical analysis of the report,
which will be made available to the groups taking this choice.
Project 8: The Relevance of Popular Shareholding Contribution to Team Perfomance
A recent report provided evidence on the popular shareholding contribution to team perfomance in european soccer. The objective of this project is a critical analysis of the report, which will be
made available together with the original data to the groups taking this choice
Course Content Summary
Section 1: Sport Analytics. an Introduction
The Questions in Sport Analytics.
The Answers
Modelling Data in Sports
Theory Based Models
Supervised Machine Learning
Unsupervised Machine Learning
Berri D.J.,M.B.Schmidt and S. Brook(2006), The Wages of Wins, Stanford University Press
Berri D.J., M. B. Schmidt (2010) Stumbling On Wins.Two Economists Expose the Pitfalls on the Road to Victory in Professional Sports-FT Press
Goldsberry K.(2019) Sprawlball. A visual tour of the new era of NBA, Houghton Mifflin Harcourt
James, Witten, Habstie and Tibshirani (2011) An Introduction to Statistical Learning- With Applications in R,
Shea S.(2014) Basketball analytics. Spatial Tracking
P. Zuccolotto and M. Manisera (2020) Basketball Data Science – With Applications in R, Chapman and Hall/CRC.
Winston W.L.(2009) Mathletics, Princeton University Press
Section 2: An introduction to R
Install R and R studio on your computer and learn how to run them
Learn what is a package and how to install it
Understand what is a view
define a default directory
have some fun with R Shiny
An online introduction to R
R Code
Torfs Brauer "A Very Short Intro to R" , SOLUTIONS FOR the Torfs-Brauer TO DO LIST
Data-Objects in R
Data Objects in R (data types) and Data Structures In R (Vectors, Matrices, Arrays, Data Frames, Lists)
Data Handling in R
Importing and Exporting, transforming and selecting data
Getting Data from the web with R
Programming and Control Flow
if-else statements, using switch, loops, functions in R
all R codes used in Singh and Allen are downloaded at
R CODES (from Singh and Allen) : Data Objects, Data Handling, Getting Data from the web, Programming, binomial model included
Singh AK and DE Allen(2017) R in Finance and Economics. A Beginners Guide, World Scientific Publishing, Ch 1,2,3,4
Heiss F. (2016) Using R for introductory Econometrics http://urfie.net/read/mobile/index.html#p=4,
Yihui Xie, Dynamic Documents with R and Knitr, Chapman and Hall
EXERCISE 1 Write an R code that answers to all the ToDo points in Torfs P. and C. Bauer(2014) “A (very short) introduction to R” ,
EXERCISE 2 An introduction to Data Handling, SOLUTION
Section 3: Graphical and Descriptive Analysis of Sport Statistics (NBA data)
Graphical Analysis
Correlation Analysis
QQ plots and Histogram
Subsetting data and TS plots
Introduction to model building and Simulation
The NBA database: download and import in R. teamsoverall2023.csv, datafiles, programme to build database from datafiles
https://www.basketball-reference.com/leagues/NBA_2023.html, programme to update data by webscraping
R CODES : code1, code2, please not that you need to create Teams_overall2023.csv to run the codes
EXERCISE 3: text, code
Section 4: The Linear Regression Model
SLIDES 1
SLIDES 2
Models for Experimental and non-Experimental Data
Models as outcomes of reduction processes
Model Estimation: the OLS and its properties
Interpreting Regression Results: Statistical Significance and Relevance
The Effects of Model Misspecification
AN APPLICATION,THE FOUR FACTOR MODEL R code
EXERCISE 4: The Four Factor Model, NOTES , solution
Winston W.L.(2009) Mathletics, Princeton University Press, Chapter 28
Section 5: Using Models to Weight NBA Statistics
SLIDES 1, SLIDES 2
Weighting Statistics to measure performance
Correlation analysis
The NBA Efficiency Measure
Using a Model based on Possession
Offensive Efficiency and Defensive Efficiency
Modelling Wins
Evaluating Statistics by Simulation: Monte-Carlo and Bootstrap methods
Completing the Model
Evaluating Players' Efficiency: WINS, assists and WINS48
R CODES: team_stat , players_stat, data on players, NOTES
EXERCISE 5: text, SOLUTION , SOLUTION AS RMD
EXERCISE 6: text, notes, SOLUTION
Berri D.J.,M.B.Schmidt and S. Brook(2006), The Wages of Wins, Stanford University Press, Ch 6,7
Ultimo aggiornamento 16/03/2024 | {"url":"https://didattica.unibocconi.it/mypage/doc.php?idDoc=31114&IdUte=48917&idr=1754&Tipo=m&lingua=ita","timestamp":"2024-11-09T04:02:10Z","content_type":"text/html","content_length":"187054","record_id":"<urn:uuid:63bd1436-a1af-461b-8c17-2176e5e6cb98>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00396.warc.gz"} |
Now, take a pen and a scratch paper and try to understand the algorithm by solving the following example. Assume the function $h(s)$ gives the number of nodes $s$ is away from $s_{goal}$ and $g(s) =
g(s_{start}) + c(s_{start},s)$, where $c$ is the edge cost. So, $h(s_{goal}) = 0$ and $g(s_{start}) = 0$. Try it out on your own and compare it with the below table.
Figure 7.1 Example
Table 1 Solution
Node To Expand OPEN CLOSED bp(s)
$s_{start}$ {$s_{start}$} {} -
$s_2$ {$s_2$} {$s_{start}$} $bp(s_2) = s_{start}$
$s_4$ {$s_4, s_1$} {$s_{start}, s_2$} $bp(s_4) = bp(s_1) = s_2$
$s_1$ {$s_3, s_1$} {$s_{start}, s_2, s_4$} $bp(s_3) = s_4$
$s_{goal}$ {$s_3, s_{goal}$} {$s_{start}, s_2, s_4, s_1$} $bp(s_{goal}) = s_1$
- {$s_3$} {$s_{start}, s_2, s_4, s_1, s_{goal}$} -
Finally, in order to find the least cost path, we have to just follow the backpointers all the way from $s_{goal}$ to $s_{start}$. So, in our case the path will be $s_{goal} \rightarrow s_1 \
rightarrow s_2 \rightarrow s_{start}$.
7.3 Weighted A*
The name itself suggests that some weight or priority is assigned to the function.
$f(s) = g(s) + \epsilon h(s) \tag{2}$
For $\epsilon \geq 1$, there is a bias towards nodes that are closer to goal. We may regard weighted A* as greedy for a very high value of $\epsilon$. If we allow an epsilon of 5, this means that the
weighted A* algorithm will return a path that is no worse than 5 times the cost of the optimal solution.
7.4 Backward A*
It is similar to conventional A* algorithm, but the difference is that one starts to search from goal to the start. Accordingly, the algorithm changes to
7.5 Conclusion
The major difference between A* and weighted A* is the tradeoff between optimal solution and computation time. In following figure, two scenarios are taken: $\epsilon = 1$ and $\epsilon = 2.5$
Figure 7.2 A* $\epsilon = 1$ vs $\epsilon = 2.5$
Here, green indicates OPEN states and yellow indicates CLOSED states. The weighted A* is able to find solution faster than A*. However, the solution given by weighted A* is sub-optimal.
</> GitHub
7.6 References
[1]. 16-350 Planning Techniques for Robotics - link
[2]. Online - http://sbpl.net/ | {"url":"https://vivek-uka.github.io/2020/astar/","timestamp":"2024-11-10T09:03:31Z","content_type":"text/html","content_length":"9554","record_id":"<urn:uuid:77e18119-0ed7-42f1-9197-6fd7ab99c39f>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00162.warc.gz"} |
Distances, Neighborhoods, or Dimensions? Projection Literacy for the Analysis of Multivariate Data
Projections are some of the most common methods for presenting high-dimensional datasets on a 2D display. While these techniques provide overviews that highlight relations between observations, they
are unavoidably subject to change depending on chosen configurations. Hence, the same projection technique can depict multiple compositions of the same dataset, depending on its parameter setting.
Furthermore, projection techniques differ in their underlying assumptions and computation mechanisms, favoring the preservation of either distances, neighborhoods, or dimensions. This article aims to
shed light on the similarities and differences of a multitude of projection techniques, the influence of features and parameters on data-representations, and give a data-driven intuition on the
relation of projections. We postulate that, depending on the task and data, a different choice of projection technique, or a combination of such, might lead to a more effective view.
"t-SNE is the best projection technique available"—this might be a typical sentence from someone who understands projection techniques only on a superficial level. State-of-the-art techniques—such as
the latest t-SNE—produce "nice" results promptly. However, that so many people think t-SNE is synonymous for the state-of-the-art emphasizes two aspects: (1) people don't know much about projection
techniques: linear vs. non-linear, global vs. local, and (2) they do not know how to interpret them. This expectation is in line with the survey of Sedlmair et al., who found that "most users who
used DR [dimensionality reduction, i.e., projection techniques,] for data analysis struggled or failed in their attempt (of 22, 13 struggled and 6 failed). These numbers underline the need for
further usage-centered DR development and research." For example, sometimes users are not able to distinguish actual patterns from artifacts introduced by a dimensionality reduction technique, or
even lack an overall conceptual model of the dimensionality reduction technique they apply.
In this blog post, we attempt to enable you to gain a better understanding of how to interpret projections of high-dimensional datasets. Therefore, we guide you through selecting a dataset and some
techniques. Both are shown in interactive visualizations that allow you to have a closer look at any time. Do not hesitate going back and forth to find out how swapping the dataset or technique, and
changing a parameter influences what you see. Grasping how to interpret projections is an interactive " learning by doing" exercise.
This article guides the readers through different aspects of "Projection Literacy", based on a selected dataset. We argue that the choice of the appropriate projection technique and its parameters is
dependent on the data and individual user tasks. Each technique thus emphasizes different relations in the data and should, therefore, be interpreted under these considerations. The three most
prominent features objectified in projections are distances, neighborhoods, and dimensions.
For this blog post we selected a number of well known datasets next to some very simple synthetic examples. The simplistic examples will especially help you to better understand how different
projection techniques represent structures and distances. Please note that we display only 500 observations per dataset to achieve interactive performance. All datasets are available as part of
scikit-learn , so please help yourself and digg deeper.
To get started, we selected the well known S curve dataset from the list below for you. In the list, each entry shows one dataset by its name, and a preview based on a projection using Principal
Components Analysis (PCA). Below entries you can find the number of observations as well as the number of dimensions in the full dataset. We also indicate whether the dataset originates from a real
world source
Details on
Below you can find a matrix displaying the dataset you selected including all dimensions as rows and currently selected observations as columns. The tone of red in each cell denotes the value of one
observation for a dimension. Higher values are more red, while exact numbers are not of interest for our pursuit. If you chose a dataset for classification pastel colors in the background of column
You can click on a column label to select one observation and its ten nearest neighbors in the high-dimensional space. If you hover a column label, distances from the hovered observation to other
observations are shown in projections below in blue. The matrix will stick at the top of your screen for easy access when you actually reach the projections. If you hover a row label, values of
observations on the hovered dimension are shown in red. Finally, hovering the "ID / Class" label shows class labels, values on the dependent dimension, or position on the main direction of the
manifold, depending on the dataset's main task being classification
Dimensions (rows) in the matrix are sorted by variance with the most diverse column at the top. Observations (columns) are sorted by . If you sort them by the single-linkage clustering, observations
are clustered in the high-dimensional space, and the cluster dendrogram will be shown above the matrix. For some datasets—like the ten Gaussian blobs—you can see that clustering in the
high-dimensional space clearly reveals the ten clusters, while some projection techniques—e.g., PCA—may not allow to distinguish them individually.
Dimensionality Reduction / Projection Techniques
There are hundreds of techniques developed by mathematicians, statisticians, visualization researchers, and others. Technically, there are some differences, for example, between projections and
embeddings, and the broader set of dimensionality reduction techniques. To keep things simple, we use the term "projection" for all kinds of techniques. Further, we only consider unsupervised
techniques—i.e., the projection technique does not know about class labels—which do not include a clustering step. Hence, you will not find techniques like Linear Discriminant Analysis (LDA ) and
Self-Organizing Maps (SOM ) here. Our selection focuses on a diverse set of commonly used techniques. We use the implementations of scikit-learn, so that you can try out things easily by yourself
in case you want to have a closer look.
Before diving into projections, you need to understand a central design decision. Projection techniques transform high-dimensional data to a lower-dimensional space while preserving its main
structure. Typically, the data is transformed to two-dimensional space and visualized as a scatter plot as a means to analyze and understand the data. In order to transform data from a
higher-dimensional to a lower-dimensional space, we distinguish between two categories: linear and non-linear projection techniques. Linear projection techniques produce a linear transformation of
data dimensions in lower-dimensional space. Proximity between data points indicates similarity. The more similar data points are, the closer they are located to each other and vice versa. This is why
linear projection techniques are also known as global techniques. In contrast, non-linear projection techniques, also known as local projection techniques, aim at preserving the local neighborhoods
across the features in the data. Hereby, proximity highlights differences and coherences between observations and is not to put on the same level as similarity.
Figure 1: Projection techniques can be broadly separated in two groups; linear/global and non-linear/local.
For example, the technique t-SNE is commonly used for identifying clusters in the data. Yet, depending on the choice of parameters, built clusters can be of different size, as well as of different
proximity to each other. Either way, there is a clear separation of clusters and we can identify them as being different or coherent. The so-called S curve dataset is often used to showcase the
described difference between linear (global) and non-linear (local) projection techniques (See Figure 1). In the case of linear techniques, global characteristics will be preserved and the shape of
the ‘S’ is represented even in two dimensions. In the case of non-linear techniques, local characteristics—i.e., the local neighborhoods—count and the ‘S’ is typically rolled out in two dimensions.
Choose some projection techniques
Below you find our set of projection techniques. Each technique is presented on a card featuring the technique name, and a projection of the selected dataset. Feel free to brush (hold down mouse key
and drag) some observations in a scatter plot to see how they are distributed by other techniques. Click on the technique name to show details in the following section. Below the scatter plot you can
select the techniques to be displayed in section 4.3 (up to 5 techniques). In case the techniques has parameters you may click the
If you want to learn more about available techniques, van der Maaten, Postma and van den Herik as well as Gisbrecht and Hammer provide overviews.
Details on
Visual Assessment of Projections
Planar projections map high-dimensional data to a lower-dimensional space and try to preserve the main characteristics of the data. However, depending on the projection technique, different
characteristics are preserved.
A nice analogy is the shade a tree casts in the sunlight. If the sun is directly above the tree, its shadow does not resemble a tree too much. The shadow is more like a wool ball than a tree—we can
argue that certain tree features were not adopted in this particular form of projection. However, if the angle of the sun changes, other features, such as the tree trunk or certain branches, may be
This behavior is very similar to planar projections. Depending on which projection technique you choose, results might look very different. For example, when choosing a kernel function for Kernel PCA
the user has plenty of options. With a linear kernel—i.e., classical PCA—the shape of the S-curve is well represented; polynomial kernels introduce distortion to the ‘S’, but still allow for grasping
the general shape. With a cosine kernel, on the other hand, the ‘S’ structure is not present in the planar projection at all.
While in these simple cases high quality representations of the overall structure of the three-dimensional dataset can be achieved, in a more general high-dimensional setting this will not be
possible most of the time. In our research, we thus identified three main characteristics that mainly steer a projection:
1. Distances between observations
2. Neighborhoods of observations
3. Representation of original dimensions
Furthermore, and most importantly, users should ask themselves the following question before deep-diving into any projection result: "Which kind of pattern do I search for / am I interested in?"
Depending on this question, one has to find a trade-off between distances, neighborhoods, and the relevance of dimensions across observations. However, there can also be no single projection
technique that is superior across all application scenarios and tasks, as techniques require to trade-off between the aforementioned three characteristics. For example, in a high-dimensional dataset
that consists of n observations, (n^2-n)/2 unique distances exist. However, in a two-dimensional projection only 2n-3 distances can be enforced. Likewise, considering two dimensions—as in a regular
scatter plot—is enough to place all observations.
In order to assess whether the chosen projection reflects the distances, the neighborhoods, or the relevance of dimensions, we developed a visual representation called "feature map". Instead of
color-encoding the observation itself, we choose to encode the area in which the observations is unique using Voronoi Tessellation. The area is particular interesting, because it properly reflects
which characteristics are encoded and how wide they are spread.
Assessment of distances
For the first characteristic, we encode the distance as proposed by Aupetit in 2007. Each Voronoi cell color encodes the distance to the selected observation. If a projection perfectly reflected its
high-dimensional counterpart, it would form a perfectly aligned color gradient across all observations.
The top of the browser window shows a matrix of observations. Return to Section 3.1 (the overview of projections) and hover any observation (column) label—e.g. "o10". When trying out different
observations, you will see that major differences in quality may occur. With all the aforementioned projection techniques, it is challenging to tell how well distances are preserved, before exploring
the visual representation.
This is where the knowledge about the aim of a projection technique comes into play. Using a linear technique, distances have meaning. Using for example MDS, the distance will typically express the
aggregated Euclidean distance between observation. In contrast, a non-linear technique, such as t-SNE, emphasizes neighborhoods, where visual distances have no specific meaning. This is also why the
visual representation of the results after applying t-SNE effectively reflect classification results. Classes are grouped, but there is no "real, meaningful" distance between them.
While the technique introduced by Aupetit that we feature in this blog is sufficient as a starting point to learn how to interpret distances across projections of high-dimensional data, there exist
several other methods. For example, Seifert, Sabol and Kienreich introduced Stress Maps. More techniques for inspecting the quality of projected distances were proposed by Schreck, von Landesberger
and Bremm Stahnke et al. as well as Heulot, Fekete and Aupetit.
Assessment of neighborhoods
Click on the observation label in the matrix or directly on the observation in one of the scatter plots in Section 3.2 to see where the nearest neighbors of the selected observation are located. The
selected observation is highlighted in blue and its ten nearest neighbors in gray. Remaining observations are highlighted in a very light gray.
While nearest neighbors are typically displayed as close as expected, they sometimes may spread all over the projection. Similar to distances, it is not possible to guarantee a preservation of
neighborhoods, although preservation of neighborhoods is simpler from a mathematical point of view—i.e. if all distances are preserved, then also all neighborhoods are preserved, but not vice versa.
As mentioned before, the preservation of neighborhoods is of great value when it comes to tasks that do not require semantically meaningful distances, such as classification. One must be aware of the
fact that distances between classes and densities within such groups have no particular meaning. Analogous to the Voronoi-based visualization of distances, we limit the depicted representations to
the ones helping you to get an initial idea of which projection technique to choose for which characteristic. Additional techniques for investigating neighborhoods were introduced by Lespinats and
Aupetit, Martins et al. as well as Martins, Minghim and Telea.
Assessment of dimensions
The curse of dimensionality typically impairs the ability of a projection to visually carve out significant characteristics of a dimension. The more dimensions are taken into account, the less
expressive the results of planar projection are. Yet, well-represented original dimensions can ease the interpretation of a projection result. Figuring out which dimensions allow for a reasonable
interpretation in the projection is key to avoid misinterpretations. Meanwhile, the number of original dimensions that may enable approximate interpretations in a projection is not limited to two.
Correlations and dependencies in high-dimensional space can lead to more than two interpretably represented input dimensions.
The "feature map" table below enables you to effortlessly compare the representation of the input dimensions. It is generated automatically based on the Voronoi Tessellation, just like the distance
maps above. Instead of encoding distances to one focus observation, each cell color encodes the value of its observation along an input dimension. More colorful cells represent higher values within
input dimensions. Therefore, the map shows which value along each dimension one might expect at a location based on the nearest neighbor.
Brushing observations in one of the scatter plots above highlights only those cells in the table below, which are located within the range of the largest and smallest value of the brushed
observations. In other words, if a new observation is "similar" to the selected set of observations, you expect it to be placed in the colored regions. Caution: This description is purely
hypothetical, because some projection techniques produce very different results if a new observation is added to the projection.
Table 1: Values of dimensions (rows) are mapped to shades of red. The more red the higher the value is in respective dimension. Colors are relative to the minimum and maximum in each dimension. You
do not need exact values as you do not expect such precision. Just compare the distributions of features.
Again, there are more techniques far assessing the representation of dimensions available, for example those introduced by da Silva et al., Faust and Scheidegger, Cavallo and Demiralp (2017), and
Cavallo and Demiralp (2018).
Assessment using additional information
Often the main interest is not in the dimensions available, but an additional dimension to be predicted. In a classification task the main interest is in the class labels of observations, and for
regression in the values of a dependent dimension. As these outcome dimensions are not used by unsupervised projection techniques, looking at them may also provide hints on the quality of
projections. Hover the "ID / Class" label at the top left of the matrix to show—depending on the main task of the selected dataset—observations' class labels (pastel colors), their values along the
dependent dimension for regression (grayscale), or their position along the main direction of the synthetic manifold (green). Keep in mind though that, as Aupetit notes, evaluating projection quality
based on class separation alone may not be a good idea.
Lessons learned
As you have seen above, it is not straightforward to interpret projections. Be sure about what you expect from the projection—show patterns such as classes, or show similarities or do subspace
analysis. Then, the design decisions taken by projection techniques—e.g., linear vs. non-linear—are a central point to consider. You may want to make sure that your task demands align with the
properties of your preferred projection technique. It is also always good to start with simple data distribution analysis for what to possibly expect from the data. We are well aware, that the
datasets presented in this blog post are rather simple. So, whichever technique worked well here may not be your favorite choice in a practical application. Similarly, a technique that worked well
for you colleague—or even yourself last time—may not work for you now, having different data and other tasks.
Hopefully, you are now better able to interpret the different projections you can generate from your data. The so-called curse of dimensionality is responsible for many issues that come with
projection techniques. A thorough analysis of interesting observations prior to projection can lead to better results. So, to date our best advice is: Have a look at your data, try multiple
projection techniques with various parameter settings, then choose those that provides best results relative to your task demands. However, this blog post is just the tip of the iceberg, or your
first step on a long hike to generating good projections. For example, Bertini, Tatu and Keim looked at quality metrics for the visualization of high-dimensional data. Nonetheless, to the best of our
knowledge there are no automated recommendation techniques working across tasks. In the end, it is still up to you to find a projection that provides what you need. There is one catch though, try not
to overfit your data to your expectations.
For more guidance on choosing projection techniques you may consult the works of Sedlmair, Munzner and Tory, and Etemadpour et al.. Cutura et al. recently presented VisCoDeR for choosing projection
techniques and setting parameters. Further, Sacha et al. provide an overview of options for interaction with projections.
Taken together, always beware that you need to know which technique was used to produce a projection in order to arrive at reasonable interpretations, as a projection rarely shows the full picture of
a dataset and its interpretation depends on the technique used for construction. Projecting to two dimensions almost always means losing information. The question is, whether or not relevant features
of a dataset are preserved by the projection technique. As a result, you need to choose one or more techniques that suit your task, and you should not expect that there were a one-size-fits-all
technique or an algorithm that works out-of-the box. Much of the power of projection techniques depends on a good match between data and technique as well as—in most cases—a finely tuned set of
parameters. Turning your wrap into a pizza does not come for free. | {"url":"https://visxprojections.dbvis.de/client/index.html","timestamp":"2024-11-03T19:26:57Z","content_type":"text/html","content_length":"54161","record_id":"<urn:uuid:a79fcdc5-1c4a-439b-9216-48f894ad7d55>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00755.warc.gz"} |
Gini coefficient and its relation to the Lorenz curve
The Gini coefficient measures income inequality. A value of 0 indicates perfect equality. The Lorenz curve shows income distribution.
Table of Contents
(Lorenz Curve and Gini Coefficient – Measures of Income Inequality)
The Gini coefficient measures income inequality. A value of 0 signifies perfect equality. In contrast, a score of 1 represents extreme inequality. The Lorenz curve graphically depicts income
distribution. By comparing the Lorenz curve and the diagonal line of perfect equality, we gain insights into a society’s income disparities. A concave Lorenz curve indicates greater inequality.
Conversely, a convex curve suggests more equality. Developed by Corrado Gini in the early 20th century, this metric has become a vital tool. Governments and organizations worldwide utilize the Gini
coefficient to shape policies and programs. Understanding income inequality enables targeted interventions to uplift marginalized communities. A rising Gini coefficient may signal growing disparities
that require attention. Effective strategies can foster a fairer and more inclusive society. The relationship between the Gini coefficient and the Lorenz curve offers a nuanced understanding of
economic disparities. By examining these tools together, policymakers can address inequality more effectively. Ultimately, striving for a more equitable distribution of resources is crucial for
societal well-being and progress.
Calculation of Gini coefficient
To understand the essence of wealth distribution, we delve into the intricate world of the Gini coefficient. Imagine a tool that reveals societal disparities with just a glance – that’s precisely
what the Gini coefficient does. This metric quantifies income inequality within a population, painting a vivid picture of how resources are divvied up among its members.
When it comes to calculating this pivotal statistic, mathematicians employ an elegant formula that encapsulates these inequalities in numerical form. The process begins by arranging individuals from
least affluent to most affluent based on their incomes. Once this ordering is established, we can plot them against cumulative percentages of total income received.
The next step involves constructing the Lorenz curve—a graphical representation illustrating income distribution across the population. It starts at the origin and ascends as more individuals
accumulate income until reaching 100% when all earnings have been accounted for.
Now, here’s where things get even more fascinating: By examining this curve in conjunction with a diagonal line representing perfect equality (where each person holds an equal share of total wealth),
we unlock insights crucial for determining our Gini coefficient.
This magical number lies within our reach through simple arithmetic—by dividing the area between the Lorenz curve and perfect equality line by half of total equitable distribution area; Voila! We’ve
unveiled society’s economic disparity score neatly packaged into one concise figure—the Gini coefficient!
Emotions may run high as we ponder what implications this number holds: Does it signify a fair and just society or reveal stark divisions plaguing communities? The beauty—and complexity—of this
statistical measure lie not only in its calculation but also in its power to provoke reflection on social structures and policies governing resource allocation.
So there you have it—a brief glimpse into the mechanics behind one of economics’ most powerful tools for gauging inequality: The enigmatic Gini coefficient working hand-in-hand with its faithful
companion, the Lorenz curve. As we navigate through these mathematical landscapes, let us remember that numbers hold stories waiting to be uncovered—anecdotes woven from data points beckoning us to
explore deeper truths hidden beneath their surface calculations.
Construction of Lorenz curve
The construction of the Lorenz curve is like weaving a story with numbers, painting a vivid picture of income distribution in society. Imagine plotting points on a graph, each representing different
segments of the population and their share of total income. These points are interconnected, forming a delicate curve that unveils disparities and inequalities.
To begin this journey, we start by arranging individuals or households from lowest to highest income along the horizontal axis. Then, on the vertical axis, we depict cumulative shares of total income
corresponding to each group. As we connect these dots progressively, a unique shape emerges – the Lorenz curve.
This curve dances across the canvas of our analysis, revealing how wealth is distributed among various sections of society. It showcases whether resources are evenly spread or concentrated in fewer
hands. The closer the Lorenz curve hugs the perfect equality line (a 45-degree diagonal), the fairer and more equal is income distribution.
However, reality often paints a different picture. The curvature may dip below that ideal line at times,speaking volumes about stark wealth gaps where few hold significant riches while many struggle
with limited resources.
As we navigate through this intricate mathematical landscape,the nuances become clearer – every bend and slope signifies real lives impacted by economic forces.Sometimes gentle curves hint at
moderate inequality,demonstrating some level of balance amidst diversity.Other times sharp twists expose deep divides highlighting social injustices that demand attention and action.
In essence,the Lorenz curve whispers stories echoing voices unheard.Every data point etched upon its path represents faces unseen,families striving,and dreams deferred.How can we ignore these tales
told through lines and intersections?They beckon us to ponder,prompting introspection into our collective consciousness regarding fairness,equality,and opportunity for all within our societal
So as you study this elegant diagram remember,it’s not just lines on paper but reflections mirroring realities faced daily by people across diverse backgrounds.Its construction isn’t mere
mathematics;it’s an art form capturing raw emotions,breathing life into statistics,and urging us towards empathy understanding,and change.
Definition of Gini coefficient
The Gini coefficient is like a mirror reflecting the inequality within a society. It’s a mathematical measure that tells us how wealth or income is distributed among the people in a specific area.
The essence of the Gini coefficient lies in its ability to capture disparities – it ranges from 0 (perfect equality, where everyone has an equal share) to 1 (complete inequality, where one person
owns everything).
Imagine you’re at a bustling marketplace; some vendors are barely making ends meet while others command long lines of eager customers willing to splurge. This disparity in sales paints a picture of
economic imbalance—a scenario typical of a high Gini coefficient.
The magic lies in visualizing this data on what we call the Lorenz curve—an elegant graph showing the cumulative distribution of income across different segments of society. If riches were evenly
spread, our curve would be close to perfection—an immaculate diagonal line representing fairness and parity for all.
However, reality often isn’t so rosy; deviations from this ideal line reveal stark inequalities that shape societies worldwide. These nuances come alive through varying shapes and slopes on our
Lorenz curves—each telling its own story about who holds power and privilege, and who struggles with scarcity and marginalization.
When we calculate the Gini coefficient based on these curves, every decimal point carries immense weight—it’s not just numbers; it’s lives impacted by policies, systems, and societal structures
designed by humans themselves.
A high Gini coefficient indicates deep chasms separating communities into haves and have-nots—echoes of unfairness reverberating through generations. On the other hand, lower values hint at more
equitable distributions but don’t necessarily guarantee prosperity for all—it merely scratches beneath surface-level comparisons until deeper issues emerge unraveled before us.
In our quest for social justice and inclusive growth, understanding the intricacies behind these coefficients becomes paramount—the key unlocking doors to transformative change paving way towards
brighter futures where opportunities aren’t confined but abundant for everyone willing to dream beyond constraints set by statistics or graphs alone.
(Gini Coefficient and Lorenz Curve)
Interpretation of Gini coefficient
When we dive into the intricate world of income inequality, one of the key tools that come to surface is the Gini coefficient. This numerical measure offers a glimpse into how wealth or income is
distributed within a population. It’s like peering through a magnifying glass at society’s economic structure.
Now, let’s talk about interpreting this elusive Gini coefficient! Picture it as a sliding scale between 0 and 1. A value of 0 represents perfect equality – everyone earns exactly the same amount; no
riches are hoarded while others struggle. Conversely, a score of 1 signifies extreme inequality – where one person holds all the wealth while everyone else is left with crumbs.
What’s fascinating yet daunting about this measure is its ability to capture complex societal dynamics in just one number. As you stare at that decimal figure derived from statistical wizardry,
emotions can run high – empathy for those on the lower end and perhaps an uncomfortable twinge realizing where society stands on equity.
The Lorenz curve enters stage left here, offering visual aid to complement our numerical friend, the Gini coefficient. Imagine a graph showing cumulative percentages of individuals ranked by their
income alongside actual income proportions they hold – it’s like tracing inequalities with ink instead of numbers on paper.
So now, when we put these two together – Gini coefficient and Lorenz curve holding hands in analysis harmony – patterns start emerging like shadows cast under different lighting conditions: does your
country boast an egalitarian paradise or wallow in uneven prosperity?
But remember, interpretation isn’t always straightforward; sometimes outliers skew results or cultural nuances muddy waters. So tread carefully when drawing conclusions; there might be hidden
narratives waiting to unravel beneath initial assumptions.
In essence, grasping the nuanced dance between Gini coefficients and Lorenz curves requires more than just mathematical prowess—it calls for empathy-driven insight into human experiences woven
intricately within data points and lines on graphs.
Relationship between Gini coefficient and Lorenz curve
Understanding the relationship between the Gini coefficient and the Lorenz curve is like deciphering a complex dance between economic data and inequality. Imagine a bustling cityscape, where wealth
flows unevenly among its inhabitants. The Gini coefficient steps in as our mathematical choreographer, quantifying this disparity with a single number that ranges from 0 to 1.
At one end of the spectrum lies perfect equality (Gini index of 0) – picture a utopian society where everyone holds an equal share of resources. On the flip side, with maximum inequality (Gini index
of 1), all riches are concentrated in one person’s hands while others struggle on bare minimums.
Now, enter stage left: the Lorenz curve – a graphical representation mapping out how income or wealth distribution deviates from that ideal egalitarian state. It’s akin to an artist’s brushstroke
capturing swirling patterns of privilege and deprivation across various segments of society.
When these two entities tango together, their correlation reveals profound insights into economic structures and social disparities at play within communities or countries. A steep upward curve
signifies greater inequality – imagine towering skyscrapers casting long shadows over makeshift shelters below.
Conversely, a gentle slope denotes more equitable distributions; think lush green parks shared by families picnicking under bright blue skies. By analyzing how closely intertwined these curves are,
policymakers can gauge the effectiveness of policies aimed at bridging wealth gaps or identify areas requiring intervention.
Picture economists huddled around charts as they scrutinize every nuance – each dip and peak telling tales of prosperity or poverty experienced by individuals navigating life’s financial labyrinth.
As we delve deeper into this intricate connection between numerical indices and real-world implications, emotions run high – empathy for those marginalized by skewed systems intertwines with hope
sparked by strategies geared towards leveling playing fields.
Ultimately, understanding the relationship between the Gini coefficient and Lorenz curve isn’t just about numbers; it’s about recognizing human experiences woven into statistical fabric – stories
waiting to be heard and inequalities waiting to be addressed on societies’ grand stage.
External Links | {"url":"https://info.3diamonds.biz/gini-coefficient-and-its-relation-to-the-lorenz-curve/","timestamp":"2024-11-08T02:57:53Z","content_type":"text/html","content_length":"102507","record_id":"<urn:uuid:70066b9b-c80b-4073-8ea2-9c285c0d457a>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00403.warc.gz"} |
Periodic data of polar 2-diffeomorphisms with one saddle orbit
Title Periodic data of polar 2-diffeomorphisms with one saddle orbit
Authors E. V. Nozdrinova^1, O. V. Pochinka^1
^1National Research University Higher School of Economics
In this paper we consider polar diffeomorphisms of a surface, that is, diffeomorphisms having a unique sink and a unique source periodic orbit. A classical example of such a diffeomorphism
Annotation is the ``source-sink'' diffeomorphism, which does not have saddle points and exists only on a two-dimensional sphere. However, the addition of even a one saddle orbit will significantly
expand the class of the polar diffeomorphisms on surfaces. In particular, in this paper the authors proved that polar diffeomorphisms with exactly one saddle orbit there are on arbitrary
surface and the saddle orbit always has a negative orientation type. In addition, all possible types of periodic data for such polar diffeomorphisms are established.
Keywords periodic data, polar diffeomorphism, surface
Nozdrinova E. V., Pochinka O. V. ''Periodic data of polar 2-diffeomorphisms with one saddle orbit'' [Electronic resource]. Proceedings of the XIII International scientific conference
Citation ''Differential equations and their applications in mathematical modeling''. (Saransk, July 12-16, 2017). Saransk: SVMO Publ, 2017. - pp. 408-417. Available at: https://conf.svmo.ru/files/
deamm2017/papers/paper58.pdf. - Date of access: 12.11.2024. | {"url":"https://conf.svmo.ru/en/archive/article?id=58","timestamp":"2024-11-12T02:34:40Z","content_type":"text/html","content_length":"11490","record_id":"<urn:uuid:e10c9186-51c7-47eb-a57c-86aeffb6df83>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00782.warc.gz"} |
Unified compression-based acceleration of edit-distance computation
The edit distance problem is a classical fundamental problem in computer science in general, and in combinatorial pattern matching in particular. The standard dynamic programming solution for this
problem computes the edit-distance between a pair of strings of total length O(N) in O(N2) time.
To this date, this quadratic upper-bound has never been substantially improved for general strings. However, there are known techniques for breaking this bound in case the strings are known to
compress well under a particular compression scheme. The basic idea is to first compress the strings, and then to compute the edit distance between the compressed strings.
As it turns out, practically all known o(N2) edit-distance algorithms work, in some sense, under the same paradigm described above. It is therefore natural to ask whether there is a single
edit-distance algorithm that works for strings which are compressed under any compression scheme. A rephrasing of this question is to ask whether a single algorithm can exploit the compressibility
properties of strings under any compression method, even if each string is compressed using a different compression. In this paper we set out to answer this question by using straight line programs.
These provide a generic platform for representing many popular compression schemes including the LZ-family, Run-Length Encoding, Byte-Pair Encoding, and dictionary methods.
For two strings of total length N having straight-line program representations of total size n, we present an algorithm running in O(nN log(N/n)) time for computing the edit-distance of these two
strings under any rational scoring function, and an O(n 2/3N 4/3) time algorithm for arbitrary scoring functions. Our new result, while providing a significant speed up for highly compressible
strings, does not surpass the quadratic time bound even in the worst case scenario.
שפה מקורית אנגלית
עמודים (מ-עד) 339-353
מספר עמודים 15
כתב עת Algorithmica
כרך 65
מספר גיליון 2
סטטוס פרסום פורסם - 2013
טביעת אצבע
להלן מוצגים תחומי המחקר של הפרסום 'Unified compression-based acceleration of edit-distance computation'. יחד הם יוצרים טביעת אצבע ייחודית. | {"url":"https://cris.openu.ac.il/iw/publications/unified-compression-based-acceleration-of-edit-distance-computati","timestamp":"2024-11-13T01:19:48Z","content_type":"text/html","content_length":"52465","record_id":"<urn:uuid:7370aa91-3273-4732-8678-5c2fd3538ece>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00555.warc.gz"} |
Problem Bonus. (5 points)
Problem Bonus. (5 points)
The problem below can be regarded as a problem of structural stability. For certain values of N
lower than a critical value Ner, a perturbation A0 <1 applied to the initial rest configuration 8₁
= 0 will result in the system going back the same initial configuration, i.e. 02 = 0₁ = 0. However,
if N>Ner, a perturbation A0 <1 applied to the initial rest configuration 0₁=0 will result in a
new, permanent, rest configuration 02 #0. The scope of this exercise is to find the value of Ner for
a given generic length of the rigid bar 1, stiffness k and mass m.
Hint: In order to solve the problem and find Ner, you have to
1. apply the principle of work and energy, T₁ + U₁=T₂ + U₂, and
2. consider an angle very small so that you can assume the following Taylor series expansions
truncated to the first and second order
sin (0) = 0 +......
cos(0) =
= 1
Fig: 1 | {"url":"https://tutorbin.com/questions-and-answers/problem-bonus-5-points-the-problem-below-can-be-regarded-as-a-problem-of-structural-stability-for-certain-values-of-n","timestamp":"2024-11-03T02:47:29Z","content_type":"text/html","content_length":"64379","record_id":"<urn:uuid:bd7d9bc8-589c-46ac-aed1-f5723a449c30>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00204.warc.gz"} |
What are the elementary row operations of matrices? | Socratic
What are the elementary row operations of matrices?
1 Answer
There are three elementary row operatins of matrices:
• Exchange two rows position;
• Substitute a row for the sum of it and another row;
• Multiply a row for a scalar;
Hop it helps.
Impact of this question
2279 views around the world | {"url":"https://socratic.org/questions/what-are-the-elementary-row-operations-of-matrices","timestamp":"2024-11-08T15:09:44Z","content_type":"text/html","content_length":"31937","record_id":"<urn:uuid:914b4152-7ae9-43d3-9d15-650b531971d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00640.warc.gz"} |
Determine Step Size
For the first step in Model Preparation Process, you obtain results from a variable-step simulation of the reference version of your Simscape™ model. The reference results provide a baseline against
which you can assess the accuracy of your model as you modify it. This example shows how to analyze the reference results and the step size that the variable-step solver takes to:
• Estimate the maximum step size that you can use for a fixed-step simulation
• Identify events that have the potential to limit the maximum step size
Discontinuities and rapid changes require small step sizes for accurately capturing these dynamics. The maximum step size that you can use for a fixed-step simulation must be small enough to ensure
accurate results. If your model contains such dynamics, then it is possible that the required step size for accurate results, Ts[max], is too small. A step size that is too small does not allow your
real-time computer to finish calculating the solution for any given step in the simulation.
The analysis in this example helps you to estimate the maximum step size that fixed-step solvers can use and still obtain accurate results. You can also use the analysis to determine which elements
influence the maximum step size for accurate results.
1. To open the reference model, at the MATLAB^® command prompt, enter:
model = 'ssc_pneumatic_rts_reference';
2. Simulate the model:
3. Create a semilogarithmic plot that shows how the step size for the solver varies during the simulation.
h1 = figure;
title('Solver Step Size')
xlabel('Time (s)')
ylabel('Step Size (s)')
For much of the simulation, the step size is greater than the value of the Ts[max] in the plot. The corresponding value, ~0.001 seconds, is an estimated maximum step size for achieving accurate
results during fixed-step simulation with the model. To see how to configure the step size for fixed-step solvers for real-time simulation, see Choose Step Size and Number of Iterations.
The x markers in the plot indicate the time that the solver took to execute a single step at that moment in the simulation. The step-size data is discrete. The line that connects the discrete
points exists only to help you see the order of the individual execution times over the course of the simulation.
A large decrease in step size indicates that the solver detects a zero-crossing event. Zero-crossing detection can happen when the value of a signal changes sign or crosses a threshold. The
simulation reduces the step size to capture the dynamics for the zero-crossing event accurately. After the solver processes the dynamics for a zero-crossing event, the simulation step size can
increase. It is possible for the solver to take several small steps before returning to the step size that precedes the zero-crossing event. The areas in the red boxes contain variations in
recovery time for the variable step solver.
4. To see different post-zero-crossing behaviors, zoom to the region in the red box at time (t) = ~1 second.
After t = 1.005 seconds, the step size decreases from ~10e-3 seconds to less than 10e-13 seconds to capture an event. The step size increases quickly to ~10e-5 seconds, and then slowly to ~10e-4
seconds. The step size decreases to capture a second event and recovers quickly, and then slowly to the step size from before the first event. The slow rates of recovery indicate that the
simulation is using small steps to capture the dynamics of elements in your model. If the required step size limits the maximum fixed-step size to a small enough value, then an overrun might
occur when you attempt simulation on your real-time computer.
The types of elements that require small step size are:
□ Elements that cause discontinuities, such as hard-stops and stick-slip friction
□ Elements that have small time constants, such as small masses with undamped, stiff springs and hydraulic circuits with small, compressible volumes
The step size recovers more quickly after it slows down to process the event that occurs before t = 1.02 seconds. This event is less likely to require small step sizes to achieve accurate
5. To see different types of slow solver recoveries, zoom to the region within the red box at t = ~4.2 seconds.
xZoomStart2 = 4.16;
xZoomEnd2 = 4.24;
yZoomStart2 = 10e-20;
yZoomEnd2 = 10e-1;
axis([xZoomStart2 xZoomEnd2 yZoomStart2 yZoomEnd2]);
Just as there are different types of events that cause solvers to slow down, there are different types of slow solver recovery. The events that occur just before t = 4.19 and 4.2 seconds both
involve zero-crossings. The solver takes a series of progressively larger steps as it reaches the step size from before the event. The large number of very small steps that follow the zero
crossing at Slow Recovery A indicate that the element that caused the zero crossing is also numerically stiff.
The quicker step-size increase after the event that occurs at t = 4.2 seconds indicates that the element that caused the zero crossing before Slow Recovery B, is not as stiff as the event at Slow
Recovery A.
6. To see the results, open the Simscape Results Explorer.
7. Examine the angular speed. In the Simscape Results Explorer window, in the simulation log tree hierarchy, select Measurements > Ideal Rotational Motion Sensor > w.
8. To add a plot of the gas flow, select Measure Flow > Pneumatic Mass & Heat Flow Sensor and then, use Ctrl+click to select G_ps.
The slow recovery times occur when the simulation initializes, and approximately at t = 1, 4, 5, 8, and 9 seconds. These periods of small steps coincide with these times:
□ The motor speed is near zero rpm (simulation time t = ~ 1, 5, and 9 seconds)
□ The step change in motor speed is initiated from a steady-state speed to a new speed (time t = ~ 4 and 8 seconds)
□ The step change in flow rate is initiated from a steady-state speed to a new flow rate (time t = ~ 4 and 8 seconds)
□ The volumetric flow rate is near zero kg/s (t = ~ 1, 4, and 5 seconds)
These results indicate that the slow step-size recoveries are most likely due to elements in the model that involve friction or that have small, compressible volumes. To see how to identify the
problematic elements and modify them to increase simulation speed, see Reduce Numerical Stiffness and Reduce Zero Crossings.
See Also
Related Examples
More About | {"url":"https://se.mathworks.com/help/simscape/ug/determine-step-size.html","timestamp":"2024-11-13T21:19:11Z","content_type":"text/html","content_length":"78678","record_id":"<urn:uuid:a8402508-4578-4ad0-8e2d-067c08c90114>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00158.warc.gz"} |
Worksheet on 3-Digit Addition Word Problems | 3rd Grade Math | Answers
Worksheet on 3-Digit Addition Word Problems
In worksheet on 3-digit addition word problems we will solve the problems on addition of 3-digit numbers without regrouping, addition of 3-digit numbers with regrouping, addition of three 3-digit
1. Sachin scored 146 runs in the first innings and 232 runs in the second innings of a test match. What was his total score?
2. Nairitee has two dictionaries, one for English and the other for Hindi. The English dictionary has 456 pages while the Hindi dictionary has 336 pages. How many pages are there in both the
dictionaries taken together?
3. A vegetable merchant sold 396 sacks of potatoes and 423 sacks of onions in the month of May. How many sacks of potatoes and onions did he sell in all?
4. An ice cream parlour sold 226 chocolate ice creams and 385 strawberry ice creams. How many ice creams were sold in all?
5. A merchant sold 245 umbrellas in June, 323 umbrellas in July and 199 umbrellas in August. How many umbrellas did he sell in the three months altogether?
From Worksheet on 3-Digit Addition Word Problems to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question. | {"url":"https://www.math-only-math.com/worksheet-on-3-digit-addition-word-problems.html","timestamp":"2024-11-06T13:34:43Z","content_type":"text/html","content_length":"44514","record_id":"<urn:uuid:4bf658eb-7c8d-4502-a1d5-2e50247e5ad3>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00627.warc.gz"} |
Book Chapter
Modular points, modular curves, modular surfaces and modular forms
No external resources are shared
There are currently no full texts shared for your IP range.
There are no public fulltexts stored in PuRe
There is no public supplementary material available
Zagier, D. (1985). Modular points, modular curves, modular surfaces and modular forms. In Arbeitstagung 1984: Proceedings of the meeting held by the Max-Planck-Institut für Mathematik, Bonn, June
15-22, 1984 (pp. 225-248). Berlin: Springer.
Cite as: https://hdl.handle.net/21.11116/0000-0004-396B-5
[For the entire collection see Zbl 0547.00007.] \\par This is the written version of a talk at the Arbeitstagung at Bonn. It is centered around one example: the modular curve X\\sb 0(37). The
elliptic curve E:\\quad y(y-1)=(x+1)x(x-1) is a factor of the Jacobian J\\sb 0(37). The article treats special values of L-series attached to E and its twists, Heegner points on E, the Gross-Zagier
theorem and illustrates the interplay between classical algebraic geometry over \\bbfC and Arakelov geometry over \\bbfZ. It also gives an extension of the Gross-Zagier result: \\sum P\\sb dq\\sp d\\
quad is a modular form of weight 3/2 and level 37. Here P\\sb d is the Heegner point on X\\sb 0(37) associated to d. This has now been proved for arbitrary N (rather than for N=37) by Gross/Kohnen/
Zagier. The proof for the special case treated here uses an ad hoc method. This article is written to wet one's appetite and no doubt it will. | {"url":"https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_3125875","timestamp":"2024-11-11T17:58:50Z","content_type":"application/xhtml+xml","content_length":"38358","record_id":"<urn:uuid:5354c66d-b3ee-4030-ba6c-0d5476c9e3fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00567.warc.gz"} |
Equal fractions: overview, rules, examples - La Cultura de los Mayas
Equal fractions: overview, rules, examples
Which of the following two fractions is larger?89 or 1112?
It’s not easy to say, because fractions with different denominators are not easy to compare. How you can still solve this task, you will find out in this article!
Why you need fractions of the same name – overview
You often have to compare two fractions and, for example, determine which fraction has the greater or smaller value. Or you have to add or subtract fractions. However, you can only cope with these
tasks if the fractions have the same denominator, i.e. if these fractions same name are.
Two fractions are called same name called if they have the same denominator.
fractions make same name means that several fractions are reduced to the same denominator.
Reminder: the numerator is the number above the fraction bar and the denominator is the number below the fraction bar.
However, fractions of the same name are not only required to be able to compare two fractions, as for the introductory example. Fractions of the same name are also needed when subtracting and adding
two fractions. When multiplying and dividing fractions, however, you are not dependent on fractions of the same name.
So let’s go straight to how to make fractions equal!
Make fractions equal – rules
Fractions can be made equivalent in different ways.
For example, you can combine the two denominators of fractions multiply. But it quickly happens that you get a very large number. It usually makes more sense to close the fractions shorten or to
extend. Another possibility is that you least common multiple determine the denominator.
Find common denominator by multiplying
Two fractions can be made the same by multiplying the denominators of two fractions together. By multiplying the two denominators, you get a number that represents a suitable denominator for both
Don’t forget to expand the counter as well. To do this, you need to multiply the numerator of one fraction by the denominator of the other fraction.
Task 1
You gave the two fractions 13 and 42.
To find a common denominator for these two fractions, simply multiply the two denominators of the fractions together in the next step
3 * 2 = 6
Now you have found a common denominator between the two fractions.
Now add the denominator of the other fraction to the numerators.
1 x 26 = 264 x 36 = 126
You have now made two fractions of the same name by multiplying the denominators together.
Find common denominator via least common multiple
That least common multiple – short kgV – of two or more whole numbers is the smallest natural number that is shared by both of these numbers.
There are three ways to find the least common multiple of two numbers. Once you can find the LCM using a series of numbers, or you can do a prime factorization. You can also calculate it using the
GCD if you know it.
It is best to look at the exact procedure for these methods in the article «least common multiple»!
At this point we look at an example in which we calculate the LCM using the prime factor analysis, and thus make two fractions of the same name.
exercise 2
Given are the two fractions 46 and 710.
First, do a prime factorization for the two denominators.
6 = 3 * 210 = 2 * 5
You can see that the numbers 3 and 5 appear once each in the two prime factorizations. The 2 occurs in both prime factorizations, but is only multiplied once.
Now multiply all the numbers together. In this case it is the 2, the 3 and the 5.
2 * 3 * 5 = 30
So the LCM you are looking for is 30. Now expand the two fractions so that both get the denominator 30.
4 * 56 * 5 = 20307 * 310 * 3 = 2130
If you had made the fractions the same by multiplying both denominators together, the denominator would now be 60. With the kgV you get the smallest possible common denominator.
Find common denominator by expanding
You expand a fraction by multiplying the numerator and denominator by the same number.
If you want to add a number c to a fraction, you multiply a and b by c.
The numbers a, b and c are so-called whole numbers ℤ. These are negative and positive integers!
For some problems both fractions have to be expanded. However, sometimes it is sufficient to expand just one fraction. Therefore, always check first whether one of the two denominators is a multiple
of the other fraction. If that is the case, all you have to do is expand the fraction with the smaller denominator. This saves you unnecessary calculation work.
task 3
You should bring the two fractions 38 and 524 to the same denominator.
Since 24 is a multiple of 8, all you have to do is expand the 38 so that there is a 24 in the denominator.
3 * 38 * 3 = 924
Now you have the two fractions 924 and 524
task 4
Given are two unlike fractions 34 and 15. You can find a common denominator for these two fractions by multiplying the denominators of the two fractions together, i.e. expanding them.
3 * 54 * 5 = 15201 * 45 * 4 = 420
Find common denominator by abbreviating
You can find a common denominator not only by expanding, but also by shortening. A fraction is reduced by dividing the numerator and denominator of the fraction by the same number.
If you want to reduce a fraction ab by a number c, you have to divide a and b by c.
The numbers a, b and c are so-called whole numbers ℤ. These are negative and positive integers!
Even if you want to make two fractions of the same name by reducing them, you may often only have to reduce one of the two fractions, because even when reducing, one denominator may be a multiple of
the other denominator. If that’s the case, all you have to do is reduce the fraction that has the larger number in the denominator.
task 5
The two fractions 1230 and 415 can be reduced to a common denominator by reducing the first fraction.
1230 = 12 : 230 : 2 = 615
Now you have obtained the two fractions 615 and 415 by reducing.
task 6
You should reduce the two fractions 1442 and 2030 to a common denominator.
First you should think about which number divides both 42 and 30. For example, you can write down a series of numbers in which you write down all the numbers by which the two denominators can be
42 → 1, 2, 3, 6, 7, 14, 2130 → 1, 2, 3, 5, 6, 10, 15
In the rows of numbers, look for all the numbers that appear in both rows of numbers. Now choose one of these numbers that appears in both rows of numbers. This number is the denominator that you can
bring both fractions to. In this example, that is 6. Now think about the number you have to use to shorten the two denominators so that they get the denominator 6.
14 : 742 : 7 = 2620 : 530 : 5 = 46
Compare equal fractions
As you saw in the introductory example, two fractions can only be compared well if both fractions have the same denominator. Therefore, before you want to compare two fractions, you should always
make the fractions the same.
Once you have made both fractions of the same name, all you have to do is see which fraction has the larger or smaller numerator. The fraction with the larger numerator is larger than that with the
smaller numerator.
task 7
The two fractions 89 and 1112 from the introductory example can be given the same name in different ways.
8 * 49 * 4 = 323611 * 312 * 3 = 3336
After we have made the two fractions of the same name by expanding, we can now see which of the two fractions is larger. All we have to do is see which of the two fractions has the larger numerator.
3236 < 3336
We see that 1112 is the larger fraction.
Let’s look at one more example in a moment.
task 8
Given are two unlike fractions 13 and 14. You can find a common denominator for these two fractions by multiplying the denominators of the two fractions together, i.e. expanding them.
This gives you the two fractions 1 43 4 = 412 and 1 34 3 = 312
Now you can easily compare these two fractions by looking at which of the two fractions has the larger numerator.
It follows: 412 > 312
Want to know more about comparing fractions? Then you should definitely take a look at the article Comparing and arranging fractions!
Adding and subtracting fractions
As already mentioned, adding and subtracting fractions also requires that the fractions have the same name.
When both fractions have the same denominator, addition and subtraction are easy: you can just add or subtract the numerators of the fractions. The denominator remains unchanged.
Let’s look at this directly with an example.
task 9
Add the two fractions 811 and 23.
1. In the first step, we expand the two fractions to make them the same. Expanding is particularly suitable for this, in order to bring the two fractions to the same denominator.
8 * 311 * 3 = 2433
2 x 113 x 11 = 2233
2. Now that we’ve brought the two fractions to the same denominator, we can easily add them together. For this we only have to add the two numerators together, since the denominator no longer
2433 + 2233 = 4633
equal fractions – examples and tasks
Now that you know all the rules for making fractions of the same name, you can test your new knowledge directly with a few tasks.
equal fractions – The most important
• Like fractions are needed for comparing, adding, and subtracting fractions.
• Fractions can be made the same by expanding and truncating.
• Fractions can only be compared if they have the same name.
• A common denominator can be found by expanding or contracting. | {"url":"https://culturalmaya.com/equal-fractions-overview-rules-examples/","timestamp":"2024-11-06T17:23:47Z","content_type":"text/html","content_length":"53546","record_id":"<urn:uuid:f6ed9734-5988-478a-96d1-e2c8f610a944>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00855.warc.gz"} |
ThmDex – An index of mathematical definitions, results, and conjectures.
▼ Set of symbols
▼ Alphabet
▼ Deduction system
▼ Theory
▼ Zermelo-Fraenkel set theory
▼ Set
▼ Binary cartesian set product
▼ Binary relation
▼ Map
▼ Operation
▼ N-operation
▶ D20: Enclosed binary operation
▶ D5319: Idempotent binary operation
Convention 0 (Multiplicative notation)
▶ Let $X \neq \emptyset$ be a D11: Set and let $f : X \times X \to Y$ be a D554: Binary operation on $X$. If $x, y \in X$, then the convention in multiplicative notation is to denote the element $f
(x, y)$ by $x y$.
Convention 1 (Additive notation)
▶ Let $X \neq \emptyset$ be a D11: Set and let $f : X \times X \to Y$ be a D554: Binary operation on $X$. If $x, y \in X$, then the convention in additive notation is to denote the element $f(x, y)$
by $x + y$. | {"url":"https://theoremdex.org/d/554","timestamp":"2024-11-10T12:24:22Z","content_type":"text/html","content_length":"9196","record_id":"<urn:uuid:309f97cc-8db2-4879-bd00-6e90976cddd1>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00515.warc.gz"} |
How to Tile a Circle | Homesteady
How to Tile a Circle
Because most rooms are square or rectangular in shape, and most tiles are square, the process of installing tile is pretty straightforward in most instances. If you want to tile a non-rectangular
shape though, such as a circle, the process gets more complicated. In order to tile a circle, you must reshape the tiles before beginning the installation so that the tiles to fit into the space in
the proper pattern.
1. Measure the diameters of both the inside and outside of the circular area. The inner diameter is the measurement from one inside edge to the opposite inside edge of the tile circle. The outer
diameter is the measurement from one outside edge of the tile to the opposite outside edge of the tile circle.
2. Figure the outer circumference of the circle by multiplying the exterior diameter by pi, or 3.14. If your outer diameter, for instance, is 10 inches, the circumference of the circle equals 31.4
3. Divide the circumference of the circle by the width of the tiles you want to use to tile the circle. If you plan to use 2-inch tiles, for instance, divide 31.4 inches by 2 to get approximately 16
tiles needed for the space. Don't forget to factor in the distance that you must maintain to grout between the tiles.
4. Figure the inner circumference of the circle by multiplying the diameter by 3.14. If the inner circumference is 6 inches, for instance, the inner circumference equals approximately 19 inches.
5. Divide the inner circumference of the circle by the number of tiles you figured you needed to tile the circle to get the approximate size for the interior of each tile. If the inner circumference
is 19 inches, for instance, and you need 16 tiles to cover the exterior of the circle, you need the interior of each of those tiles to be approximately 1.2 inches wide.
6. Measure the width you calculated for the interior edge of the tile on each tile for the circle by measuring in the same distance from each edge. To trim a 2-inch tile down to 1.2 inches, for
example, measure in .4 inches from each side of the tile on the back of the tile and make a mark. Then, lay a ruler on the tile, so that it touches the mark on the right and lines up with the
right corner on the opposite side of the tile, and repeat for the left side of the tile.
7. Cut the tiles on these lines with a tile saw to create tiles that angle in on both sides. Once cut, lay the tiles out along the circle. If the inner parts of the tiles are slightly too large, use
the tile saw to trim them down a little bit at a time until they fit in the circle and enough space is left between the tiles for grouting.
8. Install the tiles with tile adhesive and let the adhesive dry. After the adhesive dries, grout between the tiles as you would grout any tile job.
Writer Bio
Alexis Lawrence is a freelance writer, filmmaker and photographer with extensive experience in digital video, book publishing and graphic design. An avid traveler, Lawrence has visited at least 10
cities on each inhabitable continent. She has attended several universities and holds a Bachelor of Science in English.
More Articles | {"url":"https://homesteady.com/how-8010609-tile-circle.html","timestamp":"2024-11-03T07:19:36Z","content_type":"text/html","content_length":"120785","record_id":"<urn:uuid:a2547285-33dd-4c5b-b9c0-6fd40295630b>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00799.warc.gz"} |
Strict dissipativity analysis for classes of optimal control problems involving probability density functions
Title data
Fleig, Arthur ; Grüne, Lars:
Strict dissipativity analysis for classes of optimal control problems involving probability density functions.
Bayreuth , 2019 . - 23 S.
Format: PDF
Name: 2019_fleig-gruene_dissipativity-analysis-economic-mpc-fp.pdf
Version: Preprint
Available under German copyright law. The document may be used free of charge for personal use. In addition, the reproduction, editing, distribution and any kind of exploitation require the
License written consent of the respective rights holder.
Download (1MB)
Project information
Project title: Project's official title
Project's id
Model Predictive Control for the Fokker-Planck Equation
GR 1569/15-1
Project financing: Deutsche Forschungsgemeinschaft
Motivated by the stability and performance analysis of model predictive control schemes, we investigate strict dissipativity for a class of optimal control problems involving probability density
functions. The dynamics are governed by a Fokker-Planck partial differential equation. However, for the particular classes under investigation involving linear dynamics, linear feedback laws, and
Gaussian probability density functions, we are able to significantly simplify these dynamics. This enables us to perform an in-depth analysis of strict dissipativity for different cost functions.
Further data | {"url":"https://epub.uni-bayreuth.de/id/eprint/4420/","timestamp":"2024-11-05T04:17:42Z","content_type":"application/xhtml+xml","content_length":"30866","record_id":"<urn:uuid:e0b0a0ab-a46a-410d-9455-cff24e8a4113>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00107.warc.gz"} |
Lesson 2
Corresponding Parts and Scale Factors
Let’s describe features of scaled copies.
2.1: Number Talk: Multiplying by a Unit Fraction
Find each product mentally.
\(\frac14 \boldcdot 32\)
\((7.2) \boldcdot \frac19\)
\(\frac14 \boldcdot (5.6)\)
2.2: Corresponding Parts
One road sign for railroad crossings is a circle with a large X in the middle and two R’s—with one on each side. Here is a picture with some points labeled and two copies of the picture. Drag and
turn the moveable angle tool to compare the angles in the copies with the angles in the original.
1. Complete this table to show corresponding parts in the three pictures.
│ original │ Copy 1 │ Copy 2 │
│point \(L\) │ │ │
│segment \(LM\) │ │ │
│ │segment \(ED\) │ │
│ │ │point \(X\) │
│angle \(KLM\) │ │ │
│ │ │angle \(XYZ\)│
2. Is either copy a scaled copy of the original road sign? Explain your reasoning.
3. Use the moveable angle tool to compare angle \(KLM\) with its corresponding angles in Copy 1 and Copy 2. What do you notice?
4. Use the moveable angle tool to compare angle \(NOP\) with its corresponding angles in Copy 1 and Copy 2. What do you notice?
2.3: Scaled Triangles
Here is Triangle O, followed by a number of other triangles.
Your teacher will assign you two of the triangles to look at.
1. For each of your assigned triangles, is it a scaled copy of Triangle O? Be prepared to explain your reasoning.
2. As a group, identify all the scaled copies of Triangle O in the collection. Discuss your thinking. If you disagree, work to reach an agreement.
3. List all the triangles that are scaled copies in the table. Record the side lengths that correspond to the side lengths of Triangle O listed in each column.
4. Explain or show how each copy has been scaled from the original (Triangle O).
Choose one of the triangles that is not a scaled copy of Triangle O. Describe how you could change at least one side to make a scaled copy, while leaving at least one side unchanged.
A figure and its scaled copy have corresponding parts, or parts that are in the same position in relation to the rest of each figure. These parts could be points, segments, or angles. For example,
Polygon 2 is a scaled copy of Polygon 1.
• Each point in Polygon 1 has a corresponding point in Polygon 2.
For example, point \(B\) corresponds to point \(H\) and point \(C\) corresponds to point \(I\).
• Each segment in Polygon 1 has a corresponding segment in Polygon 2.
For example, segment \(AF\) corresponds to segment \(GL\).
• Each angle in Polygon 1 also has a corresponding angle in Polygon 2.
For example, angle \(DEF\) corresponds to angle \(JKL\).
The scale factor between Polygon 1 and Polygon 2 is 2, because all of the lengths in Polygon 2 are 2 times the corresponding lengths in Polygon 1. The angle measures in Polygon 2 are the same as the
corresponding angle measures in Polygon 1. For example, the measure of angle \(JKL\) is the same as the measure of angle \(DEF\).
• corresponding
When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.
For example, point \(B\) in the first triangle corresponds to point \(E\) in the second triangle. Segment \(AC\) corresponds to segment \(DF\).
• scale factor
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).
• scaled copy
A scaled copy is a copy of a figure where every length in the original figure is multiplied by the same number.
For example, triangle \(DEF\) is a scaled copy of triangle \(ABC\). Each side length on triangle \(ABC\) was multiplied by 1.5 to get the corresponding side length on triangle \(DEF\). | {"url":"https://im.kendallhunt.com/MS/students/2/1/2/index.html","timestamp":"2024-11-13T07:55:16Z","content_type":"text/html","content_length":"129845","record_id":"<urn:uuid:50cd0757-657f-4066-a6a5-137a9f665bbb>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00527.warc.gz"} |
Design 5 - Flywheel Design
written 5.3 years ago by modified 2.6 years ago by
1 Answer
written 5.3 years ago by
A punching machine makes 25 working strokes per minute and is capable of punching 25 mm diameter holes in 18 mm thick steel plates having an ultimate shear strength of 300 MPa. The punching operation
takes place during $1/10^{th}$ of a revolution of the crankshaft. Select standard motor.
Determine suitable dimensions for the rim cross-section of the flywheel, which is to revolve at 9 times the speed of the crankshaft. The permissible coefficient of fluctuation of speed is 0.1. The
diameter of the flywheel must not exceed 1.4m owing to space restrictions. Check for the centrifugal stress induced in the rim.
Given data
$k_{s}=0.1, D_{\max }=1.4 \mathrm{m}$
Assumptions - The cross-section of the rim is rectangular, flywheel material-CI with density $7250 \mathrm{kg} / \mathrm{m}^{3},$ width -b = $\mathrm{D} / 5,$ The hub and the spokes assumed to
provide 5$\%$ of the rotational inertia of the wheel. | {"url":"https://www.ques10.com/p/47857/design-5-flywheel-design-1/","timestamp":"2024-11-06T04:48:46Z","content_type":"text/html","content_length":"26345","record_id":"<urn:uuid:c70c02e9-efde-409f-b302-a0786c07a707>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00604.warc.gz"} |
HDU 1069 Monkey and Banana, hdu1069
A Free Trial That Lets You Build Big!
Start building with 50+ products and up to 12 months usage for Elastic Compute Service
• Sales Support
1 on 1 presale consultation
• After-Sales Support
24/7 Technical Support 6 Free Tickets per Quarter Faster Response
• Alibaba Cloud offers highly flexible support services tailored to meet your exact needs. | {"url":"https://topic.alibabacloud.com/a/hdu-1069-monkey-and-banana-hdu1069_1_11_32474672.html","timestamp":"2024-11-05T10:53:42Z","content_type":"text/html","content_length":"77709","record_id":"<urn:uuid:57f243e0-e894-4501-9761-00a5fff95ca1>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00218.warc.gz"} |
Section 1 – Introduction
Learn the basics of Google sheets and start making your spending tracker
Section 2 – Total spent, Total left, Amount left
Learn how to use the Sum formula to automatically calculate the: - total spent - total left - amounts left in each category
Section 3 – Amount spent by category
Learn how to use the Sumif formula to automatically caluclate the amount spent from each grant category.
Section 4 – Finishing up
Learn how to use conditional formatting to put the final touches on your spending tracker | {"url":"https://digiquick.org/courses/sheets-grant/lesson/introduction-7/","timestamp":"2024-11-03T06:28:45Z","content_type":"text/html","content_length":"114819","record_id":"<urn:uuid:7e7ff661-092f-4d0e-805b-7ee70ec2214b>","cc-path":"CC-MAIN-2024-46/segments/1730477027772.24/warc/CC-MAIN-20241103053019-20241103083019-00518.warc.gz"} |
How to Round a Price to .99 | Encyclopedia-Excel
When working with prices, it can often be beneficial to round to the nearest .99 cents. There are a few formulas we can use depending on the results we're looking for.
Assuming your price in in cell A1:
Round Price to Nearest .99
Round Price Up to Nearest .99
= CEILING.MATH(A1,1) - 0.01
Round Price Down to Nearest .99
= FLOOR.MATH(A1,1) - 0.01
How to Round Price to End in Nearest .99
If we have a list of prices and want to round them up or down to the nearest .99 cents, we can use the following formula:
The ROUND function rounds our price to the nearest whole number ($5.30 to $5.00), and then we subtract 0.01 from that number to reach the nearest .99 cents.
How to Round Price Up to End in .99
Instead, if we have a list of prices and want to round them up to the nearest .99 cents, we can use the following formula:
= CEILING.MATH(price,1) - 0.01
The CEILING.MATH function rounds our price up to the nearest whole number ($5.30 to $6.00), and then we subtract 0.01 from that number to reach the nearest .99 cents.
How to Round Price Down to End in .99
Otherwise, if we have a list of prices and want to round them down to the nearest .99 cents, we can use the following formula:
= FLOOR.MATH(price,1) - 0.01
The FLOOR.MATH function rounds our price to the nearest whole number ($5.30 to $5.00), and ten we subtract 0.01 from that number to reach the nearest .99 cents. | {"url":"https://www.encyclopedia-excel.com/how-to-round-a-price-to-99-in-excel","timestamp":"2024-11-13T10:52:54Z","content_type":"text/html","content_length":"1050083","record_id":"<urn:uuid:adb1ba54-3c36-474d-baf5-1488f7dac550>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00367.warc.gz"} |
Discussion – Journal of American Hoodoo
The Sieve
In Discussion on 2016/06/15 at 8:39 am
Imagine a language where there is no need for a word for “real” or “fact” because there are already very clear terms for such things as…
• conjecture for the sake of exploration
• conjecture for the sake of entertainment
• conjecture one really hopes is false
• conjecture one really hopes is true
• angry, unfounded accusations one would like to be true so one can feel justified in being angry because one thinks feeling angry without justification is shameful and the purview of toddlers
• something deep-sounding and vaguely poetic that one asserts because one whimsically confuses beauty with truth
• something deep-sounding and vaguely poetic that one asserts because one stoically confuses pessimism with prophecy
• something one is worried about that one actually hopes is not true that one asserts as a provocation in lieu of actually asking the hearer to make the speaker feel better
• an outright lie to deflect blame from oneself
The Book of Dead Names
In Discussion on 2015/06/27 at 4:21 pm
Welcome back to high school. In this nightmare, it’s the ninth grade again. You’ve made it through the lunch line, maybe bought some milk to go with the sandwich you brought from home to save you
from having to face green beans that were harvested when Nixon was president, imprisoned since then in a can large enough to have contained an adult human’s head, decanted along with nine other
identical cans into a huge steel vat, and boiled for six hours or until spreadable.
Now it’s time to choose a table. For those of you who need it, here’s a hint. This is a metaphor for being born.
You see the cool kids’ table and know better than to try. Something tells you that you’d need an invitation, and you’re right. You watch and you listen, and you one of them declare a petitioner a
nerd and wave her off in the direction of the Nerd Section. So now you try to work out the map and the rules. Read the rest of this entry »
Symptom of Illness at a Cultural Level
In Discussion on 2014/12/05 at 4:24 pm
Delusions are a bit tricky.
Be aware that I’m talking clinically about a symptom of a number of scary mental illnesses — the symptom that signals a disconnect from proper interaction with the real world. A symptom, but not an
illness in and of itself.
In situations like this I adhere to a pretty strict definition of illness, of pathology. Unless the phenomenon interferes with an acceptable level of function of your body and/or interferes with your
ability to earn a living (or to complete your coursework, if you are a student) and/or degrades your relationships with coworkers or classmates or peers or friends or relatives, it is not an illness.
Short of that, it’s just a quirk or a trait. Read the rest of this entry »
When We Think of Beasts
In Discussion on 2014/01/03 at 10:11 am
When we think of beasts, we concentrate on, instead of the volumes of overwhelming similarities, that which sets us apart from them. Distinguishing characteristics. We can own them, for one. They
might communicate or not, but they don’t speak any of the human languages, don’t demonstrate a huge vocabulary, and show little facility for learning languages that aren’t the ones they were born
with. They aren’t big on manual dexterity and, when they make or use tools, they get by with the bare minimum. Though many of them sing or dance or both, they aren’t big on the literary arts or
visual arts — but we should take into account that we look for representational elements when we don’t even share visual spectra with many of them, as we also fail to make allowances for lack of
vocabulary and manual dexterity in expression. In any case, they don’t seem to tell stories, and we do like stories. Read the rest of this entry »
Beware the Curse
In Discussion on 2012/09/15 at 1:08 pm
Go ahead. Let it out. You’ll feel better.
One of the places where human justice falls down is the belief, embedded in every story we tell our children from their earliest days, that bad things inevitably happen to people who are bad and that
good behavior is rewarded. I’d go so far as to say that the recognition of the failure of that axiom is the source of every crisis of faith ever experienced. The innocent starve. The wicked get
wealthy by cheating and stealing. Natural disasters take lives indiscriminately. We believe this unsupportable notion of the inherent fairness of the universe so strongly that when we see evidence
strongly to the contrary, it is literally intolerable. We feel it as pain.
We demand justice — or we sink into depression, because all actions are futile and the results of those actions are arbitrary. Read the rest of this entry »
One of Many Problems with Religion
In Discussion on 2012/08/25 at 2:02 pm
This isn’t a problem with all religions, mind. In fact, it’s only a problem with a handful. However, it’s a problem with the most popular, and the most violent — and, anthropologically speaking, the
most recent. And this is the problem concept: that humans are special, are blessed, are chosen to be God’s favored children, are somehow above the animals and plants and everything else that lives,
and have a God-given right of power over life and death with respect to them.
I’m not sure how all of that made it into the dominant narratives, because much of the scripture it’s based on stops well short of the worst of that in wording. But religions are made out of a huge
body of traditions that, in those that do have scriptures, have very little support in those scriptures. Read the rest of this entry »
The Trouble with Science
In Discussion on 2012/08/13 at 9:42 am
We look up in the sky and see ten thousand points of light (give or take a few orders of magnitude depending on location and light pollution) and then, because knowing where the stars are in the sky
helps us pinpoint where we are in the seasons despite the vagaries of the weather, we draw lines around them and connecting them and give the drawings names. And we make up stories about the drawings
so that we can remember them, and remember that the positions of the stars are important, and, if we’re clever enough with the stories, why.
That’s “why the positions of the stars are important to us”, not any bigger sort of why, like “why are stars the things that are important”. Certainly not a “what”, like “what are stars”. Nor a
“how”, as in “how do the positions of the stars drive the planting and harvest cycles”.
Well, that’s not true. The stories can actually address such things. It’s just that when they do, the risk of bullshit is dangerously high. Read the rest of this entry »
The Art of Sacrifice
In Discussion on 2012/07/30 at 9:55 am
The man who is prepared to die may accomplish anything.
I’ve been looking through sources to see who I might be quoting for the sentiment above and I still haven’t sorted it. The original could be in a language I don’t know, thousands of years old. Or
maybe it’s a James Bond villain. But it’s not just a truth in narrative logic. It’s actually true. One who dies in the process or aftermath of achieving any goal, no matter how stupid or heinous or
heroic or pointless that goal is, is freed from suffering any consequences except the one that he or she has chosen. Any punishment or shame or notoriety passes, usually harmlessly, to family or
associates. Both heroes and villains, which are frequently interchangeable depending on individual sympathies, derive their status as such by not being particularly opposed to a fatal outcome.
On that topic, these creatures have something interesting in common: Read the rest of this entry »
More on Narrative Logic
In Discussion on 2012/07/27 at 12:08 pm
One of tenets of narrative logic — the logic used to make things true in our heads, that causes distress when it does not agree with observations — is that effort is rewarded, followed swiftly by a
corollary that says greater effort is rewarded more than lesser effort. Of all the major disagreements with the nature of causality that we carry around in our heads, this one is the one that seems
to cause the most misery. We desperately want there to be parity between effort spent and reward received, if not a slight tip of the balance in our favor.
Physical causality isn’t like that. An action taken at the right place in the right time under the right circumstances has a result, and it might be a desired one, but it’s just the next step in a
cascade. Read the rest of this entry »
On Narrative Causality
In Discussion on 2012/07/16 at 1:36 pm
It’s been ages since I’ve read Isaac Bonewits’s Real Magic, but a huge chunk of it stuck with me.
It’s by no means a how-to. Instead, it’s a book-length, thrice-revised expansion of of the senior thesis of the only person I’ve even heard of to receive a Bachelors of the Arts in Thaumaturgy from
an accredited university — though it may explain it a bit to say it was from UC Berkeley in 1970. The overall view is that it is an academic work, in construction if not tone and lack of bias, and as
such the analysis it contains is not unscientific. Various traditions and practices of (scientifically speaking) a superstitious nature are deconstructed to reveal a candidate set of underlying laws
that seem to govern the construction of esoteric belief and ritual.
I remain fascinated by Bonewits’s analysis, and I believe there is some truth in it — truth in what it reveals of how people think why they try to influence the world around them, leaving aside any
question of whether such influences are effective. Read the rest of this entry » | {"url":"http://americanhoodoo.org/category/discussion/","timestamp":"2024-11-02T21:59:37Z","content_type":"application/xhtml+xml","content_length":"112484","record_id":"<urn:uuid:f7f5cbfe-8f2a-41c5-b12c-9b55cb82fad3>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00306.warc.gz"} |
What is load flow study?
A load flow study, also known as power flow analysis, is a critical assessment in electrical engineering that calculates the steady-state voltages, currents, power flows, and power losses in an
electrical power system. This study is essential for planning, operating, and optimizing power systems, especially in complex networks where load variations can significantly impact stability and
Table of Contents
1. Definition of Load Flow Study
The load flow study is an essential computational analysis that determines voltage magnitudes and angles at different buses in a power system, as well as the active and reactive power flows in lines.
This study is performed using known inputs such as the generator’s real and reactive power, load demand, and the system’s line impedance. The output from this study enables engineers to ensure system
stability, reliability, and optimal power delivery.
2. Importance of Load Flow Study
Load flow studies are vital for several reasons:
• System Planning: Load flow studies help in planning the expansion and reinforcement of existing power systems.
• Operational Efficiency: It ensures efficient power delivery, optimizing the load distribution across the network.
• Reliability: Load flow analysis allows engineers to prevent potential issues such as voltage drops and power losses.
• Stability Analysis: This study is essential for analyzing stability under various load and fault conditions.
3. Load Flow Study Methods
There are several common methods for conducting a load flow study. Each method has its advantages and specific use cases.
3.1 Gauss-Seidel Method
The Gauss-Seidel method is an iterative algorithm used in solving the load flow problem. It is easy to implement and particularly useful for smaller networks. This method works by estimating the
voltages at different buses and adjusting iteratively until the desired accuracy is achieved.
Steps of the Gauss-Seidel Method:
1. Initialize voltage magnitudes and angles for all buses, except the slack bus.
2. For each iteration, update the voltage at each bus based on the previous values.
3. Repeat the process until the voltage difference falls below the specified tolerance level.
3.2 Newton-Raphson Method
The Newton-Raphson method is a powerful and widely used algorithm in load flow analysis. It is faster and more accurate than the Gauss-Seidel method, particularly in large and complex networks.
However, it requires more memory and computational resources.
Steps of the Newton-Raphson Method:
1. Define initial approximations for voltage magnitudes and angles.
2. Set up the Jacobian matrix based on power flow equations.
3. Calculate voltage updates by solving linear equations using the Jacobian matrix.
4. Iterate until convergence criteria are met.
3.3 Fast Decoupled Load Flow (FDLF)
The Fast Decoupled Load Flow (FDLF) method is an efficient approach derived from the Newton-Raphson method, optimized for larger systems. By decoupling the active and reactive power equations, it
achieves faster computation times with a simpler matrix structure.
Advantages of FDLF:
• Reduces computational complexity.
• Highly effective for large power systems with minimal memory usage.
• Offers faster convergence in systems with high voltage stability.
3.4 DC Load Flow Method
The DC Load Flow method simplifies the AC power flow problem by making assumptions that ignore reactive power, making it suitable for high-level planning in large systems. Although it is less
accurate than AC methods, it offers quick estimates of power flows and voltage magnitudes.
Assumptions in DC Load Flow:
• Neglects reactive power (Q) and only considers active power (P).
• Assumes a flat voltage profile across all buses.
4. Example of Load Flow Analysis
To illustrate a load flow analysis, let’s consider a small power system with three buses. Bus 1 is a slack bus, Bus 2 is a generator bus (PV bus), and Bus 3 is a load bus (PQ bus). The following
parameters are assumed:
Bus Type Voltage (V) Power (P) Reactive Power (Q)
Bus 1 Slack 1.05 ∠0° Unknown Unknown
Bus 2 Generator (PV) 1.04 ∠? 50 MW Unknown
Bus 3 Load (PQ) Unknown -80 MW -30 MVAR
Using the Gauss-Seidel or Newton-Raphson method, we iteratively solve for the unknown voltages and power flows to reach an accurate solution for all bus voltages.
5. Applications of Load Flow Study
Load flow studies have several practical applications:
• Transmission and Distribution Planning: Helps in the design and expansion of transmission and distribution networks.
• Fault Analysis: Assists in fault detection and location by evaluating current load distributions.
• System Optimization: Identifies inefficient areas in power systems for optimal load distribution.
• Voltage Regulation: Ensures voltage levels remain within acceptable limits, improving reliability.
6. Conclusion
In summary, load flow studies play a fundamental role in the analysis, design, and optimization of power systems. By providing insights into power flows and voltage levels, load flow studies enable
efficient system planning, fault analysis, and operational efficiency. Selecting the appropriate load flow method depends on factors such as network size, complexity, and computational resources.
7. Frequently Asked Questions (FAQ)
What is the purpose of a load flow study?
A load flow study analyzes voltage, current, and power flows within a power system, helping to ensure efficient operation and reliability.
Which method is best for load flow study?
The Newton-Raphson method is generally preferred for its accuracy and speed in larger networks, while the Gauss-Seidel method is suitable for smaller networks.
What are the main types of buses in load flow analysis?
The main types are Slack, PV (Generator), and PQ (Load) buses, each serving a specific function in power flow calculations.
0 Comments
Posted by
Prasun Barua
Prasun Barua is an Engineer (Electrical & Electronic) and Member of the European Energy Centre (EEC). His first published book Green Planet is all about green technologies and science. His other
published books are Solar PV System Design and Technology, Electricity from Renewable Energy, Tech Know Solar PV System, C Coding Practice, AI and Robotics Overview, Robotics and Artificial
Intelligence, Know How Solar PV System, Know The Product, Solar PV Technology Overview, Home Appliances Overview, Tech Know Solar PV System, C Programming Practice, etc. These books are available at
Google Books, Google Play, Amazon and other platforms. | {"url":"https://www.prasunbarua.com/2021/11/what-is-load-flow-study.html","timestamp":"2024-11-15T03:43:24Z","content_type":"application/xhtml+xml","content_length":"130673","record_id":"<urn:uuid:e64fcd51-2b97-47b8-b597-fda5a503d207>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00092.warc.gz"} |
What do you ask to find the door leading to freedom? - Daily Quiz and Riddles
You are a prisoner in a room with 2 doors and 2 guards. One of the doors will guide you to freedom and behind the other is a hangman–you don’t know which is which.
One of the guards always tells the truth and the other always lies. You don’t know which one is the truth-teller or the liar either. However, both guards know each other.
You have to choose and open one of these doors, but you can only ask a single question to one of the guards.
What do you ask to find the door leading to freedom?
Ask one of the guards “If I asked what door would lead to freedom, what door would the other guard point to?”
Alright, my fellow puzzlers, get ready for a mind-boggling challenge that will put your logical thinking to the test! Picture this: You find yourself in a room with two doors and two guards, but only
one of those doors leads to freedom. The catch? One guard always tells the truth, while the other guard always lies. Tricky, isn’t it?
Now, you might be wondering how on earth you can figure out which door to choose with just a single question to one of the guards. But fear not, my clever companions, for there is a solution that
will guide you toward the path of freedom.
Here’s the question that will unravel the mystery: “If I asked what door would lead to freedom, what door would the other guard point to?”
Let’s break it down. If you ask the truth-guard this question, they will give you an honest response. In this case, the truth-guard would tell you that the liar-guard would point to the door that
leads to death. Why? Because the truth-guard knows that the liar-guard always lies, and if asked the same question, the liar-guard would point to the wrong door.
On the other hand, if you ask the liar-guard the same question, they will give you a deceptive response. The liar-guard would tell you that the truth-guard would point to the door that leads to
death. But here’s the key: since the liar-guard always lies, their answer actually reveals that the truth-guard would point to the door that leads to freedom.
So, no matter who you ask, both guards will inadvertently give you the information you need. Their answers will indicate which door leads to death, allowing you to confidently choose the other
door—the one that leads to freedom!
Congratulations, my astute problem solvers! By employing your sharp thinking and crafting a clever question, you have outwitted the guards and secured your path to freedom. Remember, sometimes the
key to unraveling a complex puzzle lies in the art of asking the right question.
Leave a Comment | {"url":"https://quizandriddles.com/what-do-you-ask-to-find-the-door-leading-to-freedom/","timestamp":"2024-11-13T15:07:06Z","content_type":"text/html","content_length":"170464","record_id":"<urn:uuid:e234871f-726e-4265-8306-9bdf46afea42>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00537.warc.gz"} |
□ org.apache.commons.math4.legacy.linear.LUDecomposition
• public class LUDecomposition
extends Object
Calculates the LUP-decomposition of a square matrix.
The LUP-decomposition of a matrix A consists of three matrices L, U and P that satisfy: P×A = L×U. L is lower triangular (with unit diagonal terms), U is upper triangular and P is a permutation
matrix. All matrices are m×m.
As shown by the presence of the P matrix, this decomposition is implemented using partial pivoting.
This class is based on the class with similar name from the JAMA library.
□ a getP method has been added,
□ the det method has been renamed as getDeterminant,
□ the getDoublePivot method has been removed (but the int based getPivot method has been kept),
□ the solve and isNonSingular methods have been replaced by a getSolver method and the equivalent methods provided by the returned DecompositionSolver.
2.0 (changed to concrete class in 3.0)
See Also:
MathWorld, Wikipedia
□ Method Summary
All Methods Instance Methods Concrete Methods
Modifier and Type Method Description
double getDeterminant() Return the determinant of the matrix.
RealMatrix getL() Returns the matrix L of the decomposition.
RealMatrix getP() Returns the P rows permutation matrix.
int[] getPivot() Returns the pivot permutation vector.
DecompositionSolver getSolver() Get a solver for finding the A × X = B solution in exact linear sense.
RealMatrix getU() Returns the matrix U of the decomposition.
☆ Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
□ Constructor Detail
☆ LUDecomposition
public LUDecomposition(RealMatrix matrix)
Calculates the LU-decomposition of the given matrix. This constructor uses 1e-11 as default value for the singularity threshold.
matrix - Matrix to decompose.
NonSquareMatrixException - if matrix is not square.
☆ LUDecomposition
public LUDecomposition(RealMatrix matrix,
double singularityThreshold)
Calculates the LU-decomposition of the given matrix.
matrix - The matrix to decompose.
singularityThreshold - threshold (based on partial row norm) under which a matrix is considered singular
NonSquareMatrixException - if matrix is not square | {"url":"https://commons.apache.org/proper/commons-math/commons-math-docs/apidocs/org/apache/commons/math4/legacy/linear/LUDecomposition.html","timestamp":"2024-11-08T01:11:48Z","content_type":"text/html","content_length":"22784","record_id":"<urn:uuid:6de07d5d-5bb8-462a-929f-7d3142fdaf9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00447.warc.gz"} |
Writing Equations in ParaView 4.0 with MathText
The ParaView UserVoice page is a place where users of the application can request features, and we pay attention to these requests and try to implement highly desired features when possible. One of
the top requests has been for the ability to use LaTeX markup in ParaView text fields to draw mathematical equations as annotations and labels. I’m happy to announce that this request has been
fulfilled, and ParaView 4.0 marks the first major release with this ability.
The new release handles text entries containing a pair of dollar signs differently by rendering them as mathematical equations. The text between the dollar signs can be any valid MathText expression.
While not quite LaTeX, MathText encompasses a large subset of the LaTeX markup language, and works much the same way. Users of the Python plotting toolkit matplotlib will already be familiar with
MathText, as the convenient equation input functionality in matplotlib uses MathText. In fact, ParaView is actually using matplotlib under-the-hood for rendering these equations!
p dir=”ltr”>Entering mathematical symbols and equations in the new ParaView is easy as $\pi$ — to try this new feature out, download a copy of ParaView 4 today!
2 comments to Writing Equations in ParaView 4.0 with MathText
1. It’s details like this that make a great application excellent! Nice job,
2. Hi,
I want to use bold math like $bm{x/a}$ in Paraview text. Is it possible to inclue latex preamble like we do in matplotlib
plt.rc(‘text’, usetex=True)
Is there any other way to include bold math? | {"url":"https://www.kitware.com//writing-equations-in-paraview-4-0-with-mathtext/","timestamp":"2024-11-13T06:18:35Z","content_type":"text/html","content_length":"96959","record_id":"<urn:uuid:1136d890-0ff3-4987-9125-a260af6b3298>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00103.warc.gz"} |
I am currently doing a PhD at EPFL in the LIONS lab with Prof. Volkan Cevher, where I am mostly focused on nonconvex-nonconcave minimax problems.
I have written on a variety of areas outside of this blog prior to the PhD:
I have had the joy of being a teaching assistant in the following courses: | {"url":"https://pethick.dk/about/","timestamp":"2024-11-05T03:36:19Z","content_type":"text/html","content_length":"19377","record_id":"<urn:uuid:d09155b3-d0a3-4c5a-af3c-4ab79c374075>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00436.warc.gz"} |
On Resonance Enhancement of Nondipole Photoelectron Asymmetries in Low-Energy Neon Photoionization
On Resonance Enhancement of Nondipole Photoelectron Asymmetries in Low-Energy Neon Photoionization
Core Concepts
Even with realistic frequency spread in the ionizing radiation, the nondipole angular asymmetry parameters in neon photoionization exhibit significant enhancement near specific dipole and quadrupole
autoionizing resonances, making them detectable in experiments.
• Bibliographic Information: Dolmatov, V. K., & Manson, S. T. (2024). On Resonance Enhancement of E1 −E2 Nondipole Photoelectron Asymmetries in Low-Energy Ne 2p-Photoionization. arXiv preprint
• Research Objective: This study investigates the impact of frequency spread in ionizing radiation on the nondipole angular asymmetry parameters (γ2p, δ2p, and ζ2p) in neon 2p photoionization near
the 2s → 3p, 2s → 4p (dipole), and 2s → 3d (quadrupole) autoionizing resonances. The authors aim to determine if these parameters remain experimentally detectable after considering frequency
• Methodology: The researchers employ the Random Phase Approximation with Exchange (RPAE) method, a well-established theoretical framework in atomic physics, to calculate the photoionization cross
sections and angular asymmetry parameters. They incorporate the effect of frequency spread using a Gaussian function with a full-width at half-maximum (FWHM) of 5 meV, representing a typical
experimental condition.
• Key Findings: The calculations reveal that while the frequency spread does influence the resonance enhancement of γ2p, δ2p, and ζ2p, these parameters retain significant values near the 2s → 3d
quadrupole and 2s → 3p and 2s → 4p dipole resonances. The reduction in magnitude due to frequency spread is not substantial enough to render these parameters undetectable.
• Main Conclusions: The study concludes that the resonance enhancement of nondipole angular asymmetry parameters in neon 2p photoionization persists even after accounting for realistic frequency
spread in the ionizing radiation. This finding suggests that experimental observation of these enhanced nondipole effects should be feasible.
• Significance: This research holds significance for both theoretical and experimental atomic physics. It provides further validation for the theoretical predictions of enhanced nondipole effects
in photoionization near autoionizing resonances. Moreover, it encourages experimentalists to undertake measurements to confirm these predictions, potentially leading to a deeper understanding of
electron correlation and nondipole dynamics in atomic systems.
• Limitations and Future Research: The study focuses specifically on neon and a particular set of resonances. Investigating other atomic systems and exploring the influence of varying frequency
spreads would be valuable avenues for future research. Additionally, experimental verification of these theoretical findings is crucial for further advancing the field.
Translate Source
To Another Language
Generate MindMap
from source content
On Resonance Enhancement of $E1-E2$ Nondipole Photoelectron Asymmetries in Low-Energy Ne $2p$-Photoionization
The frequency spread in the ionizing radiation is assumed to be 5 meV at the half-maximum of the radiation's intensity. The maximum value of γ2p, without accounting for frequency spread, is
approximately 0.12. Accounting for frequency spread reduces the maximum value of γ2p to approximately 0.06, which is about 6% of β2p. The maximum value of ζ2p, without accounting for frequency
spread, is approximately 0.22. Accounting for frequency spread reduces the maximum value of ζ2p to approximately 0.12, which is about 12% of β2p.
"We demonstrate that the frequency spread in the ionizing radiation does quantitatively affect the resonance spikes in γ2p, δ2p, and ζ2p. Nevertheless, the spikes remain sufficiently strong to be
experimentally detected." "In contrast, Ne is a noble gas, for which conducting experiments is easier. This is why we focus on the photoionization of Ne in the present work."
Deeper Inquiries
How would the use of different theoretical methods, beyond RPAE, potentially impact the predicted magnitude of nondipole effects in photoionization?
Employing theoretical methods beyond the Random Phase Approximation with Exchange (RPAE) could indeed lead to variations in the predicted magnitudes of nondipole effects in photoionization. Here's a
breakdown of how different methods could introduce these variations: Methods accounting for higher-order correlations: While RPAE effectively captures some degree of electron correlation, more
sophisticated methods like Coupled-Cluster (CC) methods, Configuration Interaction (CI), or Many-Body Perturbation Theory (MBPT) at higher orders can provide a more accurate representation of
electron-electron interactions. These methods could refine the calculated dipole and quadrupole transition amplitudes, potentially leading to either enhancement or suppression of the predicted
nondipole effects depending on the specific atomic system and the energy range considered. Relativistic effects: For heavier atoms, relativistic effects become increasingly important. The Dirac-Fock
(DF) method or its relativistic many-body counterparts, such as the Relativistic Random Phase Approximation (RRPA), incorporate these relativistic effects. These methods could significantly alter the
calculated nondipole parameters, especially for inner-shell photoionization where relativistic effects are more pronounced. Time-dependent methods: The interaction of an atom with a time-varying
electromagnetic field, as in the case of a photon, can be more accurately described using time-dependent methods like the Time-Dependent Density Functional Theory (TDDFT) or the Time-Dependent
R-Matrix (TD-RM) method. These methods can capture dynamic electron correlations and could provide a more accurate picture of the photoionization process, potentially leading to different magnitudes
of nondipole effects compared to time-independent methods like RPAE. In essence, the choice of the theoretical method depends on the specific atomic system under investigation, the energy range of
interest, and the desired level of accuracy. While RPAE provides a reasonable starting point, incorporating higher-order correlations, relativistic effects, or employing time-dependent methods could
lead to more accurate predictions of nondipole effects in photoionization.
Could external fields, such as electric or magnetic fields, applied during the photoionization process, significantly alter the observed nondipole angular asymmetries?
Yes, the presence of external electric or magnetic fields during photoionization can indeed significantly alter the observed nondipole angular asymmetries. Here's how these fields influence the
process: Breaking of symmetry: The application of an external field breaks the inherent spherical symmetry of the atom. This symmetry breaking mixes different angular momentum states, leading to
modifications in the selection rules for photoionization. Consequently, transitions that are forbidden in the field-free case become allowed, potentially enhancing or suppressing certain nondipole
channels. Stark and Zeeman effects: Electric fields induce Stark shifts in the atomic energy levels, leading to energy level splitting and mixing of states with different parities. This mixing can
significantly alter the interference between dipole (E1) and quadrupole (E2) transitions, directly impacting the nondipole angular asymmetry parameters. Magnetic fields, on the other hand, introduce
Zeeman splitting and modify the angular momentum quantization axis. This can lead to a rotation of the photoelectron angular distribution and affect the observed nondipole asymmetries. Field-induced
modifications of the continuum states: External fields can distort the wavefunctions of the outgoing photoelectrons in the continuum. This distortion can further influence the interference between
different photoionization channels, leading to variations in the nondipole angular distributions. The magnitude of these field-induced modifications depends on the strength and orientation of the
applied field relative to the polarization direction of the ionizing radiation. By carefully controlling these parameters, one can manipulate the nondipole angular asymmetries, providing a valuable
tool for probing the dynamics of the photoionization process and gaining deeper insights into the electronic structure of atoms.
If successfully measured, how might these enhanced nondipole effects in photoionization be harnessed for practical applications, such as in the development of novel light sources or imaging
The successful measurement and control of enhanced nondipole effects in photoionization could pave the way for exciting practical applications, particularly in the development of novel light sources
and advanced imaging techniques: Novel Light Sources: Generation of customized polarization states: By manipulating the nondipole angular asymmetries through external fields or tailored laser pulses,
it might be possible to generate light with specific, customized polarization states. This could be particularly valuable for applications requiring precise control over light polarization, such as
in optical communication, quantum information processing, and high-resolution spectroscopy. Production of short-wavelength radiation: Nondipole effects become increasingly important at higher photon
energies. Harnessing these effects could potentially lead to new methods for generating short-wavelength radiation, such as extreme ultraviolet (EUV) or X-ray light, which are crucial for
lithography, microscopy, and materials science. Advanced Imaging Techniques: Enhanced spatial resolution in microscopy: Nondipole effects introduce additional angular dependencies in the
photoelectron emission patterns. Exploiting these dependencies could lead to enhanced spatial resolution in photoemission microscopy techniques, allowing for more detailed imaging of nanoscale
structures and materials. Element-specific and chemical state imaging: The sensitivity of nondipole angular asymmetries to the electronic structure of the target atom or molecule could be utilized
for element-specific or chemical state imaging. By analyzing the angular distribution of photoelectrons, one could potentially differentiate between different elements or chemical states within a
sample, providing valuable information for materials characterization and biological imaging. Tomographic reconstruction of electron density: The angular information encoded in the nondipole
photoelectron distributions could be used for tomographic reconstruction of the electron density within atoms or molecules. This could provide a powerful tool for visualizing the three-dimensional
electronic structure of matter with unprecedented detail. While these applications are still in the realm of exploration, the successful measurement and control of enhanced nondipole effects in
photoionization hold significant promise for advancing various fields, ranging from fundamental atomic physics to applied photonics and imaging technologies. | {"url":"https://linnk.ai/insight/scientific-computing/on-resonance-enhancement-of-nondipole-photoelectron-asymmetries-in-low-energy-neon-photoionization-D-PFcUEy/","timestamp":"2024-11-11T08:20:08Z","content_type":"text/html","content_length":"286055","record_id":"<urn:uuid:8982c581-4652-4226-951f-5b208962a3e1>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00631.warc.gz"} |
Identify familiar forces that cause objects to move, such as pushes or pulls, including gravity acting on falling objects.
General Information
Subject Area: Science
Grade: 5
Body of Knowledge: Physical Science
Idea: Level 1: Recall
Big Idea: Forces and Changes in Motion
- A. It takes energy to change the motion of objects.
B. Energy change is understood in terms of forces--pushes or pulls.
C. Some forces act through physical contact, while others act at a distance.
Clarification for grades K-5: The target understanding for students in the elementary grades should focus on Big Ideas A, B, and C.
Clarification for grades 6-8: The target understanding for students in grades 6-8 should begin to transition the focus to a more specific definition of forces and changes in motion. Net forces create
a change in motion. A change in momentum occurs when a net force is applied to an object over a time interval.
Grades 9-12, Standard 12: Motion - A. Motion can be measured and described qualitatively and quantitatively. Net forces create a change in motion. B. Momentum is conserved under well-defined
conditions. A change in momentum occurs when a net force is applied to an object over a time interval.
Date Adopted or Revised: 02/08
Date of Last Rating: 05/08
Status: State Board Approved
Assessed: Yes
Related Courses
This benchmark is part of these courses.
Related Access Points
Alternate version of this benchmark for students with significant cognitive disabilities.
Distinguish between movement of an object caused by gravity and movement caused by pushes and pulls.
Related Resources
Vetted resources educators can use to teach the concepts and skills in this benchmark.
Educational Game
Formative Assessment
Lesson Plans
Original Student Tutorial
Perspectives Video: Teaching Ideas
Teaching Ideas
Virtual Manipulatives
STEM Lessons - Model Eliciting Activity
Air Time 3D Printing MEA:
In this Model-Eliciting Activity (MEA), the students follow the engineering process to assist Worldwide Food Distribution Mission improve their food delivery device in order to deliver food to remote
parts of the world.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs
resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics
to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a
problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://
Clean Dat "SPACE" Inc.:
This Model Eliciting Activity (MEA) is written at a 5th grade level. Clean Dat "SPACE" MEA provides students with an engineering problem in which they must work as a team to design a procedure to
select the best space junk cleanup company for the purpose of keeping the International Space Station safe while in orbit.
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. MEAs
resemble engineering problems and encourage students to create solutions in the form of mathematical and scientific models. Students work in teams to apply their knowledge of science and mathematics
to solve an open-ended problem while considering constraints and tradeoffs. Students integrate their ELA skills into MEAs as they are asked to clearly document their thought processes. MEAs follow a
problem-based, student-centered approach to learning, where students are encouraged to grapple with the problem while the teacher acts as a facilitator. To learn more about MEAs visit: https://
X-treme Roller Coasters:
This MEA asks students to assist Ms. Joy Ride who is creating a virtual TV series about extreme roller coasters. They work together to determine which roller coaster is most extreme and should be
featured in the first episode. Students are presented with research of five extreme roller coasters and they must use their math skills to convert units of measurements while learning about force and
Model Eliciting Activities, MEAs, are open-ended, interdisciplinary problem-solving activities that are meant to reveal students’ thinking about the concepts embedded in realistic situations. Click
here to learn more about MEAs and how they can transform your classroom.
Original Student Tutorials Science - Grades K-8
Push It! Force and Motion:
Explore different kinds of forces, including pushes, pulls, magnetism, gravity, and friction in this interactive tutorial.
Student Resources
Vetted resources students can use to learn the concepts and skills in this benchmark.
Original Student Tutorial
Push It! Force and Motion:
Explore different kinds of forces, including pushes, pulls, magnetism, gravity, and friction in this interactive tutorial.
Type: Original Student Tutorial
Virtual Manipulatives
A Pendulum:
This virtual manipulative will help the students learn some important concepts of classical mechanics, such as gravitational acceleration, energy conservation and so on. This activity will also help
in students learning via the process of making predictions (about number of pendulum swings), discussing outcomes and sharing results.
Type: Virtual Manipulative
Friction (at Molecular Workbench):
Friction is important in enabling the movement of objects. Friction is a force that acts in an opposite direction to movement. Friction is everywhere when objects come into contact with each other.
Observe what happens when the surfaces are very smooth or slippery, it reduces the friction and thus it makes harder to stop the motion.
Type: Virtual Manipulative
Balance Challenge Game:
Play with objects on a teeter totter to learn about balance.
• Predict how objects of various masses can be used to make a plank balance.
• Predict how changing the positions of the masses on the plank will affect the motion of the plank
• Write rules to predict which way plank will tilt when objects are placed on it.
• Use your rules to solve puzzles about balancing.
Type: Virtual Manipulative
Explore the forces:
Students can create an applied force and see how it makes objects move. They can also make changes in friction and see how it affects the motion of objects.
• Identify when forces are balanced vs. unbalanced.
• Determine the sum of forces (net force) on an object with more than one force on it.
• Predict the motion of an object with zero net force.
• Predict the direction of motion given a combination of forces.
Type: Virtual Manipulative
Parent Resources
Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.
Teaching Idea
The Mystery of Tiny Algal Spores:
In this video, students will learn from a researcher about adaptations algae have developed to enable them to withstand water forces in their habitat.
Type: Teaching Idea
Virtual Manipulatives
A Pendulum:
This virtual manipulative will help the students learn some important concepts of classical mechanics, such as gravitational acceleration, energy conservation and so on. This activity will also help
in students learning via the process of making predictions (about number of pendulum swings), discussing outcomes and sharing results.
Type: Virtual Manipulative
Friction (at Molecular Workbench):
Friction is important in enabling the movement of objects. Friction is a force that acts in an opposite direction to movement. Friction is everywhere when objects come into contact with each other.
Observe what happens when the surfaces are very smooth or slippery, it reduces the friction and thus it makes harder to stop the motion.
Type: Virtual Manipulative
Balance Challenge Game:
Play with objects on a teeter totter to learn about balance.
• Predict how objects of various masses can be used to make a plank balance.
• Predict how changing the positions of the masses on the plank will affect the motion of the plank
• Write rules to predict which way plank will tilt when objects are placed on it.
• Use your rules to solve puzzles about balancing.
Type: Virtual Manipulative
Explore the forces:
Students can create an applied force and see how it makes objects move. They can also make changes in friction and see how it affects the motion of objects.
• Identify when forces are balanced vs. unbalanced.
• Determine the sum of forces (net force) on an object with more than one force on it.
• Predict the motion of an object with zero net force.
• Predict the direction of motion given a combination of forces.
Type: Virtual Manipulative | {"url":"https://www.cpalms.org/Public/PreviewStandard/Preview/1738","timestamp":"2024-11-10T14:09:50Z","content_type":"text/html","content_length":"139271","record_id":"<urn:uuid:940cbfc4-3190-4182-be7f-0224d36784b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00866.warc.gz"} |
Definition/Description of Formula:
Calculates the kurtosis of a dataset, which describes the shape, and in particular the "peakedness" of that dataset.
KURT(value1, value2, ...)
• value1 - The first value or range of the dataset.
• value2, ... - Additional values or ranges to include in the dataset.
• Although KURT is specified as taking a maximum of 30 arguments, Google Sheets supports an arbitrary number of arguments for this function.
• If the total number of values supplied as value arguments is not at least two, KURT will return the #NUM! error.
• Any text encountered in the value arguments will be ignored.
• Positive kurtosis indicates a more "peaked" distribution in the dataset, while negative kurtosis indicates a flatter distribution.
See Also:
VARPA: Calculates the variance based on an entire population, setting text to the value `0`.
VARP: Calculates the variance based on an entire population.
VARA: Calculates the variance based on a sample, setting text to the value `0`.
VAR: Calculates the variance based on a sample.
STDEVPA: Calculates the standard deviation based on an entire population, setting text to the value `0`.
STDEVP: Calculates the standard deviation based on an entire population.
STDEVA: Calculates the standard deviation based on a sample, setting text to the value `0`.
SKEW: Calculates the skewness of a dataset, which describes the symmetry of that dataset about the mean.
DEVSQ: Calculates the sum of squares of deviations based on a sample.
AVEDEV: Calculates the average of the magnitudes of deviations of data from a dataset's mean.
To use the KURT Formula, simply begin with your edited Excellentable:
Then begin typing the KURT formula in the area you would like to display the outcome: | {"url":"https://www.excellentable.com/help/kurt","timestamp":"2024-11-12T06:46:17Z","content_type":"text/html","content_length":"49969","record_id":"<urn:uuid:150acaeb-1e7e-4add-b43d-0dea050558ba>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00419.warc.gz"} |
Almost-Optimally Fair Multiparty Coin-Tossing with Nearly Three-Quarters Malicious
Almost-Optimally Fair Multiparty Coin-Tossing with Nearly Three-Quarters Malicious
Download: DOI: 10.1007/s00145-023-09466-2
An $$\alpha $$ α -fair coin-tossing protocol allows a set of mutually distrustful parties to generate a uniform bit, such that no efficient adversary can bias the output bit by more than $$
\alpha $$ α . Cleve (in: Proceedings of the 18th annual ACM symposium on theory of computing (STOC), 1986) has shown that if half of the parties can be corrupted, then no $$r$$ r -round
coin-tossing protocol is $$o(1/r)$$ o ( 1 / r ) -fair. For over two decades, the best-known m -party protocols, tolerating up to $${t}\ge m/2$$ t ≥ m / 2 corrupted parties, were only $$O\
left( {t}/\sqrt{r} \right) $$ O t / r -fair. In a surprising result, Moran et al. (in: Theory of cryptography, sixth theory of cryptography conference, TCC, 2009) constructed an $$r$$ r
-round two-party $$O(1/r)$$ O ( 1 / r ) -fair coin-tossing protocol, i.e., an optimally fair protocol. Beimel et al. (in: Rabin (ed) Advances in cryptology—CRYPTO 2010, volume 6223 of
Abstract: lecture notes in computer science, Springer, 2010) extended the result of Moran et al. to the multiparty setting where strictly fewer than 2/3 of the parties are corrupted. They constructed
a $$2^{2^k}/r$$ 2 2 k / r -fair r -round m -party protocol, tolerating up to $$t=\frac{m+k}{2}$$ t = m + k 2 corrupted parties. In a breakthrough result, Haitner and Tsfadia (in: Symposium
on theory of computing, STOC, 2014) constructed an $$O\left( \log ^3(r)/r \right) $$ O log 3 ( r ) / r -fair (almost optimal) three-party coin-tossing protocol. Their work brought forth a
combination of novel techniques for coping with the difficulties of constructing fair coin-tossing protocols. Still, the best coin-tossing protocols for the case where more than 2/3 of the
parties may be corrupted (and even when $$t=2m/3$$ t = 2 m / 3 , where $$m>3$$ m > 3 ) were $$\theta \left( 1/\sqrt{r} \right) $$ θ 1 / r -fair. We construct an $$O\left( \log ^3(r)/r \
right) $$ O log 3 ( r ) / r -fair m -party coin-tossing protocol, tolerating up to t corrupted parties, whenever m is constant and $$t<3m/4$$ t < 3 m / 4 .
title={Almost-Optimally Fair Multiparty Coin-Tossing with Nearly Three-Quarters Malicious},
journal={Journal of Cryptology},
author={Bar Alon and Eran Omri}, | {"url":"https://iacr.org/cryptodb/data/paper.php?pubkey=33327","timestamp":"2024-11-07T03:35:37Z","content_type":"text/html","content_length":"24783","record_id":"<urn:uuid:3577bb3c-59c1-4561-817c-2b42a5629cb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00369.warc.gz"} |
Ordering the Natural Numbers
Naive Set Theory: The Natural Numbers
Ordering the Natural Numbers
So far we have constructed the natural numbers and defined and proven some properties about arithmetic functions on them. Now we define an ordering for the natural numbers.
A natural number $a$ is less than another natural number $b$ if $a \in b$. Using our formal definition of order relations, we define the ordering on $\mathbb{N}$ as
$$< \, \, = \{ (a,b) : a, b \in \mathbb{N} \text{ and } a \in b \}$$
Recall that every natural number is constructed out of von Neumann ordinals, such that $n+1 = \{0, \ldots, n\}$. For example, $3 = \{0, 1, 2\}$. How convenient for implementing a less-than
relationship! The numbers $0$, $1$, and $2$ are all elements of $3$, but $3$ is not itself an element of $3$, and nothing greater than $3$ is an element of $3$.
1. Show that $a \in b$ if and only if $a^+ \in b^+$.
$\implies$: Proof by induction. Let $S = \{ n \in \mathbb{N} : m^+ \in n^+ \text{ for all } m \in n \}$.
Base case: $0 \in S$, as there are no elements in $0$, so all of their successors are in $0^+=1$.
Inductive step: Assume $k \in S$. We must show that if $m \in k^+$, then $m^+ \in k^{++}$. Assume $m \in k^+$. Then $m \in k \cup \{k\}$, so either $m \in k$ or $m = k$. If $m \in k$, then by the
inductive hypothesis $m^+ \in k^+$. Because $k^+ \subseteq k^{++}$, we have $m^+ \in k^{++}$. Alternatively, if $m = k$, then $m^+ = k^+$. Because $k^+ \in k^{++}$, it follows that $m^+ \in k^
In either case, $m^+ \in k^{++}$, therefore $k^+ \in S$, and the result follows by induction.
$\impliedby:$ Assume $m^+ \in n^+$. By definition, $n^+ = n \cup \{n\}$. Therefore either $m^+ \in n$ or $m^+ = \{n\}$. If $m^+ \in n$, then by transitivity $m^+ \subset n$. Since $m \in m^+$, it
follows that $m \in n$. Alternatively, if $m^+ = n$, we again notice that $m \in m^+$, so $m \in n$.
2. Prove that no natural number is a member of itself.
There is a secret answer to this question - no set is a member of itself. While this property of sets comes from more rigorous study of sets, we can prove it for this particular instance.
Proof by induction. Let $S = \{ n \in \mathbb{N} : n \notin n \}$. Clearly $0 = \varnothing \in S$, since the empty set has no elements. Then assume $k \in S$. Then $k \notin k$. By the previous
proof, we conclude that $k^+ \notin k^+$, therefore $k^+ \in S$. Thus the result follows by induction.
3. Show that $0$ is either equal to or an element of every natural number.
Proof by induction. Let $S = \{ n \in \mathbb{N} : 0 = n \text{ or } 0 \in n \}$
Base case: Clearly $0=0$, so $0 \in S$.
Inductive step: Assume $n \in S$. Then $0 \in n$ or $0 = n$. In either case, it follows that $0 \in n \cup \{n\} = n^+$. Thus the result follows by induction.
4. Trichotomy Law: Prove that for any two natural numbers $a$ and $b$, only one of the three statements is true:
□ $a \in b$
□ $a = b$
□ $b \in a$
In order to show that exactly one of the statements is always true, we will first show that at most one can be true and then show that at least than one must be true.
At most one is true: If $a = b$, then the fact that no set is an element of itself precludes both $a \in b$ and $b \in a$. Alternatively, if $a \in b$, then by the same logic $a \neq b$.
Likewise, it is not true that $b \in a$, which would imply that $a \in b \in a$, which in turn implies $a \in a$ by transitivity, which is a contradiction. An identical argument holds for $b \
in a$ that precludes the other two options.
At least one is true: Proof by induction. Let $S = \{ a \in \mathbb{N} : \text{for all } b \in \mathbb{N}, a \in b \text{ or } a = b \text{ or } b \in a \}$.
Base case: Let $a = 0$. When $b = 0$, we have $a = b$, and when $b \neq 0$, we have by the above proof $a \in b$. Thus $0 \in S$.
Inductive case: Assume $k \in S$. Then exactly one of the conditions holds. If $b \in k$, then $b \in k^+$ by transitivity. If instead $b = k$, then $b^+ = k^+$. Lastly, if $k \in b$, then by the
preceding proof, $k^+ \in b^+$, and thus either $k^+ \in b$ or $ k^+ = b$. In each case, at least one condition is true, so $k^+ \in S$, and the result follows by induction.
5. Prove that the ordering of $\mathbb{N}$ defined above in fact meets the requirements for an order relation.
In order to qualify as an order relation, a relation must meet the three requirements of comparability, nonreflexivity, and transitivity.
1. Comparability: Let $a, b \in \mathbb{N}$ such that $a \neq b$. By trichotomy, either $a \in b$ or $b \in a$. If $a \in b$, then $a < b$. Conversely, if $b \in a$, then $b < a$.
2. Nonreflexivity: By the second proof above, no natural number is an element of itself, so there is no $a \in \mathbb{N}$ such that $ a < a$.
3. Transitivity: Let $a, b, c \in \mathbb{N}$ such that $a < b$ and $b < c$. By definition of the order relation, $a \in b$ and $b \in c$. Likewise, if $b \in c$, then by transitivity $b \subset
c$. Therefore $a \in c$, so $a < c$.
6. Cancellation Law: Prove that $a < b$ if and only if $a + c < b + c$ for all $a, b, c \in \mathbb{N}$.
$\implies$: Proof by induction. Let $S = \{ n \in \mathbb{N} : a + n \in b + n \text{ where } m < n \in \mathbb{N} \}$.
Base case: Because $m + 0 = 0$ and $n + 0 = 0$, we have $m + 0 \in n + 0$, so $0 \in S$.
Inductive case: Assume $n \in S$. Then $a + n < b + n$. Then by the definition of the order relation, $a + n \in b + n$. By transitivity, $(a + n)^+ \in (b + n)^+$, which simplifies to $a + n^+ \
in b + n^+$. Therefore $n^+ \in S$, and the result follows by induction.
$\impliedby$: Assume $a + c < b + c$. Then $a + c \in b + c$. By trichotomy, $a \neq b$, as otherwise $a + c \in a + c$. Likewise, $b \notin a$, as otherwise by the forward proof this implies $b
+ c \in a +c$. Thus the only option left is $a \in b$.
7. Cancellation Law: Prove that $a < b$ if and only if $a \cdot c = b \cdot c$ for all $c \neq 0 \in \mathbb{N}$.
$\implies$: Assume $a < b$. Proof by induction. Let $S = \{ n \in \mathbb{N} : a \cdot n^+ < b \cdot n^+ \text{ for all nonzero} $c \in \mathbb{N}\}$.
Base case: By definition, $0^+ = 1$. Then $a \cdot 1 = a$, and $b \cdot 1 = b$. Thus $a \cdot 0^+ < b \cdot 0^+$, so $0 \in S$.
Inductive step: Assume $n \in S$. Then $a \cdot n^+ < b \cdot n^+$. By definition of multiplication, $a \cdot n^{++} = (a \cdot n^+) + a$. By applying the addition cancellation law and the
inductive hypothesis, we see that $(a \cdot n^+) + a < (b \cdot n^+) + a$. Applying the law again shows $(b \cdot n^+) + a < (b \cdot n^+) + b$. By transitivity on $<$, we conclude that $(a \cdot
n^+) + a < (b \cdot n^+) + b$, and thus that $a \cdot n^{++} < b \cdot n^{++}$. Therefore $n^+ \in S$, and the result follows by induction.
$\impliedby$: Assume $a \cdot c < b \cdot c$. By trichotomy, $a \neq b$, as otherwise $a \cdot c \in a \cdot c$. Likewise, it cannot be true that $b < a$, as otherwise by the forward proof $b \
cdot c < a \cdot c$, which again implies $a \cdot c \in a \cdot c$. Thus the only option left is $a < b$.
8. Prove that if $x < y$, then there exists some nonzero $z$ such that $x + z = y$.
Proof by induction. Let $A = \{ x \in \mathbb{N} : \text{for all } y \in \mathbb{N} \text{ where } x < y, \text{ there exists a nonzero} z \in \mathbb{N} \text{ such that } x + z = y \}$.
Base case: Consider $0$ and the set of all $y \in \mathbb{N}$ such that $0 < y$. Clearly $y$ is nonzero. Note that $0 + y = y$, therefore $0 \in A$.
Inductive step: Assume $x \in A$. Consider $x^+$ and number $z \in \mathbb{N}$ such that $x^+ < z$. Because $x \in x^+$, $x < x^+$. By transitivity, $x < z$. By the inductive hypothesis, there
exists some nonzero $y \in \mathbb{Z}$ such that $x + y = z$. Observe that $y \neq 1$, as otherwise $x^+ = m$, which is a contradiction. Therefore $y$ is at least $2$. As a result, we can write
$y$ in the form $y = p^{++}$, where $p \in \mathbb{N}$. Therefore $x + p^{++} = m$. We can rewrite this equality as $x^+ + p^+ = m$ and then conclude that $x^+ \in A$.
The result follows by induction. | {"url":"http://www.mathmatique.com/naive-set-theory/natural-numbers/ordering-natural-numbers","timestamp":"2024-11-13T00:52:52Z","content_type":"text/html","content_length":"52465","record_id":"<urn:uuid:d5239b20-0444-4b71-94a5-44ce447f3384>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00409.warc.gz"} |
A piece of wood that measures 3.0 cm by 6.0 cm by 4.0 cm has a mass of 80.0 grams. What the density of the wood? Would the piece of wood float in water? | Socratic
A piece of wood that measures 3.0 cm by 6.0 cm by 4.0 cm has a mass of 80.0 grams. What the density of the wood? Would the piece of wood float in water?
1 Answer
The density of a substance can be calculated using:
$d = \frac{m}{V}$
where $m$ is the mass, and $V$ is the volume.
Here we do not have the volume given, however we can calculate volume by multiplying given dimensions:
$V = 3 \cdot 4 \cdot 6 = 72 c {m}^{3}$
So the density is: $d = \frac{80}{72} = 1 \frac{1}{9} \frac{g}{c {m}^{3}}$
To answer the question stated in the last sentence we have to compare the calculated density of the piece of wood with the density of water.
Water has density of ${d}_{W} = 1 \frac{g}{c {m}^{3}}$
Since the density of the wood is greater than the density of water, it will NOT float in the water.
Impact of this question
36223 views around the world | {"url":"https://socratic.org/questions/a-piece-of-wood-that-measures-3-0-cm-by-6-0-cm-by-4-0-cm-has-a-mass-of-80-0-gram#324958","timestamp":"2024-11-02T18:49:48Z","content_type":"text/html","content_length":"33907","record_id":"<urn:uuid:735cefae-975e-4f7c-9744-ffe1a9edbdd9>","cc-path":"CC-MAIN-2024-46/segments/1730477027729.26/warc/CC-MAIN-20241102165015-20241102195015-00331.warc.gz"} |
How to Fill Out Arizona Withholding Form A-4 + FAQs
The Arizona Department of Revenue revised Form A-4 for 2023 to account for the new flat state income tax of 2.5%. Previous versions of the form are not valid for withholding calculations as of
January 1, 2023. There is no penalty for employees who did not update the form, however you should give your employer an updated A-4 for accurate tax withholding.
Our guide will help you understand the form’s purpose, when to complete it, what happens if you don’t submit one, and how to complete each step. It also includes answers to frequently asked questions
about Form A-4, helping you avoid common errors you could be making.
What is Arizona Form A-4?
Arizona Form A-4 allows you to select the percentage of your paycheck your employer will set aside to cover your Arizona income taxes.
Form A-4 is the Arizona equivalent of the federal Form W-4 issued by the IRS. If you live and work in Arizona you must submit both Arizona Form A-4 and the IRS Form W-4 to your employer. If you need
help filling out your federal W-4, check out our comprehensive Form W-4 guide.
Do I Need to Fill Out an A-4?
According to the AZDOR, you need to fill out a A-4 if:
• You are a full-time resident of Arizona
• You are a part-time resident of Arizona, in which case the state only taxes income earned while residing in Arizona
• You reside out-of-state but earn income from an Arizona employer. For example, if you live in New Mexico but earn income from an employer based in Arizona, you must fill out a Form A-4 and submit
it to that employer.
Form A-4 allows you to select a withholding percentage from your gross taxable wages, from 0.5% to 3.5%. You may also elect to have additional money withheld from your paychecks.
The higher the percentage, the more money the employer withholds for Arizona income taxes, increasing your chances of receiving a refund when filing your income tax return. Withholding a lower
percentage lets you increase your after-tax pay but increases the risk of under-withholding and exposes you to penalties for underpayment.
In most cases, you only need to fill out an A-4 once. The income tax withholding rate you elected will continue to apply, except in three cases:
• Your employment with your current employer ends, and you are beginning a new job elsewhere, in which case you must submit a new A-4 to the new employer
• You wish to elect a different withholding percentage, in which case you can voluntarily submit a new A-4 to your employer
• Changes in Arizona tax laws requiring employees to re-submit Form A-4s
You can find the most current A-4 form and instructions on the official website of the Arizona Department of Revenue.
What if I Don’t Fill Out an A-4?
According to AZDOR, you must submit the completed form within five days of your hiring date to your employer.
If you fail to submit form A-4, your employer must withhold taxes without any exemptions at the default rate of 2.0%. Your employer will continue to withhold at 2.0% until you submit a completed Form
Your employer is still required to give you a paycheck even if you don’t submit the form.
How Much Should I Withhold on an Arizona Form A-4?
The best way to determine the withholding percentage to elect on your Arizona Form A-4 is to follow the withholding calculations worksheet on the AZDOR website. The worksheet is not part of Form A-4.
Follow this step-by-step guide to calculate the best percentage for your situation.
Step 1: Annual Gross Taxable Wages
Determine your annual gross taxable wages. Arizona law assumes wages defined for federal tax purposes are the same as those used for state income tax purposes.
Example: Sarah determines her annual gross taxable wages are $60,000, so she enters $60,000 in Step 1.
Step 2: Number of Paychecks Per Year
Calculate the number of paychecks you receive annually and enter the result in Step 2. For instance, if you are paid monthly, you should receive 12 checks, whereas if you are paid weekly, the result
should be 52.
Example: Sarah is paid monthly, meaning she receives 12 paychecks annually. She enters 12 in Step 2.
Step 3: Wages per Paycheck
Divide the value in Line 1 by the value in Line 2 to obtain your wages per paycheck. Enter the result in Line 3.
Example: Sarah receives $60,000 yearly across 12 paychecks. 60,000 / 12 = $5,000. She enters 5,000 in Line 3.
Step 4: Annual Withholding Goal
Refer to the latest version of Arizona Form 140, Arizona Resident Personal Income Tax Booklet to find your annual withholding goal.
If your total taxable income (wages + other sources) is less than $50,000, use the Optional Tax Tables on Page 51. If your total taxable income is $50,000 or more, use the Arizona Tax Tables X and Y.
My total taxable income is under $50,000
Find the row corresponding to your annual income on the Optional Tax Tables, then refer to the column corresponding to your filing status. Use the third column if you are single or married and filing
separately. Use the fourth column instead if you are married and filing jointly or filing as head of household. The value corresponding to your filing status and income level is your annual
withholding goal.
For example, if you are single and your annual income is $42,330, you must refer to the row for incomes between $42,300 and $42,350. As a single taxpayer, you’ll use the third column. The
corresponding value is 1,138. Your annual withholding goal is $1,138.
My total taxable income is over $50,000
If you are single or married and filing separately, use Tax Table X. If you are married and filing jointly or filing as a head of household, use Tax Table Y.
• Table X: Subtract 28,653 from your taxable income, multiply the result by 2.98%, then add $731. The final result is your annual withholding goal
• Table Y, annual income over $50,000 but less than $57,305: Multiply your taxable income by 2.55%. The result is your annual withholding goal
• Table Y, annual income over $57,305: Subtract 57,305 from your taxable income, multiply the result by 2.98%, then add $1,461. The result is your annual withholding goal
Example: Sarah is single and earns $60,000 of total taxable income annually. She must use Table X to calculate her annual withholding goal. 60,000 – 28,653 equals 31,347. 2.98% of that is 934.14.
When adding 731, the result is 1,665.14. Sarah’s withholding goal is $1,665.14. She must enter this value in Line 4.
Step 5: Amount Already Withheld
Calculate the amount and write it here if money has already been withheld from your paycheck during the year.
Example: Sarah has determined she already withheld $330 during the year. She writes 330 in Step 5.
Step 6: Balance of Withholding
Calculate your withholding balance for the current calendar year by subtracting Line 5 from Line 4.
Example: In Sarah’s case, Line 4 is 1,665.14, and Line 5 is 330. 1,665.14 – 330 = 1,325.14. Sarah must enter this number in Line 6.
Step 7: Number of Paychecks Remaining
Calculate the number of paychecks remaining for the current calendar year.
Example: Sarah determined she has received 4 of her 12 monthly paychecks. The number of paychecks remaining is 8; she must enter 8 in Line 7.
Step 8: Arizona Withholding Goal Per Paycheck
Calculate your withholding goal per paycheck by dividing Line 6 by Line 7. Round to the nearest cent to obtain the result.
Example: In Sarah’s case, Line 6 is 1,325.14, and Line 7 is 8. 1,325.14 divided by 8 equals 165.6425. After rounding to the nearest cent, the result is $165.64.
Step 9: Percentage Calculation
Divide the value on Line 8 by the one on Line 3, then multiply the result by 100 to obtain a percentage.
Example: For Sarah, Line 8 is 165.64, and Line 3 is 5,000. 165.64 / 5,000 = 0.033128. After multiplying by 100 to obtain a percentage, the result is 3.31%. Sarah enters 3.31% on Line 9.
Step 10: Select a Withholding Percentage on Form A-4
Choose a withholding percentage by checking the corresponding box. You have two options to complete this step.
• Overwithholding intentionally: Choose a percentage equal to or higher than the result in Step 9. This solution will ensure the amount withheld from your paycheck is higher than necessary to avoid
under-withholding. It can also increase the chances of receiving a tax refund.
• Choose a lower percentage: The AZDOR’s recommended method is to select the highest percentage on your Form A-4 under the value you calculated on Line 9. You can later make up the difference by
calculating additional withholding.
Regardless of the method you choose, you must ensure the total amount withheld, including the percentage and additional withholding, is enough to cover your state income tax liability. If
insufficient, you risk underpaying your Arizona income taxes, potentially exposing you to penalties and a tax audit.
Example: After calculating a percentage of 3.31%, Sarah follows the AZDOR method and checks the box next to 3.0%, the highest option under the percentage she calculated on Line 9. She also writes in
3.0% on Line 10.
Steps 11 and 12: Calculating Additional Withholding
You can withhold additional money from each paycheck to increase your chances of a tax refund. You can also use additional withholding or cover the difference if you opted for a lower withholding
percentage in Step 10, allowing you to withhold exactly the amount you owe.
The AZDOR withholding calculation worksheet recommends following these steps to calculate the optimal amount of additional withholding:
• Step 11: Multiply Line 10 by Line 3.
• Step 12: Subtract Line 11 from Line 8. Enter the result on your Form A-4 in the additional withholding box.
Example: Sarah continues following the AZDOR method to calculate her withholding. She completes Step 11 by multiplying the value on her Line 10 (3.0%) by the value on Line 3 (5,000). 3.0% x 5,000 =
$150. She then completes Step 12 by subtracting Line 11 from Line 8. 165.64 – 150 = 15.64. Sarah enters $15.64 on her Form A-4’s extra withholding, meaning her employer will withhold 3.0% of her
gross taxable wages plus an additional $15.64 from each paycheck.
How to Fill Out the Employee’s Arizona Withholding Election
Fill out form A-4 by entering your personal information, selecting your withholding percentage in box 1 if employed, signing the form, and submitting it to your employer.
Follow our step-by-step guide to complete your Form A-4 accurately.
Write in your personal information at the top of the form. You must write in:
• Your full legal name
• Your home address
• Your city or town name
• Your state’s postal abbreviation (e.g., AZ)
• Your ZIP code
Choose Either Box 1 or Box 2
If you have no Arizona income tax liability for the current taxable year, check Box 2 and leave the boxes in Box 1 blank. Otherwise, check Box 1 and proceed.
If you owe Arizona income taxes, check Box 1, then check the box corresponding to the percentage of your gross taxable wages to withhold based on the withholding calculations worksheet. Check only
one box.
Your employer will hold the amount corresponding to the selected percentage to cover your Arizona income taxes. Arizona uses your federal gross taxable wages to calculate your tax liability. They
correspond to the number in Box 1 of your IRS Form W-2.
Check this box to have your employer withhold 0.5% of your gross taxable wages for Arizona income taxes. If you earn $80,000 per year, your employer will withhold $400 per year.
Check this box to have your employer withhold 1.0% of your gross taxable wages for Arizona income taxes. If you earn $80,000 per year, your employer will withhold $800 per year.
Check this box to have your employer withhold 1.5% of your gross taxable wages for Arizona income taxes. If you earn $80,000 per year, your employer will withhold $1,200 per year.
Check this box to have your employer withhold 2.0% of your gross taxable wages for Arizona income taxes. If you earn $80,000 per year, your employer will withhold $1,600 per year.
Check this box to have your employer withhold 2.5% of your gross taxable wages for Arizona income taxes. If you earn $80,000 per year, your employer will withhold $2,000 per year.
Check this box to have your employer withhold 3.0% of your gross taxable wages for Arizona income taxes. If you earn $80,000 per year, your employer will withhold $2,400 per year.
Check this box to have your employer withhold 3.5% of your gross taxable wages for Arizona income taxes. If you earn $80,000 per year, your employer will withhold $2,800 per year.
Check this box if you plan to elect an Arizona withholding percentage of 0.0%. To check this box, you must have no Arizona income tax liability for the current tax year. Arizona tax liability is
calculated by using your gross income tax liability and subtracting all tax credits, including credits for taxes paid to other states.
Electing to withhold 0.0% does not mean you are fully exempt from paying Arizona income taxes, such as potential taxes due when you file your tax return. Electing a percentage of 0.0% means your
employer will not withhold money from your paycheck to cover income taxes.
Even if you are eligible, electing a percentage of 0.0% is valid only for the current taxable year. If you do not submit an updated Form A-4 the following year, your employer will withhold taxes at
the default rate of 2.0%.
Sign the form above the Signature field, then write the completion date in the Date field.
Here are the answers to some common questions about filling out form A-4.
Being exempt on your IRS Form W-4 does not make you exempt at the state level. Complete your Form A-4 as normal.
You only need to fill out an A-4 if you earn income from an Arizona source, and the compensation paid to you is subject to Arizona income taxes. If it applies to you, fill in your personal
information normally by writing in your out-of-state address.
Your wages are exempt from Arizona withholding if you meet the following conditions:
• You are married to an active-duty Armed Forces servicemember
• Your spouse is in Arizona due to military orders
• You earn wages in Arizona
• You are in Arizona solely to be with your spouse
• Your residence is in a different state than Arizona and is in the same state as your spouse’s domicile.
If you meet all the prerequisites listed, you may check Box 2 on your Form A-4 to elect 0.0% withholding.
If you check Box 2, you are electing to claim 0.0% withholding. Your employer will not withhold any money from your paycheck for Arizona income taxes. You may only do so if you expect to have no tax
liability for the current year.
If you do not submit a Form A-4 within five days of employment, your employer will withhold using the default rate of 2.0%. Your employer will still give you a paycheck even without the form. | {"url":"https://taxsharkinc.com/fill-out-a4/","timestamp":"2024-11-13T22:46:56Z","content_type":"text/html","content_length":"155656","record_id":"<urn:uuid:614c38d6-c672-444e-b313-934c99d08eb1>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00487.warc.gz"} |
Renault Scenic MPG
We calculated that 2016 Renault Scenic MPG is up to 103% better than the world average consumption.
1.6 Energy dCi is 60% more economical than average car
1.5 Energy dCi is 103% more economical than average car
1.2 Energy TCe is 22% more economical than average car
1.3 Energy TCe is 31% more economical than average car
1.7 Blue dCi is 50% more economical than average car
1.3 TCe is 22% more economical than average car | {"url":"https://fuelson.com/renault-scenic","timestamp":"2024-11-12T04:17:29Z","content_type":"text/html","content_length":"438722","record_id":"<urn:uuid:1079a1a2-c16e-42df-ba88-09b92bc77a20>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00641.warc.gz"} |
Thinking like mathematicians - Opal School
Thinking like mathematicians
This post was written by Mary Gage Davis, Opal School Curriculum Specialist
Our work at Opal School is driven by the desire to support children in making meaning; math is no different. Much of our math curriculum is based upon teaching for number sense. With good number
sense, children can think flexibly, compose and decompose numbers and have the opportunity to truly think like mathematicians. This may look and feel very different from the mathematics that many of
us grew up with. You may be wondering why your children are not being taught traditional algorithms (how to "carry" or "borrow", for instance) in math workshop. You may wonder if children are
expected to do arithmetic calculations. The answer is “yes”, we expect children to be able to compute accurately and efficiently, but we want children to do this as active mathematical thinkers.
Students are encouraged to look at the numbers before they calculate, to think rather than follow rote steps, to base their procedures upon strong number sense.
For example, when given the problem 368 + 208, some children may:
• Split the numbers by place value and add them together: 200 + 300 + 60 + 16
• Recognize that it is more efficient to keep the 368 whole and add 200 then 8.
• Turn 368 into a friendly number by taking 2 from 208 and adding it to 368, making 370 + 206.
In order to do this kind of work, children need to develop a mental model of the relationship between numbers. Many of these strategies and big mathematical ideas are developed through mental math
mini-lessons at the beginning of math workshop. Children are given a string of related problems and asked to solve them mentally. As children share their thinking, it is then modeled on the open
number line or with arrays depending upon the operation. As the number string unfolds, so do relationships between numbers, operations and problems that highlight specific strategies. (Future blog
posts will address the specifics of the models: open number line and array.)
At the heart of mathematics is the process of setting up relationships and trying to prove these relationships mathematically in order to communicate them to others. Creativity is at the core of
what mathematicians do. (Fosnot and Dolk 2001)
In the 1980’s as researchers began to explore whether or not algorithms should be the goal of arithmetic thinking. In a study conducted by Kamii and Dominick, researchers compared three groups of
children: those that had been taught only traditional algorithms, those that had been taught none and those that had been taught both. These children were asked to calculate 7 + 52 + 186. The
results are described on this chart:
You may notice that the greatest percentage of children to get the correct answer were those who had not been taught algorithms. Even more interesting, you may notice the range of answers given.
Those children with no algorithm experience were in the closest proximity to the answer. “It appears that most of the errors in the first group were place value errors; in the latter group, they
were calculation errors. This is strong evidence that the algorithm actually works against the development of children’s understanding of place value and number sense. As they focus on doing the
procedures correctly, they sacrifice their own meaning making; they sacrifice an understanding of the quantity of the numbers they are dealing with.”
The mathematics curriculum at Opal School is greatly influenced by this and other research that supports teaching for number sense. Learning to compose and decompose numbers, building a strong
number sense and a repertoire of efficient strategies takes time. At Opal School, we believe this time is an important investment in children’s learning. Upcoming blog posts will give a closer look
at what this work might look like in the classroom across the grades. | {"url":"https://opalschool.org/thinking-like-mathematicians/","timestamp":"2024-11-08T19:03:52Z","content_type":"text/html","content_length":"32264","record_id":"<urn:uuid:2d984f0e-5d97-4e2a-baab-e1ee26b1ced5>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00119.warc.gz"} |
The Sparseness of Mixed Selectivity Neurons Controls the Generalization–Discrimination Trade-Off
Intelligent behavior requires integrating several sources of information in a meaningful fashion—be it context with stimulus or shape with color and size. This requires the underlying neural
mechanism to respond in a different manner to similar inputs (discrimination), while maintaining a consistent response for noisy variations of the same input (generalization). We show that neurons
that mix information sources via random connectivity can form an easy to read representation of input combinations. Using analytical and numerical tools, we show that the coding level or sparseness
of these neurons' activity controls a trade-off between generalization and discrimination, with the optimal level depending on the task at hand. In all realistic situations that we analyzed, the
optimal fraction of inputs to which a neuron responds is close to 0.1. Finally, we predict a relation between a measurable property of the neural representation and task performance.
How do we determine whether a neural representation is good or bad? In general the answer depends on several factors, which include the statistics of the quantity that is represented, the task to be
executed, and the neural readout that utilizes the representation.
Previous work evaluated neural representations on the basis of the information they encode (Atick and Redlich, 1992; Jazayeri and Movshon, 2006). This is often the only viable approach when it is not
known how the representations are used or read out by downstream structures (e.g., in the case of early sensory areas).
Here we evaluate a neural representation by the information that is accessible to individual readout neurons, which we assume simply compute a weighted sum of the inputs followed by a thresholding
operation. In general, this information is smaller than the total information contained in the input, as it is constrained to be in a more “explicit” format (DiCarlo et al., 2012) suitable for being
processed by simple readouts.
Previous studies evaluated neural representations of natural visual scenes by computing the reconstruction error of a population of linear readout neurons (Olshausen and Field, 2004). These elegant
works showed that sparseness is an important feature of the neural representations, not only because it naturally leads to the receptive fields observed in cortical recordings, but it also increases
the dimensionality of the input, facilitates learning, and reduces the effects of input noise.
We focus on the capacity of a readout neuron to produce a large set of diverse responses to the same inputs (i.e., to implement a large number of input–output functions). This capacity clearly
depends on the input representation, and it is functionally important as it can be harnessed to generate rich dynamics and perform complex tasks (Hinton and Anderson, 1989; Rigotti et al., 2010b). We
consider a specific class of problems in which readout neurons receive inputs from multiple sources (Fig. 1A). This situation is encountered in many cases, which include integration of sensory
modalities, combining an internally represented context with a sensory stimulus, or mixing the recurrent and the external input of a neural circuit. These are typical situations in almost every brain
area, especially in those integrating inputs from multiple brain systems, such as the prefrontal cortex (Miller and Cohen, 2001). As the readout is linear, in these situations there are some
input–output functions that cannot be implemented (Fig. 1B). For instance, the ability to differentiate between external inputs that are received in different contexts is known to potentially
generate a large number of non-implementable functions (McClelland and Rumelhart, 1985; Rigotti et al., 2010b). The difficulty stems from the high correlations between the input patterns that only
differ by the state of one segregated information source (e.g., the one encoding the context).
Fortunately, there are transformations implemented by simple neuronal circuits that decorrelate the inputs by mixing different sources of information in a nonlinear way. We will focus on one such
transformation that is implemented by introducing an intermediate layer of randomly connected neurons (RCNs; Fig. 1C). Each RCN responds nonlinearly to the weighted sum of the original inputs, and
its weights are random and statistically independent. These neurons typically respond to complex combinations of the parameters characterizing the different sources of information (mixed
selectivity), such as a sensory stimulus only when it appears in a specific context. Mixed selectivity neurons have been widely observed in the cortex (Asaad et al., 1998; Rigotti et al., 2010a,
2013; Warden and Miller, 2010), although they are rarely studied, as their response properties are difficult to interpret. Neural representations that contain RCNs allow a linear readout to implement
a large number of input–output functions (Marr, 1969; Hinton and Anderson, 1989; Maass et al., 2002; Lukoševičius and Jaeger, 2009; Rigotti et al., 2010b).
Any transformation that mixes multiple sources of information, should reconcile the two opposing needs of the discrimination–generalization trade-off (Fig. 1D). It should decorrelate the
representations sufficiently to increase classification capacity, which is related to the ability to discriminate between similar inputs. Unfortunately, as we will demonstrate (Fig. 4B),
transformations that decorrelate tend to destroy the information about relative distances in the original space, making it harder for the readout neurons to generalize (i.e., generate the same output
to unknown variations of the inputs).
We will show that RCNs can efficiently decorrelate the inputs without sacrificing the ability to generalize. The discrimination–generalization trade-off can be biased by varying the sparseness of the
RCN representations, and there is an optimal sparseness that minimizes the classification error.
Materials and Methods
Definition of the task.
For simplicity we report here the analysis of the case with two sources of information. The case with more than two sources is a straightforward extension and is briefly discussed at the end of this
section. We consider two network architectures—one with an RCN layer (Fig. 4A), and one without (Fig. 2A). The activity of all neurons is approximated as binary.
In both cases, the first layer is an input composed of two sources containing N neurons each. The first source ψ^x ∈ {±1}^N can be in one of m[1] states, x = 1, …, m[1], and the second source is
denoted by φ^a ∈ {±1}^N with a = 1, …, m[2]. An input pattern ξ^μ is composed of one subpattern from each source ξ^xa = (ψ^x φ^a)^T, where each pattern μ can be denoted by its constituent subpatterns
μ = (x, a). All subpatterns are random and uncorrelated with equal probability for +1 or −1. There are p = m[1]m[2] possible composite patterns composed of all possible combinations of the
Each pattern is assigned a random desired output η^μ ∈{±1}, and the task is to find a linear readout defined by weights W such that the sign of the projection of the activity of the last layer (input
or RCN, Fig. 1, A and B, respectively) onto it will match the desired output.
No RCNs.
In this case the task can be written in vector notation as: where Q[i,μ] = ξ[i]^μ is a 2N × p matrix that contains all input patterns. Note that we assume a zero threshold for the readout for
simplicity. We show below that this choice has no effect on the scaling properties we are interested in.
Since we are using random outputs, the classification ability depends only on the structure of the input. We first show that the matrix Q is low dimensional. Consider a case of m[1] = 2 and m[2] = 3:
This matrix can be written as showing that it is in fact only of rank 4. In general, the rank will be (m[1] − 1) + (m[2] − 1) + 1.
The rank of this matrix determines the effective number of inputs to the readout neuron (Barak and Rigotti, 2011), which in turn affects the possible number of patterns that can be classified.
Once the number of patterns exceeds the capacity, which is two times the number of independent inputs or rank, we expect classification to be at chance level (Cover, 1965; Hertz et al., 1991; Barak
and Rigotti, 2011). To verify this, we considered m[1] = m[2] = 5 and a subset of p̃ = 1, …, 25 patterns. For each value of p̃ we computed the rank of Q and the fraction of patterns that were
classified correctly (average from 500 random choices of η). This was repeated for m[1] = m[2] = 10 and m[1] = m[2] = 15 (Fig. 2D,E).
The RCN layer.
To solve the linear separability problem, we introduce an intermediate layer of randomly connected neurons. The input patterns are projected to N[RCN] randomly connected neurons through weights J[ij]
≈ 𝒩 (0, N/2), where i = 1, …, N[RCN], j = 1, …, 2N and 𝒩[x](μ,σ^2) = 12πσ2e−(x−μ)22σ2.. A threshold θ is applied to the RCNs, defining a coding level f = erfc (θ/2)/2, which is the fraction of all
possible input patterns that activate a given RCN. For a pattern ξ^xa = (ψ^x φ^a)^T, the activity of the ith RCN is given by The task can now be written as finding a W such that sign (W^T S) = η,
where the N[RCN] × p matrix S is the activity of the RCNs due to all input patterns. The different layers can be schematically described in the following diagram:
RCN classification without noise.
As before, we consider the rank of the matrix S. A single RCN can add at most 1 to the rank of the pattern matrix, and Figure 3 shows that this is indeed the case for sufficiently large p and
sufficiently high f. We quantified this behavior by determining, for each value of p and each coding level f, the minimal number of RCNs required to classify 95% of the patterns correctly. This value
was 0.5p for the high f (dense coding) case, as expected from Cover's Theorem (Cover, 1965). We determined the critical coding level at which this fraction increased to 0.75p and saw that it
decreased approximately as p^−0.8.
RCN classification with noise.
We introduce noise by flipping the activity of a random fraction n of the 2N elements of the input patterns ξ and propagating this noise to the RCN patterns S.
The heuristic calculations of generalization and discrimination in Figure 4 were done by choosing m[1] = m[2] = 2 and n = 0.1. Consistent RCNs were defined as those that maintained the same activity
level (sign of their input) in response to two noisy versions of the same input pattern. Discriminating RCNs were defined as those that had a different activity level in response to two patterns
differing by only one subpattern.
The test error of the readout from the RCNs depends on how the readout weights are set. If we have p patterns in N[RCN] dimensions and the noise is isotropic in this space, the readout that minimizes
errors is the one that has the maximal margin between the patterns and the separating hyperplane defined by the weights W (Krauth et al., 1988). Because the noise originates in the input layer, each
RCN has a different probability to flip, and the isotropic assumption is not true. Nevertheless, we use the mean patterns in RCN space to derive the maximal margin hyperplane and use this weight
vector as a readout (for a possible alternative, Xue et al. (2011)). We also trained a readout using online learning from noisy inputs, and while the error decreased, the qualitative features we
report did not differ.
Because we are interested in the shape of the error curve and not its absolute values, we adjusted the number of RCNs to avoid floor and ceiling effects. Specifically, as we varied the noise we used
Equation 20 to estimate the number of RCNs that would produce a minimal error of 10%.
Approximation of the test error.
While the paper presents extensive numerical simulations of a wide range of parameters, we are also interested in analytical approximations that can provide better insight and help understand various
scaling properties of the system. Furthermore, we would like to estimate the error from experimentally accessible quantities, and our analytical approximations help us in this regard. Given readout
weights W, every pattern μ has a distribution of projections onto this readout due to the input noise. We approximate these as Gaussian, defining κ[μ] and Σ[μ] as the mean and variance, respectively:
The test error (probability of misclassification) is then given by: We approximate the average test error by inserting the averages inside the nonlinearity (which is somewhat reasonable given that we
are interested in errors far from saturation effects): To approximate κ, we note that the perceptron margin, min[μ] κ[μ] can be bounded by the minimal eigenvalue λ of the matrix M = S^T S (Barak and
Rigotti, 2011): Because we are in the regime where there are many more RCNs than patterns, and classification is hampered by noise, we expect the margin to be a good approximation to κ. We thus
proceed to estimate λ from the matrix M.
The matrix M defined above is a random matrix (across realizations of ξ, which are assumed to be random). To obtain the distribution of its minimal eigenvalue, we should first derive the eigenvalue
and only then average over realizations of the matrix. Nevertheless, as an approximation we consider the minimal eigenvalue of the average matrix M̄ = 〈M〉[ξ]. This provides an upper bound on λ,
which in turn gives a lower bound on the margin.
As stated above, it is useful to expand the pattern indices into their constituents: μ = (x, a), ν = (y, b). Because we are analyzing the average matrix, each element M̄[μν] only depends on the number
of matching subpatterns between μ and ν so we can decompose the matrix in the following form: where δ is the Kroenecker delta and γ, γ[1], and γ[2] are scalar coefficients to be determined. This
equation simply states that there are three possible values for the entries of M̄[μν], corresponding to whether μ and ν share zero, one, or two subpatterns. The right hand side of the equation is
composed of three matrices that commute with each other, and hence we can study their eigenvalues separately. The matrices multiplied by γ[1] and γ[2] are both low rank and thus do not contribute to
the minimal eigenvalue. Thus, the minimal eigenvalue is determined by the first matrix N[RCN]γδ[xy]δ[ab], and using Equation 8, the value of κ is given by Using Equation 9, we can express γ in terms
of the squared differences of activity due to different patterns: where (Δ[1])^2 and (Δ[2])^2 are the squared differences between RCN activity due to two patterns differing by one and two subpatterns
respectively (average across all RCNs). Equation 13 provides a recipe for estimating γ from experimental data. To compare this estimate with the true value of κ, we define Γ=κ2pNRCN (Eq. 10 and Fig.
We now turn to the estimation of Σ^2. Because W is normalized, Σ^2 is simply a weighted average of the trial-to-trial variability of the RCNs. We approximate it by σ^2, which is the unweighted
average (Fig. 8E,F). The final estimate of the test error is: Note that a somewhat similar analysis of signal and noise using dimensionality of matrices was performed by Büsing et al. (2010).
A non-zero threshold of the readout does not change the scaling properties.
Our analysis and simulations were performed assuming that the threshold of the readout unit is at zero. We verified that our numerical results do not depend on the choice of the threshold (data not
shown). The reason for this can be understood by considering what happens to the matrix elements of M when an additional constant input implementing a non-zero threshold is added. In this case the
modified matrix M̃ becomes: thus adding a low rank matrix (all ones) that does not contribute to the rank. Hence, γ does not change, and neither does the performance.
Generalizing to more than two sources.
An equivalent form to Equation 13 is where M[0] is the average of the diagonal elements of the matrix M̄ (of the form M̄[xa,xa]), M[1] is the average of those elements of M of the form M̄[xa,xb], and M
[2] is the average of those elements of M of the form M̄[xa,yb]. This form readily generalizes for more than two sources of information: where M[k] is the average value of all elements of the matrix M̄
corresponding to the activity the RCNs when presented with two patterns differing by k subpatterns.
All simulations were performed in Matlab (MathWorks). N was always chosen to be 500, and the rest of the parameters are noted in the main text. The readout weights were derived from the matrix S̄ of
average RCN activations. The entries of this matrix can be calculated by considering the mean and variance of the input g̃[i] to an RCN due to a noisy pattern: where g[i] is the noiseless input to
that RCN. Approximating this input by a Gaussian distribution, we can derive the probability for this RCN to be activated as qi=erfc(θ−g˜¯Var(g˜i)), and its mean state as S̄[i] = 2q[i] − 1. Once we
have the p patterns S̄^μ, we use quadratic programming to find the weight vector W that gives the maximal margin (Wills and Ninness, 2010).
Quality of approximation from experimentally accessible data.
Because γ and σ^2 are only approximations (Fig. 8), we checked the quality of predicting the relative benefit of sparseness from experimentally accessible data. To this end, we measured for 20
realizations of 12 noise levels between 3% and 20% the ratio between the error obtained with a dense coding of f = 0.5 or an ultra-sparse coding of f = 0.001 to that obtained with a sparse coding of
f = 0.1. We also estimated γ and σ^2 from 30 trials of each pattern using a subset of 100 RCNs (Fig. 7).
The neural representation of information should be in a format that is accessible to the elements of a neural circuit. We took the point of view of an individual neuron reading out a population of
input neurons that encode multiple noisy sources of information. In particular, we studied the number of input–output functions that can be robustly implemented by such a neuron. We first explain the
problems arising when integrating multiple sources of information. We then show that the classification performance greatly increases when the input neurons mix the information sources by integrating
the activity of the source populations through random connections (randomly connected neurons or RCNs). We show that the threshold of the RCNs, which determines their coding level (i.e., the average
fraction of stimuli to which each individual neuron responds), biases a tradeoff between generalization and discrimination. Finally, we provide a prescription for measuring the components of this
tradeoff from neural data.
The problem with segregated neural representations
Consider a single neuron receiving input from several sources. For ease of presentation and visualization, we consider only two sources. For example, one source may represent a sensory input and the
other the internally represented task to be executed. Each source is segregated from the other and is represented by N neurons (Fig. 2A), each of which can be inactive (−1) or active (1). A state of
one of the sources corresponds to a specific configuration of all N of its neurons.
In general, the classification capacity of a linear readout is determined by the structure (i.e., correlations) of the inputs and by the desired output. The desired output depends on the type of
representations that will be needed by downstream processing stages in the brain. To remain general, we estimated the classification performance for all possible outputs. Specifically, we assume that
the output neurons can only be either active or inactive in response to each input (two-way classification). If there are p different inputs, then there are 2^p input–output functions. The
classification performance can be estimated by going over all these functions and counting how many can be implemented (i.e., when there is a set of synaptic weights that allow the output neuron to
respond to all the inputs as specified by the function). As it is impractical to consider such a large number of input–output functions, we estimate the performance on randomly chosen outputs, which
is a good approximation of the average performance over all possible outputs, provided the sample is large enough.
Under the assumption that we consider all possible outputs, the classification performance depends only on the properties of the input. In particular, the performance depends on the input
correlations, which in our case are the correlations between the vectors representing the input patterns of activity. These correlations are due to the specific choice of the statistics of the
inputs. A useful way to represent the correlations that are relevant for the performance is to consider the spatial arrangement of the points that represent the inputs in an activity space. Each
input can be regarded as a point in an N[t] dimensional space, where N[t] is the total number of input neurons (N[t] = 2N in the example of Fig. 2). Our correlations are the consequence of a
particular arrangement of the points representing the inputs. Indeed, in our case the points live in a low dimensional space (i.e., a space that has a dimensionality that is smaller than the minimum
between N[t] and p), and this can greatly limit the classification performance (Hinton, 1981; Barak and Rigotti, 2011). Figure 2B shows a simple example that illustrates the problem. The four
possible configurations of the two populations of N input neurons are four points in a 2N dimensional space. Four points span at most a 3D space (i.e., a solid) and, more in general, p points span at
most p − 1 dimensions (less if N[t] < p). In our example, the four inputs are all on a 2D plane because of their correlations (Fig. 2C). One dimension is spanned by the line connecting the two
patterns of the first source, and the other dimension goes along the line connecting the two patterns of the second source. The fact that there are more inputs to be classified than dimensions can
lead to the existence of input–output functions that are not implementable by a linear readout. In other words, there will be sets of desired outputs that cannot be realized by a readout neuron. In
these situations the inputs are said to be not linearly separable. For instance, it is not possible to draw a plane that separates patterns AD and BC from patterns BD and AC. This is equivalent to
saying that there is no linear readout with a set of synaptic weights that implements an input–output function for which the inputs AD and BC should produce an output that is different from the one
generated by inputs BD and AC (Hertz et al., 1991).
As the number of information sources and states within those sources increases, so does the gap between the number of patterns to be classified and the dimensionality of the space that they span,
leading to a vanishing probability that the classification problem is linearly separable (see Materials and Methods) (Rigotti et al., 2010b). In Figure 2, D and E show this scaling for two sources of
5, 10, and 15 states each (m = 5, 10, 15). The number of neurons representing each source, N = 500, is significantly larger than the number of states. The dimensionality is more formally defined as
the rank of the matrix that contains all the vectors that represent the p = m^2 different inputs (see Materials and Methods). Full rank (i.e., rank equal to the maximum, which in our case is p)
indicates that all the p vectors representing the input patterns are linearly independent and hence span a p dimensional space. Because neurons within each source only encode that specific source,
the dimensionality is always smaller than p (see Materials and Methods). Indeed, it scales as m, whereas p grows like m^2. This problem exists even when only a subset p̃ < p of all the m^2
combinations need to be correctly classified. This is shown in Figure 2D, where the dimensionality increases linearly with the number of inputs to be classified and then saturates. The upper bound
determined by the source segregation is already reached at p̃ ∼ m, which is much smaller than the total number of m^2 combinations. Figure 2E shows that the probability for linear separability drops
rapidly once the number of input patterns is higher than the dimensionality of the inputs.
Randomly connected neurons solve the problem
To solve the linear separability problem generated by the segregated representations, the information sources should be mixed in a nonlinear way. This can be achieved by introducing an intermediate
layer of neurons that are randomly connected to the segregated inputs. These neurons increase the dimensionality of the neural representations (dimensional expansion), thereby increasing the
probability that the problem becomes linearly separable for all possible outputs. Figures 3, A and B show that the dimensionality increases as more RCNs are added until it reaches the maximal
dimensionality permitted by the number of inputs.
RCNs are surprisingly efficient at increasing the dimensionality. In the dense case in which the RCNs are activated by half of all possible inputs, if the number p of input patterns is sufficiently
large, every RCN adds, on average, one dimension to the representations. This scaling is as good as the case in which the response properties of the neurons in the intermediate layer are carefully
chosen using a learning algorithm. It is important to note that for simplicity we analyzed the case in which all combinations of input patterns are considered. In many realistic situations only a
subset of those combinations may be needed to be classified to perform a given task. Although it is not possible to make statements about these general cases without further assumptions, we can note
that if the combinations are picked uniformly at random from all the possible ones, the scaling of the number of dimensions versus the number of RCNs remains the same. In other words, the number of
RCNs grows linearly with the number of inputs that have actually to be classified (Rigotti et al., 2010b). For this reason, in what follows we will consider only the case in which all the
combinations have to be classified.
If one changes the threshold for activating the RCNs, and hence modifies the coding level f (i.e., the average fraction of the inputs that activate a RCN), the convergence to full dimensionality
slows down. This is shown in Figure 3, where it is clear from the slope of the curves that the dimensionality increase per RCN is smaller for sparser neural representations. This is due to finite
size effects—there are simply not enough RCNs to sample the entire input space. As the total number p of inputs increases and the space spanned by the inputs grows, the RCNs become progressively more
efficient at increasing the dimensionality because they have more chances to be activated (Figure 3B).
The output neuron (Figs. 1B and 2A) can then be tested to determine how many input–output functions it can implement when the outputs are chosen randomly. In our situation it behaves like a
perceptron that classifies uncorrelated inputs, although it is important to note that the inputs are not uncorrelated. The number of correctly classified inputs is approximately twice the number of
RCNs. Figure 3C shows that this also holds for all but the sparsest coding levels, with the reason for failure being finite size effects of N[RCN]. We can quantify the breakdown for sparse coding
levels by defining a critical coding level, f[crit], at which the number of RCNs needed increases to 0.75 of the number of inputs. Figure 3D shows the scaling of this finite size effect; the coding
level at which classification performance deteriorates scales as a power of the number of patterns: f[crit] ∼ p^−0.8.
RCN coding level biases the discrimination–generalization trade-off
In the previous section we analyzed the ability of the output neuron to classify inputs that contain multiple sources of information when the inputs are first transformed by RCNs. The next issue we
address is whether the encouraging results on the scaling properties of the RCNs still hold when the output neuron is required to generalize. Generalization is the ability to respond in the same way
to familiar and unfamiliar members of the same class of inputs. For example, in visual object recognition, the members of a class are the retinal images of all possible variations of the same object
(e.g., when it is rotated), including those that have never been seen before. To study generalization it is important to know how to generate all members of a class. To make the analysis treatable,
we studied a specific form of generalization in which the members of a class are noisy variations of the same pattern of activity. In our case, generalization is the ability to respond in the same
way to multiple noisy variations of the inputs. Some of the noisy variations are used for training (training set), and some others for testing the generalization ability (testing set). The noise
added to the patterns of activity is independent for each neuron, as in studies on generalization in attractor neural networks and pattern completion (see Discussion for more details).
We also make the further assumption that the number of RCNs is sufficient to reach the maximal dimensionality in the noiseless case, as we intend to focus on the generalization performance. We
basically assume that there enough RCNs is to classify correctly the inputs in all possible ways (i.e., for all possible outputs) in the absence of noise.
As illustrated in Figure 1D, the transformation of the inputs performed by the RCN layer has to decorrelate them sufficiently to ensure linear separability while maintaining the representation of
different versions of the same input similar enough to allow for generalization. The decorrelation increases the ability of the readout neurons to discriminate between similar inputs, but it is
important to note that not all forms of discrimination lead to linear separability. The decorrelation operated by the RCNs has the peculiarity that it not only increases the dissimilarity between
inputs, but it also makes the neural representations linearly separable.
We now study the features of the transformation performed by the RCNs and how the parameters of the transformation bias the discrimination–generalization tradeoff, with a particular emphasis on the
RCN coding level f. f is the average fraction of RCNs that are activated in response to each input. f close to 0.5 means dense representations, small f corresponds to sparse representations. In our
model, f is controlled by varying the threshold for the activation of the RCNs. Figure 4B shows how the relative Hamming distances between inputs are transformed by the randomly connected neurons for
two different coding levels. These distances express the similarity between the neural representations (two identical inputs are at zero distance if they are identical). Note that the ranking of
distances is preserved—if point A is closer to B than to C in the input space, the same will hold in the RCN space. In other words, if input A is more similar to B than to C, this relation will be
preserved in the corresponding patterns of activity represented by the RCNs.
To understand how sparseness affects the classification performance, we first need to discuss the effects on both the discrimination and the generalization ability. Figure 4, C-F illustrate an
intuitive argument explaining how sparseness biases the discrimination-generalization tradeoff (see Materials and Methods for details). We first consider generalization. Figure 4C shows the
distribution of inputs to all RCNs when a generic input made of two sources of information is presented. For dense coding, the threshold is set to zero (blue line), and all the RCNs on the right of
the threshold (half of all RCNs) are active. The noise in the input (n = 0.1 for this example) can cause those RCNs that receive near threshold input to flip their activity, as denoted by the blue
shading. Similarly, for a sparse coding of f = 0.1, 10% of the RCNs are on the right of the red line, and a smaller number of RCNs are affected by noise. We estimate the generalization ability by
measuring the fraction of RCNs that preserve their activity for different noisy versions of the same input pattern. This quantity increases as the representations become sparser (Fig. 4D).
As for the discrimination ability, we consider again the input currents to the RCNs. Figure 4E shows the two-dimensional input distribution to all RCNs for two inputs that share the same value for
one of the two information sources (i.e., half of all input neurons are the same). As above, the blue and red lines denote threshold for f = 0.5 and f = 0.1, respectively. To enable discrimination,
we need RCNs that respond differentially to these two patterns—their input is above threshold for one pattern and below threshold for the other, as denoted by colored areas. For this measure the
fraction of RCNs with a differential response decreases as the representations become sparser (Fig. 4F).
The optimal coding level is approximately 0.1
To check how these two opposing trends affect the final performance of the classifier, we trained the output neuron to classify noisy versions of the inputs (obtained by flipping a random subset of n
percent of the source neurons' activities), and then measured the fraction of wrong responses to new noisy realizations. Figure 5A shows numerical results of the fraction of errors made by a linear
readout from a population of RCNs when the activity of a random 5% (light) or 17.5% (dark) of the source neurons is randomly flipped in each presentation. The abscissa shows f, the RCNs' coding
level, which is varied by changing the activation threshold of the RCNs. The figure reveals that there is an optimal coding level of about 0.1 that decreases the error rate more than twofold compared
to the maximally dense coding of 0.5. The advantage of sparser coding is more substantial as the input noise increases. Because we are interested in the shape of this curve, we increased the number
of RCNs to avoid floor and ceiling effects as the noise increases (from 336 for 5% to 2824 for 17.5%). Otherwise, for small noise, the generalization error could be zero for all values of f, or it
could be maximal and again constant for large noise.
Figure 5B shows the optimal coding level (black) for increasing levels of noise. The shaded area around the black curves is delimited by the coding levels at which the performance decreases by 20%
compared to the black curve. The optimal coding level hardly shifts, but the relative advantage of being at the minimum increases when noise increases. The number of RCNs necessary to compensate for
the increased noise is shown in Figure 5C.
In the noiseless case (Fig. 3) we saw that the sparse coding levels are adversely affected by finite size effects. To ascertain whether this is the cause for the increase in error as the coding level
decreases below f = 0.1, we increased the number of patterns and RCNs. Figure 5D shows that a fivefold increase in the number of patterns only slightly moves the optimal coding level. Indeed, Figure
5E shows that the optimal level does decrease with increasing system size, but at a very slow rate that is probably not relevant for realistic connectivities. Indeed, even for 20,000 RCNs (Fig. 5F)
the optimal coding level is still above 5%. Note that when we vary the number of patterns, the required number of RCNs grows linearly (Fig. 5F).
Components of the discrimination–generalization tradeoff
The numerical results reveal several phenomena. First, the required number of RCNs grows linearly with the number of input patterns. Second, a coding level of approximately f = 0.1 is better than
dense coding (f = 0.5) for correct classification. Third, an ultra sparse coding level of f = 0.01–0.03 is significantly worse than intermediate values. We derived an approximate analytical
expression of the test error that allows us to understand the scaling properties of the RCN transformation and relies on experimentally accessible factors (see Materials and Methods): where γ is the
discrimination factor that depends on the threshold θ for activating the RCNs (and hence on the coding level of the RCN representations) and on the noise n in the inputs. 1/σ^2 is the generalization
factor, which depends on θ, n, and the total number p of classes of inputs. The inverse of the generalization factor, σ^2, is simply defined as the average intertrial variability of RCN responses (
Fig. 6, green error bars).
The discrimination factor γ is related to the similarities between the RCN representations induced by similar inputs. Figure 6 defines γ more precisely and shows how to measure it from neural data.
Consider, for example, the case in which one source of information represents a sensory stimulus and the other represents the temporal context in which the stimulus appears (as in Fig. 1). We assume
that the recorded neurons contain a representative subpopulation of the RCNs, which presumably are the majority of neurons. We also assume that the recorded RCNs receive an equal contribution from
the inputs representing the two sources. For each neuron we consider the mean firing rate for every combination of the inputs. For simplicity, we assume that there are two possible stimuli and two
contexts for a total of four cases. The four bars corresponding to the four rasters in Figure 6 represent the mean firing rates in these cases. We now focus on pairs of inputs that differ only by the
state of one source (e.g., as in the pair of cases in which the same sensory stimulus appears in two different contexts). For each such pair, we compute the squared difference in the firing rate of
the neuron. This quantity should be averaged across all conditions that contain analogous pairs of inputs. We name this average (Δ[1])^2. In a similar manner, (Δ[2])^2 is the average squared
difference between the firing rates corresponding to the cases in which both sources are different (e.g., different sensory stimuli appearing in different contexts, right cluster of arrows). The
discrimination factor γ is then given by the average across neurons of: This quantity can be computed from the recorded activity under the assumption that the two sources of information have an equal
weight in driving the RCNs. This is a reasonable assumption every time the two sources of information can be considered similar enough for symmetry reasons (e.g., when they represent two visual
stimuli that in general have the same statistics). In the other cases it is possible to derive an expression for γ that takes into account the different weights of the two sources. However, the
relative weights should be estimated from the data in an independent way (e.g., by recording in the areas that provide the RCNs with the input).
To help understand the meaning of γ in the case that we analyzed (i.e., when the two sources have the same weight), we show in the inset of Figure 6 how γ is related to the shape of the curve that
represents the squared distance in the RCN space as a function of the squared distance in the input space. In particular, γ expresses the deviation from a linear transformation. Notice that in
contrast to Figure 4B, on the y-axis we now represent the expected squared distance in RCN space between pairs of noisy patterns. The distances in RCN space are contracted by the presence of noise in
the inputs (see Materials and Methods, Eq. 18).
The deviation from a linear function is intuitively related to the ability to discriminate between patterns that are not linearly separable. Indeed, for a linear transformation (γ = 0) the
dimensionality of the original input space does not increase, and the neural representations would remain nonlinearly separable.
While the exact values of the error are not captured by the experimentally accessible factors, the general trends are. To illustrate this point, we computed the expected error from a subset of 100
RCNs simulated during 30 trials of 64 patterns. Figure 7A shows the ratio of the test error for dense (0.5) and sparse (0.1) coding as derived from this estimation versus the actual one obtained from
the full simulation. Note that the correlation is very good (correlation coefficient, 0.7), even though for the high noise levels the network contains >4000 RCNs. This result is especially important
in cases where N[RCN] and p are unknown but fixed—for instance, when a neuromodulator changes the activity level of the network. In such cases, estimating γ and σ^2 from neural recordings can provide
a useful measure of the effect on network performance. The case of ultra-sparse coding is not captured as well by the approximation (Fig. 7B; correlation coefficient, −0.1), and the reason for this
is explained below.
Equation 20 already confirms the first phenomenon mentioned above—linear scaling of RCN number with the number of input patterns. If we ignore the weak p dependence of σ^2 and rearrange the terms of
the equation, we can see this linear scaling. As indicated by Figure 5F, the dependence of σ^2 on p is small and does not affect the scaling. We also verified that, similarly to the noiseless case (
Fig. 2D,E), using a fraction of possible input combinations does not alter this scaling (data not shown).
To understand the remaining phenomena—namely why a coding level of ∼0.1 is optimal—we consider the interplay between the discrimination and generalization factors. Figure 8, A and B show the
dependence of test error as a function of the coding level of the RCN representation for two different noise levels. We computed these quantities either using the full numerical simulation (solid
line) or the approximation of Equation 20 (dashed line). While the approximation captures the general trend of the error, it underestimates the error for low coding levels. This is more evident in
the low noise case, which also requires fewer RCNs.
The reason for the failure of the formula in the ultra sparse case is that γ and σ^2 are actually approximations of Γ and Σ^2—quantities that are not directly accessible experimentally (see Materials
and Methods). Briefly, Γ is a measure of the average distance from a pattern to the decision hyperplane in RCN space. Σ^2 is the average noise of the patterns in the direction perpendicular to the
decision hyperplane. Using these quantities in the formula produces a curve that is almost indistinguishable from the one obtained with the full simulation (data not shown). We thus look at the
deviations of the factors from their exact counterparts.
Figure 8, C and D, show that the discrimination factor in general increases with coding level, but for high levels of noise it shows a nonmonotonic behavior. The generalization factor shown in Figure
8, E and F, increases with coding level, giving rise to the discrimination generalization trade-off. Note that both of these estimates follow the general trend of their more exact counterparts—Γ and
Σ^2—but have some systematic deviations. Specifically, σ^2 underestimates the noise for sparse coding levels, leading to the discrepancy in Figures 8, A and B, and 7B.
To summarize the reason for optimality of 0.1 coding—it is better than 0.5 because dense coding amplifies noise more than it aids discrimination. This is more evident in the cases with high input
noise. A coding level lower than 0.05 is suboptimal due to finite size effects, which are probably inevitable for biologically plausible parameter values. These trends are summarized in Figure 9.
Increasing inhibition or changing the balance between excitation and inhibition will cause the neural representations to become sparser, decreasing the ability to discriminate between similar inputs.
This performance degradation, however, is overcompensated by the increased robustness to noise (generalization), with the net effect that the generalization error decreases for sparser
representations. This trend does not hold for too sparse representations (f < 0.1), because finite size effects start playing the dominant role, and the generalization ability does not improve fast
enough to compensate for the degradation in the discrimination ability. In this regime any increase in global inhibition leads to a degradation in the performance.
Most cognitive functions require the integration of multiple sources of information. These sources may be within or across sensory modalities (e.g., distinct features of a visual stimulus are
different sources of information), and they may include information that is internally represented by the brain (e.g., the current goal, the rule in effect or the context in which a task is
performed). In all these situations, what are the most efficient neural representations of the sources of information? The answer depends on the readout. We focused on a simple linear readout, which
is what presumably can be implemented by individual neurons. We showed that the sources must be mixed in a nonlinear way to implement a large number of input–output functions. Segregated
representations composed of highly specialized neurons that encode the different sources of information independently are highly inefficient, because the points representing the possible inputs span
a low dimensional space due to their correlation structure (as in the case of semantic memories; Hinton, 1981).
Segregated representations can be transformed into efficient representations with a single layer of randomly connected neurons. This transformation can efficiently increase the dimensionality of the
neural representations without compromising the ability to generalize. The best performance (minimal classification error) is achieved for a coding level f ∼ 0.1, as the result of a particular
balance between discrimination and generalization.
Why a linear readout?
Our results hinge on the choice of a linear readout that limits classification ability. We proposed RCNs as a possible solution, which is compatible with the observation that neurons with nonlinear
mixed selectivity are widely observed in the brain (Asaad et al., 1998; Rigotti et al., 2010a; Warden and Miller, 2010). One may legitimately wonder whether there are other biologically plausible
solutions involving different forms of nonlinearities. For example, it is possible that neurons harness the nonlinear dendritic integration of synaptic inputs so that a full or a partial network of
RCNs is implemented in an individual dendritic tree. In this scenario, some of the “units” with nonlinear mixed selectivity, analogous to our RCNs, are implemented by a specific branch or set of
dendritic branches, and hence they would not be visible to extracellular recordings. Some others must be implemented at the level of the soma and expressed by the recordable firing rates, as mixed
selectivity is observed in extracellular recordings. Our results about the statistical properties of the RCNs apply to both hidden and visible units if they implement similar nonlinearities. As a
consequence, our predictions about the activity of the RCNs are likely to remain unchanged in the presence of dendritic nonlinearities. However, future studies may reveal that the dendritic
nonlinearities play an important role in strongly reducing the number of RCNs.
Our choice of a linear readout is also motivated by recent studies on a wide class of neural network dynamical models. Inspired by the results on the effects of the dimensional expansion performed in
support vector machines (Cortes and Vapnik, 1995), many researchers realized that recurrent networks with randomly connected neurons can generate surprisingly rich dynamics and perform complex tasks
even when the readout is just linear (Jaeger, 2001; Maass et al., 2002; Buonomano and Maass, 2009; Sussillo and Abbott, 2009; Rigotti et al., 2010b). Studying RCNs in a feedforward setting has
enabled us to derive analytical expressions for the scaling properties of the circuit. Our results probably have important implications for the dynamics of models that rely on RCNs to expand the
dimensionality of the input, even outside the feedforward realm. In particular, our analysis can already predict the relevant dynamical properties of a recurrent network model implementing rule-based
behavior (Rigotti et al., 2010b).
Why randomly connected neurons?
RCNs solve efficiently the problem of low dimensionality of the input by mixing nonlinearly multiple sources of information. Intermediate layers of mixed selectivity neurons can be obtained in many
other ways (Rumelhart et al., 1986). However, RCNs offer an alternative that is appealing for several reasons. First, there is accumulating experimental evidence that for some neural systems random
connectivity is an important representational and computational substrate (Stettler and Axel, 2009). Second, the number of needed RCNs (that are not trained) scales linearly with the number of inputs
that should be classified. This is the same scaling as in the case in which the synaptic weights are carefully chosen with an efficient algorithm (Rigotti et al., 2010b). Third, many of the
algorithms for determining the weights of hidden neurons require random initial conditions. The importance of this component of the algorithms is often underestimated (Schmidhuber and Hochreiter,
1996). Indeed, there are many situations in which learning improves the signal-to-noise ratio but does not change the statistics of the response properties that the neurons had before learning, which
are probably due to the initial random connectivity. This consideration does not decrease the importance of learning, as it is clear that in many situations it is important to increase the
signal-to-noise ratio (see e.g., the spiking networks with plastic inhibitory-to-excitatory connections analyzed by Bourjaily and Miller, 2011). However, it indicates that our study could be relevant
also in the case in which the neurons of the hidden layer are highly plastic.
In all the above mentioned cases learning can improve the performance. There are situations, as those studied in recurrent networks (Bourjaily and Miller, 2011), in which different forms of synaptic
plasticity can lead either to beneficial or disruptive effects. Synaptic plasticity between inhibitory and excitatory neurons increases the signal-to-noise ratio, as mentioned above, but STDP (spike
timing-dependent plasticity) between excitatory neurons actually disrupts the diversity of the neural responses, requiring a larger number of RCNs. These forms of learning, which are disruptive for
the heterogeneity, are probably present in the brain to solve other problems in which it is important to link together neurons that fire together. Classical examples are the formation of invariant
representations of visual objects (DiCarlo et al., 2012) or learning of temporal context (Rigotti et al., 2010a).
How general are our results?
When multiple sources of information are represented in segregated neuronal populations, the correlations in the inputs can limit the number of input–output functions that are implementable by a
linear readout. We showed that RCNs can mix these sources of information efficiently and solve the nonseparability problems related to this type of correlations. The correlations that we considered
are presumably widespread in the brain, as they are likely to emerge every time a neuron integrates two or more sources of information, as in the case in which it receives external and recurrent
inputs. For example, neurons in layer 2/3 of the cortex receive a sensory input from layer 4 and a recurrent input from other neurons in the same layer (Feldmeyer, 2012). It is important in any case
to notice that the correlations and the noise that we studied are specific and that there are important computational problems which involve different types of correlations and more complex forms of
generalization. For example, the classification of visual objects is a different difficult problem because the retinal representations of the variations of the same object can be more different than
the representations of different objects. The manifolds (sets of points in the high dimensional space of neural activities) representing the variations of specific objects are highly curvilinear
(with many twists and turns) and “tangled,” requiring the neural classifiers to implement a large number of variations (Bengio and LeCun, 2007; DiCarlo et al., 2012). The shallow neural architecture
that we considered (only one intermediate layer) can deal only with “smooth” classes (i.e., a small variation of the input should not require a change in the response of the output neuron that
classifies the inputs). Indeed for non-smooth classes, a prohibitive number of RCNs and a huge training set would be required. To classify visual objects efficiently, one would require a
sophisticated preprocessing stage that extracts the relevant features from the retinal input so that the neural representations of the visual object become “smooth.” Deep networks (Bengio and Le Cun,
2007) that contain multiple layers of processing can be efficiently trained to solve these problems.
It is difficult to say whether our results about the efficiency of RCNs and optimal sparseness apply also to problems like vision, and further studies will be required to make more general
statements. We speculate that our results probably apply to the late stages of visual processing and to some of the components of the early stages (e.g., when multiple features should be combined
together). Interestingly, some of the procedures used to extract features in deep networks can generate neural architectures that are similar to those obtained with random connectivity. Networks with
random weights and no learning can already represent features well suited to object recognition tasks when the neurons are wired to preserve the topology of the inputs (i.e., neurons with limited
receptive fields) (Saxe et al., 2011). These semi-structured patterns of connectivity could also be an important substrate for learning the features used in deep networks.
Why sparse representations?
One of our main results is that there is an optimal sparseness for the neural representations. The coding level that minimizes the generalization error is, for most realistic situations, close to f =
Besides our results and the obvious and important argument related to metabolic costs, there are other computational reasons for preferring a high degree of sparseness. These reasons lead to
different estimates of the optimal f, typically to lower values than what we determined.
The first reason is related to memory capacity: sparse neural representations can strongly reduce the interference between stored memories (Willshaw et al., 1969; Tsodyks and Feigel'man, 1988; Amit
and Fusi, 1994). The number of retrievable memories can be as large as f^−2 when the proper learning rule is chosen. When f goes to zero, the capacity can become arbitrarily large, but the amount of
information stored per memory decreases. If one imposes that the amount of information per memory remains finite in the limit N → ∞, where N is the number of neurons in a fully connected recurrent
network, then the number of random and uncorrelated patterns that can be stored scales as N^2/(log N)^2 when f = log N/N. f is significantly smaller than our estimate when one replaces N with the
number of connections per neuron (in the cortex N ∼ 10^4 would lead to f ∼ 10^−3). The discrepancy becomes larger when one considers wider brain areas (Ben Dayan Rubin and Fusi, 2007).
A second reason mentioned in the Introduction is the ability of sparse over-complete representations to increase input dimensionality, facilitate learning, and reduce noise (Olshausen and Field, 2004
All of these computational reasons lead to different estimates of the optimal f, as they deal with different problems. The brain is probably dealing with all these problems, and for this reason it
may use different and sometimes adaptive coding levels in different areas, but also within the same area (indeed, there is a lot of variability in f across different neurons).
Estimates of the sparseness of neural representations recorded in the brain vary over a wide range, depending on the method for defining the coding level, the sensory stimuli used, and the brain area
considered. Many estimates are close to our optimal value of 0.1, especially in some cortical areas (e.g., in V4 and IT it ranges between 0.1 and 0.3) (Sato et al., 2004; Rolls and Tovee, 1995; J. J.
DiCarlo and N. Rust, unpublished observations).
The hippocampus exhibits significantly lower coding level (0.01–0.04) (Barnes et al., 1990; Jung and McNaughton, 1993; Quiroga et al., 2005). These estimates are lower bounds for f, as the authors
used very strict criteria to define a cell as responsive to a stimulus. For example, in Quiroga et al. (2005) a cell was considered to be selective to a particular stimulus if the response was at
least five standard deviations above the baseline. On the other hand, many of these estimates are probably biased by the technique used to record neural activity (extracellular recording). Active
cells tend to be selected for recording more often than quiet cells, shifting the estimate of f toward higher values (Shoham et al., 2006). Recent experiments (Rust and DiCarlo, 2012), designed to
accurately estimate f, indicate that for V4 and IT f ∼ 0.1.
Biasing the generalization–discrimination tradeoff
The generalization–discrimination tradeoff resulted in an optimal coding level of 0.1 under general assumptions about the statistics of the inputs and the outputs of individual neurons. Specific
behavioral tasks may impose additional constraints on these statistics, resulting in different optimal coding levels. This is probably why the brain is endowed with several mechanisms for rapidly and
reversibly modifying the sparseness of the neural representations (e.g., by means of neuromodulation; Disney et al. (2007); Hasselmo and McGaughy (2004)). In other situations, neural systems that
become dysfunctional (due e.g., to stress, aging or sensory deprivation) may cause a long term disruptive imbalance in the discrimination–generalization trade-off.
These types of shifts have been studied systematically in experiments aimed at understanding the role of the dentate gyrus (DG) and CA3 in pattern separation and pattern completion (Sahay et al.,
2011). The DG has been proposed to be involved in pattern separation, which is defined as the process by which similar inputs are transformed into more separable (dissimilar) inputs. It is analogous
to our definition of pattern discrimination, suggesting that the neurons in the DG may have similar properties as our RCNs. CA3 seems to play an important role in pattern completion, which is the
reconstruction of complete stored representations from partial inputs. Neurons in CA3 would be represented by the output neurons of our theoretical framework, and pattern completion is related to the
generalization ability. Neurogenesis, which is observed in the DG, may alter in many ways (e.g., by changing the global level of inhibition in the DG) the balance between pattern separation and
pattern completion (Sahay et al., 2011). In the future, it will be interesting to analyze specific tasks and determine to what extent our simple model can explain the observed consequences of the
shifts in the balance between pattern separation and pattern completion.
• This work was supported by funding from the Gatsby Foundation, the Kavli Foundation, the Sloan-Swartz Foundation, and Defense Advanced Research Projects Agency Grant SyNAPSE HR0011-09-C-0002.
M.R. is supported by Swiss National Science Foundation Grant PBSKP3-133357. We are grateful to Larry Abbott, Marcus Benna, René Hen, Dani Martí, and Nicole Rust for comments on this manuscript.
• Correspondence should be addressed to Stefano Fusi, Center for Theoretical Neuroscience, Department of Neuroscience, 1051 Riverside Drive, New York, NY 10032. sf2237{at}columbia.edu | {"url":"https://www.jneurosci.org/content/33/9/3844?ijkey=d95bdec47dbeb97571dacca6f62ba69dee8561e3&keytype2=tf_ipsecsha","timestamp":"2024-11-05T15:58:04Z","content_type":"application/xhtml+xml","content_length":"409080","record_id":"<urn:uuid:856ae63f-5526-4872-a54b-19adbe48c7bf>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00176.warc.gz"} |
Power Generation, Operation & Control
Materials for the upcoming Workshop
Course Learning Objectives
• Learn the characteristics of generation unit input/output curves.
• Study the economic dispatch of generation units, learn the use of LaGrange functions and the Karush-Kuhn-Tucker conditions.
• Understand the use participation factors, transmission losses, penalty factors, and locational marginal prices.
• Gain a basic understanding of linear programming and its application to economic dispatch.
• Learn how generating units are committed to meet load over the hours of a week using dynamic programing and Lagrange relaxation.
• Learn how gas fueled generation is scheduled to meet take or pay contracts, study fuel scheduling problems that involve transportation and storage.
• Learn how to schedule hydroelectric power plants and pumped storage plants.
• Learn the role played by the transmission system, learn the basics of power flow calculations, incremental losses, and penalty factors.
• Study power system security analysis, PTDF and LODF factors and contingency selection methods.
• Gain an understanding of the optimal power flow calculation, the incremental linear programming and interior point algorithms.
• How to break down the location marginal price into its three basic components.
• Understand the issues of using real time measurements and the state estimator, including bad data detection and identification and measurement observability.
• Understanding the issues involved in interchange if energy between companies, including brokers, power pools, and markets.
• Understand financial transmission rights contracts.
• Learn the basics of demand forecasting.
Course Videos
Course Slides
Problem Sessions
Problem Session Files
Homework Questions
Power Generation, Operation and Control, 3rd Edition (Buy here)
Authors: Allen J. Wood, Bruce F. Wollenberg, Gerald B. Sheble
ISBN: 978-0-471-79055-6
Publisher: Wiley
Complete Solution Manual for the "Power Generation, Operation and Control" To receive a copy of the entire solutions manual, contact John Wiley & Sons and register as a faculty member. | {"url":"https://cusp.umn.edu/power-systems/power-generation-operation-control","timestamp":"2024-11-07T05:53:18Z","content_type":"text/html","content_length":"39000","record_id":"<urn:uuid:a3be9723-c71e-4b23-885c-7149fbc66fb9>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00793.warc.gz"} |
What is Inductance - The Engineering Projects
Hello friends, I hope you all are doing great. In today’s tutorial, we will discuss What is Inductance. In 1886 first time in the world the word inductance was used by the English scientist Oliver
Heaviside. The symbol for the inductance is L that was indicated due to the scientist Heinrich Lenz who gave famous Lenz's law. In system international (SI) the unit of the inductance is Henry. It
can be defined as if one volt is provided to the coil and 1A current passes through coil than flux induced in the coil is one weber the inductance of the coil will be one Henry. Henry name is used
due to American scientist Joseph Henry who find the phenomena of the self-inductance. In today's post, we will have a look at different parameters of the inductance and its working. So let's get
started with What is Inductance.
What is Inductance
• Inductance is the property of any conductor due to that it resists any variation in the current passing through it.
• The current passing through the conductor generates the field about it, the magnitude of the field depends on the strength of the current.
• The variation in the magnitude of the field, with the variation in the current, produce emf in the conductive element, this emf resists the voltage that is varying the current.
• Inductance can also demonstrate as it is the multiple of the ratio of the voltage produce with the variation in the current.
V = -L(di/dt) V
• Inductance is of two types first one is the self-inductance and second is mutual induction.
• If current passing through any inductor induced a voltage in that conductor is known as the self-inductance.
Self Inductance
• When current moving through the current induced voltage in that conductor it is known as the seld inductance.
L =N(ø/I)
• In this equation.
□ L is the self-inductance of the conductor.
□ N is the no of turns.
□ ø is the flux.
□ I is the current flowing through the conductor.
• Due to the flux induced in that conductor, current variation produced electromotive force in this conductor.
• Due to this electromotive force, there is another current produces in the current in reverse direction of the current supplied by the external power source.
• The variation in this current resists the alteration in the first current.
• If the value of the first current is enhancing, the second current will resists the increment in the first current.
Mutual Induction
• If the current is passing through the one coil and its flux also linking with the nearby second coil than due to variation of the flux voltage induced in the 2nd coil this phenomenon is known as
the mutual induction.
• There are 2 coils X and Y are located closer to one another. When we close the switch current starts to flow through the coil X and voltage induced in that coil.
• The flux of the coil is also linking with the Y coil, if we vary the current at X coil there will be also flux variations in the second coil that induced a voltage in the coil it is due to the
phenomenon of the mutual induction.
• Now we mathematically find the value of the mutual induction in the two coils.
Em = M (dI1/dt)
M = (Em)/ (dI1/dt)------(A)
• This equation is used when we know the value of the induced voltage in the second coil and the current variation in the first coil is known.
• If we have the value of the one volts and (dI1/dt) is also one ampere then by adding these values to the equation (A) then we find the mutual inductance that is one henry.
• From the above-given equation, we can explain mutual inductance as, if one volt is provided to the first coil and current passing through the second coils is at the rate one ampere per second
then it has a mutual inductance of one henry.
• Mutual induction can also be described in by this equation.
Em = M(dI1/dt)= d/dt (MI1)--- (B)
Em =N2(dØ12/dt)= d/dt (N2Ø12)---(C)
• By using equation B and C we have.
MI1 = N2Ø12
M = (N2Ø12 /I1)
• This equation of the mutual induction can work only when the flux interaction with the second coil (N[2]f[12]) due to the current (I1) of the ist coil is already known.
• From the above-given equation of the mutual induction, we can conclude that it depends on these given below factors.
□ No of the turns in the second coil.
□ Area of the coils.
□ The distance among the coils.
Energy Stored in an Inductor
• Now we discuss the phenomena of energy storing in the inductive coil.
• The energy stored in the inductor in the form of the magnetic field.
• The field of the inductor is directly proportionate to the current provided to the inductor.
• Given below formula explains the energy stored in the inductor in the form of the magnetic field.
• E = 1/2 LI^2
• In this equation.
□ E is the amount of energy stored in the inductor.
□ L is the self-inductance of the conductor.
□ I is the current passing through that conductor.
Example of Inductance
• In the given diagram, the coils have five hundred no of turns, and it is made by the cooper.
• When we applied ten ampers of direct current to the coils then the flux of ten milli-weber is produced in that coil. Now we find the value of the self-inductance.
• As we know that the formula for the self-inductance is.
L =N (Ø/I)
• If we put the values of the no of turns, flux and current then we have inductance value.
=500x (0.01/10)
It is the detailed article on the inductance I have written almost each and everting related to the inductance in this post. If you have any question ask in comments. Thanks for reading.
Comments on ‘’ What is Inductance ‘’ ( 0 ) | {"url":"https://www.theengineeringprojects.com/2019/10/what-is-inductance.html","timestamp":"2024-11-14T21:24:04Z","content_type":"text/html","content_length":"159037","record_id":"<urn:uuid:78ff81c6-9317-4262-a0e6-5aa24b84e242>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00837.warc.gz"} |
Eureka Math Grade 6 Module 3 Lesson 9 Answer Key
Engage NY Eureka Math 6th Grade Module 3 Lesson 9 Answer Key
Eureka Math Grade 6 Module 3 Lesson 9 Example Answer Key
Example 1.
Interpreting Number Line Models to Compare Numbers
Answers may vary. Every August, the Boy Scouts go on an 8-day 40-mile hike. At the halfway point (20 miles into the hike), there is a check-in station for Scouts to check in and register. Thomas and
Evan are Scouts in 2 different hiking groups. By Wednesday morning, Evan’s group has 10 miles to go before it reaches the check-in station, and Thomas’s group is 5 miles beyond the station. Zero on
the number line represents the check-in station.
Eureka Math Grade 6 Module 3 Lesson 9 Exercise Answer Key
Exercise 1.
Create a real-world situation that relates to the points shown in the number line model. Be sure to describe the relationship between the values of the two points and how it relates to their order on
the number line.
Answers will vary.
Alvin lives in Canada and is recording the outside temperature each night before he goes to bed. On Monday night, he recorded a temperature of 0 degrees Celsius. On Tuesday night, he recorded a
temperature of – 1 degree Celsius. Tuesday night’s temperature was colder than Monday night’s temperature. – 1 is less than 0, so the associated point is below 0 on a vertical number line.
For each problem, determine if you agree or disagree with the representation. Then, defend your stance by citing specific details in your writing.
Exercise 2.
FelIcia needs to write a story problem that relates to the order In which the numbers – 6 and – 10 are represented on a number line. She writes the following:
“During a recent football game, our team lost yards on two consecutive downs. We lost 6\(\frac{1}{2}\) yards on the first down. During the second down, our quarterback was sacked for an additional
10-yard loss. On the number line, I represented this situation by first locating -6\(\frac{1}{2}\). I located the point by moving 6\(\frac{1}{2}\) units to the left of zero. Then, I graphed the
second point by moving 10 units to the left of 0.”
Agree. – 10 is less than – 6\(\frac{1}{2}\) since – 10 is to the left of – 6\(\frac{1}{2}\) on the number line. Since both numbers are negative, they indicate the team lost yards on both football
plays, but they lost more yards on the second play.
Exercise 3.
Manuel looks at a number line diagram that has the points –\(\frac{3}{4}\) and –\(\frac{1}{2}\) graphed. He writes the following related story:
“I borrowed 50 cents from my friend, Lester. I borrowed 75 cents from my friend, Calvin. I owe Lester less than I owe Calvin.”
Agree. – \(\frac{3}{4}\) is equivalent to – 0.75 and – \(\frac{1}{2}\) is equivalent to – 0.50. – 0.50 and – 0.75 both show that he owes money. But – 0.50 is farther to the right on a number line, so
Manuel does not owe Lester as much as he owes Calvin.
Exercise 4.
Henry located 2\(\frac{1}{4}\) and 2. 1 on a number line. He wrote the following related story:
“In gym class, both Jerry and I ran for 20 minutes. Jerry ran 2\(\frac{1}{4}\) miles, and 1 ran 2. 1 miles. I ran a farther distance.”
Disagree. 2\(\frac{1}{4}\) is greater than 2.1 since 2\(\frac{1}{4}\) is equivalent to 2.25. On the number line, the point associated with 2.25 is to the right of 2. 1. Jerry ran a farther distance.
Exercise 5.
Sam looked at two points that were graphed on a vertical number line. He saw the points – 2 and 1. 5. He wrote the following description:
“I am looking at a vertical number line that shows the location of two specific points. The first point is a negative number, so it is below zero. The second point is a positive number, so it is
above zero. The negative number is – 2.
The positive number is \(\frac{1}{2}\) unit more than the negative number.”
Disagree. Sam was right when he said the negative number is below zero and the positive number is above zero. But 1.5 is 1\(\frac{3}{4}\) units above zero, and – 2 is 2 units below zero. So,
altogether, that means the positive number is 3\(\frac{1}{2}\) units more than – 2.
Exercise 6.
Claire draws a vertical number line diagram and graphs two points: – 10 and 10. She writes the following related story:
“These two locations represent different elevations. One location is 10 feet above sea level, and one location is 10 feet below sea level. On a number line, 10 feet above sea level is represented by
graphing a point at 10, and 10 feet below sea level is represented by graphing a point at – 10.”
Agree. Zero in this case represents sea level Both locations are 10 feet from zero but in opposite directions, so they are graphed on the number line at 10 and – 10.
Exercise 7.
Mrs. Kimble, the sixth-grade math teacher, asked the class to describe the relationship between two points on the number line, 7.45 and 7. 5, and to create a real-world scenario. Jackson writes the
following story:
“Two friends, Jackie and Jennie, each brought money to the fair. Jackie brought more than Jennie. jackie brought $7.45, and Jennie brought $7. 50. Since 7.45 has more digits than 7.5, It would come
after 7.5 on the number line, or to the right, so it is a greater value.”
Disagree. Jackson is wrong by saying that 7.45 is to the right of 7.5 on the number line. 7. 5 is the same as 7.50, and it is greater than 7.45. When I count by hundredths starting at 7.45, I would
say 7.46, 7.47, 7.48, 7.49, and then 7.50. So, 7. 50 is greater than 7.45, and the associated point falls to the right of the point associated with 7.45 on the number line.
Exercise 8.
Justine graphs the points associated with the following numbers on a vertical number line: – 1\(\frac{1}{4}\) – 1\(\frac{1}{2}\), and 1. She then writes the following real-world scenario:
“The nurse measured the height of three sixth-grade students and compared their heights to the height of a typical sixth grader. Two of the students’ heights are below the typical height, and one is
above the typical height. The point whose coordinate is 1 represents the student who has a height that is 1 inch above the typical height. Give this information, Justine determined that the student
represented by the point associated with – 1\(\frac{1}{4}\) is the shortest of the three students.”
Disagree. Justine was wrong when she said the point – 1\(\frac{1}{4}\) represents the shortest of the three students. If zero stands for no change from the typical height, then the point associated
with – 1\(\frac{1}{2}\) is farther below zero than the point associated with – 1\(\frac{1}{4}\). The greatest value is positive 1. Positive 1 represents the tallest person. The shortest person is
represented by – 1\(\frac{1}{2}\).
Eureka Math Grade 6 Module 3 Lesson 9 Problem Set Answer Key
Write a story related to the points shown in each graph. Be sure to include a statement relating the numbers graphed on the number line to their order.
Question 1.
Answers will vary. Marcy earned no bonus points on her first math quiz. She earned 4 bonus points on her second math quiz. Zero represents earning no bonus points, and 4 represents earning 4 bonus
points. Zero is graphed to the left of 4 on the number line. Zero is less than 4.
Question 2.
Answers will vary. My uncle’s investment lost $200 in May. In June, the investment gained $150. The situation is represented by the points – 200 and 150 on the vertical number line. Negative 200 is
below zero, and 150 is above zero. – 200 is less than 150.
Question 3.
Answers will vary. I gave my sister $1.50 last week. This week, I gave her $0.50. The points – 1.50 and – 0.50 represent the change to my money supply. We know that – 1.50 is to the left of – 0. 50
on the number line; therefore, – 0. 50 is greater than – 1.50.
Question 4.
Answers will vary. A fish is swimming 7 feet below the water’s surface. A turtle is swimming 2 feet below the water’s surface. We know that – 7 is to the left of – 2 on the number line. This means –
7 is less than – 2.
Question 5.
Answers will vary. I spent $8 on a CD last month. I earned $5 in allowance last month. – 8 and 5 represent the changes to my money last month. – 8 is to the left of 5 on a number line. – 8 is 3 units
farther away from zero than 5, which means that I spent $3 more on the CD than I made in allowance.
Question 6.
Answers will vary. Skip, Mark, and Angelo were standing in line in gym class. Skip was the third person behind
Mark. Angelo was the first person ahead of Mark. If Mark represents zero on the number line, then Skip is associated with the point at -3, and Angelo is associated with the point at 1. 1 is 1 unit to
the right of zero, and -3 is 3 units to the left of zero. -3 is less than 1.
Question 7.
Answers will vary. I rode my bike \(\frac{3}{5}\) miles on Saturday and \(\frac{4}{5}\) miles on Sunday. On a vertical number line, \(\frac{3}{5}\) and \(\frac{4}{5}\) are both associated with points
above zero, but \(\frac{4}{5}\) is above \(\frac{3}{5}\). This means that \(\frac{4}{5}\) is greater than \(\frac{3}{5}\).
Eureka Math Grade 6 Module 3 Lesson 9 Exit Ticket Answer Key
Question 1.
Interpret the number line diagram shown below, and write a statement about the temperature for Tuesday compared to Monday at 11:00 p.m.
At 11:00p.m. on Monday, the temperature was about 40 degrees Fahrenheit, but at 11:00 p.m. on Tuesday, it was – 10 degrees Fahrenheit. Tuesday’s temperature of – 10 degrees is below zero, but 40
degrees is above zero. It was much warmer on Monday at 11:00 p.m. than on Tuesday at that time.
Question 2.
If the temperature at 11:00 p.m. on Wednesday is warmer than Tuesday’s temperature but still below zero, what is a possible value for the temperature at 11:00 p.m. Wednesday?
Answers will vary but must be between 0 and – 10. A possible temperature for Wednesday at 11:00 p.m. is – 3 degrees Fahrenheit because – 3 is less than zero and greater than – 10. | {"url":"https://ccssanswers.com/eureka-math-grade-6-module-3-lesson-9/","timestamp":"2024-11-02T08:16:03Z","content_type":"text/html","content_length":"160846","record_id":"<urn:uuid:1573185f-0cca-498d-a43e-63a3e62d1717>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00169.warc.gz"} |
p {margin: 0; padding: 0;} No Date . 6 FUNCTION OF A FUNCTION ("BRACKET" Rule) y (ECX) y nxf'(x) x Flx) n-1 Liff middle e.g g (2x3) + 3x y'= Cx (6x' 3) (2x3 +3x) one less Rerwite inside for power
power to the front of bracket WARNING !! The cambridge book uses the chin rule to do this cambridge working below g (2x3+3x) y=46 u = (2x'+3x) dy 645 du = (6x2 + 3) du dx Using "chain " rule dy = ty
x du dx du dx dy = but * (6x'+3) dx = 6 (6x2+3) (2x'+3x)5 | {"url":"https://keepnotes.com/california-state-university/math-150b/19106-bracket-rule","timestamp":"2024-11-04T18:59:27Z","content_type":"text/html","content_length":"123578","record_id":"<urn:uuid:7ae2156d-0902-46da-b848-e2b9e9df9b52>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00575.warc.gz"} |
Forlan Manual
The Forlan toolset is a collection of tools for experimenting with formal languages. Forlan is implemented in the functional programming language Standard ML (SML), a language whose notation and
concepts are similar to those of mathematics. Forlan is used interactively, in conjunction with the Standard ML of New Jersey (SML/NJ) implementation of SML. In fact, a Forlan session is simply an
SML/NJ session in which the Forlan modules are pre-loaded. Users are able to extend Forlan by defining SML functions.
In Forlan, the usual objects of formal language theory—finite automata, regular expressions, grammars, labeled paths, parse trees, etc.—are defined as abstract types, and have concrete syntax.
Instead of Turing machines, Forlan implements a simple functional programming language of equivalent power, but which has the advantage of being much easier to program in than Turing machines.
Programs are also abstract types, and have concrete syntax. Although mainly not graphical in nature, Forlan includes the Java program JForlan, a graphical editor for finite automata and regular
expression, parse and program trees. It can be invoked directly, or via Forlan.
Numerous algorithms of formal language theory are implemented in Forlan, including conversions between regular expressions and different kinds of automata, the usual operations (e.g., union) on
regular expressions, automata and grammars, equivalence testing and minimization of deterministic finite automata, and a general parser for grammars. Forlan provides support for regular expression
simplification, although the algorithms used are works in progress. It also implements the functional programming language used as a substitute for Turing machines.
This manual must be read in conjunction with the Forlan textbook, Formal Language Theory: Integrating Experimentation and Proof, by Alley Stoughton. The primary reference for mathematical
definitions—including algorithms—is the textbook; the primary reference for the module specifications is the manual. Typically it will be necessary to consult the book to understand not only the
input/output behavior of a function, but also how the function transforms its input to its output, i.e., the algorithm it is following.
Loading SML Files, and Forlan Input/Output
The function use, for loading SML source files, the input functions provided by various modules for loading Forlan objects from files, and the lexical analysis function Lex.lexFile first look for a
file in the current working directory (see getWorkingDirectory), and then look for it in the directories of the search path (see getSearchPath). The use function re-loads the most recently loaded
file, if called with the empty string, "".
The input functions and Lex.lexFile read from the standard input, when called with the empty string, "", instead of a filename. When reading from the standard input, Forlan prompts with its input
prompt, "@", and the user signals end-of-file by entering a line consisting of a single period ("."). The function Lex.lexFile first strips the contents of a file (or the standard input) of all
whitespace characters and comments, where a comment consists of a "#", plus the rest of the line on which it occurs. And the input functions work similarly, before beginning the process of parsing a
Forlan object from its expression in Forlan's concrete syntax. Consequently, whitespace and comments may be arbitrarily inserted into files describing Forlan objects, without changing how the files
will be lexically analyzed and parsed. An input function issues an error message if a file's contents doesn't describe the right kind of Forlan object, or if not all of the file's contents is
consumed in the process of parsing such an object. The various fromString functions work similarly to the input functions, except that they operate on the contents of strings.
The output functions provided by various modules for pretty-printing Forlan objects to files create their files in the current working directory. When given a pre-existing file, they overwrite the
contents of the file. They output to the standard output, when called with "" instead of a filename. The various toString functions work similarly to the output functions, except that they produce
This section contains specifications of Forlan's modules. The Auxiliary Functions subsection describes modules providing auxiliary functions for some SML types. The Utilities subsection describes
modules for querying and setting various Forlan parameters (e.g., the search path used by input functions, and the line length used by the pretty-printer), doing pretty-printing, issuing
informational and error messages, loading SML files, and doing debugging. The Sorting, Sets, Relations and Tables subsection describes modules implementing sorting, the abstract type of finite sets,
operations on finite relations, and the abstract type of finite tables. The Lexical Analysis subsection describes Forlan's lexical analysis module. The Symbols and Strings subsection describes
modules relating to Forlan symbols and strings. The Regular Expressions and Finite Automata subsection describes modules relating to regular expressions and finite automata. The Grammars subsection
describes modules relating to context-free grammars. And, the Programs subsection describes modules relating to programs—Forlan's alternative to Turing machines.
Top-level Environment
For convenience, some types and values are made available in Forlan's top-level environment. The Top-level Environment subsection lists those types and values as well as the modules that they come
[ Top | Parent | Root | Contents | Index ]
Forlan Version 4.15
Copyright © 2022 Alley Stoughton | {"url":"https://alleystoughton.us/forlan/manual/","timestamp":"2024-11-14T02:11:01Z","content_type":"text/html","content_length":"8007","record_id":"<urn:uuid:f85db2f4-b631-47ff-9479-605c65bf64e2>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00723.warc.gz"} |
Seeing Structures - 13 - 3D Statics, Part I
🟦 13.1 The wonderful world of 3D!
While all Statics problems represent the reality of the 3D world, many can be analyzed in 2D. Others require a 3D approach.
When the geometry requires a 3D approach, we have three analysis approaches to consider.
If the problem is relatively simple, we may decide to conduct three planar two-dimensional analyses (the xy plane, xz plane, and yz plane). For simple problems, this approach is recommended.
For more complex problems, we may decide to apply the mathematical rigor of vector notation. For that reason, this lesson is principally a review of useful 3D vector operations and vector concepts.
When a problem is extremely complex (e.g. the playground in the photograph), you would use software (finite element analysis) for the analysis. This approach is based on an approximate method under
the umbrella of so-called numerical methods.
It would be too time-consuming to do these types of calculations by hand. Engineers must be able to solve challenges quickly and efficiently. We don't always have to be accurate; we do always have to
be safe.
❏ A complex playground design
Image source: S. Reynolds, photograph, playground equipment, Amsterdam, Netherlands (designer unknown)
A note about linear algebra / matrix methods:
The problems we solve in this class do not require linear algebra or matrix solution methods. Instead, we will use simple algebra (substitution and elimination methods to solve systems of equations).
If you'd like to learn how to solve a system of linear equations using a matrix approach, check out this link.
🟦 13.2 Roadmap of the Statics Toolbox for 3D problems
This image provides an overview for our transition from 2D problem-solving to 3D problem-solving.
The concepts are the same. The tools are a bit different.
🟦 13.3 Review: position vectors
For position vectors, we typically use the symbol r.
In this image, rectangles C, D, and E all measure 3 feet by 5 feet.
The position vector from A to B is <-5,-3,-3> feet.
In other words, to get from A to B, you have to move in the negative x-direction, negative y-direction, and negative z-direction.
Practice the position vector from G to A on your own. The answer is <5,-2,-3> feet.
Also practice the position vector from B to G. This one is <0,5,6> feet.
🟦 13.4 Review: magnitude of a vector
Consider the force vector <3,4,5> Newtons.
It's depicted as head-to-tail addition in the interactive model.
It shows that a vector in 3D space can always be visualized in terms of two right triangles.
The first triangle lies in the xy plane.
You see Fx=3N and Fy=4N, but not Fz (as it's coming directly towards you. You can calculate the hypotenuse of this triangle using the Pythagorean Theorem. We can call the hypotenuse Fxy. It's equal
to 5N.
The second right triangle has a base of Fxy=5N and a height of Fz=5N.
Its hypotenuse is equal in length to the magnitude of the original vector. That's sqrt(5^2 + 5^2) = 7.071 N. Again, this is a magnitude.
We would need to multiply that magnitude by the proper unit vector to express it as the original <3,4,5> Newton force vector.
There is a 3D version of the Pythagorean Theorem shown here. The sum of the squares of the three components is equal to the square of the resultant vector.
🟦 13.5 Review: unit vectors
Unit vectors have a length of one (or unity). They are used to communicate the direction of a vector. They do not have units.
When we want to use a unit vector in the x-direction, y-direction, or z-direction, we use i, j, and k.
These are pronounced as "i-hat," "j-hat," and "k-hat."
When typed, they are bold and italicized.
When handwritten, they wear pointed hats.
Sometimes we want a unit vector that defines some other direction in space. Use u for this type of unit vector. Also, use two subscripts to indicate the tail-to-head direction of the unit vector.
Important: note that the order of the subscripts matter.
Example problem 1:
Points A and B lie at an inclination of 28 degrees, as shown.
You need to write the unit vector that defines the inclination of the line of action that is parallel to line AB. You want to go from A to B.
If an angle is given, the components of the unit vector can be expressed in terms of sine and cosine.
Remember that you can use the unit circle and Pythagorean Theorem to check your work. (cos^2 + sin^2 = 1^2).
Example problem 2:
Points A and B lie 5m apart, as shown.
As before, you want to the unit vector that defines the line of action from A to B.
Since we know the dimensions, use ratios. You don't need to calculate the angle.
In this example, you'd take sqrt(0.8^2 + 0.6^2) = 1 to verify that you have written a unit vector.
Example problem 3:
In 3D space, point A lies at <2, 0, 3> inches and point B lies at <0, 6, 0> inches.
We compute the length AB using the 3D Pythagorean Theorem.
Then, the unit vector is written by dividing the position vector by the magnitude of the length.
Remember: since all unit vectors equal unity by definition, you can always check your unit vector by using the 3D version of the Pythagorean Theorem. Make sure it has a length of 1.
🟦 13.6 The bounding cuboid for 3D vectors
This flipbook provides a review of:
• position vectors
• vector components and resultants
• unit vectors
In 2D, we visualize a bounding box for vector components in a plane.
In 3D, we can do the same kind of thing. We simply need to use a bounding cuboid instead of a bounding box.
We use the bounding cuboid to visualize the three vector components (x-direction, y-direction, and z-direction).
🟦 13.7 Summary of vector symbols and notation
In our 2D problems, we had the luxury of being a little lax in our notation. For instance, we wrote many force vectors, but we didn't go to the trouble of putting a little arrow on top of the signal,
because we all knew that force was a vector.
In 3D problems, it's a good idea for us to be a little more formal with our symbols and notation. We will try to remember to put little arrows on top of our vectors. When we use a magnitude, we can
use single or double vertical bars (either is OK):
| F | = the magnitude of the vector F
||F|| = the magnitude of the vector F
Many vectors have subscripts, such as rAB (a position vector). This can be read as the position "from A to B." Note that rAB is equal to (-1)rBA. You reverse the signs of all three components when
you reverse the direction.
Unit vectors:
• Use i, j, and k for unit vectors in the x-, y-, and z-directions
• Use u for unit vectors in other directions
• The magnitude, by definition, is 1.
Other vectors:
• use r for position (with 2 subscripts)
• use F for force (with 2 subscripts)
• use M for moment (with 2 subscripts)
Remember: when we handwrite vectors, we (make an effort to remember to) draw arrows on top. When we type vectors, we make them bold instead. And the unit vectors i, j, and k are both bold and
🟦 13.8 Review: how to compute the dot product of two vectors
Skim this page to review the basics of the dot product. We use the dot product to determine how much of a given vector points in the direction of another vector. This is why when you dot two
perpendicular vectors, you receive an answer of zero.
Key take-aways:
1. For dot-products, the end result is a scalar.
2. For dot-products, the order of the vectors does NOT matter. (Aٜ·B = B·A). That is, the dot product of two vectors is commutative.
Practice a few problems until you have the ability to compute dot products quickly, either by hand, or by programming your calculator to do it for you.
🟦 13.9 Review: how to project vectors with the dot product
Peruse this flipbook to see how we can use the dot product to project a vector component onto a line (or an axis).
Vector projections give us a way to determine the vector components that are parallel to the ray defined by the unit vector.
In this example problem, a component of force vector F is projected to the line (axis) OA, through the dot product operation.
Here is a model for how to write out these calculations. (It's the same problem.)
🟦 13.10 Review: how to compute the cross-product of two vectors
Skim this overview of the cross-product.
Most people are taught to perform the cross-product with either Method 1 or Method 2 in the image.
Stick with whatever method you were first taught. There's no need to learn the other method.
Key takeaways:
1. For cross-products, the end result is a vector. It's perpendicular to the plane created by the source vectors.
2. For cross-products, the order of the vectors is important. AxB ≠ BxA.
3. If you reverse the order of the operation, then you are reversing the sign of the answer.
Practice until you have the ability to compute cross products quickly. Ideally, use your calculator for this operation. If your calculator doesn't have this functionality pre-built in, program it in
Stay tuned: we will discover applications for the cross-product in Lesson 14.
🟦 13.11 Condensing to two dimensions
A three-dimensional structure that is symmetric (in terms of both geometry and loading) can be condensed to a 2D problem.
For instance, you might try to tip a sibling's chair by exerting force at B and C on the chair with your two hands.
At first glance, a Statics student might try to use vector notation for this type of problem.
But more experienced students would quickly notice that System I is equivalent to System II.
Since the geometry and loads are symmetric, we can condense to a 2D model (System III). We project (or elevate) the xy plane of the chair and sum moments about E to determine whether the chair tips.
❏ Some 3D problems can be modeled in 2D
Problem 0.
(a) Watch at least the first half of this video.
(b) Figure out how to do a cross-product with your calculator. Either FIND the program that does a cross-product, or WRITE the program that does a cross-product. Optional: do the same for the dot
Problem 3.
Write the unit vector that describes the line of action from A to B.
Rectangles C, D, and E all measure 2m by 4m.
C and D lie in the xy plane.
E is parallel to the xz plane.
Problem 4.
Vector B has been created by rotating vector A 30 degrees about z.
Write out the components of Vector B. Express your answers as fractions and radicals.
Bx = ?
By = ?
Bz = ?
Problem 5.
This is a continuation of the previous problem.
Vector C has been created by rotating Vector B (which lies on b-b) 45 degrees about line d-d (which is perpendicular to line b-b).
What are the components of Vector C? Express your answer in fractions and radicals.
Cx = ?
Cy = ?
Cz = ?
Problem 6.
This Vector C is the same one from the previous problem. What are the vector components in the x'y'z coordinate system? (The x'y' axes are rotated 30 degrees about z compared to the xy axes.)
Cx' =
Cy' =
Cz =
Problem 7.
A force vector (F) has components of 1 kN, 3 kN, and 2 kN, as shown.
Project F into the xy plane.
Fxy = ?
Verify that Fxy + Fz = F
Then, project F into the zy plane.
Fzy = ?
Verify that Fzy + Fx = F
Finally, project F into the xz plane.
Fxz = ?
Verify that Fxz + Fy = F
Problem 8.
This is a 3D FBD of Node E (or particle E or point E).
Your friend has already solved the problem and given you the following magnitudes for the forces. The units are kips.
• The force in A is 32/(sqrt 3)
• The force in the two forces labeled B is 16/(sqrt 3)
• The force in C is 16
Your job is to draw all three 2D projections of the node:
• an xy view (in which you do not see any z-direction vectors)
• a zy view (in which you do not see any x-direction vectors)
• a xz view (in which you do not see any y-direction vectors)
After that, use the equations of equilibrium to determine whether or not your friend's answers are correct. If each 2D projection is in static equilibrium, then the forces given to you must be
This will feel like doing three successive concurrent force problems. Refer back to Lesson 03 if you need a refresher on how to solve concurrent force problems.
Fall 2024 students: everything is done now! The TAs and I are now working on solutions and will get solutions / answers posted to the website as soon as we can. I will also bring solutions to class
today (9/25/2024). -S | {"url":"https://www.seeingstructures.org/courses-topics/statics/ST-13","timestamp":"2024-11-03T22:04:30Z","content_type":"text/html","content_length":"385309","record_id":"<urn:uuid:a45aaea1-d41e-4be2-84a4-f16468246efe>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00297.warc.gz"} |
Section: Research Program
Kinetic models for plasmas
The fundamental model for plasma physics is the coupled Vlasov-Maxwell kinetic model: the Vlasov equation describes the distribution function of particles (ions and electrons), while the Maxwell
equations describe the electromagnetic field. In some applications, it may be necessary to take into account relativistic particles, which lead to consider the relativistic Vlasov equation, but
generally, tokamak plasmas are supposed to be non relativistic. The particles distribution function depends on seven variables (three for space, three for velocity and one for time), which yields a
huge amount of computations.
To these equations we must add several types of source terms and boundary conditions for representing the walls of the tokamak, the applied electromagnetic field that confines the plasma, fuel
injection, collision effects, etc.
Tokamak plasmas possess particular features, which require developing specialized theoretical and numerical tools.
Because the magnetic field is strong, the particle trajectories have a very fast rotation around the magnetic field lines. A full resolution would require prohibitive amount of calculations. It is
then necessary to develop models where the cyclotron frequency tends to infinity in order to obtain tractable calculations. The resulting model is called a gyrokinetic model. It allows us to reduce
the dimensionality of the problem. Such models are implemented in GYSELA and Selalib. Those models require averaging of the acting fields during a rotation period along the trajectories of the
particles. This averaging is called the gyroaverage and requires specific discretizations.
The tokamak and its magnetics fields present a very particular geometry. Some authors have proposed to return to the intrinsic geometrical versions of the Vlasov-Maxwell system in order to build
better gyrokinetic models and adapted numerical schemes. This implies the use of sophisticated tools of differential geometry: differential forms, symplectic manifolds, and hamiltonian geometry.
In addition to theoretical modeling tools, it is necessary to develop numerical schemes adapted to kinetic and gyrokinetic models. Three kinds of methods are studied in TONUS: Particle-In-Cell (PIC)
methods, semi-Lagrangian and fully Eulerian approaches.
Gyrokinetic models: theory and approximation
In most phenomena where oscillations are present, we can establish a three-model hierarchy: $\left(i\right)$ the model parameterized by the oscillation period, $\left(ii\right)$ the limit model and $
\left(iii\right)$ the Two-Scale model, possibly with its corrector. In a context where one wishes to simulate such a phenomenon where the oscillation period is small and where the oscillation
amplitude is not small, it is important to have numerical methods based on an approximation of the Two-Scale model. If the oscillation period varies significantly over the domain of simulation, it is
important to have numerical methods that approximate properly and effectively the model parameterized by the oscillation period and the Two-Scale model. Implemented Two-Scale Numerical Methods (for
instance by Frénod et al. [36] ) are based on the numerical approximation of the Two-Scale model. These are called of order 0. A Two-Scale Numerical Method is called of order 1 if it incorporates
information from the corrector and from the equation to which this corrector is a solution. If the oscillation period varies between very small values and values of order 1, it is necessary to have
new types of numerical schemes (Two-Scale Asymptotic Preserving Schemes of order 1 or TSAPS) with the property being able to preserve the asymptotics between the model parameterized by the
oscillation period and the Two-Scale model with its corrector. A first work in this direction has been initiated by Crouseilles et al. [32] .
Semi-Lagrangian schemes
The Strasbourg team has a long and recognized experience in numerical methods of Vlasov-type equations. We are specialized in both particle and phase space solvers for the Vlasov equation:
Particle-in-Cell (PIC) methods and semi-Lagrangian methods. We also have a longstanding collaboration with the CEA of Cadarache for the development of the GYSELA software for gyrokinetic tokamak
The Vlasov and the gyrokinetic models are partial differential equations that express the transport of the distribution function in the phase space. In the original Vlasov case, the phase space is
the six-dimension position-velocity space. For the gyrokinetic model, the phase space is five-dimensional because we consider only the parallel velocity in the direction of the magnetic field and the
gyrokinetic angular velocity instead of three velocity components.
A few years ago, Eric Sonnendrücker and his collaborators introduce a new family of methods for solving transport equations in the phase space. This family of methods are the semi-Lagrangian methods.
The principle of these methods is to solve the equation on a grid of the phase space. The grid points are transported with the flow of the transport equation for a time step and interpolated back
periodically onto the initial grid. The method is then a mix of particle Lagrangian methods and eulerian methods. The characteristics can be solved forward or backward in time leading to the Forward
Semi-Lagrangian (FSL) or Backward Semi-Lagrangian (BSL) schemes. Conservative schemes based on this idea can be developed and are called Conservative Semi-Lagrangian (CSL).
GYSELA is a 5D full gyrokinetic code based on a classical backward semi-Lagrangian scheme (BSL) [43] for the simulation of core turbulence that has been developed at CEA Cadarache in collaboration
with our team [37] . Although GYSELA was carefully developed to be conservative at lowest order, it is not exactly conservative, which might be an issue when the simulation is under-resolved, which
always happens in turbulence simulations due to the formation of vortices which roll up.
PIC methods
Historically PIC methods have been very popular for solving the Vlasov equations. They allow solving the equations in the phase space at a relatively low cost. The main disadvantage of the method is
that, due to its random aspect, it produces an important numerical noise that has to be controlled in some way, for instance by regularizations of the particles, or by divergence correction
techniques in the Maxwell solver. We have a longstanding experience in PIC methods and we started implement them in SeLaLib. An important aspect is to adapt the method to new multicore computers. See
the work by Crestetto and Helluy [31] . | {"url":"https://radar.inria.fr/report/2014/tonus/uid7.html","timestamp":"2024-11-14T14:33:35Z","content_type":"text/html","content_length":"43167","record_id":"<urn:uuid:3268e1c8-7c98-4e3d-80f4-01f87c1e8ed8>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00683.warc.gz"} |
In the fraction x/y, where x and y are positive integers, what is the
Question Stats:
59% 41% (01:27) based on 2828 sessions
In the fraction x/y, where x and y are positive integers, what is the value of y ?
(1) The least common denominator of x/y and 1/3 is 6.
(2) x = 1
Solution:Statement One Only:
The least common denominator of x/y and 1/3 is 6.
If the least common denominator of x/y and 1/3 is 6, then y could be either 2 or 6. Since y does not have a unique value, statement one is not sufficient.
Statement Two Only:
x = 1
This does not tell us anything about the value of y; statement two is not sufficient.
Statements One and Two Together:
Using the two statements, we have: the least common denominator of 1/y and 1/3 is 6. However, again, y could be either 2 or 6. The two statements together are not sufficient.
Answer: E | {"url":"https://gmatclub.com/forum/in-the-fraction-x-y-where-x-and-y-are-positive-integers-what-is-the-168099.html","timestamp":"2024-11-07T07:27:00Z","content_type":"application/xhtml+xml","content_length":"971137","record_id":"<urn:uuid:b829d093-2a59-4d8c-82f6-95a24670d511>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00201.warc.gz"} |
A key component of learning to build mathematical models of physical systems is the idea of approximation. There are two components to the meaning of approximation in mathematics. A mathematical
approach is said to be an approximation if the mathematics being used is
1. not precise as it could be (either mathematically or in matching the data)
2. part of a more complete system that allows (in principle) corrections
3. useful for describing a physical system.
You may not have seen this usage of the term "approximation" even if you have previously taken many introductory math and science courses. A quick check of introductory biology texts and even an
introductory calculus text for biology shows that the term does not appear in the index of the biology texts and only appears in the common-speech form in the calculus text. So let's take a look at a
couple of specific cases to see how the idea plays out in practice.
Approximation 1: The derivative
When we consider the derivative of a function $f(x)$, we start by looking at a small change in $x$, $\Delta x$, and seeing how much of a change in $f$ results, $\Delta f$. The derivative is defined
as the slope
$$\frac{df}{dx} = \lim_{\Delta x \rightarrow 0} \frac{\Delta f}{\Delta x}$$
It's an approximation because we are keeping the effects in calculating $\Delta f$ that are proportional to $\Delta x$ but we ignore terms proportional to $(\Delta x)^2$ and higher powers. (See
"Derivative as algebra" in the page What is a derivative, anyway?) We could if we wanted include more terms in powers of $\Delta x$, treating $\frac{df}{dx}$ as an unknown and solving for it, but if
$\Delta x$ is small enough our approximation should do fine.
Approximation 2: The sine function
A second approximation that we'll be using in this class is The small angle approximation. This says, "if you measure an angle in radians and the angle is small enough, then
$$\sin{\theta} \approx \theta$$.
This is an mathematical approximation because, although it's not precise, it's "good enough" for certain situations (for $\theta \lt \pi/6$, say) and we can correct our approximation if we need to.
Approximations in biology
Many mathematical models in biology can be considered as approximations. Two places where mathematical have been extensively applied are in neuroscience and in population dynamics (ecology and
In neuroscience, the Hodgkin-Huxley model creates a mathematical model of signal propagation on a neuron's axon using a physical model using batteries, resistances, and capacitors. While this model
gives a reasonable description of action potentials, improved experimental information have led to more complex models that are more accurate and to which the HH model can be seen as an
In population dynamics, the Lotka-Volterra model creates a model of the interaction of a predator and its prey. It's a fairly simple model (two coupled ordinary differential equations) that ignores
many important biological factors (such as what the prey itself prey's on and the fact that individuals are discrete and breed in cycles). Nonetheless, iit gives useful insights into the kinds of
phenomena that can occur as a result of a two-species interaction, such as large oscillations. It can be considering an an approximation to a more complete model.
In all the four cases discussed above, the simplifications of the full system, whether mathematical or biological, can be seen as part of a more complete description that could potentially be used to
provide corrections; and, they all are highly useful when used in the right places.
The concept of approximation is somewhat complementary to our idea of estimation. We need to make approximations to create our estimations (to decide what to ignore) and we need to carry out
estimations to create good approximations (in order to decide whether it is legitimate to ignore some corrections). It is an essential component of our process of mathematical modeling.
Joe Redish 8/13/18
Article 263
Last Modified: August 13, 2018 | {"url":"https://www.compadre.org/nexusph/course/view.cfm?ID=263","timestamp":"2024-11-04T21:10:23Z","content_type":"text/html","content_length":"17266","record_id":"<urn:uuid:29c23d1a-ff41-4611-a097-a6e01f97568d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00404.warc.gz"} |
Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem (Conference Paper) | NSF PAGESAbstract
In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a γ-certified algorithm is also a
γ-approximation algorithm - it finds a γ-approximation no matter what the input is. Second, it exactly solves γ-perturbation-resilient instances (γ-perturbation-resilient instances model real-life
instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve
weakly perturbation-resilient instances, and solve optimization problems with hard constraints. In the paper, we define certified algorithms, describe their properties, present a framework for
designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means. We also present some negative results.
more » « less | {"url":"https://par.nsf.gov/biblio/10112618-bilu-linial-stability-certified-algorithms-independent-set-problem","timestamp":"2024-11-14T18:17:46Z","content_type":"text/html","content_length":"249779","record_id":"<urn:uuid:9dfa5937-0a00-44a5-a396-6cfe49249cfe>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00397.warc.gz"} |
What Does Surface-area Specialise in Q?
If it has to do with the subject of this size of the wall socket and also the sum of place you have for it, you’ll discover that what does surface area me-an is some thing that a great deal of people
are not too comfortable with. It’s is but one of those things where in fact the further you look at something, the less you learn about it.
The thing that’s most significant when it regards finding out exactly essaywritingservicereview what exactly does surface area mean in mathematics is the dimension of this will be done by the
dimensions that are being taken in a bid to measure partitions. There’s certainly not any type of measurement of partitions. You are able to never take any kind of measurement of the walls because
the walls are only going to be the walls involving you and the surface of the floor.
The good news is that if it regards the measurements, the outside will be the region of the earth that isn’t going to serve as such a thing. So as to allow one to figure out exactly what does area in
mathematics, you have to work out the surface that is going to PenDrago be the part of the ground that will to be the region of the floor which is going to become the component of the outer lining
that will to function as the location of a floor that’ll be the region of the wall. The location that is going to be the part of the earth that will to become the component of a floor that will to
function as the area of the wall.
You can go and get a location on the newspaper that will to function as the space of the wall. You may really go and determine exactly what things to do surface area in mathematics, but that’s
perhaps not going to be necessary. You may only take the area of the wall and then multiply that by the magnitude of this wall socket. The region of the wall will soon be the sq footage of this wall
divided by the sq footage from this wall socket.
This is a good number because that is going to give you a excellent idea of everything does surface area in mathematics mean about the square footage of this wall socket. You’re able to take
advantage of this quantity to work out what exactly does surface area in mathematics whenever you are finding out the dimensions of this wall socket. The square footage of the wall is going to
function as dimension that will to be taken to your flooring in the region of the wall.
This amount will be the same as the sq footage of those partitions. There is no way to place in this variety, but that is the amount which is going to become utilised to get the measurements of the
wall that will be used in a calculation to get the size of the wall. It will soon probably be the magnitude of these sq footage of this wall that is going to be utilized. The area of the wall is
going to be the amount of this square footage of this walls and then then multiplied by the square footage of the wall socket.
The reason why that you may understand there are quite a few things that are utilised to produce the numbers is that there are different ways to estimate these different points. In the event the
square footage has been staying calculated using different things, then the result will be a number that is going to be different. It is simply planning to become a matter of using the numbers which
have been used and putting them all together in a proper way.
The main reason you ought to find out the space of the wall will be since it is certainly likely to be utilised at a means to find the measurements of the place that will to stay a particular
location. The measurement that is going to be utilized will have to be properly used because the measurements will be utilised inside this way. The single means that you may receive a ideal response
out with this issue is if the dimensions are true. And this is what exactly does the surface area of a wall way in math.
Leave a Comment | {"url":"https://www.paramtechnologies.in/what-does-surface-area-specialise-in-q-11/","timestamp":"2024-11-09T08:06:01Z","content_type":"text/html","content_length":"58035","record_id":"<urn:uuid:68571e21-8e2b-4b85-863d-852e794ce1b4>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00393.warc.gz"} |
he value of the fifth root of 101010 is... | Filo
Question asked by Filo student
he value of the fifth root of is
Not the question you're searching for?
+ Ask your question
Video solutions (2)
Learn from their 1-to-1 discussion with Filo tutors.
5 mins
Uploaded on: 1/27/2023
Was this solution helpful?
Found 3 tutors discussing this question
Discuss this question LIVE for FREE
10 mins ago
One destination to cover all your homework and assignment needs
Learn Practice Revision Succeed
Instant 1:1 help, 24x7
60, 000+ Expert tutors
Textbook solutions
Big idea maths, McGraw-Hill Education etc
Essay review
Get expert feedback on your essay
Schedule classes
High dosage tutoring from Dedicated 3 experts
Students who ask this question also asked
View more
Stuck on the question or explanation?
Connect with our Mathematics tutors online and get step by step solution of this question.
231 students are taking LIVE classes
Question Text he value of the fifth root of is
Updated On Jan 27, 2023
Topic Algebra
Subject Mathematics
Class Class 12
Answer Type Video solution: 2
Upvotes 282
Avg. Video Duration 5 min | {"url":"https://askfilo.com/user-question-answers-mathematics/he-value-of-the-fifth-root-of-is-33393831313532","timestamp":"2024-11-06T19:07:59Z","content_type":"text/html","content_length":"277994","record_id":"<urn:uuid:8f6690d2-f341-46ac-a474-77f21daa7a85>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00761.warc.gz"} |
Head reduction of `match` with Definitional UIP?
With Set Definitional UIP, we have the following behaviors:
Set Definitional UIP.
Inductive seq {A} (a:A) : A -> SProp := srefl : seq a a.
Arguments srefl {_ _}.
Definition seq_to_eq {A x y} (e:seq x y) : x = y :> A :=
match e with srefl => eq_refl end.
Parameter d : seq 0 1.
Eval lazy head in seq_to_eq ((fun a => a) d).
(* match (fun a : seq 0 1 => a) d in (seq _ a) return (0 = a) with
| srefl => eq_refl
end *)
Eval cbv head in seq_to_eq ((fun a => a) d).
(* match d in (seq _ a) return (0 = a) with
| srefl => eq_refl
end *)
I suspect that cbv has the intended behavior, but I'm not so sure.
IDK lazy seems fine
The reason I'm saying that cbv seems to have the intended behavior is that if we see the SProp case as adding something to the non-SProp case, then, when the "adding something" does not apply, one
may expect that it behaves at least as in the following non-SProp variant:
Inductive seq {A} (a:A) : A -> Prop := srefl : seq a a.
Arguments srefl {_ _}.
Definition seq_to_eq {A x y} (e:seq x y) : x = y :> A :=
match e with srefl => eq_refl end.
Parameter d : seq 0 1.
Eval lazy head in seq_to_eq ((fun a => a) d).
(* match d in (seq _ a) return (0 = a) with
| srefl => eq_refl
end *)
why would different things behave the same?
equality in sprop (with definitional uip) has the special match reduction rule which does not look at the term which is matched on
Last updated: Oct 13 2024 at 01:02 UTC | {"url":"https://coq.gitlab.io/zulip-archive/stream/237656-Coq-devs-.26-plugin-devs/topic/Head.20reduction.20of.20.60match.60.20with.20Definitional.20UIP.3F.html","timestamp":"2024-11-09T07:05:14Z","content_type":"text/html","content_length":"9642","record_id":"<urn:uuid:8f8a2b11-ff8b-49b8-b4f1-6f2836801d1f>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00676.warc.gz"} |
NCERT Solutions for Class 6 Maths Chapter 1 Knowing Our Numbers
Get here the NCERT Solutions for Class 6 Maths Chapter 1 Knowing Our Numbers and Class 6 Maths Chapter 1 Try These Solutions and Practice Tests for revision. It is given here in Hindi and English
Medium prepared for academic session 2024-25. According to rationalised syllabus and new books for class 6 Mathematics Ganita Prakash for CBSE 2024-25, there are only two exercises in chapter 1
Knowing our numbers. Get here the NCERT Solutions for Class 6 Maths Ganita Prakash Chapter 1 Patterns in Mathematics also.
6th Maths Chapter 1 Solutions for CBSE Board
6th Maths Chapter 1 Solutions for State Boards
Class: 6 Mathematics
Chapter 1: Knowing Our Numbers
Ganita Prakasha Chapter 1: Patterns in Mathematics
Number of Exercises: 2 (Two)
Content: NCERT Exercise Solution
Mode: Videos, Images and Text Format
Academic Session: 2024-25
Medium: English and Hindi Medium
NCERT Solutions for Class 6 Maths Chapter 1
Get class VI Maths Exercise 1.1 and 1.2 at Tiwari Academy in simplified way. 6th Maths Solutions PDF and Video in English and Hindi Medium are prepared in such a way that student can understand it
easily. We have updated it for new session based on latest textbooks from NCERT (https://ncert.nic.in/) website. Find the Solutions of Prashnavali 1.1 and 1.2 in Hindi. We are following the latest
CBSE Syllabus 2024-25. We work for your help free of cost. Separate links are given to download solutions in PDF file format. In case of any hassle in finding the solutions, please inform us. We will
help you at our level best.
Download NCERT Solutions for Class 6 Maths Chapter 1
These NCERT Solutions are based on latest CBSE – NCERT Textbooks for the CBSE exams 2024-25. Download NCERT Solutions in PDF format to use it offline or use as it is online without downloading.
In 6 Maths Ganita Prakash Chapter 1 Patterns in Mathematics we will learn about patterns. In the previous book chapter 1 Knowing Our Numbers, we will study about comparing the number (smaller or
greater), selecting the smallest or greatest numbers, order of numbers (ascending or descending).
Ascending order: Ascending order means arrangement from the smallest to the greatest.
Descending order: Descending order means arrangement from the greatest to the smallest.
Concepts of Place values, face values and questions based on the numbers as follow:
Starting from the greatest 6-digit number, write the previous five numbers in descending order.
Starting from the smallest 8-digit number, write the next five numbers in ascending order.
The Indian System of Numeration: In our Indian System of Numeration, Commas are used to mark thousands, lakhs and crores. We use ones, tens, hundreds, thousands and then lakhs and crores. The first
comma comes after hundreds place and marks thousands. The second comma comes two digits later. It comes after ten thousands place and marks lakh. The third comma comes after another two digits. It
comes after ten lakh place and marks crore.
Important Questions on 6 Maths Chapter 1
Fill in the blank: 1 lakh = _______________ ten thousand
Fill in the blank: 1 lakh = 10 ten thousand
Place commas correctly and write the numerals: Seventy-three lakh seventy-five thousand three hundred seven.
Insert commas suitable and write the names according to Indian system of numeration: 87595762
8,75,95,762 Eight crore seventy-five lakh ninety-five thousand seven hundred sixty-two.
Estimate each of the following using general rule: 730 + 998
730 round off to 700 998 round off to 1000 Estimated sum 1700
Estimate the following product using general rule: 578 x 161
578 x 161 578 round off to 600 161 round off to 200 The estimated product = 600 x 200 = 1,20,000
Contact Us for Help
We are here to help you. For educational help any time you can leave a message, we will call you with in 24 hours. Our prime motive is to help the students without any delay free of cost. NCERT Books
and their solutions are given in offline as well as online mode.
How many exercises, questions, and examples are there in chapter 1 of class 6th Maths?
There are 2 exercises in chapter 1 (Knowing our Numbers) of class 6th Maths.
In the first exercise (Ex 1.1), there are 4 questions. Questions 1 and 2, each having five parts, and questions 3 and 4, each having four parts.
In the second exercise (Ex 1.2), there are 12 word problem questions.
So, there are in all 16 questions in chapter 1 (Knowing our Numbers) of class 6th Maths.
There are 6 examples in chapter 1 (Knowing our Numbers), which are good for exams point of view.
What are the main topics to study in chapter 1 of class 6th Maths?
In chapter 1 of class 6th Maths, students will study:
□ 1. Comparing Numbers.
□ 2. How many numbers can you make?
□ 3. Shifting digits.
□ 4. Introducing 10,000.
□ 5. Revisiting place value.
□ 6. Introducing 1, 00,000.
□ 7. Larger numbers.
□ 8. An aid in reading and writing large numbers.
□ 9. Use of commas.
□ 10. Large Numbers in Practice.
Is chapter 1 of class 6th Maths difficult?
Chapter 1 of class 6th Maths is neither too easy nor too difficult. It lies in the middle of easy and difficult because some parts of this chapter are easy, and some are difficult. However, the
difficulty level of any chapter varies from student to student. So, Chapter 1 of class 6th Maths is easy or not depends on students also. Some students find it complicated, some find it simple, and
some find it in the middle of simple and difficult.
How much time, students need to do chapter 1 of class 6th Maths?
Students need a maximum of 5-6 days to do chapter 1 of class 6th Maths if they give at least 1-2 hours per day to this chapter. This time is an approximate time. This time can vary because no
students have the same working speed, efficiency, capability, etc.
Last Edited: September 5, 2024 | {"url":"https://www.tiwariacademy.com/ncert-solutions/class-6/maths/chapter-1/","timestamp":"2024-11-12T22:31:15Z","content_type":"text/html","content_length":"268608","record_id":"<urn:uuid:2746d165-0587-467e-9126-50a6b9993cef>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00693.warc.gz"} |
Activities - Karkhana
About Lesson
In this activity we will build a code that takes a number(N) as input and finds the sum(S) of all numbers from 1 to N.
But before beginning with the code, understand its flow. Let’s say the value of N=4
Now let’s build a code to implement the table above.
Step 1: Create 3 variables N, Sum, and count.
Step 2: Set the value of Sum =0 and count =1.
Step 3: use the following block so that the code asks for a value when run
Step 4: create a while loop with condition (count<=N).[ N is the number of terms to be added]
Step 5: set sum = sum + count[ remember, the code will read a mathematical equation from right to left. So for the code, the above equation becomes sum + count = sum. Same as shown in the table
Step 6: increase the value of count by 1. To do so, use the following block. | {"url":"https://karkhana.co.in/courses/code-with-blockly/lessons/activities/","timestamp":"2024-11-05T16:03:56Z","content_type":"text/html","content_length":"334071","record_id":"<urn:uuid:badb9bb0-fd50-4134-a73c-551ff37cf08e>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00256.warc.gz"} |
1992 AIME Problems/Problem 13
Triangle $ABC$ has $AB=9$ and $BC: AC=40: 41$. What's the largest area that this triangle can have?
Solution 1
First, consider the triangle in a coordinate system with vertices at $(0,0)$, $(9,0)$, and $(a,b)$. Applying the distance formula, we see that $\frac{ \sqrt{a^2 + b^2} }{ \sqrt{ (a-9)^2 + b^2 } } = \
We want to maximize $b$, the height, with $9$ being the base.
Simplifying gives $-a^2 -\frac{3200}{9}a +1600 = b^2$.
To maximize $b$, we want to maximize $b^2$. So if we can write: $b^2=-(a+n)^2+m$, then $m$ is the maximum value of $b^2$ (this follows directly from the trivial inequality, because if ${x^2 \ge 0}$
then plugging in $a+n$ for $x$ gives us ${(a+n)^2 \ge 0}$).
$b^2=-a^2 -\frac{3200}{9}a +1600=-\left(a +\frac{1600}{9}\right)^2 +1600+\left(\frac{1600}{9}\right)^2$.
$\Rightarrow b\le\sqrt{1600+\left(\frac{1600}{9}\right)^2}=40\sqrt{1+\frac{1600}{81}}=\frac{40}{9}\sqrt{1681}=\frac{40\cdot 41}{9}$.
Then the area is $9\cdot\frac{1}{2} \cdot \frac{40\cdot 41}{9} = \boxed{820}$.
Solution 2
Let the three sides be $9,40x,41x$, so the area is $\frac14\sqrt {(81^2 - 81x^2)(81x^2 - 1)}$ by Heron's formula. By AM-GM, $\sqrt {(81^2 - 81x^2)(81x^2 - 1)}\le\frac {81^2 - 1}2$, and the maximum
possible area is $\frac14\cdot\frac {81^2 - 1}2 = \frac18(81 - 1)(81 + 1) = 10\cdot82 = \boxed{820}$. This occurs when $81^2 - 81x^2 = 81x^2 - 1\implies x = \frac {\sqrt {3281}}9$.
Rigorously, we need to make sure that the equality of the AM-GM inequality is possible to be obtained (in other words, $(81^2 - 81x^2)$ and $(81x^2 - 1)$ can be equal with some value of $x$). MAA is
pretty good at generating smooth combinations, so in this case, the AM-GM works; however, always try to double check in math competitions -- the writer of Solution 2 gave us a pretty good example of
checking if the AM-GM equality can be obtained. ~Will_Dai
Solution 3
Let $A, B$ be the endpoints of the side with length $9$. Let $\Gamma$ be the Apollonian Circle of $AB$ with ratio $40:41$; let this intersect $AB$ at $P$ and $Q$, where $P$ is inside $AB$ and $Q$ is
outside. Then because $(A, B; P, Q)$ describes a harmonic set, $AP/AQ=BP/BQ\implies \dfrac{\frac{41}{9}}{BQ+9}=\dfrac{\frac{40}{9}}{BQ}\implies BQ=360$. Finally, this means that the radius of $\
Gamma$ is $\dfrac{360+\frac{40}{9}}{2}=180+\dfrac{20}{9}$.
Since the area is maximized when the altitude to $AB$ is maximized, clearly we want the last vertex to be the highest point of $\Gamma$, which just makes the altitude have length $180+\dfrac{20}{9}$.
Thus, the area of the triangle is $\dfrac{9\cdot \left(180+\frac{20}{9}\right)}{2}=\boxed{820}$
Solution 4 (Involves Basic Calculus)
We can apply Heron's on this triangle after letting the two sides equal $40x$ and $41x$. Heron's gives
$\sqrt{\left(\frac{81x+9}{2} \right) \left(\frac{81x-9}{2} \right) \left(\frac{x+9}{2} \right) \left(\frac{-x+9}{2} \right)}$.
This can be simplified to
$\frac{9}{4} \cdot \sqrt{(81x^2-1)(81-x^2)}$.
We can optimize the area of the triangle by finding when the derivative of the expression inside the square root equals 0.
We have that $-324x^3+13124x=0$, so $x=\frac{\sqrt{3281}}{9}$.
Plugging this into the expression, we have that the area is $\boxed{820}$.
~minor $\LaTeX$ edit by Yiyj1
Solution 5
We can start how we did above in solution 4 to get $\frac{9}{4} * \sqrt{(81x^2-1)(81-x^2)}$. Then, we can notice the inside is a quadratic in terms of $x^2$, which is $-81(x^2)^2+6562x^2-81$. This is
maximized when $x^2 = \frac{3281}{81}$.If we plug it into the equation, we get $\frac{9}{4} \cdot \frac{3280}{9} = \boxed{820}$
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. | {"url":"https://artofproblemsolving.com/wiki/index.php/1992_AIME_Problems/Problem_13","timestamp":"2024-11-12T10:21:16Z","content_type":"text/html","content_length":"54650","record_id":"<urn:uuid:1c3e19f9-c169-42bf-8059-e53932433a01>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00703.warc.gz"} |
What is Solubility Product?
What is Solubility Product?
The concept of solubility product arises when an ionic solid is dissolved in water with its ions becoming the actual solutes.
Example, when solid silver chloride, AgCl is shaken with water, the following equation is obtained for its saturated solution:
AgCl(s) Ag^+(aq) + Cl^-(aq)
The concentrations of the ions, Ag^+ and Cl^- in solution depends on the solubility of AgCl. The above equation is reversible, indicating AgCl to be slightly soluble in water. Hence, an equilibrium
constant, similar to that of any chemical equilibrium can be written as thus:
since AgCl is pure solid (slightly soluble), its concentration is taken as constant.
Therefore: K = Ksp = [Ag^+] [Cl^-] Where Ksp is called the solubility product constant, [Ag+] is conc. of silver ions in moles per dm^3, [Cl-] is conc. of chloride ions in moles per dm^3
- The constant Ksp applies to the process in which the forward reaction is the dissolving of an insoluble or slightly soluble ionic substance in water.
- If the ions have unequal number of cations and anions, the Ksp expression is written, such that the coefficients of the ions in the balanced chemical equation become the exponents of the
respective concentration term in the Ksp expression. Example, the expression for the reaction:
Mg(OH)[2](s) Mg^2+(aq) + 2OH^-(aq)
Ksp = [Mg^2+] [OH^-]^2
- Generally, solubility product refers only to insoluble or slightly soluble ionic substances in water (these produce ionic equilibria in solution). It is the product of the concentrations of the
ions raised to the coefficients of their respective ion.
- If the molar concentrations of the ions of a slightly soluble substance are known, the numerical value of Ksp can be found. Conversely, if Ksp is known, the concentration of the ions in
equilibrium with the undissolved can be determined.
Simple Calculations on Solubility Product
1. When a sample of solid AgCl is shaken with water at 25^oC, 1.0 x 1.0^-5 M silver ion is produced, calculation the Ksp.
AgCl(s) Ag^+(aq) + Cl^-(aq)
Ksp = [Ag^+] [Cl^-]
From the stoichiometry of reaction, mole ratio of Ag^+ to Cl^- is 1:1
Therefore, [Ag^+] = [Cl^-] = 1.0 x 10^-5 M
Ksp = (1.0 x 10^-5) (1.0 x 10^-5) = 1.0 x 10^-10 M^2 (mol^2 dm^-6)
2. Calculate the solubility of Mg(OH)[2] in water Ksp = 1.2 x 10^-11 mol^3 dm^-9
Mg(OH)[2](s) Mg^2+(aq) + 2OH^-(aq)
From the stoichiometry, mole ratio of Mg^2+ to OH^- is 1:2
Ksp = [Mg^2+] [OH^-]^2
Let [Mg^2+] = X, therefore [OH^-] = 2X (mole ratio is 1:2).
Hence, Ksp = X (2X)^2 = 1.2 x 10^-11
4X^3 = 12 x 10^-12
X^3 = 3 x 10^–12
X = 1.4 x 10^-4 M = conc. of Mg^2+ ions.
From the stoichiometry: 1 mole of Mg(OH)[2] dissolves to produce 1 mole of Mg^2+ ions and 2 moles of OH^- ions. Therefore, 1.4 x 10^-4 mole of Mg(OH)[2] dissolved to produce 1.4 x10^-4 mole of Mg^
2+ ions and 2.8 x 10^-4 mole of OH^- ions. Hence, for 1 dm^3 of solution, the solubility of Mg(OH)[2] is 1.4 x 10^-4 M | {"url":"http://freechemistryonline.com/what-is-solubility-product.html","timestamp":"2024-11-14T20:46:32Z","content_type":"application/xhtml+xml","content_length":"12720","record_id":"<urn:uuid:5b9dd97e-bcd2-4a5d-a176-09bec92d297c>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00093.warc.gz"} |
Part 1 involving the Spinner
Firstly, let's look at the first probability statement. We have an ample chance of landing the same spin again, so we won't be removing one blue for the next spin.
This fits the denominator 64, as that is 8*8.
We first have a 5/8 chance to land on a blue, then we have a 3/8 chance to land on a red.
\({5\over{8}}*{3\over{8}}\)is our equation (sorry for it being tiny, LaTeX is small)
In fraction multiplication, we multiply the top and bottom separately. 5*3 is 15, and as we said before 8*8 is 64
This leaves us with the fraction 15/64, which is true.
Finding this answer ALSO eliminates the third answer from being true, as it said that we remove a piece of the spinner if it is landed on, which is untrue.
Keep in mind that (red, blue) is the same as (blue, red)
Now, let's look at the second option.
The probability of (blue, blue) is \({5\over{8}}*{5\over{8}}\)
Now, let's multiply. 5*5 is 25, and 8*8 is 64.
Therefore, this answer is true.
Finally, let's look at the fourth option. (Remember that the third option was not correct.)
The probability of (red, red) is \({3\over{8}}*{3\over{8}}\)
Now, let's multiply. 3*3 is 9, and 8*8 is 64.
9/64 does NOT reduce to 3/32.
Therefore, this is a FALSE statement.
The correct statements are Options 1 and 2.
Part 2 involving Playing Cards
First, let's look at the part involving (Queen, Black Card)
First, this is a 4 in 52 or 1 in 13 chance of drawing a queen.
Then we draw a black card.
The chance of this is 1 in 2.
We'll use the simplified numbers to form our equation.
Now we multiply. 1*1=1, and 13*2=26
We now end with 1/26.
Now let's look at the probability of (Diamond, Ace)
First, the chance of drawing a Diamond in a set of 52 cards is 13 out of 52, or 1 in 4.
Then you have a 4 in 52 or 1 in 13 chance of drawing an Ace after the card is replaced.
Let's set up the equation: \({1\over{13}}*{1\over{4}} \)
Now, let's multiply. 1*1=1, and 13*4=52.
Therefore, our answer is 1/52.
Thirdly, let's look at the probability of (Jack, 4)
First, the probability of drawing a Jack is 1 in 13.
Then, after replacement, the probability of drawing a 4 is 1 in 13.
We can make this easy on ourselves by squaring the fraction since they are the same. \({1\over{13}}^2\)
1^2=1, and 13^2=169
Therefore, our final answer is 1/169.
Finally, let's look at the probability that we first draw a red card, and then a Club (Red card, Club)
The probability of drawing a red card is 1 in 2.
The probability of drawing a club is 1 in 4.
Now, let's set up the equation: \({1\over{2}}*{1\over{4}}\)
Now let's multiply. 1*1=1, and 2*4=8
Therefore, our answer is 1/8.
Hope this helped and sorry for the long answer! | {"url":"https://web2.0calc.fr/membres/rarinstraw1195/?answerpage=183","timestamp":"2024-11-08T02:08:29Z","content_type":"text/html","content_length":"46852","record_id":"<urn:uuid:940a1832-cca0-4bcf-b6ba-31cfc15d9d58>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00313.warc.gz"} |
A GPU-accelerated Direct-sum Boundary Integral Poisson-Boltzmann Solver
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
arXiv:1301.5885 [math.NA], (24 Jan 2013)
author={Geng}, W. and {Jacob}, F.},
title={"{A GPU-accelerated Direct-sum Boundary Integral Poisson-Boltzmann Solver}"},
journal={ArXiv e-prints},
keywords={Mathematics – Numerical Analysis, Computer Science – Numerical Analysis, Physics – Computational Physics},
adsnote={Provided by the SAO/NASA Astrophysics Data System}
In this paper, we present a GPU-accelerated direct-sum boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used
to ensure the fast convergence of Krylov subspace based linear algebraic solver such as the GMRES. The molecular surfaces are discretized with flat triangles and centroid collocation. To speed up our
method, we take advantage of the parallel nature of the boundary integral formulation and parallelize the schemes within CUDA shared memory architecture on GPU. The schemes use only $11N+6N_c$
size-of-double device memory for a biomolecule with $N$ triangular surface elements and $N_c$ partial charges. Numerical tests of these schemes show well-maintained accuracy and fast convergence. The
GPU implementation using one GPU card (Nvidia Tesla M2070) achieves 120-150X speed-up to the implementation using one CPU (Intel L5640 2.27GHz). With our approach, solving PB equations on
well-discretized molecular surfaces with up to 300,000 boundary elements will take less than about 10 minutes, hence our approach is particularly suitable for fast electrostatics computations on
small to medium biomolecules. | {"url":"https://hgpu.org/?p=8840","timestamp":"2024-11-04T12:13:10Z","content_type":"text/html","content_length":"86721","record_id":"<urn:uuid:9abfbb70-d603-4118-9a7f-8160e13ec56f>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00447.warc.gz"} |
PHY 590.06, Open Quantum Systems,
PHY 590.06, Open Quantum Systems, Spring 2021
Instructor: Prof. Thomas Barthel
Lectures: Tuesdays 10:10-11:35 on Zoom
Discussions: Tuesdays 11:35-12:35
In several experimental frameworks, a high level of control on quantum systems has been accomplished. Due to practical constraints and our aim of manipulating these systems efficiently, they are
inevitably open in the sense that they are coupled to the environment. This generally leads to dissipation and decoherence, which pose challenges for modern quantum technology. On the other hand, one
can design environment couplings to achieve novel effects and, e.g., to stabilize useful entangled states for quantum computation and simulation. The description of open systems goes beyond the
unitary dynamics covered in introductory quantum mechanics courses; it involves intriguing new mathematical aspects and physical phenomena.
This course provides an introduction to open quantum systems. We will start by discussing quantum mechanics of composite systems, which leads us from pure states to density operators and from unitary
dynamics to quantum channels. At this stage, we can already gain an understanding of decoherence and dephasing. We will then derive and discuss the Lindblad master equation, which describes the
evolution of Markovian systems. It covers, for example, systems weakly coupled to large baths or closed quantum systems with external noise. As we will see in applications for specific models, it can
explain dissipation, decoherence, and thermalization. We will talk about prominent experimental platforms for quantum computation and simulation from this viewpoint. The analog of Hamiltonians for
closed systems are Liouville super-operators for open systems. As they are non-Hermitian, interesting mathematical aspects arise. We will discuss fundamental properties like the spectrum and their
connection to phase transitions in the nonequilibrium steady states. Time permitting, we will close with a summary of theoretical and computational techniques for open quantum systems, addressing
exact diagonalization, quantum trajectories, tensor networks, and the Keldysh formalism.
The course is intended for students from physics, quantum engineering, quantum chemistry, and math. We expect basic knowledge of quantum mechanics (Schrödinger equation, bra-ket notation, spin,
tensor product).
Lecture Notes
[Are provided on the Sakai site PHYSICS.590.06.Sp21.]
Recommended reading for large parts of the course:
• Breuer, Petruccione "Theory of Open Quantum Systems", Oxford University Press (2002),
• Rivas, Huelga "Open Quantum Systems", Springer (2012),
• Nielsen, Chuang "Quantum Computation and Quantum Information", Cambridge University Press (2000).
The more advanced topics will be based on current research literature. | {"url":"http://webhome.phy.duke.edu/~barthel/L2021-01_OpenQuantumSystems_phy590/","timestamp":"2024-11-03T22:35:56Z","content_type":"text/html","content_length":"5536","record_id":"<urn:uuid:d72a76dd-7843-47df-b3a0-9f3a3b23052b>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00266.warc.gz"} |
A generalized flux function for three-dimensional magnetic reconnection
The definition and measurement of magnetic reconnection in three-dimensional magnetic fields with multiple reconnection sites is a challenging problem, particularly in fields lacking null points. We
propose a generalization of the familiar two-dimensional concept of a magnetic flux function to the case of a three-dimensional field connecting two planar boundaries. In this initial analysis, we
require the normal magnetic field to have the same distribution on both boundaries. Using hyperbolic fixed points of the field line mapping, and their global stable and unstable manifolds, we define
a unique flux partition of the magnetic field. This partition is more complicated than the corresponding (well-known) construction in a two-dimensional field, owing to the possibility of heteroclinic
points and chaotic magnetic regions. Nevertheless, we show how the partition reconnection rate is readily measured with the generalized flux function. We relate our partition reconnection rate to the
common definition of three-dimensional reconnection in terms of integrated parallel electric field. An analytical example demonstrates the theory and shows how the flux partition responds to an
isolated reconnection event. (C) 2011 American Institute of Physics. [doi:10.1063/1.3657424]
• chaos
• magnetic reconnection
• plasma magnetohydrodynamics
• PARALLEL ELECTRIC-FIELDS
• HAMILTONIAN-SYSTEMS
• CHAOTIC ADVECTION
• TRANSPORT
• MAPS
• DYNAMICS
• MANIFOLDS
• FLOWS
• SETS
Dive into the research topics of 'A generalized flux function for three-dimensional magnetic reconnection'. Together they form a unique fingerprint. | {"url":"https://discovery.dundee.ac.uk/en/publications/a-generalized-flux-function-for-three-dimensional-magnetic-reconn","timestamp":"2024-11-13T18:16:18Z","content_type":"text/html","content_length":"56759","record_id":"<urn:uuid:b5a592f6-8cc2-4c92-820b-55a1e50b9daa>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00719.warc.gz"} |
Jamboard Unit Review Template 5 - IntoMath
Jamboard Unit Review Template 5
This Jamboard Template will let you customize the problems you want students to work on. Pages contain boxes for problems and solutions.
There is a page for reflection where students are to share their main takeaways from the review.
Some pages contain animations, some are static.
Students can post sticky notes or use a pen to solve the problems.
Students can also upload their solutions as images.
Check out other Whiteboard Templates
Try our free Interactive Math Quizzes | {"url":"https://intomath.org/jamboard-unit-review-5/","timestamp":"2024-11-13T22:05:41Z","content_type":"text/html","content_length":"107753","record_id":"<urn:uuid:0dc7f502-0482-46c6-a11a-9804e1768a4b>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00366.warc.gz"} |
Dynamic Programming vs Memoization | The Hash Table
Dynamic Programming vs Memoization
Memoization means the optimization technique where you memorize previously computed results, which will be used whenever the same result will be needed.
Dynamic programming (DP) means solving problems recursively by combining the solutions to similar smaller overlapping subproblems, usually using some kind of recurrence relations.
Nowadays I would interpret "dynamic" as meaning "moving from smaller subproblems to bigger subproblems". (The word "programming" refers to the use of the method to find an optimal program, as in
"linear programming". People like me treat it as in software programming sometimes)
In summary, here are the difference between DP and memoization.
• DP is a solution strategy which asks you to find similar smaller subproblems so as to solve big subproblems. It usually includes recurrence relations and memoization.
• Memoization is a technique to avoid repeated computation on the same problems. It is special form of caching that caches the values of a function based on its parameters.
Memoization is the technique to "remember" the result of a computation, and reuse it the next time instead of recomputing it, to save time. It does not care about the properties of the computations.
Dynamic programming is the research of finding an optimized plan to a problem through finding the best substructure of the problem for reusing the computation results. In other words, it is the
research of how to use memoization to the greatest effect.
The name "dynamic programming" is an unfortunately misleading name necessitated by politics. The "programming" in "dynamic programming" is not the act of writing computer code, as many (including
myself) had misunderstood it, but the act of making an optimized plan or decision.
The earlier answers are wrong to state that dynamic programming usually uses memoization. Dynamic programming always uses memoization. They make this mistake because they understand memoization in
the narrow sense of "caching the results of function calls", not the broad sense of "caching the results of computations". | {"url":"https://thehashtable.com/notes/20220928-dynamic-programming-vs-memoization","timestamp":"2024-11-11T01:34:50Z","content_type":"text/html","content_length":"17766","record_id":"<urn:uuid:1840c048-0691-4da2-a51d-f8011c84f1c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00245.warc.gz"} |
An object with a mass of 4 kg is acted on by two forces. The first is F_1= < 8 N , -6 N> and the second is F_2 = < 2 N, 7 N>. What is the object's rate and direction of acceleration? | Socratic
An object with a mass of #4 kg# is acted on by two forces. The first is #F_1= < 8 N , -6 N># and the second is #F_2 = < 2 N, 7 N>#. What is the object's rate and direction of acceleration?
1 Answer
The question gives two forces in vector form.
The first step is to find the net force acting upon the object. This can be calculated by vector addition.
The sum of two vectors $< a , b >$ and $< c , d >$ is $< a + c , b + d >$.
Add the two force vectors $< 8 , - 6 >$ and $< 2 , 7 >$ to get $< 10 , 1 >$.
The next step is to find the magnitude of the vector, which is necessary to find the "size" of the force.
The magnitude of a vector $< a , b >$ is $\sqrt{{a}^{2} + {b}^{2}}$.
The "size" of the force is $\sqrt{{10}^{2} + {1}^{2}} = \sqrt{101} \setminus \text{N}$.
According to Newton's second law of motion, the net force acting upon an object is equal to the object's mass times its acceleration, or ${F}_{\text{net}} = m a$. The net force on the object is $\
sqrt{101} \setminus \text{N}$, and its mass is $4 \setminus \text{kg}$. The acceleration is #(sqrt(101)\ "N")/(4\ "kg")=sqrt(101)/4\ "m"/"s"^2~~2.5\ "m"/"s"^2#.
Newton's first law of motion also states that the direction of acceleration is equal to the direction of its net force. The vector of its net force is $< 10 , 1 >$.
The angle "theta" of a vector $< a , b >$ is $\tan \left(\theta\right) = \frac{b}{a}$.
The angle $\theta$ of the direction of this vector is $\tan \left(\theta\right) = \frac{1}{10}$. Since both components of the vector are positive, the angle of the vector is in the first quadrant, or
$0 < \theta < {90}^{\circ}$. Then, $\theta = \arctan \left(\frac{1}{10}\right) \approx {5.7}^{\circ}$ (the other possible value, ${185.7}^{\circ}$, is not correct since $0 < \theta < {90}^{\circ}$).
Impact of this question
4511 views around the world | {"url":"https://socratic.org/questions/an-object-with-a-mass-of-4-kg-is-acted-on-by-two-forces-the-first-is-f-1-8-n-6-n-1","timestamp":"2024-11-14T12:01:28Z","content_type":"text/html","content_length":"38255","record_id":"<urn:uuid:6dc705dc-096f-4c53-8da9-5bf1454ebfda>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00172.warc.gz"} |
Protractor Printable Free
Protractor Printable Free - It’s a standard 6″ protractor with angle measurements in both directions to find inside and outside. Just print, cut out and use as normal for an accurate free. Presenting
protractors in four different sizes, this set of printable. Here’s a pair of printable protractors available whenever you want one. Web use our free printable protractor to accurately measure an
angle. Web these free printable protractors are just the assistance you need to finish your project. Web math > geometry > angles > protractors > printable protractor. Web use this printable
protractor to measure angles when you don’t have a regular protractor available. Web have you ever needed a protractor but couldn’t find one?
Here’s a pair of printable protractors available whenever you want one. Web these free printable protractors are just the assistance you need to finish your project. It’s a standard 6″ protractor
with angle measurements in both directions to find inside and outside. Web use this printable protractor to measure angles when you don’t have a regular protractor available. Web use our free
printable protractor to accurately measure an angle. Just print, cut out and use as normal for an accurate free. Web have you ever needed a protractor but couldn’t find one? Presenting protractors in
four different sizes, this set of printable. Web math > geometry > angles > protractors > printable protractor.
Related Post: | {"url":"https://tracker.dhis2.org/printable/protractor-printable-free.html","timestamp":"2024-11-06T18:02:53Z","content_type":"text/html","content_length":"27555","record_id":"<urn:uuid:98288618-c333-49b7-967c-6667595e9592>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00287.warc.gz"} |
end behavior of a function calculator
SOLUTION The function has degree 4 and leading coeffi cient −0.5. In the odd-powered power functions, we see that odd functions of the form [latex]f\left(x\right)={x}^{n}\text{, }n\text{ odd,}[/
latex] are symmetric about the origin. We can use this model to estimate the maximum bird population and when it will occur. On the graph below there are three turning points labeled a, b and c: You
would typically look at local behavior when working with polynomial functions. “x”) goes to negative and positive infinity. End Behavior The behavior of a function as \(x→±∞\) is called the
function’s end behavior. •It is possible to determine these asymptotes without much work. End behavioris the behavior of a graph as xapproaches positive or negative infinity. find (a) a simple basic
function as a right end behavior model and (b) a simple basic function as a left end behavior model for the function. In general, you can skip the multiplication sign, so 5 x is equivalent to 5 ⋅ x.
With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Did you have an idea for improving this content? Describe in words and symbols the end behavior of
[latex]f\left(x\right)=-5{x}^{4}[/latex]. Preview this quiz on Quizizz. These turning points are places where the function values switch directions. Graphically, this means the function has a
horizontal asymptote. At the left end, the values of xare decreasing toward negative infinity, denoted as x →−∞. However, as the power increases, the graphs flatten somewhat near the origin and
become steeper away from the origin. Enter the polynomial function into a graphing calculator or online graphing tool to determine the end behavior. \(\displaystyle y=e^x- 2x\) and are two separate
problems. Retrieved from https://math.boisestate.edu/~jaimos/classes/m175-45-summer2014/notes/notes5-1a.pdf on October 15, 2018. We write as [latex]x\to \infty , f\left(x\right)\to \infty [/latex].
For these odd power functions, as x approaches negative infinity, [latex]f\left(x\right)[/latex] decreases without bound. Because the degree is even and the leading coeffi cient isf(xx f(xx Its
population over the last few years is shown below. Example—Finding the Number of Turning Points and Intercepts, https://www.calculushowto.com/end-behavior/, Discontinuous Function: Types of
Discontinuity, If the limit of the function goes to some finite number as x goes to infinity, the end behavior is, There are also cases where the limit of the function as x goes to infinity. An
example of this type of function would be f(x) = -x2; the graph of this function is a downward pointing parabola. #y=f(x)=1, . N – 1 = 3 – 1 = 2. All of the listed functions are power functions. We
can graphically represent the function. The square and cube root functions are power functions with fractional powers because they can be written as [latex]f\left(x\right)={x}^{1/2}[/latex] or
[latex]f\left(x\right)={x}^{1/3}[/latex]. As the power increases, the graphs flatten near the origin and become steeper away from the origin. Contents (Click to skip to that section): The end
behavior of a function tells us what happens at the tails; what happens as the independent variable (i.e. Describing End Behavior Describe the end behavior of the graph of f(x) = −0.5x4 + 2.5x2 + x −
1. Ex: End Behavior or Long Run Behavior of Functions. The exponent of the power function is 9 (an odd number). Describe the end behavior of the graph of [latex]f\left(x\right)=-{x}^{9}[/latex]. End
Behavior Calculator. No. With even-powered power functions, as the input increases or decreases without bound, the output values become very large, positive numbers. The table below shows the end
behavior of power functions of the form [latex]f\left(x\right)=a{x}^{n}[/latex] where [latex]n[/latex] is a non-negative integer depending on the power and the constant. To predict the end-behavior
of a polynomial function, first check whether the function is odd-degree or even-degree function and whether the leading coefficient is positive or negative. 2. I know how to find the vertical and
horizontal asypmtotes and everything, I just don't know how to find end behavior for a RATIONAL function without plugging in a bunch of numbers. There are three main types: If the limit of the
function goes to infinity (either positive or negative) as x goes to infinity, the end behavior is infinite. f(x) = x3 – 4x2 + x + 1. Use the degree of the function, as well as the sign of the
leading coefficient to determine the behavior. and the function for the volume of a sphere with radius r is: [latex]V\left(r\right)=\frac{4}{3}\pi {r}^{3}[/latex]. Step 1: Find the number of degrees
of the polynomial. At this point you can only Once you know the degree, you can find the number of turning points by subtracting 1. The End behaviour of multiple polynomial functions helps you to
find out how the graph of a polynomial function f(x) behaves. In symbolic form we write, [latex]\begin{array}{c}\text{as } x\to -\infty , f\left(x\right)\to -\infty \\ \text{as } x\to \infty , f\left
(x\right)\to \infty \end{array}[/latex]. Even and Positive: Rises to the left and rises to the right. Is [latex]f\left(x\right)={2}^{x}[/latex] a power function? 1. In symbolic form, we could write,
[latex]\text{as }x\to \pm \infty , f\left(x\right)\to \infty[/latex]. End Behavior of a Function The end behavior of a function tells us what happens at the tails; what happens as the independent
variable (i.e. On a small island coeffi cient −0.5 of a polynomial function f x. Be used to model populations of various animals, including birds the number of degrees of leading! Is [ latex ] x\to -
\infty, f\left ( x\right ) \to \infty [ /latex ] a function! Several rational functions, as the sign of the function… Preview this quiz on Quizizz graphs! A simple upward pointing parabola of a
polynomial function, not a power function is 8 ( an even ). Even-Powered power functions, as well as the power increases, the Practically Cheating Calculus Handbook, end behavior a... [ latex ] f\
left ( x\right ) [ /latex ] ( output increases. Function as \ ( \displaystyle y=e^x- 2x\ ) and are two important markers end! The zeros of the leading coefficient ( x\right ) [ /latex ] the! Function
below, a third degree polynomial, has infinite end behavior: degree and the exponent of function... Steps shown a calculator to help determine which values are the roots and perform division..., end
behavior of a function calculator even and negative: Falls to the left and Rises to the behavior numbers. Behavior, Local behavior & turning points are places where the degree of the of... Use a
calculator to help determine which values are the roots and perform synthetic division with those roots leading.. The background end-behavior asymptoteis an asymptote used to describe the end
behavior of a function that can be using. Use a calculator to help determine which values are the roots and perform synthetic division with those roots goes negative., including birds ex: end
behavior Log InorSign Up ax n 1 a =.... With even-powered power functions, as well as the coefficient is 1 ( positive ) and horizontal... Questions from an expert in the field points can be found
using N-1 as [ latex ] f\left ( )!: //math.boisestate.edu/~jaimos/classes/m175-45-summer2014/notes/notes5-1a.pdf on October 15, 2018 for each individual term % 20Behavior.htm on October 15 2018.
Calculus Handbook, the output increases without bound ( x→±∞\ ) is called an exponential function, as do polynomials. Animals, including birds somewhat near the origin and become steeper away from
the island infinity, the output without! Has infinite end behavior of the leading coefficient function contains a variable raised an... Toward negative infinity, the graphs flatten near the origin
and become steeper away the... Behavior or Long Run behavior of the leading co-efficient of the right ) \to [... Suppose a certain species of bird thrives on a small island suppose a certain species
of bird thrives a! Switch directions a power function is a simple upward pointing parabola 3, since it is by. ) = { 2 } ^ { x } [ /latex ] the additive value of leading! So 5 x is equivalent to 5 ⋅ x
we need to understand a specific type of function large positive! Left side of this is `` end behavior of a function as x approaches positive.... Help determine which values are the roots and perform
synthetic division with those roots determine., very much like that of the following polynomial function Falls to the left and to! Of f ( x ) behaves function does not match understand a specific
type of.. 2: Subtract one from the degree in the form coefficient is 1 ( positive ) and are important... Run behavior of the leading coefficient infinite end behavior of functions behavior the
behavior as numbers become and. A certain species of bird thrives on a cliff with the sun rising in the field: Falls the! From increasing to decreasing with a Chegg tutor is free three birds on
cliff..., and it 's pretty easy this module can be used to describe behavior... Three birds on a cliff with the sun rising in the field animals, including birds positive infinity in form. And left
side of this is determined by a polynomial function ) \to \infty [ /latex ] [ /latex (! Rational functions, that together cover all cases types of end behavior graph.: Falls to the left and Falls to
the left and Rises to the behavior of a polynomial function?... Is 1 ( positive ) and the leading coefficient to determine these asymptotes without much work calculator end behavior of a function
calculator. % 20Unit % 20Folder/Introduction % 20to % 20End % 20Behavior.htm on October 15, 2018 coefficient to determine end... Also use this model to estimate the maximum bird population will
disappear from the degree is highest. Steeper away from the island individual term understand the bird problem, need. An exponent is known as a coefficient as an example, a degree! Origin and become
steeper away from the origin coefficient is 1 ( positive ) and the coefficient. Is a simple upward pointing parabola = 3 – 1 = 3 – 1 = –. Years is shown below graphically, this means the function as
x approaches increases without bound division with roots... Population and when it will occur, f\left ( x\right ) [ /latex ] 30 with! Markers of end behavior polynomial end behavior describe the
behavior keep in mind a number that multiplies a base... Population will disappear from the origin and become steeper away from the.! 1: find the end behavior of the exponents for each individual
term graph is by... It 's pretty easy behavior & turning points can be found using N-1 Long... Can use words or symbols to describe how the graph of the graph of a graph as xapproaches or... We write
as [ latex ] x\to -\infty, f\left ( x\right ) [ /latex.. Has infinite end behavior of a function behave the island simple upward pointing parabola last years. ) goes end behavior of a function
calculator negative and positive infinity, the output decreases without bound = 2 it! Aspects of this is called the function has a constant base raised an! Of bird thrives on a cliff with the sun
rising in the field: n – 1 =.. We can also use this model to estimate the maximum bird population when! Functions helps you to find similarities and differences an exponent is known as the
coefficient is (... Asymptoteis an asymptote used to describe how the graph shows that as x approaches represented in the.... Behavior Loading... polynomial end behavior of a polynomial function 2:
Subtract one from the degree of function. Improving this content various animals, including birds left side of this function has a constant raised... It 's pretty easy } [ /latex ] a power function
given its equation or graph write the polynomial to! The cubic function help determine which values are the roots and perform synthetic division with those roots the of! In this module can be found
using N-1 rising in the above to! Populations of various animals, including birds ( x→±∞\ ) is called the function as! 20Unit % 20Folder/Introduction % 20to % 20End % 20Behavior.htm on October 15
2018... Pretty easy % 20Behavior.htm on October 15, 2018 and differences 5 x... As xapproaches positive or negative infinity, the output increases without bound horizontal.... Calculator will
determine the behavior of the right and left side of this function not! Degree is the additive value of the function… Preview this quiz on Quizizz infinite end of... Much work various animals,
including birds maximum bird population and when will. And become steeper away from the degree in the form this calculator will the... The field using N-1 asymptote as approaches negative infinity is
and the asymptote... } ^ { x } [ /latex ] ( output ) increases without bound or negative,. From an expert in the form for example, a third degree polynomial, has infinite end behavior, the. This
module can be represented in the form over the last few is... 4X2 + x − 1, including birds once you know the degree and the horizontal asymptote as negative! = −0.5x4 + 2.5x2 + x − 1 describing end
behavior the cubic.... A polynomial function f ( x ) behaves, since it is the highest.... Chegg Study, you can skip the multiplication sign, so 5 x is to... For each individual term the polynomial in
factored form and determine the behavior number Learn how to the! Have similar shapes, very much like that of the following polynomial function, as the power,... Horizontal asymptote as approaches
negative infinity is •An end-behavior asymptoteis an asymptote used to model populations various. Need to understand a specific type of function pointing parabola, a function as \ x→±∞\. From https:
//math.boisestate.edu/~jaimos/classes/m175-45-summer2014/notes/notes5-1a.pdf on October 15, 2018 power function latex f\left... Without much work or Long Run behavior of the given polynomial function
and n are real numbers and a known! Form and determine the end behaviour of end behavior of a function calculator polynomial functions helps you find... Graph is determined by the degree you found in
step 1: n – 1 = 3 – =! Math 175 5-1a Notes and Learning end behavior of a function calculator Retrieved from http: //jwilson.coe.uga.edu/EMAT6680Fa06/Fox/Instructional % 20Unit % %. Function behave I
describe the end behavior polynomial end behavior of a graph as xapproaches positive or negative infinity.., consider functions for area or volume Loading... polynomial end behavior of a polynomial
function 1 ( positive and. Understand the bird problem, we need to understand a specific type of function form, well. Of functions ) approaches infinity, denoted as x →−∞ a polynomial function f ( x
) = +... Two separate problems perform synthetic division with those roots horizontal asymptote as negative. Solution the function x 4 − 4 x 3 + 3 x + 25 in... | {"url":"http://boshuisnijhildenberg.nl/hybrid-flowers-xgcbcag/end-behavior-of-a-function-calculator","timestamp":"2024-11-04T21:05:06Z","content_type":"text/html","content_length":"29360","record_id":"<urn:uuid:b4e8223f-50ae-45bf-898f-9cefbdc2edf3>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00596.warc.gz"} |
Converting Fractions to Decimals
Converting Fractions to Decimals
There are some fractions whose denominator can be written as a power of 10. To convert such fractions to decimals:
1) Find an equivalent fraction with denominator 10, 100, or 1000 etc.
2) Then, counting from extreme right to left, place the decimal point in the numerator after as many digits as there are zeros in the denominator.
If the numerator has fewer digits, then add the desired number of zeros to the left of the numerator, and then place the decimal point.
For example: Let's write the fraction
as a decimal.
First, find the equivalent fraction of
having denominator as a power of 10.
We can multiply the denominator, 5, by 2 to get 10.
Now, let's write
as a decimal.
Observe that there is one zero in the denominator.
So, counting from right to left, place the decimal point in the numerator after one digit.
The fraction
can be written as the decimal 0.8.
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
The correct answer is
Remember :
The smallest number is the one that comes first while counting.
Solution :
To arrange the given numbers in order from smallest to greatest, find the smallest number among all the given numbers.
21 is the smallest number. | {"url":"https://members.turtlediary.com/quiz/converting-fractions-to-decimals.html?app=1?...","timestamp":"2024-11-12T18:28:04Z","content_type":"text/html","content_length":"167379","record_id":"<urn:uuid:003f09f9-2dc5-4381-b28e-7534d7a01266>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00496.warc.gz"} |
Calculating the length of a coastline
You must be registered to use this feature. Sign in or register.
The River Nile and Infinite Systems
Calculating the length of a coastline
Benoît Mandelbrot Mathematician
So, Richardson was for me a very important point, because first of all it came back to simple shapes: coastlines. Everybody knows coastlines. There was not one geographer of my acquaintance;
therefore I had no vested interest in geography. Besides Richardson had provided all his examples, to which I added, of coastlines or geographical lines measured by different people giving entirely
different results, like the border between Spain and Portugal which Portugal has claimed is thirty percent longer than what Spain claims. So, the indeterminacy of these quantities, the fact I'd been
drumming in the context of economics that the quantities underlying their most important theories were in fact ill-defined, that indeterminancy had moved all over. It had not been eliminated by
physical cleanliness, but it had moved into the domain of physics. In the domain of geophysics or geology, which is physical sciences if not physics proper, one had quantities which were totally
ill-defined and for which one could not give a number like length; one had to deal with them differently. And so I published this paper first in Science, called How long is the coast of Britain?
which became well known. It was published in 1967. It took very long time publishing, and so could I say that there was much actual graphics in it? No. The graphics were very primitive, I just drew
by the hand some classical constructions by Koch and others, and variants, and argued that any sensible person would consider a coastline as being made of capes and bays and then smaller capes and
bays and so on, and unless something drastic is true, for which there's no evidence, the length is going to increase more and more as you go into details. I thought that this was a great insight; of
course I was totally wrong; it was absolutely well-known to everybody, but it was, how to say, suppressed. I'm told by scholars of Greek history that Greek sailors sailing from Athens towards the
western Mediterranean were reporting a very different length for Sardinia's coast, saying if you go on our ship it's so much, but if you go on, say, the little lifeboat, then it's much longer, and if
you walk along it, it's much longer. They knew it. "So which length do you want?" they asked the Admiralty. The Admiralty did not know. The idea of area, of length and so on, became separate at that
time, but certainly by the '60s, the power of school books, the power of school teachers, the power of people who barely understood the bare rudiments of mathematics was such that length was
something which was very entrenched, and also very fragile. Infinite variance was not as real as infinite length. Infinite dependence, as I was supposing for these errors, was absolutely not real.
Nobody can think simultaneously of these things. One can see the coastline. And so this paper and several papers that followed began fractals, by introducing specifically the eye into scientific
research. I would say that '67, '68 were the critical years for that.
Benoît Mandelbrot (1924-2010) discovered his ability to think about mathematics in images while working with the French Resistance during the Second World War, and is famous for his work on fractal
geometry - the maths of the shapes found in nature.
Title: Calculating the length of a coastline
Listeners: Daniel Zajdenweber Bernard Sapoval
Daniel Zajdenweber is a Professor at the College of Economics, University of Paris.
Bernard Sapoval is Research Director at C.N.R.S. Since 1983 his work has focused on the physics of fractals and irregular systems and structures and properties in general. The main themes are the
fractal structure of diffusion fronts, the concept of percolation in a gradient, random walks in a probability gradient as a method to calculate the threshold of percolation in two dimensions, the
concept of intercalation and invasion noise, observed, for example, in the absorbance of a liquid in a porous substance, prediction of the fractal dimension of certain corrosion figures, the
possibility of increasing sharpness in fuzzy images by a numerical analysis using the concept of percolation in a gradient, calculation of the way a fractal model will respond to external stimulus
and the correspondence between the electrochemical response of an irregular electrode and the absorbance of a membrane of the same geometry.
Duration: 3 minutes, 58 seconds
Date story recorded: May 1998
Date story went live: 24 January 2008 | {"url":"https://www.webofstories.com/play/benoit.mandelbrot/55;jsessionid=2C74EEAA911522A9BF847AE3ED4997E9","timestamp":"2024-11-09T18:53:09Z","content_type":"application/xhtml+xml","content_length":"56732","record_id":"<urn:uuid:f333a1b9-a337-42f5-b79a-89c6dcf78be9>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00251.warc.gz"} |
Speed Distance Time Calculator
Speed distance time calculator helps to calculate the speed at which you're traveling over a certain distance in a specific amount of time.
“The ratio of the distance traveled by an object to the time is known as its speed”
It is the scaler quantity and according to system international, the exact unit for speed are termed meters per second \(ms^{-1}\) and the dimension of speed is \( M^{0}L^{1}T^{-1}\) The unit of
speed depends on the units that you select but this speed distance time calculator display additional units. Along with the SI unit, this tool calculates the speed in inch per second, mile per
second, foot per second, kilometer per second, centimeter per second, and yard per second.
Speed Formula:
Speed= Distance / Time or S = d / t
Also, you can use the velocity calculator to determine the speed precisely.
“The total measurement of the length between two points is known as distance”
It is also a scalar quantity, the exact units used to define distance are termed meters and the dimension is L. But the resulting unit will depend on the unit of speed value. Along with the SI units,
the calculator will finds the distance covered at a particular speed over a certain period of time in the form of Inches, Mile, Foot, Kilometer, Centimeter, and Yard.
Distance Formula:
Distance = speed × time Or d = s × t
For convenience, you can also get the assistance of a distance calculator. Simply, input the required values and get the distance calculated in seconds.
“The measurement of period during which an action, process, or condition takes place”
It is also a scalar quantity and the actual units used for time are expressed as seconds and the dimension of time is T. Along with the SI unit, the calculator finds the time in Years, Months, Days,
Hours, Minutes, and hh:mm:ss
Time Formula:
The time formula of an object is expressed as:
Time= Distance/speed Or T = D / S
For more information, refer to the source SplashLearn.
How To Calculate Speed, Distance, and Time?
Let’s move on to resolving a couple of examples with the help of time and distance calculator to clarify your concept in more depth!
Speed Example:
A car covered a distance of 150 km per hour. Determine the speed of the car in m/s.
The distance covered by a car in meters = 150×1000m = 150000m Time taken by car in seconds = 60×60 = 3600 seconds
To find:
The speed of the car =?
Using the speed formula, Speed = Distance/Time = 150000/3600 = 41.66 m/sec
Distance Example:
Calculate the distance of a truck traveling at a constant speed of 60 m/s in 90 seconds.
Speed of the truck = 60 m/s Time is taken to cover by the truck = 80 sec
To find:
Distance covered by the truck=?
Using the formula Distance= speed * time = 60m/s * 80 s = 4800
Time Example:
A train covered a distance of 130 km at a speed of 50km/hr. Use the free time formula to calculate time with speed and distance taken by the train.
Distance covered by a train = 130 km Speed of a train = 50 km/hr
To find:
The time is taken by the train=?
Using Formula of Time, Time = Distance/Speed = 120/60 = 2 hr | {"url":"https://calculator-online.net/speed-distance-time-calculator/","timestamp":"2024-11-13T14:57:36Z","content_type":"text/html","content_length":"67369","record_id":"<urn:uuid:4db736e7-217a-443f-a07d-4e65966d14d7>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00033.warc.gz"} |
1. Introduction
Evaporation involves the conversion of a solvent into vapor, which is then removed from a solution or slurry. In most cases, water is the solvent used in evaporation systems. This process entails
vaporizing a portion of the solvent to create a concentrated solution, thick liquor, or slurry [
Dairy manufacturers commonly employ concentration techniques to produce dairy products with higher levels of dry matter, increased value, reduced volume, and extended shelf-life [
]. Lowering the water activity and reducing transportation and storage costs are key benefits of dehydrating dairy products. This process involves converting a liquid product into a dry powder by
removing nearly all the available water. However, dairy products are sensitive to heat, and their functional properties and digestibility can be negatively affected by excessive heat during the
dehydration process. Therefore, a single water removal method cannot consistently achieve optimal performance. It thus, becomes necessary to employ multiple processing steps tailored to the specific
properties of the material being processed, while considering both product quality considerations and processing costs [
Typically, the main steps involved in the production of milk powder include standardisation, homogenisation, pasteurisation, evaporation and drying [
]. Evaporation is a significant step in milk-powder production plants, serving not only to concentrate milk to the desired viscosity for subsequent spray drying but also to reduce the energy
requirements during the spray drying process. In the evaporation stage, sterilized milk is concentrated under vacuum conditions at temperatures ranging from 40 to 70°C. This process leads to a
significant increase in the total solids content, which typically rises from around 13% to 50% [
]. Vacuum conditions are used to mitigate the negative effects of heat on heat-sensitive milk components like fats and to prevent the degradation of essential nutrients, such as vitamins.
Milk powder production consists of many thermal processes, including evaporation and drying, and is responsible for 15% of the total energy use in the dairy industry [
]. In France, approximately 25% of the total energy consumption in food industry is attributed to the dairy sector [
]. Therefore, energy consumption attributed to the process of evaporation has a substantial impact on the cost of milk powder production.
To this end, there are several energy consumption optimization strategies in the evaporation of milk that have been discussed in the literature. The ones that are examined in this work involve the
most commonly used in the industrial practice, such as the Mechanical Vapor Recompression (MVR) and the Thermal Vapor Recompression (TVR) technologies.
• In the evaporator unit, the excess heat generated by the secondary steam from the evaporator is typically released as waste heat. However, this waste heat can be effectively utilized to preheat
the feed. An important feature of MVR technology is the utilization of the secondary steam cycle. MVR employs a mechanical fan, typically powered by electricity, to recompress low-pressure vapor
to a slightly higher pressure and temperature.
• On the other hand, TVR utilizes a thermos-compressor, which employs high-pressure vapor to recompress low-pressure vapor to a slightly higher pressure and temperature. Numerous studies have
demonstrated that multi-effect evaporation reduces energy consumption costs by enhancing steam economy. This is achieved by utilizing the secondary steam generated by the preceding effect as the
heat source for the subsequent effect [
Jebson and Chen [
] assessed the effectiveness of falling film evaporators used in the New Zealand dairy sector for concentrating whole milk by calculating the kg steam utilized to kg water evaporated ratio and the
heat transfer coefficient of each evaporator pass. The steam consumption of full and skim milk was comparable. Schuck et al. [
] introduced a methodology for assessing and comparing the energy consumption involved in the production of dairy and feed powders at various stages of the dehydration process. The findings of the
study revealed that the energy consumption for fat-filled and demineralized whey powders were 9.072 kJ/kg and 15.120 kJ/kg, respectively.
Walmsley et al. [
] conducted a study on applying Pinch Analysis to an industrial milk evaporator case study to quantify the potential energy savings. By the appropriate placement of mechanical vapor recompression in
a new improved two-effect milk evaporation system design, a 78% reduction to steam (6397 kW) at the expense of 16% (364 kW) more electricity use was achieved and the emissions reduction was 3416 t CO
-e/y. Srinivasan et al. [
] studied the energy efficiency at India’s largest milk processing plant and proposed retrofits for improving the plant’s sustainability. The results reveal that exergy efficiency of certain units is
very low (<20%) while significant improvements in energy efficiencies can be achieved through simple, low-cost retrofits to these units. Moejes [
] studied the possibilities of upcoming milk processing technologies such as membrane distillation, monodisperse-droplet drying, air dehumidification, radio frequency heating, and radio frequency
heating paired with renewable energy sources such as solar thermal systems. It was illustrated that the combination of developing technologies has the potential to cut operational energy consumption
for milk powder manufacturing by up to 60%.
Zhang et al. [
] conducted a study by simulating a "pseudo" milk composition using hypothetical components in a commercial process simulator. The purpose of their work was to model a falling film evaporator
commonly (FFE) employed in milk powder production plants. The study demonstrated that commercial process simulators have the ability to accurately simulate dairy processes. Building upon this
research, Munir et al. [
] further enhanced the capabilities of commercial process simulators, providing valuable insights for practicing engineers to identify potential process improvements in the dairy industry. Bojnourd
et al. [
] developed two types of dynamic models for an industrial four-effect falling-film evaporator used to condense whole milk: lumped and distributed. The findings indicate that while the distributed
model demonstrates slightly better predictive capabilities compared to the lumped model, the latter outshines in terms of performance due to its simpler structure and significantly reduced simulation
time. Zhang et al. [
] developed models for two commonly used types of milk powder evaporators: a conventional five-effect FFE without MVR and a three-effect evaporator with MVR. Heat-recovery processes were incorporated
into the models to enable a comparison of energy consumption between the two processes. The results revealed that a three-effect FFE with MVR could achieve a 60% reduction in energy consumption
compared to a conventional five-effect evaporator.
Gourdon and Mura [
] created a modeling tool based on experimental correlations built at industrial scale circumstances. The complicated interaction between the generated vapor and the liquid flow is included in their
model. The results show that pressure drop is important in evaporator performance because of its influence on saturation temperature. Daz-Ovalle et al. [
] provided a set of dynamic models to study fouling of falling-film evaporators by considering fouling thickness, film thickness, temperature, and solids mass percentage. Hu et al. [
] developed a model for a water-to-water FFE simulation, which was employed in water vapor heat pump systems. That study focused on an existing FFE with four working tubes.
Bouman et al. [
] conducted experiments with a one-tube evaporator using whole and skim milk to ascertain the heat transfer and pressure decrease in evaporator tubes. Based on the findings, a computer software was
created to optimize the design of multistage falling-film evaporators for dairy products. Silveira et al. [
] investigated the evaporation of water and skim milk using a pilot-scale, single-stage falling-film evaporator. In comparison to water, a thicker and slower film was formed at the end of the skim
milk concentration procedure. They concluded that the behavior of a product during the evaporation process cannot be predicted solely by the overall heat transfer coefficient, and that a wide range
of information, such as residence time distribution, product viscosity, and surface tension, is required to understand the evaporation process. Gourdon et al. [
] examined the flow behavior of a dairy product under falling film evaporation. The effects of varying dry solids concentrations, flow rates, and driving temperature changes were investigated. All
three factors were shown to have a significant impact on the flow characteristics. Mura et al. [
] investigated the absolute vapor flow pressure losses during dairy product falling film evaporation using an experimental internal tube evaporator setup, adjusting the co-flow input velocity and
product dry solid content. They concluded that pressure losses are strongly dependent on the co-flowing vapor rate for dry solid content between 13 and 40%.
Wijck et al. [
] conducted an evaluation of tools used for dynamic modeling and supervisory multivariable control design of multiple-effect falling-film evaporators. They specifically focused on the NIZO
four-effect evaporator as a case study. The research aimed to achieve improved process operation for industries, leading to economic benefits such as increased yield, enhanced product quality,
reduced energy consumption, and minimized material waste. Sharma et al. [
] created an Excel-based multi-objective optimization tool based on the elitist non-dominated sorting genetic algorithm (NSGA-II) and tested it on benchmark tasks. It is then used for multi-objective
optimization (MOO) of the design of a falling-film evaporator system for milk concentration, which includes a pre-heater, evaporator, vapor condenser, and steam jet ejector. Haasbroek et al. [
], conducted a study utilizing historical data from a FFE to develop models for control purposes, without requiring knowledge of the plant's physical dimensions. The results indicated that fuzzy
predictive controllers and LQR control exhibited the highest performance, followed by cascade control, and finally, PI control. Galvản-Ángeles et al. [
] analyzed a thermo-compression evaporation method for milk. The suggested tool considers the cost optimization of the evaporation system while incorporating thermo-physical parameters of the
foodstuff as a function of composition and temperature. The results revealed that the evaporation economy is proportional to the percentage of recycled steam and the location of the effect that
recycles the steam, and inverse to the thermodynamic efficiency of the thermo-compressor.
This work aims at optimizing the milk evaporation process by minimizing steam consumption under the operational and product quality constraints. To this end, 5 different Cases of evaporator layouts
are examined using a global system analysis (GSA) and an advanced optimization approach. GSA is employed to revise decisions to improve system robustness, validate the process and reduce parameter
uncertainty. An uncertainty analysis allowed the investigation of the impact of design and operational decisions and environmental inputs on Key Performance Indicators (KPIs). Afterwards, Cases 1-4
are optimized under 5 different scenarios, either minimizing the cost of steam while maximizing product yield or minimizing the total annualized cost which is applicable to a new plant design or a
capacity expansion option. Steam economy, energy consumption profile and heat transfer areas are assessed and compared. Moreover, this study investigates to what extend switching from milk powders to
new products known as milk concentrates effects the energy consumption in the evaporation process. Thus, Cases 1-4 are optimized under different end-product specifications (30, 35 or 50% solid
content). Finally, it is evaluated whether the use of MVR or TVR is more economical for the evaporation process, based on current steam and electricity prices, economic trends, and costs of steam
generated from renewable energy sources.
The remaining of the article is structured as follows:
Section 2
describes the material composition along with some of its properties.
Section 3
provides a detailed presentation of the mathematical model that describes the operation of a falling film evaporator. Sectio 4 presents the studied evaporator layouts. The attained global system
analysis and optimization results of each studied Case/evaporator layout are presented and thoroughly discussed in
Section 5
. Finally,
Section 6
summarizes the concluded remarks emerging from the study.
3. Process model
The concentration of milk in this study is focused on falling film evaporators. In this process, the milk, which is close to its boiling point, is introduced in a uniform manner at the upper section
of the inner surface of a tube. These tubes are arranged side by side, fixed in place, and surrounded by a jacket. As the milk descends within each tube, it forms a thin film and undergoes boiling as
a result of the heat exchanged with the steam. The concentrated liquid is collected at the lower part of the equipment, while the remaining portion is separated from the steam in a subsequent
separator. In evaporators equipped with multiple effects, the concentrated liquid is pumped to the next stage, while the steam serves the purpose of heating the subsequent [
A dynamic model of a simple falling-film evaporator is developed in this Section. The model examines the flash calculations of liquid entering the evaporator, its distribution through distributor
plates, and its evaporation as it flows downwards through pipes. The model has been developed based on the following assumptions:
• It is assumed that the product immediately reaches its boiling temperature once it passes above the distributor plate, with either steam flashing or condensing.
• All the heating steam condenses at its saturation temperature. The heat released during condensation is utilized to preheat and evaporate the feed stream.
• The pressure difference on the steam side is insignificant, thus resulting in a constant temperature on the steam side.
• The unit is assumed to be perfectly mixed at all times, implying there are no spatial variations in intensive properties within it.
• A steady state balance is applied to the tube flow, but the mass hold-up of liquid on the distributed plate is calculated.
3.1. Mass balance
The rate of change of the mass hold-up,
, of any given species in phase
in the unit is given by Equation (4):
$d M i , p d t = ∑ j = 1 N i n l e t F j , p i n · w i , j , p i n − F p o u t · w i , p o u t + ∑ p n ≠ p ∈ P R p / p n , i , ∀ i ∈ I , ∀ p ∈ P$
$F j , p i n$
is the mass flowrate of the
feed stream in phase
$w i , j , p i n$
is the mass fraction of species
in the
feed stream in phase
$F p o u t$
is the mass flowrate of material phase
in the outlet stream,
$w i , p o u t$
is the mass fraction of species
in the outlet stream of phase
$R p / p n , i$
is the rate of mass transfer between the phase
and the phase
Assuming that the unit is perfectly mixed, the composition of any outlet stream is the same as the composition within the unit:
$w i , p o u t = w i , p , ∀ i ∈ I , ∀ p ∈ P$
$w i , p$
is the mass fraction of species
in phase
in the unit.
The composition of material within the unit is given by Equations (6),(7):
$w i , p = M i , p M t o t a l , p , ∀ i ∈ I , ∀ p ∈ P$
$M t o t a l , p = ∑ i ∈ I M i , p , ∀ i ∈ I , ∀ p ∈ P$
$M p , t o t a l$
is the total holdup of material in phase
within the unit.
3.2. Energy balance
The rate of accumulation of enthalpy,
, within the unit is given by Equations (8),(9):
$d H d t = ∑ j = 1 N i n l e t ∑ p ∈ P ( F j , p i n · h j , p i n ) − ∑ p ∈ P ( F p o u t · h p o u t ) − R p / p n , i h p / p n , i + Q t r a n s$
$H = ∑ p ∈ P ( M t o t a l , p · h p )$
$h p$
is the specific enthalpy of the material in the unit in phase
$h j , p i n$
is the specific enthalpy of the
feed stream in phase
$h p o u t$
is the specific enthalpy of the outlet stream in phase
$h p / p n , i$
is the enthalpy of phase change between phase
and phase
$Q t r a n s$
is the enthalpy transferred into the unit through boundary due to heat loss, heating and so forth.
Due to the assumption that the unit is perfectly mixed, the specific enthalpy of each of the outlet streams is equal to the specific enthalpy of the material within the unit:
The specific enthalpy is assumed to be a function of the composition and temperature,
, of the material within the unit:
$h p = h p ( T , T r e f , w i , p ) , ∀ p ∈ P$
3.3. Heat transfer rates
The total heat transfer rate,
$Q t r a n s$
, is the sum of energy transfer by steam,
$Q h e a t i n g$
and energy loss,
$Q l o s s$
, to the environment:
$Q t r a n s = Q h e a t i n g + Q l o s s$
$Q h e a t i n g = U A Δ Τ l m$
$Δ T l M = Δ Τ 1 − Δ Τ 2 l n ( Δ Τ 1 Δ Τ 2 )$
Where $U$ is the overall heat transfer coefficient, $A$ is the heat transfer area, $N$ is the number of tubes, $L$ is the length of the tubes and finally $Δ Τ$ and $Δ Τ l m$ are the linear and
logarithmic temperature differences, respectively.
3.4. Flow through distribute plate
The distribute plate is an inserted plate that distributes product stream evenly around the inner periphery of the evaporator tubes. Aside from the liquid holes that allow liquid to pass through,
there are also openings to let steam flow through the plate. Two designs for vapor transport through the distribute plate are considered:
• Plate with upstanding vapor tubes.
• Plate with an upstanding rim at the edges of the circular plate and a ring-shaped gap between plate and evaporator body.
The vapor flowing downwards or upwards through the distribute plate causes a vapor pressure drop, while the liquid flowing downwards through the distribute plate and liquid holdup above the plate
cause a liquid pressure drop.
The vapor flow through plate,
$F ˙ v a p$
, for plate design with upstanding vapor tubes is expressed by the following relation.
$F ˙ v a p = N v a p t u b e · π 4 d v a p t u b e 2 ρ v a p u v a p$
$Δ P v a p = 1 + f f , v a p L v a p t u b e d v a p t u b e · 1 2 ρ v a p u v a p 2$
$N v a p t u b e$
is the number of vapour uprising tubes,
$d v a p t u b e$
is the diameter of each uprising tube,
$L v a p t u b e$
is the length of each tube,
$ρ v a p$
is the vapor density above plate,
$f f , v a p$
is the vapor friction factor and
$Δ P v a p$
is the vapor pressure drop.
The vapor flow through plate for plate design with upstanding rim at the edges is expressed by Equations (18)-(19):
$F ˙ v a p = π 4 ( d o u t 2 − d i n 2 ) ρ v a p u v a p$
$Δ P v a p = 1 + f f , v a p h r i m d o u t − d i n · 1 2 ρ v a p u v a p 2$
$d o u t , d i n$
are the diameter of the evaporator body and the plate respectively and
$h r i m$
is the height of the outer rim.
The liquid flow through plate,
$F ˙ l i q$
, is expressed by Equations (20)-(21):
$F ˙ l i q = N l i q h o l e · π 4 d l i q h o l e 2 ρ l i q u l i q$
$Δ P l i q = 1 + f f , l i q h p l a t e d l i q h o l e · 1 2 ρ l i q u l i q 2$
$N l i q h o l e$
is the number of liquid holes,
$d l i q h o l e$
is the diameter of liquid holes,
$h p l a t e$
is the thickness of the distribute plate,
$ρ l i q$
is the liquid density above plate,
$v l i q$
is the liquid flow velocity through plate,
$f f , i l q$
is the liquid friction factor and
$Δ P l i q$
is the liquid pressure drop.
3.5. Liquid flow through pipes
The wetting rate,
, of a falling film evaporator tube is correlated to the liquid mass flow of the tube,
$Φ m$
and the tube diameter
$D t u b e$
The Reynolds number,
$R e$
, in the falling film evaporation used in empirical relations is defined by Equation (23):
Where $μ$is the liquid dynamic viscosity.
The liquid characteristic length,
$l c$
, is given as:
Where $ρ$is the liquid density and $g$ is the gravity acceleration.
The film thickness,
, is calculated by Equation (25):
$δ = 2.4 μ 2 ρ 2 g 1 3 R e 1 3 = 1.34 l c R e 1 3 R e < 400 0.302 3 μ 2 ρ 2 g 1 3 R e 8 15 = 0.436 l c R e 8 15 R e ≥ 400$
Assuming that the film thickness is small compared to the tube diameter, the liquid velocity downwards on the tube,
, and the residence time,
, for liquid flow to the bottom are calculated based on Equations (26)-(27):
3.6. Boiling point elevation
The boiling point elevation of a solution can be calculated by combining the laws of Raoult and Clausius Clapeyron:
$Δ Τ = − R T w a t e r 2 H e v a p M w , w a t e r l n ( X w a t e r )$
Where $R$ is the gas constant, $T w a t e r$ is the boiling temperature of water, $H e v a p$ is heat of evaporation, $M w$ is molar weight and $X w a t e r$ is the molar fraction of water in the
It is assumed that fat does not dissolve in the solvent and therefore does not affect the boiling point. The molar fraction of water in the fat-free product can therefore be calculated:
$X w a t e r = f m , w a t e r M w , w a t e r ∑ i ∈ c o m p f a t f r e e f m , i M w , i$
Where $f m$ denotes for the mass fraction and $c o m p f a t f r e e$ refers to all but fat components.
3.7. Liquid friction factor
The most widely used correlations for liquid friction factor is the Wallis correlation for annular falling film and it is incorporated to the model for better prediction of the liquid friction factor
$f f l i q = 0.05 · 1 + 300 · δ D i$
3.8. Energy cost
In order to compare the energy consumption of TVR and MVR, the variable
Energy cost
is used, defined by Equation (31). When employing TVR technology, the annual cost is given as:
$E n e r g y c o s t = S t e a m u n i t c o s t · S t e a m f l o w r a t e · h o u r s p e r y e a r$
For MVR technology, the annual energy cost is expressed as:
$E n e r g y c o s t = E l e c t r i c i t y u n i t c o s t · P o w e r c o n s u m p t i o n · h o u r s p e r y e a r$
The current price of steam based on natural gas is 20.01
/t [
]. According to Eurostat [
], the electricity unit cost for non-household users in 2023 is 0.21
/kWh. The operating hours per year are considered as 7,920h/year, thus variable
Energy cost
refers to annual energy cost.
6. Conclusion
In this work, five different and industrially relevant milk evaporator Cases have been studied using a model-based approach. For each Case study various conclusions were drawn regarding the
simulation, model validation, global system analysis and process optimization. In this section, the results are discussed in terms of comparing TVR and MVR.
Comparing Cases 2, 3 and 4 for current steam and electricity prices, when processing 1,000 kg/h raw milk the most economical option includes 3 evaporator effects with TVR (Case 3) to meet the desired
50% product dry mass content. The same figure is reported for optimization Scenarios 2 and 3. However, Case 4 indicates the most significant reduction in the annual cost when reducing the product
specification to 30 or 35% dry mass content.
It is worth mentioning that current high electricity prices (0.21$/kWh) lead to Case 4 being the most unprofitable choice. When producing product with 35% dry mass content, only a 11% reduction in
the unit electricity price leads to Case 4 being more cost effective than Case 2 with only a single evaporator effect. Simultaneous reduction by 7% of electricity price along with a 5% increase of
gas-based steam price would also lead to Case 4 being the most profitable option among these Cases.
Regarding the maximum values of product yield in Cases 2-4 (Scenario 4), Case 2 can achieve slightly higher values than Cases 3 and 4 (0.28 as opposed to 0.25 respectively). Moreover, for a new plant
design, the minimum total annualized cost is achieved in Case 4 which includes a single evaporator effect with MVR, thus indicating that the capital fixed cost in such processes has the dominant
contribution on the total annualized cost compared to the operating costs.
Cases 1 and 2 can be compared as they both include 2 evaporator effects with TVR, the former relevant to a plant scale and the latter to a pilot scale. The annual steam cost seems to have a
relatively linear relationship with capacity while lower product yield values can be achieved in Case 1 when producing products with 50% dry mass content.
Overall, switching from milk powder production to milk concentrates results to a reduction in the annual cost from 10.75 to 44% depending on the Case under consideration. Furthermore, a forecasted
reduction of biomass-based steam cost by only 20% (or more) leads to lower values of the annual expenditure in all Cases, than that of the currently used NG-based steam. As predictions indicate a
rise in natural gas’ price, renewable-based steam would potentially become more and more competitive. Finally, assuming a simultaneous increase in the price of NG-based steam by 10% and a reduction
of biomass-based steam by 10%, the former one is no longer the most economically attractive solution.
Figure 1. (a) Case 1: 2 stage FFE layout with TVR technology with vapor recycling from 1^st to 1^st stage, (b) Case 2: 2 stage FFE layout with TVR technology with vapor recycling from 2^nd to 1^st
stage, (c) Case 3: 3 stage FFE layout with TVR technology with vapor recycling from 3rd to 1st stage, (d) Case 4: 1 stage FFE layout with MVR technology, (e) Case 5: 4 stage FFE with TVR technology
with vapor recycling from 2nd to 1st stage.
Figure 2. Case 1 uncertainty analysis results: (a) Concentrate water mass fraction versus feed temperature and TVR discharge pressure, (b) Steam economy versus TVR suction ratio and feed temperature,
(c) Steam mass flowrate versus suction ratio and TVR discharge pressure, (d) Concentrate pressure versus TVR discharge pressure and feed temperature, (e) FFE1 product dry mass fraction versus TVR
suction ratio and discharge pressure.
Figure 3. Case 2 uncertainty analysis results: (a) Steam economy versus TVR suction ratio and feed temperature, (b) FFE1 product dry mass fraction versus feed temperature and discharge pressure, (c)
Steam annual cost versus feed temperature and TVR discharge pressure.
Figure 4. Case 3 uncertainty analysis results: (a) Steam economy versus feed temperature and suction ratio, (b) Dry mass fraction of product after FFE1 versus suction ratio and TVR discharge
pressure, (c) Dry mass fraction of product after FFE2 versus suction ratio and TVR discharge pressure, (d) Water mass fraction of concentrate versus TVR suction ratio and TVR discharge pressure, (e)
Steam annual cost versus TVR suction ratio and TVR discharge pressure.
Figure 5. Case 4 uncertainty analysis results: (a) Water mass fraction of concentrate versus feed temperature and MVR compression ratio, (b) Steam annual cost versus feed temperature and compression
ratio, (c) Concentrate pressure versus MVR compression ratio and feed temperature.
Figure 6. Case 5 uncertainty analysis results: (a) Water mass fraction of concentrate versus feed temperature and split fraction, (b) Yield versus feed temperature and split fraction, (c) Steam
annual cost versus TVR suction ratio and split fraction.
Whole milk components [24] Weight (%) Whole milk components [5] Weight (%)
[24] [5]
Water $87.4$ Water $87$
Carbohydrates $4.9$ Fat $4$
Proteins $3.5$ Protein $3.4$
Fat $3.5$ Lactose $4.8$
Ash $0.7$ NaCl $0.4$
KCl $0.4$
Property Equation [23]
Specific heat capacity $c p = ∑ j c p j x j$ $( 1 )$
Thermal conductivity $k = ρ ∑ j κ j x j ρ j$ $( 2 )$
Density $1 ρ = ∑ j x j ρ j$ $( 3 )$
Property Value [23] Value [5] Simulated data
Density $1020 k g / m 3$ $1021 k g / m 3$ $1017.9 k g / m 3$
$Specific heat capacity c p$ $3849 J / k g K$ $3830 J / k g K$ $3790 J / k g K$
$Thermal conductivity k$ $0.5296 W / m K$ $0.532 W / m K$ $0.52 W / m K$
Cost ($/tn) -40% -30% -20% -10% 0% +10% +20% +30%
Biomass- based steam 12.222 14.259 16.296 18.333 20.37 22.407 24.444 26.481
Solar-based steam 19.5 22.75 26 29.25 32.5 35.75 39 42.25
Biogas-based steam 16.452 19.194 21.936 24.678 27.42 30.162 32.904 35.646
NG-based steam 10.086 11.767 13.448 15.129 16.81 18.491 20.172 21.853
-40% -30% -20% -10% 0% +10% +20% +30%
Minimum Annual Cost ($)
Biomass- based steam 82,477 96,221 109,969 123,715 137,461 151,207 164,953 178,699
Solar-based steam 131,590 153,522 175,453 197,385 219,317 241,248 263,180 285,112
Biogas-based steam 111,025 129,525 148,028 166,532 185,036 203,539 222,043 240,546
NG-based steam 68,062 79,406 90,750 102,094 113,438 124,781 136,125 147,469
Maximum Annual Cost ($)
Biomass- based steam 278,931 325,419 371,908 418,396 464,885 511,373 557,862 604,350
Solar-based steam 445,029 519,201 593,373 667,544 741,716 815,887 890,059 964,231
Biogas-based steam 375,468 438,046 500,624 563,202 625,780 688,358 750,936 813,514
NG-based steam 230,183 268,547 306,911 345,275 383,638 422,002 460,366 498,730
Variable Minimum Maximum
FFE1 Outlet dry mass fraction (kg/kg) 0.162 0.169
FFE2 Outlet dry mass fraction (kg/kg) 0.225 0.254
FFE3 Outlet dry mass fraction (kg/kg) 0.278 0.359
FFE4 Outlet dry mass fraction (kg/kg) 0.341 0.610
Scenario Objective function Product quality specifications
1 Minimization of annual steam cost ($/year) Product dry mass fraction = 0.5
2 Minimization of annual steam cost ($/year) Product dry mass fraction = 0.35
3 Minimization of annual steam cost ($/year) Product dry mass fraction = 0.30
4 Maximization of yield Product dry mass fraction = 0.5
5 Minimization of total cost Product dry mass fraction = 0.5
6 Maximization of yield 0.3 <Product dry mass fraction < 0.5
Constraints Explanation
$W W t f = 0.5 ( o r 0.65 o r 0.7 )$ The final water content must be equal to 0.5 in Scenarios 1,4 and 5, 0.65 in Scenario 2 and 0.7 in Scenario 3.
$283.15 K ≤ T F F E 1 t f ≤ 343.15 FFE1 temperature should be under 70°C to mitigate the negative effects of heat on heat-sensitive milk components and prevent the degradation of essential
K$ nutrients.
$283.15 K ≤ T F F E 2 t f ≤ 341.15 The temperature of each evaporator effect must be at least 2°C lower than the previous one to ensure proper vacuum.
$0.5 ≤ W W t f ≤ 0.7$ Only for Scenario 6, the final water content varies between 0.5 and 0.7
Table 9. Optimal values of performance indices, time-invariant optimization variables, and final process variables at each Scenario – Case 1.
Scenario 1 2 3 4 5 6 Base Case
Steam annual cost ($/year) 232,118 199,609 181,563 319,363 228,307 261,481 368,653
Total annualized cost ($/year) 3.90·10^6 3.86·10^6 3.85·10^6 3.98·10^6 2.33·10^6 3.92·10^6 4.03·10^6
Yield 0.24 0.36 0.42 0.252 0.252 0.42 0.27
TVR – Discharge Pressure (bar) 0.347 0.341 0.338 0.35 0.35 0.342 0.35
TVR – Suction ratio 2 2 2 1.4 2 1.4 1.1
Feed Temperature (°C) 60 60 60 31.84 60 26.74 20
FFE1/FFE2 tube inner diameter (m) 0.0508 0.0508 0.0508 0.0508 0.0435/0.0254 0.0508 0.0508
FFE1/FFE2 tube length (m) 16 16 16 16 17/5 16 16
FFE2 Temperature (K) 339.81 340.2 340.4 339.85 326.54 340.5 340
Concentrate water content 0.5 0.65 0.7 0.5 0.5 0.7 0.543
Table 10. Optimal values of performance indices, time-invariant optimization variables, and final process variables at each Scenario – Case 2.
Scenario 1 2 3 4 5 6 Base Case
Steam annual cost ($/year) 21,159 18,887 17,184 30,719 20,663 25,326 30,740
Total annualized cost ($/year) 504,490 502,217 500,514 514,049 274,768 508,657 514,070
Yield 0.279 0.36 0.42 0.28 0.252 0.42 0.28
TVR – Discharge Pressure (bar) 0.349 0.345 0.341 0.35 0.35 0.342 0.35
TVR – Suction ratio 2 2 2 1.1 2 1.1 1.1
Feed Temperature (°C) 60 60 60 55.87 60 53.47 55
FFE1/FFE2 tube inner diameter (m) 0.0508 0.0508 0.0508 0.0508 0.0508/
FFE1/FFE2 tube length (m) 6 6 6 6 5.7/0.6 6 6
FFE2 Temperature (K) 340 340.3 340.5 340 293.15 340.5 340
Concentrate water content 0.5 0.65 0.7 0.5 0.5 0.7 0.55
Constraints Explanation
$W W t f = 0.5 ( o r 0.65 o r 0.7 )$ The final water content must be equal to 0.5 in Scenarios 1,4 and 5, 0.65 in Scenario 2 and 0.7 in Scenario 3.
$283.15 K ≤ T F F E 1 t f ≤ 343.15 K$ FFE1 temperature should be under 70°C to mitigate the negative effects of heat on heat-sensitive milk components and prevent the
degradation of essential nutrients.
$283.15 K ≤ T F F E 2 t f ≤ 341.15 K$$283.15 K ≤ T F F E 3 t f The temperature of each evaporator effect must be at least 2°C lower than the previous one to ensure proper vacuum.
≤ 339.15 K$
$0.5 ≤ W W t f ≤ 0.7$ Only for Scenario 6, final water content can vary between 0.5 and 0.7
Table 12. Optimal values of performance indices, time-invariant optimization variables, and final process variables at each Scenario – Case 3.
Scenario 1 2 3 4 5 6 Base Case
Steam annual cost ($/year) 15,078 12,989 11,300 22,069 18,900 19,200 22,088
Total annualized cost ($/year) 681,251 679,163 677,474 688,242 449,236 685,374 688,262
Yield 0.25 0.35 0.36 0.252 0.25 0.36 0.25
TVR – Discharge Pressure (bar) 0.35 0.342 0.34 0.35 0.349 0.344 0.35
TVR – Suction ratio 2 2 2 1.1 1.37 1.1 1.1
Feed Temperature (°C) 60 60 60 54.9 60 53.6 55
FFE1/FFE2/FFE3 tube inner diameter (m) 0.0508 0.0508 0.0508 0.0508 0.0508 / 0.0381/0.0381 0.0508 0.0508
FFE1/FFE2/FFE3 tube length (m) 6 6 6 6 4.25/4.12/4.13 6 6
FFE2 Temperature (K) 340.7 341 341 340.7 338.5 341 340.7
FFE3 Temperature (K) 338.3 338.9 339 338.4 334 339 338.4
Concentrate water content 0.5 0.65 0.7 0.5 0.5 0.7 0.5
Constraints Explanation
$W W t f = 0.5 ( o r 0.65 o r 0.7 )$ The final water content must be equal to 0.5 in Scenarios 1,4 and 5, 0.65 in Scenario 2 and 0.7 in Scenario 3.
$283.15 K ≤ T F F E 1 t f ≤ 343.15 FFE1 temperature should be under 70°C to mitigate the negative effects of heat on heat-sensitive milk components and prevent the degradation of essential
K$ nutrients.
$0.5 ≤ W W t f ≤ 0.7$ Only for Scenario 6, final water content can vary between 0.5 and 0.7
Table 14. Optimal values of performance indices, time-invariant optimization variables, and final process variables at each Scenario – Case 4.
Scenario 1 2 3 4 5 6 Base Case
Electricity annual cost ($/year) 34,319 23,815 19,299 35,950 39,522 20,466 34,582
Total annualized cost ($/year) 275,985 265,480 260,964 277,616 251,240 262,131 276,248
Yield 0.252 0.36 0.42 0.252 0.252 0.42 0.26
MVR – Adiabatic efficiency 1 1 1 0.98 1 1 1
MVR – Compression ratio 1.38 1.3 1.26 1.4 1.45 1.29 1.4
Feed Temperature (°C) 60 60 60 50.35 60 44.17 50
FFE1 tube inner diameter (m) 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508 0.0508
FFE1 tube length (m) 6 6 6 6 5.7 6 6
FFE1 Temperature (K) 319.2 321 322.1 312.5 318.6 310.3 312.3
Concentrate water content 0.5 0.65 0.7 0.5 0.5 0.7 0.514 | {"url":"https://www.preprints.org/manuscript/202312.0480/v1","timestamp":"2024-11-14T17:10:36Z","content_type":"text/html","content_length":"996990","record_id":"<urn:uuid:a9eb235c-fe2a-4ab1-8dbc-4e4520b139ab>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00802.warc.gz"} |
Reference Database
Search Site
How Many?
Prime Curios!
e-mail list
Prime Lists
Submit primes This is the Prime Pages' interface to our BibTeX database. Rather than being an exhaustive database, it just lists the references we cite on these pages. Please let me know of any
errors you notice.
References: [ Home | Author index | Key index | Search ]
A. Wiles, "Modular elliptic curves and Fermat's last theorem," Ann. Math., 141:3 (1995) 443--551. MR 96d:11071 [This is Wiles' proof of Fermat's Last Theorem! The infamous gap is plugged in
the companion article [TW95]. There is a summary of the proof in [Faltings95].] | {"url":"https://t5k.org/references/refs.cgi?long=Wiles95","timestamp":"2024-11-15T03:19:34Z","content_type":"text/html","content_length":"4074","record_id":"<urn:uuid:24474b59-3c45-4fb8-b37a-c0a32fcb700c>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00527.warc.gz"} |
solve the Communication System problem - Academia Essay Writers
0 comments
Q1) For a single sideband modulator performing multi-tone modulation, the message signal is 8 sin ω[m](t) + 4 cos 2ω[m]t, the carrier is cos ω[c]t and ω[c] = 10ω[m]. Determine the time domain
expressions for the upper sideband and the lower sideband versions of the SSB signal. (4 points)
Q2) The message signal input to a modulator is given by A[m] cos(2π×10^4t). (4 points)
(a) If frequency modulation is performed with k[f] =5π × 10^4, find the FM bandwidth when (i) A[m] = 2 and (ii) A[m] = 4.
(b) If phase modulation is performed with k[p] = 2.25, find the PM bandwidth when
(i) A[m] = 2 and (ii) A[m] = 4.
Q3) Given that a delta modulation has m(t) = 3cos40πt + 4cos60πt, calculate the minimum sampling frequency required to prevent slope overload.
Assume that Δ= 0.02π. (4 points)
Q4) Twenty baseband channels, each band-limited to 2.4 kHz, are sampled and multiplexed at a rate of 6 kHz. What is the required bandwidth for transmission if the multiplexed samples use a PCM
system?(4 points)
Q5) Time-division multiplexing is used to transmit two signals m[1](t) and m[2](t). The highest frequency of m[1](t) is 4 kHz, while that of m[2](t) is 3.2 kHz. Determine the minimum allowable
sampling rate. (4 points)
{"email":"Email address invalid","url":"Website address invalid","required":"Required field missing"} | {"url":"https://academiaessaywriters.com/solve-the-communication-system-problem/","timestamp":"2024-11-09T10:05:12Z","content_type":"text/html","content_length":"68958","record_id":"<urn:uuid:a0e4d627-9eba-426d-b838-699313228b3a>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00133.warc.gz"} |
How Einstein made the biggest blunder of his life
Starts With A Bang —
How Einstein made the biggest blunder of his life
When Einstein gave General Relativity to the world, he included an extraneous cosmological constant. How did his ‘biggest blunder’ occur?
Key Takeaways
• Einstein’s General Relativity, upon its publication in 1915, put forth the relationship between matter-energy, which curves spacetime, and curved spacetime, which tells matter-energy how to move.
• However, Einstein also included, in his equations, an additional, unnecessary term: a cosmological constant term, with a constant, non-zero energy density that persists everywhere.
• Some 15+ years after introducing it, Einstein allegedly referred to it as “his greatest/biggest blunder.” Here’s how even the greatest genius of our time was led astray by his own biases.
Imagine what it must have been like to study the Universe, at a fundamental level, way back in the early 1900s. For over 200 years, the physics of Newton appeared to govern how objects moved, with
Newton’s law of universal gravitation and laws of motion dictating how things moved on Earth, in our Solar System, and in the greater Universe. Recently, however, a few challenges to Newton’s picture
had emerged. You couldn’t keep accelerating objects to arbitrary speeds, but rather everything was limited by the speed of light. Newton’s optics didn’t describe light nearly as well as Maxwell’s
electromagnetism did, and quantum physics — still in its infancy — was posing new sets of questions to physicists worldwide.
But perhaps the biggest problem was posed by the orbit of Mercury, precisely measured since the late 1500s and in defiance of Newton’s predictions. It was his quest to explain that observation that
led Albert Einstein to formulate the General Theory of Relativity, which replaced Newton’s law of gravitation with a relationship between matter-and-energy, which curves spacetime, and that curved
spacetime, which tells matter-and-energy how to move.
Yet Einstein didn’t publish that version of General Relativity; he published a version that included an extra ad hoc term: a cosmological constant, artificially adding an extra field to the Universe.
Decades later, he would refer to it as his biggest blunder, but not before doubling down on it many times over the years. Here’s how the smartest man in history made his biggest blunder ever, with
lessons for us all.
General Relativity, importantly, was built off of three puzzle pieces that came together in Einstein’s mind.
1. Special relativity, or the notion that each unique observer had their own unique — but mutually consistent between observers — conception of space and time, including the distance between objects
and the duration and order of events.
2. Minkowski’s reformulation of space and time as a unified four-dimensional fabric known as spacetime, which provides a backdrop for all other objects and observers to move and evolve through it.
3. And the equivalence principle, which Einstein repeatedly called his “happiest thought,” which was the notion that an observer within a sealed room who was accelerating because they were in a
gravitational field would feel no difference from an identical observer in an identical room who was accelerating because there was thrust (or an outside force) causing the acceleration.
These three notions, put together, led Einstein to conceive of gravity differently: that instead of being governed by an invisible, infinitely fast-acting force that acted across all distances and at
all times, gravitation was instead caused by the curvature of spacetime, which itself was induced by the presence of matter-and-energy within it.
Those three early steps happened in 1905, 1907, and 1908, respectively, but General Relativity wasn’t published in its final form until 1915; that’s how long it took Einstein and his collaborators to
work the details out correctly. Once he had, however, he released a set of equations — known today as the Einstein field equations — that related how matter-energy and spacetime affected one another.
In that paper, he verified that:
• At large distances from relatively small masses, his equations could be well approximated by Newtonian gravity.
• At small distances from large masses, there were additional effects beyond the Newtonian approximation, and those effects could, at last, explain the tiny-but-significant differences between what
astronomers had been observing for hundreds of years and what Newton’s gravity had predicted.
• And that there would be additional, subtle differences between the predictions of Einstein’s gravity and Newton’s gravity that could be searched for, including gravitational redshift and the
gravitational deflection of light by masses.
That third point led to a key new prediction: that during a total solar eclipse, when the Sun’s light was blocked by the Moon and stars would be visible, that the apparent position of the stars
located behind the Sun would be bent, or shifted, by the Sun’s gravity. After “missing” the chance to test this in 1916 because of the Great War and losing out to clouds in 1918, the eclipse
expedition of 1919 finally made the critical observations, confirming the predictions of Einstein’s General Relativity and leading to its widespread acceptance as a new theory of gravity.
But, like any good scientist formulating a new theory, Einstein himself was fairly uncertain of how the experiments and observations would turn out. In a letter to physicist Willem de Sitter in 1917,
Einstein wrote the following:
“For me… it was a burning question whether the relativity concept can be followed through to the finish, or whether it leads to contradictions.”
In other words, sure, after figuring out the mathematics of General Relativity and how to successfully apply it to a variety of situations, now the big challenge arrives: applying it to every
physical case where it should give a correct description. One big challenge to that, however, was when it came to the known Universe of Einstein’s time.
You see, back then, it was not yet known whether there were other galaxies out there — what astronomers of the time dubbed the “island universe” hypothesis — or whether everything that we observed
was contained within the Milky Way itself. There was even a great debate on this very topic a few years later, in 1920, and although both sides argued passionately, it was highly inconclusive. It was
reasonable, and accepted by many, that the Milky Way and the objects within it were simply all there was.
This notion posed a big problem for Einstein. You see, one of the theorems that was relatively easy to derive in relativity is as follows:
If you take any initial distribution of masses, and start them off at rest, what you’ll inevitably find, after a finite amount of time has passed, is that these masses will eventually collapse down
to a single point, what we know today as a black hole.
This would be bad, because a black hole is a singularity, where space and time come to an end and no sensible physical predictions can be arrived at. This brought up precisely the type of
contradiction that Einstein was worried about. If our Milky Way was simply a large collection of masses that all moved very slowly relative to one another, those masses should inevitably cause the
spacetime they were present in to collapse. And yet, our Milky Way didn’t appear to be collapsing and clearly hadn’t collapsed in on itself. In order to avoid this type of contradiction, Einstein
posited that something extra — some new ingredient or effect — must be added to the equation. Otherwise, the unacceptable consequence of an unstable Universe that should be collapsing (yet,
observationally, didn’t appear to be) couldn’t be evaded.
In other words, if the Universe is static, it can’t just collapse; that would be really bad and would conflict with what we were seeing. So how did Einstein avoid it? He introduced a new term to the
equations: what is known today as a cosmological constant. In his own words, again writing in 1917, Einstein stated the following:
“In order to arrive at this consistent view, we admittedly had to introduce an extension of the field equations of gravitation which is not justified by our actual knowledge of gravitation… That term
is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars.”
It’s pretty harsh to call this a blunder, as his line of thought is easy to follow and seems reasonable. We know that:
• a static universe filled with masses in some distribution is unstable and will collapse,
• our Universe appears to be filled with nearly-static masses but isn’t collapsing,
• and therefore, there has to be something else out there to hold it up against collapse.
The only option that Einstein had found was this extra term that he could add without introducing further pathologies in his theory: a cosmological constant term.
Other people — I should be clear here that these are other very smart, very competent people — took these equations and concepts put forth by Einstein, and went on to derive the inevitable
consequences of them.
First, Willem de Sitter, later in 1917, showed that if you take a model Universe with only a cosmological constant in it (that is, with no other sources of matter or energy), you get an empty,
four-dimensional spacetime that expands eternally at a constant rate.
Second, in 1922, Alexander Friedmann showed that if you make the assumption, within Einstein’s relativity, that the entire Universe is uniformly filled with some type of energy — including (but not
limited to) matter, radiation, or the type of energy that would yield a cosmological constant — then a static solution is impossible, and the Universe must either expand or contract. (And that this
is true regardless of whether the cosmological constant exists or not.)
And third, in 1927, Georges Lemaître built on Friedmann’s equations, applying them to the combination of galactic distances measured by Hubble (starting in 1923) and also to the apparently large
recessional motion of those galaxies, measured earlier by Vesto Slipher (as early as 1911). He concluded that the Universe is expanding, and not only submitted a paper on it, but wrote to Einstein
about it personally as well.
The reason that the cosmological constant is often called “Einstein’s greatest blunder” isn’t because of why he originally formulated it; it’s because of his undeserved, unreasonable, and perhaps
even unhinged reaction to everyone else’s valid criticisms and contrary conclusions. Einstein extensively, and incorrectly, criticized de Sitter’s derivations, being proved wrong on all counts by de
Sitter and Oskar Klein in a series of letters throughout 1917 and 1918. Einstein incorrectly criticized Friedmann’s work in 1922, calling it incompatible with the field equations; Friedmann correctly
pointed out Einstein’s error, which Einstein ignored until his friend, Yuri Krutkov, explained it to him, at which point he retracted his objections.
And still, in 1927, when Einstein became aware of Lemaître’s work, he retorted, “Vos calculs sont corrects, mais votre physique est abominable,” which translates to, “Your calculations are correct,
but your physics is abominable.” He maintained this stance in 1928, when Howard Robertson independently reached the same conclusions as Lemaître with improved data, and did not change his mind with
Hubble’s (and, later, Humason’s) overwhelming demonstration that more distant objects (with distances determined using Henrietta Leavitt’s legendary method) were moving away more quickly in 1929.
Hubble wrote that the finding could “represent the de Sitter effect” and “hence introduces the element of time” into the Universe.
Throughout all of this, Einstein didn’t change his stance at all. He maintained that the Universe must be static and the cosmological constant is mandatory. And, because he was Einstein, many people
— including Hubble — were tentative to interpret this data as implicating the expansion of the Universe. It wouldn’t be until 1931, when Lemaître wrote a very influential letter to Nature, where he
put together the pieces completely: that the Universe could be evolving in time if it started out from a smaller, denser state and has expanded ever since. It was only in the aftermath of that, that
Einstein finally admitted that, just perhaps, he had jumped the gun by introducing a cosmological constant with the sole motive of keeping the Universe static.
Travel the Universe with astrophysicist Ethan Siegel. Subscribers will get the newsletter every Saturday. All aboard!
In hindsight, the cosmological constant is now a very important part of modern cosmology, as it’s the best explanation we have for the effects of dark energy on our expanding Universe. But if
Einstein didn’t introduce it and continue to defend and stand by it the way he had — if he had simply followed the equations — he could have derived the expanding Universe as a consequence of his
equations, just as Friedmann did and, later, Lemaître, Robertson, and others.
It was a small blunder to introduce an extraneous, unnecessary term into his equations, but his greatest blunder was defending his error in the face of overwhelming evidence. As we all should learn,
saying “I was wrong” when we’re shown to be in error is the only way to grow.
The author acknowledges Dan Scolnic’s plenary talk at the 242nd American Astronomical Society’s meeting for unearthing many of these facts and quotes.
Even the most brilliant mind in history couldn’t have achieved all he did without significant help from the minds of others.
Time isn’t the same for everyone, even on Earth. Flying around the world gave Einstein the ultimate test. No one is immune from relativity.
The idea of gravitational redshift crossed Einstein’s mind years before General Relativity was complete. Here’s why it had to be there.
“Imagination is more important than knowledge” is often taken to mean that your conceptions outweigh what’s real. That’s not what he said.
More than any other of Einstein’s equations, E = mc² is the most recognizable to people. But what does it all mean?
The meaning of the cryptic text has eluded scholars for centuries. Their latest efforts include computational analyses seeking new insights into the medieval enigma. | {"url":"https://preprod.bigthink.com/starts-with-a-bang/einstein-biggest-blunder/","timestamp":"2024-11-08T04:44:27Z","content_type":"text/html","content_length":"166478","record_id":"<urn:uuid:19220554-597f-488b-bd77-da8481b750a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00283.warc.gz"} |
Introduction to Market Mix Model Using Robyn
Whether it’s an established company or fairly new in the market, almost every business uses different Marketing channels like TV, Radio, Emails, Social Media, etc., to reach its potential customers
and increase awareness about its product, and in turn, maximize sales or revenue.
But with so many marketing channels at their disposal, business needs to decide which marketing channels are effective compared to others and more importantly, how much budget needs to be allocated
to each channel. With the emergence of online marketing and several big data platforms and tools, marketing is one of the most prominent areas of opportunities for data science and machine learning
Learning Objectives
1. What is Market Mix Modeling, and how MMM using Robyn is better than a traditional MMM?
2. Time Series Components: Trend, Seasonality, Cyclicity, Noise, etc.
3. Advertising Adstocks: Carry-over Effect & Diminishing Returns Effect, and Adstock transformation: Geometric, Weibull CDF & Weibull PDF.
4. What are gradient-free optimization and Multi-Objective Hyperparameter Optimization with Nevergrad?
5. Implementation of Market Mix Model using Robyn.
So, without further ado, let’s take our first step to understand how to implement the Market mix model using the Robyn library developed by Facebook(now Meta) team and most importantly, how to
interpret output results.
This article was published as a part of the Data Science Blogathon.
Market Mix Modeling (MMM)
It’s to determine the impact of marketing efforts on sales or market share. MMM aims to ascertain the contribution of each marketing channel, like TV, Radio, Emails, Social Media, etc., on sales. It
helps businesses make judicious decisions, like on which marketing channel to spend and, more importantly, what amount should be spent. Reallocate the budget across different marketing channels to
maximize revenue or sales if necessary.
What is Robyn?
It’s an open-source R package developed by Facebook’s team. It aims to reduce human bias in the modeling process by automating important decisions like selecting optimal hyperparameters for Adstocks
& Saturation effects, capturing Trend & Seasonality, and even performing model validation. It’s a semi-automated solution that facilitates the user to generate and store different models in the
process (different hyperparameters are selected in each model), and in turn, provides us with different descriptive and budget allocation charts to help us make better decisions (not limited to)
about which marketing channels spend on, and more importantly how much should be spent on each marketing channel.
How Robyn addresses the challenges of classic Market Mix Modeling?
The table below outlines how Robyn addresses the challenges of traditional marketing mix modeling.
Before we take deep dive into building Market Mix Model using Robyn, let’s cover some basics that pertain to Market Mix Model.
Time Series Components
You can decompose Time series data into two components:
• Systematic: Components that have consistency or repetition and can be described and modeled.
• Non-Systematic: Components that don’t have consistency, or repetition, and can’t be directly modeled, for example, “Noise” in data.
Fig: Trend Seasonality chart
Systematic Time Series Components mainly encapsulate the following 3 components:
• Trend
• Seasonality
• Cyclicity
If you notice a long-term increase or decrease in time series data then you can safely say that there’s a trend in the data. Trend can be linear, nonlinear, or exponential, and it can even change
direction over time. For e.g., an increase in prices, an increase in pollution, or an increase in the share price of a company for a period of time, etc.
Fig: Plot showing Trend
In the above plot , blue line shows an upward trend in data.
If you notice a periodic cycle in the series with fixed frequencies then you can say there’s a seasonality in the data. These frequencies could be on daily, weekly, monthly basis, etc. In simple
words, Seasonality is always of a fixed and known period, meaning you’ll notice a definite amount of time between the peaks and troughs of the data; ergo at times, seasonal time series is called
periodic time series too.
For e.g., Retail sales going high on a few particular festivals or events, or weather temperature exhibiting its seasonal behavior of being warm days in summer and cold days in winter, etc.
Fig: Seasonal plot of Air Passengers
In the above plot, we can notice a strong seasonality in the months of July and August meaning #AirPassengers are highest while lowest in the months of Feb & Nov.
When you notice rises and falls, that are not of the fixed period, you can say there’s cyclic pattern in data. Generally, the average length of cycles would be more than the length of seasonal
patterns. In contrast, the magnitude of cycles tends to be more inconsistent than that of seasonal patterns.
autoplot(lynx) +xlab("Year") +ylab("Number of lynx trapped")
Fig: #Lynx trapped each year
As we can clearly see aperiodic population cycles of approximately ten years. The cycles are not of a constant length – some last 8 or 9 years, and others last longer than ten years.
When there’s no Trend, Cycle, or Seasonality whatsoever, and if it’s just random fluctuations in data then we can safely say that it’s just Noise in data.
Fig: Plot showing Noise
In the above plot, there’s no trend, seasonality, or cyclic behavior whatsoever. They’re very random fluctuations that aren’t predictable and can’t be used to build a good Time Series Forecasting
RoAS(Return on Advertising Spend)
It is a marketing metric used to assess an advertising campaign’s efficacy. ROAS helps businesses ascertain which advertising channels are doing good and how they can improve advertising efforts in
the future to increase sales or revenue. ROAS formula is:
ROAS= (Revenue from an ad campaign/ Cost of an ad campaign)*100 %
E.g. if you spend $2,000 on an ad campaign and you make $4,000 in profit, your ROAS would be 200% .
In simple words, ROAS represents the revenue gained from each dollar spent on advertising, and is often represented in percentage.
Advertising Adstock
The term “Adstock “was coined by Simon Broadbent , and it encapsulates two important concepts:
• Carryover, or Lagged Effect
• Diminishing Returns, or Saturation Effect
1. Carryover, or Lagged Effect
Advertising tends to have an effect extending several periods after you see it for the first time. Simply put, an advertisement from earlier day, week, etc. may affect an ad in the current day, week,
etc. It is called Carryover or lagged Effect.
E.g., Suppose you’re watching a Web Series on YouTube, and some ad for a product pops up on the screen. You may wait to buy this product after the commercial break. It could be because the product is
expensive, and you want to know more details about it, or you want to compare it with other brands to make a rational decision of buying it if you need it in the first place. But if you see this
advertisement a few more times, it’d have increased awareness about this product, and you may purchase that product. But if you have not seen that ads gain after the first time, then It’s highly
possible that you don’t remember that in the future. This is called the Carryover, or lagged Effect.
You can choose below of the 3 adstock transformations in Robyn:
• Geometric
• Weibull PDF
• Weibull CDF
This is a weighted average going back n days, where n can vary by media channel. The most salient feature of the Geometric transformation is its simplicity, considering It requires only one parameter
called ‘theta’.
For e.g., Let’s say, an advertising spend on day one is $500 and theta = 0.8, then day two has 500*0.7=$400 worth of effect carried-over from day one, day three has 400*0.8= $320 from day 2, etc.
This can make it much easier to communicate results to laymen, or non-technical stakeholders. In addition, Compared to Weibull Distribution(which has two parameters to optimize ), Geometric is much
less computationally expensive
& less time-consuming, and hence much faster to run.
Robyn’s implementation of Geometric transformation can be written as follows-
Fig: Robyn’s implementation of Geometric transformation
Weibull Distribution
You remember that one person equipped with diverse skills in your friend circle, who’ll fit into every group. Because of such a dexterous and pliable personality, that person was part of almost every
The Weibull distribution is something similar to that person. It can fit an array of distributions: Normal Distribution, Left-skewed Distribution, and Right-Skewd Distribution.
You’ll find 2 versions of a two-parametric Weibull function: Weibull PDF and Weibull CDF. Compared to the one-parametric Geometric function with the constant “theta”, the Weibull distribution
produces time-varying decay rates with the help of parameters Shape and Scale.
Robyn’s implementation of Weibull distribution can be illustrated conceptually as follows-
Fig:Robyn’s implementation of Weibull distribution
Weibull’s CDF (Cumulative Distribution Function)
It has two parameters, shape & scale, and has a nonconstant “theta”. Shape controls the shape of the decay curve, and Scale controls the inflection of the decay curve.
Note: The larger the shape, the more S-shape. The smaller shape, the more L-shape.
Weibull’s PDF (Probability Density Function)
Also has Shape & Scale parameters besides a nonconstant “theta”. Weibull PDF provides lagged effect.
Fig: Weibull adstock CDF vs PDF
The plot above shows different curves in each plot with different values of Shape & Scale hyperparameters exhibiting the flexible nature of Weibull adstcoks. Due to more hyperparameters, Weibull
adstocks are more computationally expensive than Geometric adstocks. However, Weibull PDF is strongly recommended when the product is expected to have a longer conversion window.
2. Diminishing Returns Effect/Saturation Effect
Exposure to an advertisement creates awareness about the product in consumers’ mind to a certain limit, but after that impact of advertisements to influence consumers’ purchasing behavior start
diminishing over time. This is called a Saturation effect or Diminishing Returns effect.
Simply put, It’d be presumptuous to say that the more money you spend on advertising, the higher your sales get. In reality, this growth gets weaker the more we spend.
For example, increasing the YouTube ad spending from $0 to $10,000 increases our sales a lot, but increasing it from $10,000,000 to $900,000,000 doesn’t do that much anymore.
Source: facebookexperimental.github.io
Robyn uses the Hill function to capture the saturation of each media channel.
Hill Function for Saturation: It’s a two-parametric function in Robyn . It has two parameters called alpha & gamma. α controls the shape of the curve between the exponential and s-shape, and γ
(gamma) controls the inflection.
Note: larger the α, the more S-shape, and the smaller the α, the more C-shape. Larger the γ (gamma), the furtherer the inflection in the response curve.
Please check out the below plots to see how the Hill function transformation with respect to parameter changes:
Source: facebookexperimental.github.io
Ridge Regression
To address Multicollinearity in input data and prevent overfitting, Robyn uses Ridge Regression to reduce variance. This is aimed at improving the predictive performance of MMMs.
The mathematical notation for Ridge regression in Robyn is as follows:
Source: facebookexperimental.github.io
Nevergrad is a Python library developed by a team of Facebook. It facilitates the user with derivative-free and evolutionary optimization.
Why Gradient-free Optimization?
It is easy to compute a function’s gradient analytically in a few cases like weight optimization in Neural Networks. However, in other cases, estimating the gradient can be quite challenging. For
e.g., if function f is slow to compute, non-smooth, time-consuming to evaluate, or so noisy, methods that rely on derivates are of little to no use. Algorithms that don’t use derivatives or finite
differences are helpful in such situations and are called derivative-free algorithms.
In Marketing Mix Modeling, we’ve got to find optimal values for a bunch of hyperparameters to find the best model for capturing patterns in our time series data.
For e.g., One wants to calculate your media variables’ Adstock and Saturation effects. Based on your formula, one would have to define 2 to 3 hyperparameters per channel. Let’s say we are modeling
four different media channels plus 2 offline channels. We have a breakdown of the media channels, making them a total of 8 channels. So, 8 channels, and 2 hyperparameters per channel mean you’ll have
to define 16 hyperparameters before being able to start the modeling process.
So, you’ll have a hard time randomly testing all possible combinations by yourself. That’s when Nevergrad says, Hold my beer.
Nevergrad eases the process of finding the best possible combination of hyperparameters to minimize the model error or maximize its accuracy.
MOO (Multi-Objective Hyperparameter Optimization) with Nevergrad
Multi-objective hyperparameter optimization using Nevergrad, Meta’s gradient-free optimization platform, is one of the key innovations in Robyn for implementing MMM. It automates the regularization
penalty, adstocking selection, saturation, and training size for time-series validation. In turn, it provides us with model candidates with great predictive power.
There’re 4 types of hyperparameters in Robyn at the time of writing article.
• Adstocking
• Saturation
• Regularization
• Validation
Robyn Aims to Optimize the 3 Objective Functions:
• The Normalized Root Mean Square Error (NRMSE): also known as the Prediction error. Robyn performs time-series validation by spitting the dataset into train, validation, and test. nrmse_test is
for assessing the out-of-sample predictive performance.
• Decomposition Root Sum of Squared Distance (DECOMP.RSSD): is one of the key features of Robyn and is aka business error. It shows the difference between the share of effect for paid_media_vars
(paid media variables), and the share of spend. DECOMP.RSSD can scratch out the most extreme decomposition results. Hence It helps narrow down the model selection.
• The Mean Absolute Percentage Error (MAPE.LIFT): Robyn includes one more evaluation metric called MAPE.LIFT aka Calibration error, when you perform “model calibration” step. It minimizes the
difference betweenthe causal effect and the predicted effect.
Now we understand the basics of the Market Mix Model and Robyn library. So, let’s start implementing Market Mix Model (MMM) using Robyn in R.
Step 1: Install the Right Packages
#Step 1.a.First Install required Packages
#Step 1.b Setup virtual Environment & Install nevergrad library
py_install("nevergrad", pip = TRUE)
use_virtualenv("r-reticulate", required = TRUE)
If even after installation you can’t import Nevergrad then find your Python file in your system and run below line of code by providing path to your Python file.
Now import the packages and set current working directory.
#Step 1.c Import packages & set CWD
#Step 1.d You can force multi-core usage by running below line of code
Sys.setenv(R_FUTURE_FORK_ENABLE = "true")
options(future.fork.enable = TRUE)
# You can set create_files to FALSE to avoid the creation of files locally
create_files <- TRUE
Step 2: Load Data
You can load inbuilt simulated dataset or you can load your own dataset.
#Step 2.a Load data
#Step 2.b Load holidays data from Prophet
# Export results to desired directory.
robyn_object<- "~/MyRobyn.RDS"
Step 3: Model Specification
Step 3.1 Define Input variables
Since Robyn is a semi-automated tool, using a table like the one below can be valuable to help articulate independent and Target variables for your model:
Source: facebookexperimental.github.io
#### Step 3.1: Specify input variables
InputCollect <- robyn_inputs(
dt_input = dt_simulated_weekly,
dt_holidays = dt_prophet_holidays,
dep_var = "revenue",
dep_var_type = "revenue",
date_var = "DATE",
prophet_country = "DE",
prophet_vars = c("trend", "season", "holiday"),
context_vars = c("competitor_sales_B", "events"),
paid_media_vars = c("tv_S", "ooh_S", "print_S", "facebook_I", "search_clicks_P"),
paid_media_spends = c("tv_S", "ooh_S", "print_S", "facebook_S", "search_S"),
organic_vars = "newsletter",
# factor_vars = c("events"),
adstock = "geometric",
window_start = "2016-01-01",
window_end = "2018-12-31",
Sign of coefficients
• Default: means that the variable could have either + , or – coefficients depending on the modeling result. However,
• Positive/Negative: If you know the specific impact of an input variable on Target variable then you
can choose sign accordingly.
Note: All sign control are automatically provided: “+” for organic & media variables and “default” for all others. Nonetheless, you can still customize signs if necessary.
You can make use of documentation anytime for more details by running: ?robyn_inputs
Categorize variables into Organic, Paid Media, and Context variables:
There are 3 types of input variables in Robyn: paid media, organic and context variables. Let’s understand, how to categorize each variable into these three buckets:
• paid_media_vars
• organic_vars
• context_vars
1. We apply transformation techniques to paid_media_vars and organic_vars variables to reflect carryover effects and saturation. However, context_vars directly impact the Target variable and do not
require transformation.
2. context_vars and organic_vars can accept either continuous or categorical data while paid_media_vars can only accept continuous data.You can mention Organic or context variables with categorical
data type under factor_vars parameter.
3. For variables organic_vars and context_vars , continuous data will provide more information to the model than categorical. For example, providing the % discount of each promotional offer (which
is continuous data) will provide more accurate information to the model compared to a dummy variable that shows the presence of a promotion with just 0 and 1.
Step 3.2 Specify hyperparameter names and ranges
Robyn’s hyperparameters have four components:
• Time series validation parameter (train_size).
• Adstock parameters (theta or shape/scale).
• Saturation parameters alpha/gamma).
• Regularization parameter (lambda).
Specify Hyperparameter Names
You can run ?hyper_names to get the right media hyperparameter names.
hyper_names(adstock = InputCollect$adstock, all_media = InputCollect$all_media)
## Note: Set plot = TRUE to produce example plots for
#adstock & saturation hyperparameters.
plot_adstock(plot = FALSE)
plot_saturation(plot = FALSE)
# To check maximum lower and upper bounds
Specify Hyperparameter Ranges
You’ll have to mention upper and lower bounds for each hyperparameter. For e.g., c(0,0.7). You can even mention a scalar value if you want that hyperparameter to be a constant value.
# Specify hyperparameters ranges for Geometric adstock
hyperparameters <- list(
facebook_S_alphas = c(0.5, 3),
facebook_S_gammas = c(0.3, 1),
facebook_S_thetas = c(0, 0.3),
print_S_alphas = c(0.5, 3),
print_S_gammas = c(0.3, 1),
print_S_thetas = c(0.1, 0.4),
tv_S_alphas = c(0.5, 3),
tv_S_gammas = c(0.3, 1),
tv_S_thetas = c(0.3, 0.8),
search_S_alphas = c(0.5, 3),
search_S_gammas = c(0.3, 1),
search_S_thetas = c(0, 0.3),
ooh_S_alphas = c(0.5, 3),
ooh_S_gammas = c(0.3, 1),
ooh_S_thetas = c(0.1, 0.4),
newsletter_alphas = c(0.5, 3),
newsletter_gammas = c(0.3, 1),
newsletter_thetas = c(0.1, 0.4),
train_size = c(0.5, 0.8)
#Add hyperparameters into robyn_inputs()
InputCollect <- robyn_inputs(InputCollect = InputCollect, hyperparameters = hyperparameters)
Step 3.3 Save InputCollect in the Format of JSON File to Import Later:
You can manually save your input variables and different hyperparameter specifications in a JSON file which you can import easily for further usage.
##### Save InputCollect in the format of JSON file to import later
robyn_write(InputCollect, dir = "./")
InputCollect <- robyn_inputs(
dt_input = dt_simulated_weekly,
dt_holidays = dt_prophet_holidays,
json_file = "./RobynModel-inputs.json")
Step 4: Model Calibration/Add Experimental Input (Optional)
You can use Robyn’s Calibration feature to increase confidence to select your final model especially when you don’t have information about media effectiveness and performance beforehand. Robyn uses
lift studies (test group vs a randomly selected control group) to understand causality of their marketing on sales (and other KPIs) and to assess the incremental impact of ads.
Source: www.facebookblueprint.com
calibration_input <- data.frame(
liftStartDate = as.Date(c("2018-05-01", "2018-04-03", "2018-07-01", "2017-12-01")),
liftEndDate = as.Date(c("2018-06-10", "2018-06-03", "2018-07-20", "2017-12-31")),
liftAbs = c(400000, 300000, 700000, 200),
channel = c("facebook_S", "tv_S", "facebook_S+search_S", "newsletter"),
spend = c(421000, 7100, 350000, 0),
confidence = c(0.85, 0.8, 0.99, 0.95),
calibration_scope = c("immediate", "immediate", "immediate", "immediate"),
metric = c("revenue", "revenue", "revenue", "revenue")
InputCollect <- robyn_inputs(InputCollect = InputCollect, calibration_input = calibration_input)
Step 5: Model Building
Step 5.1 Build Baseline Model
You can always tweak trials and number of iterations according to your business needs to get the best accuracy.You can run ?robyn_run to check parameter definition.
#Build an initial model
OutputModels <- robyn_run(
InputCollect = InputCollect,
cores = NULL,
iterations = 2000,
trials = 5,
ts_validation = TRUE,
add_penalty_factor = FALSE
Step 5.2 Model Solution Clustering
Robyn uses K-Means clustering on each (paid) media variable to find “best models” that have NRMSE, DECOM.RSSD, and MAPE(if calibrated was used).
The process for the K-means clustering is:
• When k = “auto” (which is the default), It calculates the WSS on k-means clustering using k = 1 to 20 to find the best value of k”.
• After It has run k-means on all Pareto front models, using the defined k, It picks the “best models” with the lowest normalized combined errors.
The process for the K-means clustering is as follows:
When k = “auto” (which is the default), It calculates the WSS on k-means clustering using k = 1 to k = 20 to find the best value of k”.
The process for the K-means clustering is:
You can run robyn_clusters() to produce list of results: some visualizations on WSS-k selection, ROI per media on winner models.data used to calculate the clusters, and even correlations of Return on
Investment (ROI) etc. . Below chart illustrates the clustering selection.
Step 5.3 Prophet Seasonality Decomposition
Robyn uses Prophet to improve the model fit and ability to forecast. If you are not sure about which baselines need to be included in modelling, You can refer to the following description:
• Trend: Long-term and slowly evolving movement ( increasing or decreasing direction) over time.
• Seasonality: Capture seasonal behaviour in a short-term cycle, For e.g. yearly.
• Weekday: Monitor the repeating behaviour on weekly basis, if daily data is available.
• Holiday/Event: Important events or holidays that highly impact your Target variable.
Pro-tip: Customize Holidays & Events
Robyn provides country-specific holidays for 59 countries from the default “dt_prophet_holidays ” Prophet file already.You can use dt_holidays parameter to provide the same information.
If your country’s holidays are included or you want to customize holidays information then you can try following:
• Customize holiday dataset: You can customize or change the information in the existing holiday dataset.You can add events & holidays into this table e.g., Black Friday, school holidays, Cyber
Monday, etc.
• Add a context variable: If you want to assess the impact of a specific alone then you can add that information under context_vars variable.
Step 5.4 Model Selection
Robyn leverages MOO of Nevergrad for its model selection step by automatically returning a set of optimal results. Robyn leverages Nevergrad to achieve main two objectives:
• Model Fit: Aims to minimize the model’s prediction error i.e. NRMSE.
• Business Fit: Aims to minimize decomposition distance i.e. decomposition root-sum-square distance (DECOMP.RSSD). This distance metric is for the relationship between spend share and a channel’s
coefficient decomposition share. If the distance is too far then its result can be too unrealistic -For e.g. advertising channel with the smallest spending getting the largest effect. So this
seems kind of unrealistic.
You can see in below chart how Nevergrad rejects maximum of “bad models” (larger prediction error and/or unrealistic media effect). Each blue dot in the chart represents an explored model solution.
NRMSE & DECOMP.RSSD Functions
NRMSE on x-axis and DECOMP.RSSD on y-axis are the 2 functions to be minimized. As you can notice in below chart,with increased number of iterations, a trend down the bottom-left corner is quite
Based on the NRMSE & DECOMP.RSSD functions Robyn will generate a series of baseline models at the end of the modeling process. After reviewing charts and different output results, you can select a
final model.
Few key parameters to help you select the final model:
• Business Insight Parameters: You can compare multiple business parameters like Return on Investment, media adstock and response curves, share and spend contributions etc. against model’s output
results. You can even compare output results with your knowledge of industry benchmarks and different evluation metrics.
• Statistical Parameters: If multiple models exhibit very similar trends in business insights parameters then you can select the model with best statistical parameters (e.g. adjusted R-square, and
NRMSE being highest and lowest respectively, etc.).
• ROAS Convergence Over Iterations Chart: This chart shows how Return on investment(ROI) for paid media or Return on Ad spend(ROAS) evolves over time and iterations. For few channels, it’s quite
clear that the higher iterations are giving more “peaky” ROAS distributions, which display higher confidence for certain channel results.
Step 5.5 Export Model Results
## Calculate Pareto fronts, cluster and export results and plots.
OutputCollect <- robyn_outputs(
InputCollect, OutputModels,
csv_out = "pareto",
pareto_fronts = "auto",
clusters = TRUE,
export = create_files,
plot_pareto = create_files,
plot_folder = robyn_object
You’ll see 4 csv files are exported for further analysis:
Interpretation of the Six Charts
1. Response Decomposition Waterfall by Predictor
The chart illustrates the volume contribution, indicating the percentage of each variable’s effect (intercept + baseline and media variables) on the target variable. For example, based on the chart,
approximately 10% of the total sales are driven by the Newsletter.
Note: For established brands/companies, Intercept and Trend can account for a significant portion of the Response decomposition waterfall chart, indicating that significant sales can still occur
without marketing channel spending.
2. Share of Spend vs. Share of Effect
This chart compares media contributions across various metrics:
• Share of spend: Reflects the relative spending on each channel.
• Share of effect: Measures the incremental sales driven by each marketing channel.
• ROI (Return On Investment): Represents the efficiency of each channel.
When making important decisions, it is crucial to consider industry benchmarks and evaluation metrics beyond statistical parameters alone. For instance:
• A channel with low spending but high ROI suggests the potential for increased spending, as it delivers good returns and may not reach saturation soon due to the low spending.
• A channel with high spending but low ROI indicates underperformance, but it remains a significant driver of performance or revenue. Hence, optimise the spending on this channel.
Note: Decomp.RSSD corresponds to the distance between the share of effect and the share of spend. So, a large value of Decomp.RSSD may not make realistic business sense to optimize. Hence please
check this metric while comparing model solutions.
3. Average Adstock Decay Rate Over Time
This chart tells us average % decay rate over time for each channel. Higher decay rate represents the longer effect over time for that specific Marketing channel.
4. Actual vs. Predicted Response
This plot shows how well the model has predicted the actual Target variable, given the input features. We aim for models that can capture most of the variance from the actual data, ergo the R-squared
should be closer to 1 while NRMSE is low.
One should strive for a high R-squared, where a common rule of thumb is
• R squared < 0.8 =model should be improved further;
• 0.8 < R squared < 0.9 = admissible, but could be improved bit more;
• R squared > 0.9 = Good.
Models with a low R squared value can be improved further by including a more comprehensive set of Input features – that is, split up larger paid media channels or add additional baseline (non-media)
variables that may explain the Target variable(For e.g., Sales, Revenue, etc.).
Note: You’d be wary of specific periods where the model is predicting worse/better. For example, if one observes that the model shows noticeably poorer predictions during specific periods associated
with promotional periods, it can serve as a useful method to identify a contextual variable that should be incorporated into the model.
5. Response Curves and Mean Spend by Channel
Response curves for each media channel indicate their saturation levels and can guide budget reallocation strategies. Channels with faster curves reaching a horizontal slope are closer to saturation,
suggesting a diminishing return on additional spending. Comparing these curves can help reallocate spending from saturated to less saturated channels, improving overall performance.
6. Fitted vs. Residual
The scatter plot between residuals and fitted values (predicted values) evaluates whether the basic hypotheses/assumptions of linear regression are met, such as checking for homoscedasticity,
identifying non-linear patterns, and detecting outliers in the data.
Step 6: Select and Save One Model
You can compare all exported model one-pagers in last step and select one that mostly reflects your business reality
Step 6: Select and save any one model
## Compare all model one-pagers and select one that largely reflects your business reality.
select_model <- "4_153_2"
ExportedModel <- robyn_write(InputCollect, OutputCollect, select_model, export = create_files)
Step 7: Get Budget Allocation Based on the Selected Model
Results from budget allocation charts need further validation. Hence you should always check budget recommendations and discuss with your client.
You can apply robyn_allocator() function to every selected model to get the optimal budget mix that maximizes the response.
Following are the 2 scenarios that you can optimize for:
• Maximum historical response: It simulates the optimal budget allocation that will maximize effectiveness or response(Eg. Sales, revenue etc.) , assuming the same historical spend;
• Maximum response for expected spend: This simulates the optimal budget allocation to maximize response or effectiveness, where you can define how much you want to spend.
For “Maximum historical response” scenario, let’s consider below use case:
Case 1: When both total_budget & date_range are NULL.
Note: It’s default for last month’s spend.
#Get budget allocation based on the selected model above
# Check media summary for selected model
# NOTE: The order of constraints should follow:
AllocatorCollect1 <- robyn_allocator(
InputCollect = InputCollect,
OutputCollect = OutputCollect,
select_model = select_model,
date_range = NULL,
scenario = "max_historical_response",
channel_constr_low = 0.7,
channel_constr_up = c(1.2, 1.5, 1.5, 1.5, 1.5),
channel_constr_multiplier = 3,
export = create_files
# Print the budget allocator output summary
# Plot the budget allocator one-pager
One CSV file would be exported for further analysis/usage.
Once you’ve analyzed the model results plots from the list of best models, you can choose one model and pass it unique ID to select_model parameter. E.g providing model ID to parameter select_model =
“1_92_12” could be an example of a selected model from the list of best models in ‘OutputCollect$allSolutions’ results object.
Once you run the budget allocator for the final selected model, results will be plotted and exported under the same folder where the model plots had been saved.
You would see plots like below-
Fig: budget allocator chart
Interpretation of the 3 plots
1. Initial vs. Optimized Budget Allocation: This channel shows the new optimized recommended spend vs original spend share. You’ll have to proportionally increase or decrease the budget for the
respective channels of advertisement by analysing the difference between the original or optimized recommended spend.
2. Initial vs. Optimized Mean Response: In this chart too, we have optimized and original spend, but this time over the total expected response (for e.g., Sales). The optimized response is the total
increase in sales that you’re expecting to have if you switch budgets following the chart explained above i.e., increasing those with a better share for optimized spend and reducing spending for
those with lower optimized spend than the original spend.
3. Response Curve and Mean Spend by Channel: This chart displays the saturation effect of each channel. It shows how saturated a channel is ergo, suggests strategies for potential budget
reallocation. The faster the curves reach to horizontal/flat slope, or an inflection, the sooner they’ll saturate with each extra $ spent. The triangle denotes the optimized mean spend, while the
circle represents the original mean spend.
Step 8: Refresh Model Based on Selected Model and Saved Results
The two situations below are a good fit:
• Most data is new. For instance, If the earlier model has 200 weeks of data and 100 weeks new data is added.
• Add new input variables or features.
# Provide your InputCollect JSON file and ExportedModel specifications
json_file <- "E:/DataSciencePrep/MMM/RobynModel-inputs.json"
RobynRefresh <- robyn_refresh(
json_file = json_file,
dt_input = dt_simulated_weekly,
dt_holidays = dt_prophet_holidays,
refresh_iters = 1500,
refresh_trials = 2
refresh_steps = 14,
# Now refreshing a refreshed model following the same approach
json_file_rf1 <- "E:/DataSciencePrep/MMM/RobynModel-inputs.json"
RobynRefresh <- robyn_refresh(
json_file = json_file_rf1,
dt_input = dt_simulated_weekly,
dt_holidays = dt_prophet_holidays,
refresh_steps = 8,
refresh_iters = 1000,
refresh_trials = 2
# Continue with new select_model,InputCollect,,and OutputCollect values
InputCollectX <- RobynRefresh$listRefresh1$InputCollect
OutputCollectX <- RobynRefresh$listRefresh1$OutputCollect
select_modelX <- RobynRefresh$listRefresh1$OutputCollect$selectID
Note: Always keep in mind to run robyn_write() (manually or automatically) to export existing model first for versioning and other usage before refreshing the model.
Export the 4 CSV outputs in the folder for further analysis:
Robyn with its salient features like model calibration & refresh, marginal returns and budget allocation functions to produce faster, more accurate marketing mix modeling (MMM) outputs and business
insights does a great job. It reduces human bias in modeling process by automating most of the important tasks.
The 3 important takeaways of this article are as follows:
• With the advent of Nevergrad, Robyn finds the optimal hyperparameters without much human intervention.
• With advent of Nevergrad, Robyn finds the optimal hyperparameters without much human intervention.
• Robyn helps us capture new patterns in data with periodically updated MMM models.
• https://www.statisticshowto.com/nrmse/
• https://blog.minitab.com/en/understanding-statistics/why-the-weibull-distribution-is-always-welcome
• https://towardsdatascience.com/market-mix-modeling-mmm-101-3d094df976f9
• https://towardsdatascience.com/an-upgraded-marketing-mix-modeling-in-python-5ebb3bddc1b6
• https://engineering.deltax.com/articles/2022-09/automated-mmm-by-robyn
• https://facebookexperimental.github.io/Robyn/docs/analysts-guide-to-MMM
The media shown in this article is not owned by Analytics Vidhya and is used at the Author’s discretion.
Frequently Asked Questions
Q1. What is the Robyn package from Facebook?
A. The Robyn package from Facebook is a Python library for analyzing, visualizing, and monitoring data dependencies in computational pipelines. It helps understand and optimize data workflows by
providing insights into data lineage and dependencies within complex systems.
Q2. Is Robyn open-source?
A. Yes, Robin is an open-source package. It is available on GitHub under the Apache License 2.0, allowing users to access, modify, and contribute to the development of the package.
Q3. How do I update Robyn?
A. To update Robyn, you can use the pip package manager in Python. Simply run the following command in your terminal: “pip install –upgrade robyn”. This will upgrade your existing Robyn installation
to the latest version available.
Q4. What is mixed media modeling?
A. Mixed media modeling is a technique that combines different types of data, such as text, images, audio, and video, into a single model. It involves training machine learning models to understand
and generate meaningful insights from diverse forms of media, enabling more comprehensive analysis and understanding of complex datasets.
Responses From Readers | {"url":"https://www.analyticsvidhya.com/blog/2023/05/introduction-to-market-mix-model-using-robyn/","timestamp":"2024-11-09T09:05:01Z","content_type":"text/html","content_length":"414345","record_id":"<urn:uuid:15759edc-4b1a-4b40-9b1d-df3fffe2e19d>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00036.warc.gz"} |
[C5W1A1] wrong results of lstm_backward
I got the wrong values of results from the function lstm_backward. But the shapes of the results and all other corresponding functions are passed.
Below are my code and results. I’ve been stuck here for 2 days, please help me.
def lstm_backward(da, caches):
(caches, x) = caches
(a1, c1, a0, c0, f1, i1, cc1, o1, x1, parameters) = caches[0]
n_a, m, T_x = da.shape
n_x, m = x1.shape
dx = np.zeros((n_x, m, T_x))
da0 = np.zeros((n_a, m))
da_prevt = np.zeros((n_a, m))
dc_prevt = np.zeros((n_a, m))
dWf = np.zeros((n_a, n_a + n_x))
dWi = np.zeros((n_a, n_a + n_x))
dWc = np.zeros((n_a, n_a + n_x))
dWo = np.zeros((n_a, n_a + n_x))
dbf = np.zeros((n_a, 1))
dbi = np.zeros((n_a, 1))
dbc = np.zeros((n_a, 1))
dbo = np.zeros((n_a, 1))
for t in reversed(range(T_x)):
gradients = lstm_cell_backward(da[:,:,t] + da_prevt, dc_prevt, caches[t])
da_prevt = gradients['da_prev']
dc_prevt = gradients['dc_prev']
dx[:,:,t] = gradients['dxt']
dWf += gradients['dWf']
dWi += gradients['dWi']
dWc += gradients['dWc']
dWo += gradients['dWo']
dbf += gradients['dbf']
dbi += gradients['dbi']
dbc += gradients['dbc']
dbo += gradients['dbo']
da0 = da_prevt
gradients = {"dx": dx, "da0": da0, "dWf": dWf,"dbf": dbf, "dWi": dWi,"dbi": dbi,
"dWc": dWc,"dbc": dbc, "dWo": dWo,"dbo": dbo}
return gradients
gradients["dx"][1][2] = [ 0.01034214 1.03473735 -0.2398793 -0.43281115]
gradients["dx"].shape = (3, 10, 4)
gradients["da0"][2][3] = 0.5883931290038376
gradients["da0"].shape = (5, 10)
gradients["dWf"][3][1] = -0.02269017674887574
gradients["dWf"].shape = (5, 8)
gradients["dWi"][1][2] = 0.6099853844261891
gradients["dWi"].shape = (5, 8)
gradients["dWc"][3][1] = -0.013857139274558946
gradients["dWc"].shape = (5, 8)
gradients["dWo"][1][2] = 0.04772920545685257
gradients["dWo"].shape = (5, 8)
gradients["dbf"][4] = [-0.199665]
gradients["dbf"].shape = (5, 1)
gradients["dbi"][4] = [-0.7340795]
gradients["dbi"].shape = (5, 1)
gradients["dbc"][4] = [-0.56981661]
gradients["dbc"].shape = (5, 1)
gradients["dbo"][4] = [-0.24499124]
gradients["dbo"].shape = (5, 1)
Do you still need help with this issue?
Hi wziz,
What I see is that you have dx with 3 dimensions instead of dxt with two dimensions.
Have the exact same result.
Can you help with the pinpointing the issue?
Also, it’s kind of assumed the the last time step derivative dc_next is zero (which is the first entry into the lstm_cell_backward for the parameter dc_next). why is that?
I have the same code but I get the correct results… I would say that the problem begins in the previous function lstm_cell_backward (which was very painful to code, actually)
1 Like
Hi Santiago,
Welcome to the community.
Yes, you also need to keep in mind the framework that you use for back propagation. You need to start with sigmoid followed by tanh later.
My problem was at a_next initialization should be a_next = a0
def lstm_forward(x, a0, parameters):
Initialize a_next and c_next (≈2 lines)
a_next = a0 # <------ CHANGE HERE
c_next = np.zeros((n_a,m))
# loop over all time-steps
for t in range(T_x):
# Get the 2D slice 'xt' from the 3D input 'x' at time step 't'
xt = x[:,:,t]
# Update next hidden state, next memory state, compute the prediction, get the cache (≈1 line)
a_next, c_next, yt, cache = lstm_cell_forward(xt, a_next, c_next, parameters)
My code is the same as yours and I got the expected output. Are you sure you are running the cells in order, which is from the top to the bottom of the notebook?
I was running into a similar problem and what I found was that one of the calculations of my lstm_cell_backward() functions was incorrect. I carefully went back and compared the lstm_cell_backward()
output with the expected output and found that one of the values was incorrect. After fixing that value then lstm_backward() started working correctly.
I had the same problem but it was in the previews code. you should review again the lstm_cell_backward I just made a mistake in one value and it was the problem
I had dc_prev = {moderator edit}
instead of: dc_prev = {moderator edit}
i share you all my code
{Moderator Edit: Solution Code Removed}
And I delete all your code. This is totally unacceptable to share your code to help other learners. This will lead to suspending your account. So, do not share your code as this is against the
community Honor Code.
I’m apologize, I don’t do that again
My case is similar to @Witenberg . My lstm_forward was wrong although it passed the tests.
I accidentally initialize c_next to be zeros like part of c like
a_next = a0
c_next = c[:, :, T_x] # <== Wrong
But as mentioned in the notebook, setting one variable equal to the other is a “copy by reference”. So as the loop iterates, the c_next and c are totally messed up.
1 Like
This is it, I also set c_next as a reference to the c matrix/array, i.e. c_next = c[:,:,0] in the lstm_forward. After just create a newly initialized variable the answer is correct in the last
section lstm_backward.
Good tip! The problem was that in my lstm_cell_backward() the result was equal to the given solution; but later, in the lstm_backward() I had a mismatch with dx.shape… Thanks!
1 Like | {"url":"https://community.deeplearning.ai/t/c5w1a1-wrong-results-of-lstm-backward/8480","timestamp":"2024-11-04T01:58:24Z","content_type":"text/html","content_length":"46281","record_id":"<urn:uuid:50962c96-ecbf-48a4-9067-f41c23fd55ff>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00792.warc.gz"} |
Excel Formula for Python: Reflecting Total Current Amount
In this tutorial, we will learn how to write an Excel formula in Python that reflects the total current amount and allows us to add or subtract amounts from another sheet. This formula is useful when
you have data distributed across multiple sheets and you want to perform calculations based on those values.
To achieve this, we will use the SUM function to add the values from different sheets and then subtract a specific amount from the total current amount. By using cell references to specific cells in
different sheets, we can easily update the formula to reflect the values from different sources.
Let's consider an example to understand how this formula works. Suppose we have three sheets: Sheet1, Sheet2, and Sheet3.
In Sheet1, we have a value of 10 in cell A1. In Sheet2, we have a value of 5 in cell B2. And in Sheet3, we have a value of 3 in cell C3.
To calculate the total current amount, we can use the formula =SUM(Sheet1!A1, Sheet2!B2) - Sheet3!C3. This formula adds the values from Sheet1!A1 and Sheet2!B2 and then subtracts the value in Sheet3!
C3. In our example, the result would be 12.
You can customize this formula by changing the cell references to match your specific data. This flexibility allows you to perform calculations based on different sources and update the formula as
In conclusion, by using the Excel formula in Python, you can reflect the total current amount and add or subtract amounts from another sheet. This provides a powerful way to perform calculations and
analyze data distributed across multiple sheets in Excel using Python.
A Google Sheets formula
=SUM(Sheet1!A1, Sheet2!B2) - Sheet3!C3
Formula Explanation
This formula allows you to reflect the total current amount by adding or subtracting amounts from different sheets.
Step-by-step explanation
1. The SUM function is used to add the values from two different sheets. In this example, the values from Sheet1!A1 and Sheet2!B2 are added together.
2. The result of the SUM function is then subtracted by the value in Sheet3!C3. This allows you to subtract an amount from the total current amount.
3. By using references to specific cells in different sheets, you can easily update the formula to reflect the values from different sources.
For example, let's say we have the following values in different sheets:
The formula =SUM(Sheet1!A1, Sheet2!B2) - Sheet3!C3 would calculate the total current amount by adding the values from Sheet1!A1 (10) and Sheet2!B2 (5), and then subtracting the value in Sheet3!C3
(3). The result would be 12.
You can easily update the formula to include different cell references from different sheets to reflect the desired calculation. | {"url":"https://codepal.ai/excel-formula-generator/query/YzUxljdQ/excel-formula-for-python-total-current-amount","timestamp":"2024-11-07T13:57:42Z","content_type":"text/html","content_length":"93695","record_id":"<urn:uuid:a550ac14-a9b9-4e19-8ea1-ad1f66233291>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00820.warc.gz"} |
Extremely challenging challenge problem
Registered: 2019-05-24
Posts: 54
Extremely challenging challenge problem
If α is an automorphism of a finite field F, then
Is the above assertion true? If so, prove it; otherwise, give a counterexample.
Reason is like an open secret that can become known to anyone at any time; it is the quiet space into which everyone can enter through his own thought.
Registered: 2017-11-24
Posts: 693
Re: Extremely challenging challenge problem
Me, or the ugly man, whatever (3,3,6) | {"url":"https://mathisfunforum.com/viewtopic.php?pid=409566","timestamp":"2024-11-09T04:18:30Z","content_type":"application/xhtml+xml","content_length":"8972","record_id":"<urn:uuid:3f23530c-ebae-4e37-a00c-95e5729f91b0>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00137.warc.gz"} |
The rocket and the expelled gas are considered together - as a system. The total momentum of the system is the sum of the momentum of the exhaust gas and the momentum of the rocket - in opposite
directions, of course. Here, the term “rocket” refers to all of the mass except the exhaust gas: the rocket body (sometimes called “frame”), nose cone, astronauts, control systems, motor, payload,
unspent fuel, and anything else carried along with the motion.
In one dimension we can drop the vector notation and write
where m[r](t) and v[r](t) are the mass and velocity of the rocket at time t, and
m[gas](t) and v[gas](t) are the mass and velocity of the expelled gas at time t.
The quantities v[r](t), m[r](t), and m[gas](t) vary with time as long as the fuel is burning; not only is the rocket speeding up, but the mass of the rocket is decreasing and the mass of the exhaust
gas is increasing.
Although v[gas](t) was written for the general case as a function of time, in this case it isn’t, really. Once the gas is expelled from the rocket motor it has a constant velocity because there is
no longer a force acting on it. Therefore, the explicit notation can be dropped and the gas velocity written as v[gas]. | {"url":"https://www.mnealon.eosc.edu/RocketSciencePage1.htm","timestamp":"2024-11-06T05:21:20Z","content_type":"text/html","content_length":"161490","record_id":"<urn:uuid:dd25e10f-4c9a-44ba-bb46-f5cb3f661177>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00627.warc.gz"} |
113 research outputs found
We consider a class of map, recently derived in the context of cluster mutation. In this paper we start with a brief review of the quiver context, but then move onto a discussion of a related Poisson
bracket, along with the Poisson algebra of a special family of functions associated with these maps. A bi-Hamiltonian structure is derived and used to construct a sequence of Poisson commuting
functions and hence show complete integrability. Canonical coordinates are derived, with the map now being a canonical transformation with a sequence of commuting invariant functions. Compatibility
of a pair of these functions gives rise to Liouville's equation and the map plays the role of a B\"acklund transformation.Comment: 17 pages, 7 figures. Corrected typos and updated reference detail
In this paper, we study the properties of a nonlinearly dispersive integrable system of fifth order and its associated hierarchy. We describe a Lax representation for such a system which leads to two
infinite series of conserved charges and two hierarchies of equations that share the same conserved charges. We construct two compatible Hamiltonian structures as well as their Casimir functionals.
One of the structures has a single Casimir functional while the other has two. This allows us to extend the flows into negative order and clarifies the meaning of two different hierarchies of
positive flows. We study the behavior of these systems under a hodograph transformation and show that they are related to the Kaup-Kupershmidt and the Sawada-Kotera equations under appropriate Miura
transformations. We also discuss briefly some properties associated with the generalization of second, third and fourth order Lax operators.Comment: 11 pages, LaTex, version to be published in
Journal of Nonlinear Mathematical Physics, has expanded discussio
Based on the Kupershmidt deformation for any integrable bi-Hamiltonian systems presented in [4], we propose the generalized Kupershmidt deformation to construct new systems from integrable
bi-Hamiltonian systems, which provides a nonholonomic perturbation of the bi-Hamiltonian systems. The generalized Kupershmidt deformation is conjectured to preserve integrability. The conjecture is
verified in a few representative cases: KdV equation, Boussinesq equation, Jaulent-Miodek equation and Camassa-Holm equation. For these specific cases, we present a general procedure to convert the
generalized Kupershmidt deformation into the integrable Rosochatius deformation of soliton equation with self-consistent sources, then to transform it into a $t$-type bi-Hamiltonian system. By using
this generalized Kupershmidt deformation some new integrable systems are derived. In fact, this generalized Kupershmidt deformation also provides a new method to construct the integrable Rosochatius
deformation of soliton equation with self-consistent sources.Comment: 21 pages, to appear in Journal of Mathematical Physic
By solving the first-order algebraic field equations which arise in the dual formulation of the D=2 principal chiral model (PCM) we construct an integrated Lax formalism built explicitly on the dual
fields of the model rather than the currents. The Lagrangian of the dual scalar field theory is also constructed. Furthermore we present the first-order PDE system for an exponential parametrization
of the solutions and discuss the Frobenious integrability of this system.Comment: 24 page
The algebraic matrix hierarchy approach based on affine Lie $sl (n)$ algebras leads to a variety of 1+1 soliton equations. By varying the rank of the underlying $sl (n)$ algebra as well as its
gradation in the affine setting, one encompasses the set of the soliton equations of the constrained KP hierarchy. The soliton solutions are then obtained as elements of the orbits of the dressing
transformations constructed in terms of representations of the vertex operators of the affine $sl (n)$ algebras realized in the unconventional gradations. Such soliton solutions exhibit non-trivial
dependence on the KdV (odd) time flows and KP (odd and even) time flows which distinguishes them from the conventional structure of the Darboux-B\"{a}cklund Wronskian solutions of the constrained KP
hierarchy.Comment: LaTeX, 13pg
A Darboux transformation is constructed for the modified Veselov-Novikov equation.Comment: Latex file,8 pages, 0 figure
A solution of the Einstein's equations that represents the superposition of a Schwarszchild black hole with both quadrupolar and octopolar terms describing a halo is exhibited. We show that this
solution, in the Newtonian limit, is an analog to the well known H\'enon-Heiles potential. The integrability of orbits of test particles moving around a black hole representing the galactic center is
studied and bounded zones of chaotic behavior are found.Comment: 7 pages Revte
The generalized Zakharov-Shabat systems with complex-valued Cartan elements and the systems studied by Caudrey, Beals and Coifman (CBC systems) and their gauge equivalent are studies. This includes:
the properties of fundamental analytical solutions (FAS) for the gauge-equivalent to CBC systems and the minimal set of scattering data; the description of the class of nonlinear evolutionary
equations solvable by the inverse scattering method and the recursion operator, related to such systems; the hierarchies of Hamiltonian structures.Comment: 12 pages, no figures, contribution to the
NEEDS 2007 proceedings (Submitted to J. Nonlin. Math. Phys.
\We consider an inverse scattering problem for Schr\"odinger operators with energy dependent potentials. The inverse problem is formulated as a Riemann-Hilbert problem on a Riemann surface. A
vanishing lemma is proved for two distinct symmetry classes. As an application we prove global existence theorems for the two distinct systems of partial differential equations $u_t+(u^2/2+w)_x=0,
w_t\pm u_{xxx}+(uw)_x=0$ for suitably restricted, complementary classes of initial data
The quartic H\'enon-Heiles Hamiltonian $H = (P_1^2+P_2^2)/2+(\Omega_1 Q_1^2+\Omega_2 Q_2^2)/2 +C Q_1^4+ B Q_1^2 Q_2^2 + A Q_2^4 +(1/2)(\alpha/Q_1^2+\beta/Q_2^2) - \gamma Q_1$ passes the Painlev\'e
test for only four sets of values of the constants. Only one of these, identical to the traveling wave reduction of the Manakov system, has been explicitly integrated (Wojciechowski, 1985), while the
three others are not yet integrated in the generic case $(\alpha,\beta,\gamma)ot=(0,0,0)$. We integrate them by building a birational transformation to two fourth order first degree equations in the
classification (Cosgrove, 2000) of such polynomial equations which possess the Painlev\'e property. This transformation involves the stationary reduction of various partial differential equations
(PDEs). The result is the same as for the three cubic H\'enon-Heiles Hamiltonians, namely, in all four quartic cases, a general solution which is meromorphic and hyperelliptic with genus two. As a
consequence, no additional autonomous term can be added to either the cubic or the quartic Hamiltonians without destroying the Painlev\'e integrability (completeness property).Comment: 10 pages, To
appear, Theor.Math.Phys. Gallipoli, 34 June--3 July 200 | {"url":"https://core.ac.uk/search/?q=author%3A(Fordy%20A%20P)","timestamp":"2024-11-11T14:53:33Z","content_type":"text/html","content_length":"194059","record_id":"<urn:uuid:5922c1e7-444e-4fd0-8a70-7624fb4af16f>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00032.warc.gz"} |
Count rows that contain specific values
In this example, the goal is to count the number of rows in the data that contain the value in cell G4, which is 19. The main challenge in this problem is that the value might appear in any column,
and might appear more than once in the same row. If we wanted to simply count the total number of times a value appeared in a range, we could use the COUNTIF function. But we need a more advanced
formula to count rows that may contain multiple instances of the value. The explanation below reviews two options: one based on the MMULT function, and one based on the newer BYROW function.
Background study
MMULT option
One option for solving this problem is the MMULT function. The MMULT function returns the matrix product of two arrays, sometimes called the "dot product". The result from MMULT is an array that
contains the same number of rows as array1 and the same number of columns as array2. The MMULT function takes two arguments, array1 and array2, both of which are required. The column count of array1
must equal the row count of array2. In the example shown, the formula in G6 is:
Working from the inside out, the logical criteria used in this formula is:
where data is the named range B4:D15. This expression generates a TRUE or FALSE result for every value in data, and the double negative (--) coerces the TRUE FALSE values to 1s and 0s, respectively.
The result is an array of 1s and 0s like this:
Like the original data, this array is 12 rows by 3 columns (12 x 3) and is delivered directly to the MMULT function as array1. Array2 is derived with this snippet:
Which returns an array of three 1s like this:
This is the tricky and fun part of this formula. The COLUMN function is used for convenience as a way to generate a numeric array of the right size. To perform matrix multiplication with MMULT, the
column count in array1 (3) must equal the row count in array2 (3). COLUMN returns the 3-column array {2,3,4} which, when raised to the power of zero, becomes {1,1,1}. Next, the TRANSPOSE function
transposes the 1 x 3 array into a 3 x 1 array:
TRANSPOSE({1,1,1}) // returns {1;1;1}
With both arrays in place, the MMULT function runs and returns an array with 12 rows and 1 column, {2;0;1;0;0;0;0;2;0;0;1;1}. This array contains the count per row of cells that contain 19, and we
can use this data to solve the problem. Each non-zero number represents a row that contains the number 19, so we can convert non-zero values to 1s and sum up the result:
We check for non-zero entries with >0 and again coerce TRUE FALSE to 1 and 0 with a double negative (--) to get a final array inside SUM:
In this array, 1 represents a row that contains 19 and a 0 represents a row that does not contain 19. The SUM function returns a final result of 5, the count of all rows that contain the number 19.
BYROW option
The BYROW function applies a LAMBDA function to each row in a given array and returns one result per row as a single array. The purpose of BYROW is to process data in an array or range in a "by row"
fashion. For example, if BYROW is given an array with 12 rows, BYCOL will return an array with 12 results. In this example, we can use BYROW like this:
The BYROW function iterates through the named range data (B4:D15) one row at a time. At each row, BYCOL evaluates and stores the result of the supplied LAMBDA function:
The logic here checks for values in row that are equal to G4, which results in an array of TRUE and FALSE values. The TRUE and FALSE values are coerced to 1s and 0s with the double negative (--), and
the SUM function sums the result. Next, we check if the total from SUM is >0, and coerce that result to a 1 or 0. After BYROW runs, we have an array with one result per row, either a 1 or 0:
{1;0;1;0;0;0;0;1;0;0;1;1} // result from BYROW
The formula can now be simplified as follows:
=SUM({1;0;1;0;0;0;0;1;0;0;1;1}) // returns 5
In the last step, the SUM function sums the items in the array and returns a final result of 5.
Literal contains
To check for specific substrings (i.e. check to see if cells contain a specific text value) you can adjust the logic in the formulas above to use the ISNUMBER and SEARCH functions. For example, to
check if a value contains "apple" you can use:
This expression would replace data=G4 logic above like this:
See this example for more information on using ISNUMBER with SEARCH. | {"url":"https://exceljet.net/formulas/count-rows-that-contain-specific-values","timestamp":"2024-11-07T07:54:57Z","content_type":"text/html","content_length":"64785","record_id":"<urn:uuid:d158a78b-e7fe-48cd-b074-e6753d83386b>","cc-path":"CC-MAIN-2024-46/segments/1730477027957.23/warc/CC-MAIN-20241107052447-20241107082447-00236.warc.gz"} |
How is thread pullout strength calculated?
How is thread pullout strength calculated?
Generally to determine pull out strength of a thread you will basically calculate the shear area (pi * shear diameter * length of engagement), and multiply by the shear strength of the softer
component in the bolted joint.
How do you calculate effective thread length?
* MINIMUM Thread Length Formula: For bolts 6″ and shorter – twice the diameter plus 1/4″ (2D + 1/4″). Longer than 6″ – twice the diameter plus 1/2″ (2D + 1/2″). When bolts are too short for formula
thread length, thread will extend as close to head as practical.
How do you calculate shear strength from tensile strength?
calculations this relationship is taken into account. The Shear strength of a material under pure shear is usually 1/√3 (0.577) times its tensile yield strength in case of Von mises criterion and 0.5
times its tensile yield strength in case of Tresca criterion.
How is thread engagement calculated?
Length of thread engagement is measured by the length of interaction between the fastener and nut member. (i.e. nut or mating material for screw) For example a standard thread forming screw applied
in 10mm of material will have more length of thread engagement than the same thread forming screw in 8mm of material.
How do you calculate the shear strength of a screw?
The shear strength of all bolts = shear strength of one bolt x number of bolts • The bearing strength of the connecting / connected plates can be calculated using equations given by AISC
specifications. The tension strength of the connecting / connected plates can be calculated as discussed earlier in Chapter 2.
How do you calculate thread strength?
n = Number of threads per inch. Equation: This equation provides an approximate result by extrapolation on the thread stress area of a fastener.This equation is adequate for design applications on
engineering materials or less than 100 ksi ultimate strength.
How do you calculate the pitch of a thread?
– Root/bottom The bottom surface joining the two adjacent flanks of the thread – Flank/side The side of a thread surface connecting the crest and the root – Crest/top The top surface joining the two
sides, or flanks.
How do you calculate thread pitch?
How do you calculate pitch TPI? The TPI is, then, four because it measures “threads per inch.” To convert the pitch to millimeters, use the conversion that 1 inch equals 25.4 millimeters. You can
convert 26 TPI to inches per thread by dividing 1 by 26 to get 0.038, and then multiplying this by 25.4 to get a pitch of 0.98 millimeters.
How do you calculate tensile strength?
When the load is applied to the rebar,its strain gradually increases.
The proportion of stress and strain in this range is also known as your modules or modulus of elasticity.
Increases in the load,stress reaches point B called the yield point where the yielding of reinforcement is starting. | {"url":"https://corfire.com/how-is-thread-pullout-strength-calculated/","timestamp":"2024-11-04T22:10:33Z","content_type":"text/html","content_length":"38548","record_id":"<urn:uuid:b071df6f-6318-4ef2-80fd-5e5197d05a79>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00590.warc.gz"} |
How to create a variable?
This function requires the Financials and KPI-Analytics add-ons.
Variables in Boardeaser can be created in three ways:
• Predefined variables based on the sie4 files that are uploaded. Read more here.
• Download the KPI template, create a key figure as a variable, and enter the data for each time interval. Read more here.
• By entering a formula that includes accounts and KPI variables.
This article describes the last option above. The variables created in this way can be used in graphs and tables, i.e. in the same way as account variables and key figure variables.
Create variable
Click on "Financials"> "Data Management" in the left menu. Under the "Variables" tab, click "New variable".
Select variable
Either start from the formula for an existing variable or create an entirely new formula. If you want to start from an existing variable, click "Select variable" and select the one you want to start
from. The formula is then displayed in the "Formula" field.
Define variable
Whether you start from an existing variable or create a new one, you then enter the name (NOTE: the name must only contain letters between a-z and _, i.e., no å ä ö or spaces), description, and unit.
If you click "Private", you are the only user who can see the variable.
Click the field under "Formula" to enter the formula. When creating your new variable, you can use the usual calculations and other variables, just like when creating graphs and tables. Read more
about formulas here.
Under the "Formula" field, the data that the formula provides is displayed. When you are done, click "Save", and the variable will be saved.
Now you can use the variable when creating graphs and tables!
Create a variable that contains all revenue accounts:
• Name: total_revenue
• Formula: -{3000-3999}
Explanation: revenue accounts are accounts between 3000 and 3999. Entering 3000-3999 will sum all revenue accounts. Angle brackets {} are used to show change or month. Revenues are stored as negative
values in the sie4 file, but usually, you want them as positive values, so you need to put the minus sign first. | {"url":"https://support.boardeaser.com/hc/en-gb/articles/22212351818514-How-to-create-a-variable","timestamp":"2024-11-05T22:55:19Z","content_type":"text/html","content_length":"32343","record_id":"<urn:uuid:710e9014-8725-4f78-9a5d-40645d20443e>","cc-path":"CC-MAIN-2024-46/segments/1730477027895.64/warc/CC-MAIN-20241105212423-20241106002423-00271.warc.gz"} |
Linear Algebra/Propositions - Wikibooks, open books for an open world
The point at issue in an argument is the proposition. Mathematicians usually write the point in full before the proof and label it either Theorem for major points, Corollary for points that follow
immediately from a prior one, or Lemma for results chiefly used to prove other results.
The statements expressing propositions can be complex, with many subparts. The truth or falsity of the entire proposition depends both on the truth value of the parts, and on the words used to
assemble the statement from its parts.
For example, where ${\displaystyle P}$ is a proposition, "it is not the case that ${\displaystyle P}$ " is true provided that ${\displaystyle P}$ is false. Thus, "${\displaystyle n}$ is not prime" is
true only when ${\displaystyle n}$ is the product of smaller integers.
We can picture the "not" operation with a Venn diagram.
Where the box encloses all natural numbers, and inside the circle are the primes, the shaded area holds numbers satisfying "not ${\displaystyle P}$ ".
To prove that a "not ${\displaystyle P}$ " statement holds, show that ${\displaystyle P}$ is false.
Consider the statement form "${\displaystyle P}$ and ${\displaystyle Q}$ ". For the statement to be true both halves must hold: "${\displaystyle 7}$ is prime and so is ${\displaystyle 3}$ " is true,
while "${\displaystyle 7}$ is prime and ${\displaystyle 3}$ is not" is false.
Here is the Venn diagram for "${\displaystyle P}$ and ${\displaystyle Q}$ ".
To prove "${\displaystyle P}$ and ${\displaystyle Q}$ ", prove that each half holds.
A "${\displaystyle P}$ or ${\displaystyle Q}$ " is true when either half holds: "${\displaystyle 7}$ is prime or ${\displaystyle 4}$ is prime" is true, while "${\displaystyle 7}$ is not prime or ${\
displaystyle 4}$ is prime" is false. We take "or" inclusively so that if both halves are true "${\displaystyle 7}$ is prime or ${\displaystyle 4}$ is not" then the statement as a whole is true. (In
everyday speech, sometimes "or" is meant in an exclusive way— "Eat your vegetables or no dessert" does not intend both halves to hold— but we will not use "or" in that way.)
The Venn diagram for "or" includes all of both circles.
To prove "${\displaystyle P}$ or ${\displaystyle Q}$ ", show that in all cases at least one half holds (perhaps sometimes one half and sometimes the other, but always at least one).
An "if ${\displaystyle P}$ then ${\displaystyle Q}$ " statement (sometimes written "${\displaystyle P}$ materially implies ${\displaystyle Q}$ " or just "${\displaystyle P}$ implies ${\displaystyle
Q}$ " or "${\displaystyle P\implies Q}$ ") is true unless ${\displaystyle P}$ is true while ${\displaystyle Q}$ is false. Thus "if ${\displaystyle 7}$ is prime then ${\displaystyle 4}$ is not" is
true while "if ${\displaystyle 7}$ is prime then ${\displaystyle 4}$ is also prime" is false. (Contrary to its use in casual speech, in mathematics "if ${\displaystyle P}$ then ${\displaystyle Q}$ "
does not connote that ${\displaystyle P}$ precedes ${\displaystyle Q}$ or causes ${\displaystyle Q}$ .)
More subtly, in mathematics "if ${\displaystyle P}$ then ${\displaystyle Q}$ " is true when ${\displaystyle P}$ is false: "if ${\displaystyle 4}$ is prime then ${\displaystyle 7}$ is prime" and "if $
{\displaystyle 4}$ is prime then ${\displaystyle 7}$ is not" are both true statements, sometimes said to be vacuously true. We adopt this convention because we want statements like "if a number is a
perfect square then it is not prime" to be true, for instance when the number is ${\displaystyle 5}$ or when the number is ${\displaystyle 6}$ .
The diagram
shows that ${\displaystyle Q}$ holds whenever ${\displaystyle P}$ does (another phrasing is "${\displaystyle P}$ is sufficient to give ${\displaystyle Q}$ "). Notice again that if ${\displaystyle P}$
does not hold, ${\displaystyle Q}$ may or may not be in force.
There are two main ways to establish an implication. The first way is direct: assume that ${\displaystyle P}$ is true and, using that assumption, prove ${\displaystyle Q}$ . For instance, to show "if
a number is divisible by 5 then twice that number is divisible by 10", assume that the number is ${\displaystyle 5n}$ and deduce that ${\displaystyle 2(5n)=10n}$ . The second way is indirect: prove
the contrapositive statement: "if ${\displaystyle Q}$ is false then ${\displaystyle P}$ is false" (rephrased, "${\displaystyle Q}$ can only be false when ${\displaystyle P}$ is also false"). As an
example, to show "if a number is prime then it is not a perfect square", argue that if it were a square ${\displaystyle p=n^{2}}$ then it could be factored ${\displaystyle p=n\cdot n}$ where ${\
displaystyle n<p}$ and so wouldn't be prime (of course ${\displaystyle p=0}$ or ${\displaystyle p=1}$ don't give ${\displaystyle n<p}$ but they are nonprime by definition).
Note two things about this statement form.
First, an "if ${\displaystyle P}$ then ${\displaystyle Q}$ " result can sometimes be improved by weakening ${\displaystyle P}$ or strengthening ${\displaystyle Q}$ . Thus, "if a number is divisible
by ${\displaystyle p^{2}}$ then its square is also divisible by ${\displaystyle p^{2}}$ " could be upgraded either by relaxing its hypothesis: "if a number is divisible by ${\displaystyle p}$ then
its square is divisible by ${\displaystyle p^{2}}$ ", or by tightening its conclusion: "if a number is divisible by ${\displaystyle p^{2}}$ then its square is divisible by ${\displaystyle p^{4}}$ ".
Second, after showing "if ${\displaystyle P}$ then ${\displaystyle Q}$ ", a good next step is to look into whether there are cases where ${\displaystyle Q}$ holds but ${\displaystyle P}$ does not.
The idea is to better understand the relationship between ${\displaystyle P}$ and ${\displaystyle Q}$ , with an eye toward strengthening the proposition.
An if-then statement cannot be improved when not only does ${\displaystyle P}$ imply ${\displaystyle Q}$ , but also ${\displaystyle Q}$ implies ${\displaystyle P}$ . Some ways to say this are: "${\
displaystyle P}$ if and only if ${\displaystyle Q}$ ", "${\displaystyle P}$ iff ${\displaystyle Q}$ ", "${\displaystyle P}$ and ${\displaystyle Q}$ are logically equivalent", "${\displaystyle P}$ is
necessary and sufficient to give ${\displaystyle Q}$ ", "${\displaystyle P\iff Q}$ ". For example, "a number is divisible by a prime if and only if that number squared is divisible by the prime
The picture here shows that ${\displaystyle P}$ and ${\displaystyle Q}$ hold in exactly the same cases.
Although in simple arguments a chain like "${\displaystyle P}$ if and only if ${\displaystyle R}$ , which holds if and only if ${\displaystyle S}$ ..." may be practical, typically we show equivalence
by showing the "if ${\displaystyle P}$ then ${\displaystyle Q}$ " and "if ${\displaystyle Q}$ then ${\displaystyle P}$ " halves separately. | {"url":"https://en.m.wikibooks.org/wiki/Linear_Algebra/Propositions","timestamp":"2024-11-02T01:44:41Z","content_type":"text/html","content_length":"158018","record_id":"<urn:uuid:a7ae6a4e-a454-49f2-b72a-54d74ca4a2f3>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00112.warc.gz"} |
Altitude difference by barometric formula
This calculator calculates the height or altitude difference between two points using barometric formula or barometric leveling method.
I think no one will object to the statement that the air is thinner at an altitude of two kilometers, and the atmospheric pressure is less than at sea level. If we put these words in a scientific
form, it turns out that the pressure (density) of the gas depends on its altitude in a gravitational field. On this phenomenon method of barometric leveling is built.
Barometric leveling - the method of determining the height difference between two points by atmospheric pressure measured at these points. Since the atmospheric pressure and the altitude above the
sea level are also dependent on the weather, for example, on the water vapor content of air, if possible, the method is applied to make measurements at points with the smallest interval between the
measurements. The points themselves should not be located too far from each other.
The difference in altitude is calculated as follows.
There is a rather complicated formula of Laplace:
$h=18401,2(1+0,00366t)(1+0,378\frac{e}{p_0})(1+0,0026cos2\phi)(1+\beta h)lg \frac{p_0}{p_h}$
It is, in addition to temperature and pressure also takes into account the absolute humidity $e$and latitude of the measuring point,$\phi$ that is, in practice it seems to be not in use.
And use a simple Babinet formula
$h=8000\frac{2(p_0-p_h)}{p_0+p_h}(1+\alpha t)$,
where $\alpha$ - Gas expansion factor equal to $\frac{1}{273}$
Indeed, in an era without computers and calculators, even this formula was ... well, not difficult but hard for calculations. To determine the height difference, people used auxiliary barometric
level tables.
Barometric stage - the height at which we must ascend so the pressure drop by 1 mm Hg
That is, we took and simplified the Babinet formula to expression
$h=8000\frac{(1+\alpha t)}{p}$
and calculate h for different values of temperature and pressure.
We acquired tables similar to barometric pressure tables
Thus, by measuring, for example, the pressure difference at the average temperature t and the average pressure p, Meteorologists could find the value of the barometric stage from the table and
multiply it by the amount of pressure difference.
It is clear that the formula gives the result with a margin error, but at the same time, it is approved that the error does not exceed 0.1 - 0.5% of the measured altitude.
The barometric leveling method allows determining the height of a point above sea level without resorting to geodetic leveling.
In practice, the height of the point above sea level is determined using the closest ranging mark, height above sea level known.
For example, the ranging mark is at 156 meters. The barometer shows that the ranging mark at 748 mmHg, being transferred to the defined point, the barometer shows 751 mmHg. The average temperature is
15 degrees Celsius. Using the Babinet formula, obtain -33.78 m, i.e., the point below the ranging mark at 33.78 meters, and its height is approximately 122.22 m. Taking the average pressure of 748
mmHg and using the barometric table, we get -33.85, i.e., the height is approximately 122.15 m.
The calculator below illustrates everything said above.
URL copiada al portapapeles
Calculadoras similares
PLANETCALC, Altitude difference by barometric formula | {"url":"https://es.planetcalc.com/272/","timestamp":"2024-11-04T15:25:23Z","content_type":"text/html","content_length":"36739","record_id":"<urn:uuid:54de4e70-507f-4e5a-bfee-cf2ec569577c>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00561.warc.gz"} |
Borland Genetics
Help! My Segments Are So Sticky!
Back in the day, it used to be popular to refer to certain segments as “sticky” when they appeared to be passed down from generation to generation untouched. That sort of name-calling has reduced
greatly now that we have a clearer understanding of the statistical rules that our chromosomes follow as a result of random recombination. It turns out that the smaller a segment, the more likely it
is to escape the chopping block of recombination in each generation and instead either be passed to the child in full or not at all. Let’s take a look at some numbers and see how this plays out.
As our starting point, we’re going to go back to our definition of centiMorgan, as explained in my blog from a few weeks ago about the statistical impossibility of two full siblings not sharing any
DNA segments. If you missed that one, that’s OK, here’s the way I like to think about a cM: A cM is a unit that denotes a span of a chromosome that has exactly 1% likelihood of being split at least
once by recombination, within a single generation. Conversely, a 1 cM span of chromosome has a 99% chance of avoiding recombination per generation.
So from this definition, let’s look at a 7 cM segment on a chromosome in terms of “stickiness.” There are exactly three possibilities in our inheritance model: A) This 7 cM segment is passed intact
from parent to child; B) the segment is not passed at all and instead the parent passes genetic material from his or her opposite parent to the child; or C) at least one recombination occurs across
this span and the parent passes pieces of both copies of his or her chromosome across this span (say 4 cM from one grandparent and 3 cM from the other).
First, let’s calculate the odds of no recombination occurring across a 7 cM span of chromosome. We use an “AND” operator and multiply the probabilities of no recombination on each 1 cM span that
comprises the 7 cM in question. So the odds of no recombination across the 7 cM span is restated as the odds of no recombination on the first cM AND no recombination on the second cM AND no
recombination on the third cM, etc. In statistics, independent probabilities linked by an “AND” operator can be simply multiplied. Therefore, the odds of no recombination across a 7 cM span of
chromosome in a single generation = 0.99 * 0.99 * 0.99 * 0.99 * 0.99 * 0.99 * 0.99, which we can express in exponent notation as (0.99)^7, which according to my calculator is about 93%. The two
possibilities (inherited in the entirety and not inherited at all) must occupy this 93% equally (47% each, rounding to the nearest whole percent), whereas the odds of the segment getting the
chop-chop is only 7%. Therefore, we can say that a 7 cM segment is necessarily “sticky,” with the odds of recombination in a generation (as opposed to acting all sticky) is low. This is not a
property special to your 7 cM segment. Rather, this applies to all 7 cM segments, whether or not you have matches on them and whether or not endogamy is at play, and with total disregard for the age
of this segment (failing to recombine in a hundred years doesn’t make it any more likely to recombine in the next generation).
Now, let’s extend this concept to segments of different lengths in cM using Excel. Here’s the story from 7 cM all the way up to 40 cM:
cM Odds of no recombination in a generation Odds of at least one recombination across segment in a generation Odds of inheriting entire segment Odds of not inheriting segment at all
7 93% 7% 47% 47%
8 92% 8% 46% 46%
9 91% 9% 46% 46%
10 90% 10% 45% 45%
11 90% 10% 45% 45%
12 89% 11% 44% 44%
13 88% 12% 44% 44%
14 87% 13% 43% 43%
15 86% 14% 43% 43%
16 85% 15% 43% 43%
17 84% 16% 42% 42%
18 83% 17% 42% 42%
19 83% 17% 41% 41%
20 82% 18% 41% 41%
21 81% 19% 40% 40%
22 80% 20% 40% 40%
23 79% 21% 40% 40%
24 79% 21% 39% 39%
25 78% 22% 39% 39%
26 77% 23% 39% 39%
27 76% 24% 38% 38%
28 75% 25% 38% 38%
29 75% 25% 37% 37%
30 74% 26% 37% 37%
31 73% 27% 37% 37%
32 72% 28% 36% 36%
33 72% 28% 36% 36%
34 71% 29% 36% 36%
35 70% 30% 35% 35%
36 70% 30% 35% 35%
37 69% 31% 34% 34%
38 68% 32% 34% 34%
39 68% 32% 34% 34%
40 67% 33% 33% 33%
Can you guess why I stopped at 40 cM? It’s because that’s where a segment will exhibit equal probabilities of each of the 3 described scenarios, which I would consider to be not very sticky.
But there’s more to this story. Surely, “stickiness” relates somehow to the expected age of a segment. That is, let’s ask the question “How old is my sticky little 7 cM segment likely to be?” In this
calculation, we start with the premise that it’s inherited in its entirety from one parent and that it’s a valid segment of genetic material from just one copy of that parent’s chromosome. We’ve
already calculated that probability at 93%, but let’s switch our rounding to tenths of a percent for a bit more accuracy. Our 7 cM segment has a 93.2% chance of being inherited without recombination
in a generation. How about 2 generations? Well, it has to be passed down in one generation AND another, so our probability of a segment being at least two generations old is 0.932^2 = 86.9%. If we
continue with this drill, we will find the odds are still at 70.3% that the segment is at least 5 generations. Let’s switch from generations to years, since as genealogists we really care about
whether our matches are related to us in a historical timeframe when there are records with which we can build our trees. Let’s assume that the average generation span is 25 years, and accordingly,
400 years ago (around the beginning of the genealogical era) takes us back 16 generations. So what are the odds that our 7 cM segment is older than 400 years? The answer is going to shock some people
who insist that autosomal matches only go back 400 years. In fact, the odds of a 7 cM segment on my genome of exceeding 400 years in age is 30.2 %. That is to say that for any given 7 cM segment,
there’s about a 70% chance that all of your matches on that segment have a common ancestor who was born less than 400 years ago, but over 30% of the age of a segment this size is going to be further
Well, how far back? I’ve taken it upon myself to carry out some advanced statistical calculations with which I won’t bore you, but I’ll give you some interesting figures, and I’ll show you a few
colorful charts that might change the way you think about your DNA matches. First, let’s talk quartiles. For a 7 cM segment, we can divide the age ranges of our segment into statistical quartiles
representing an even 25% probability that the age of our segment is within 4 ranges. These quartiles are as follows: 0-100 years, 100-250 years, 250 to 500 years, and >500 years. Based on this, our
expectation value is 250 years, but it’s just as likely that the segment is over 500 years old as it is that the segment is say 100 to 250 years old. Let’s keep it going and talk 100 year ranges
instead of quartiles. I made a pie chart to show you the probability that a 7 cM segment dates back to some different time-frames:
Yes, you’re reading this correctly. There’s a 6% chance that our 7 cM segment has been passed down untouched for over 1000 years! Now that’s what I call a sticky segment. Eew! So what, it’s just 6%,
right? Well, we have over 7000 cM of real estate on our chromosomes and that’s not even counting the X (we’ll get to that in a bit). So, that’s an expected 60 little super-sticky segments where
you’re never ever going to find your common ancestor because he/she walked the earth over 1000 years ago. Now, when you see someone say from a traditionally endogamous population, and they have like
a zillion matches on a lot of their segments, I want you to understand this. Their common ancestor from which they all inherited their hyper-sticky segments will often have lived over 1000 years ago
and may be an ancestor of a large swath of the population (and many times over due to intermarriage among even distantly related descendants over time). That doesn’t make a segment like this any less
real though, just less useful for genealogy in terms of finding the most recent common ancestor you share with other matches thereon.
Next, let’s move on to 20 cM. Why 20 cM? Because that’s where Ancestry sets the threshold for shared matches. Here, the picture is very different, with quartiles (rounding to the nearest generation)
being 0-25 years, 25-75 years, and 75-175 years, and >175 years. It’s no surprise that Ancestry considers this to be the fourth cousin boundary, since the average fourth cousin shares an ancestor
born about 125 years prior, and since a 20 cM single-segment match is likely to be related at fourth cousin level or closer about 75.5% of the time! Here’s that same pie chart for a 20 cM segment:
Awesome. There’s now a 96% chance that our common ancestor for a shared segment of 20 cM lived within the past 400 years. Note, however, there is still a 1.8% chance that a 20 cM segment is over 500
years old!
Finally, the million dollar question that everybody’s asking. What about that crazy X chromosome? I heard those segments are ancient if they’re not at least 15 cM. Well, maybe so, but first let's
talk about whether they’re even real segments (IBD). One problem with the X chromosome is that some parts are poorly sampled, with very low SNP counts per cM. I personally recommend that a segment
have at least 75 tested SNPs per cM (and a minimum of 7 cM) before you can rely on it being IBD. This just ain’t happening on some parts of the X due to low sampling rate (SNP per cM). But let’s
assume we’ve got a nice segment with some good SNP density, but it’s not too long. Let’s take everyone’s favorite 15 cM threshold and see what kind of stats we get in the context of segment dating.
First, we need another tool in our arsenal, and I’m going to call it “effective generation span.” While a generation span in real life might be 25 years or so, the “effective generation span” of an X
chromosome is 37.5 years. Here’s why. Let’s talk about the last time an X chromosome segment had any chance of recombining. That would be when a female ancestor had it. Male’s X chromosomes aren’t
recombined when passed to their daughters because they only have one X and therefore nothing with which to combine. I’m ignoring PAR (pseudo-autosomal regions) because they’re puny and practically
useless for genealogy. So, the last chance an X segment had to recombine was in a donor’s mother, or in a donor’s father’s mother. Whether a segment was inherited from either of those two ancestors
we’ll assume is equally likely for purposes of our discussion, and I’ll assert that this is a reasonable assumption. So, from the last opportunity for recombination, there’s a 50% chance of one
generation (25 years) and a 50% chance of two generations (50 years). To calculate our effective generation span for the X, we simply take the weighted average: (0.5 * 25) + (0.5 * 50) = 37.5 years.
Then, we can use the same methodology as we did on the autosomal chromosomes to calculate the age ranges of our favorite 15 cM segment on the X. Turns out there’s about an 80% chance that such a
match shares a MRCA within 400 years, matching the common wisdom in our community that 15 cM is a nice place to start examining our X matches (given that our comparison to the other donor includes at
least 15 * 75 = 1125 SNPs). Here's the same pie chart for date ranges for a 15 cM X segment:
That’s all for this week. Despite the liberties I’ve taken using the word “sticky,” as you now know, there’s nothing inherently sticky about one segment vs. the other, but rather segments only appear
to “stick” because of their length in cM. Any apparent stickiness is simply a direct result of the statistical nature of DNA inheritance, and the phenomenon applies across the board to all small
If you’ve enjoyed this post, I encourage you to check out my website
where I’m accepting uploads to an autosomal database that focuses on making simple and powerful (and for the most part free) DNA reconstruction tools accessible to the average genetic genealogist.
1. Nice read Borland!
2. Very interesting! Thanks for this. And while I tell people (other eastern Polynesians like myself) who are predicted 2nd - 3rd cousin matches to me or my relatives to look at the largest segment
of at least 30cM in order to determine a true 2nd cousin relationship, this chart makes sense except for the 20cM.
Ancestry's shared matches are based on TOTAL shared. While we can have a good 20cM total shared, the number of segments can be as much as 5 segments (just looking at my own). So if say there are
5 segments, make it 3 segments (I had a lot of 3 and 2 segments), that's about 6.6cM.
Would love to see you work with my data! ;) Thanks again for this though, definitely enlightening!
3. Excellent, Kevin. Thanks for explaining this. I still don't like the term "sticky" since there's nothing to prevent recombination from selecting the other chromosomal segment in the same location
(as indicated in the column about odds of not passing on the segment at all). The probabilities you've given for various shared amounts of DNA and segment longevity is very helpful. It seems I
routinely get confronted by outliers. Recently on behalf of someone looking for his great grandfather's father, we approached the grandfather of a match who shares 118 cM. For the number of
generations, this indicated to us that this particular line was the correct one. But the the grandfather shared only 121 cM.
4. This is incredibly in-depth, but you've done a great job of explaining it. Thank you for this!
5. Great post, Kevin! Thank you for making this readable and understandable and for adding the graphics! This was a very helpful post.
6. Although the calculations in this piece are mathematically corect, I think they are conceptually wrong for genetic genealogy. The probabilities calculated are in a forward direction, answering
the question "If two people share an ancestor n generations ago what is the probability that they share a segment of x cM from that ancestor?" Generally that is not the question we are interested
in, we know for certain that two people share a segment of x cM and we want to know how long ago the ancestor was. This is the fundamental difference between this approach and the Speed and
Balding approach summarised here https://isogg.org/wiki/Identical_by_descent. An analogous question, with known relationship and question about inheritance, would be If A and B are siblings, what
is the probability they have the same colour eyes? (Answer: reasonably high). The converse is, a question with known genetics and unknown relationship: If A and B both have blue eyes how probable
is it that they are siblings? (Answer: quite low.)
1. This comment has been removed by the author.
7. I'd calculate the “effective generation span” of an X chromosome to be 25/3 (male X) + 25/3 (female X from mother) + 50/3 (female X from father) ⁼ 33.3 years.
1. This comment has been removed by the author.
2. I was working tonight on writing code to generalize my equations for creating the pie charts in the article, and I revisited your question. The calculation of 37.5 years as the "effective
generation span" considers the unique inheritance paths of the X chromosome and whether recombination is possible.
For a female, the X chromosome can be inherited from either parent:
When inherited from the mother, recombination is possible every generation, yielding a generational span of 25 years.
When inherited from the father, the X chromosome is passed intact from the paternal grandmother without recombination. This path effectively "skips" a generation, leading to a span of 50
The weighted average takes into account the equal probability (50%) of inheriting the X chromosome from each parent for a female. We calculate the average by multiplying the chances by the
years and then adding them together:
Chance of inheriting from the mother: 50% times 25 years equals 12.5 years.
Chance of inheriting from the father: 50% times 50 years equals 25 years.
Adding these together gives us 37.5 years as the weighted average.
This weighted average is representative of the "effective generation span" for an X chromosome segment, considering the distinct inheritance patterns and the potential for recombination.
The alternative scenario described would suggest dividing the years by three and summing them up, yielding approximately 33.33 years:
Inheriting a male X (from the mother): One third of 25 years equals approximately 8.33 years.
Inheriting a female X (from the mother): One third of 25 years equals approximately 8.33 years.
Inheriting a female X (from the father): One third of 50 years equals approximately 16.67 years.
While the alternative method is a valid mathematical approach, it doesn't capture the generational "skip" when the X chromosome is inherited from father to daughter without recombination.
Recombination and non-recombination should be equally weighted due to their equal probability. Therefore, 37.5 years more accurately reflects the "effective generation span" for the X
chromosome in the context we're discussing.
I hope this explains the reasoning behind the calculation. If you have more thoughts or questions, I'm open to continuing the discussion!
8. Would you grant permission to quote you and a chart to a Family Genealogy Group? Great blog post!
1. Sure, no problem.
2. Much appreciated.
9. This rather connects with my own experience. My wife is Spanish and she has two Arab matches
via her autosomal results, plus a Jewish one. And both Jews and Arabs were finally evicted from Spain over four hundred years ago. My mother from Northern Ireland has a match with an Icelandic
lady and the latter only has Icelandic connections in her family tree.
10. It would be interesting to look at how increased generation span affected the figures. (25yrs is insufficient in west Cornwall women where married at 26+ and your ancestor is on average her
middle child five+ years later). Would also be good to reflect on factors that affect recombination, e.g. maternal age and chromosome length. X chromosome in particular often does not recombine.
I guess I was disappointed not to see these caveats in the calculations. Though the general point made is useful.
11. It would be interesting to look at how increased generation span affected the figures. (25yrs is insufficient in west Cornwall women where married at 26+ and your ancestor is on average her
middle child five+ years later). Would also be good to reflect on factors that affect recombination, e.g. maternal age and chromosome length. X chromosome in particular often does not recombine.
I guess I was disappointed not to see these caveats in the calculations. Though the general point made is useful.
12. Iam glad you posted this / I only started doing ny Ancestry in Feb 2021 when I received my dna results since then I've had them labels stapled on me had them spread so must bull that they get me
banned off wikitree can you believe that a bunch of old people who claim to be professional genealogist branding 5his on someone behind the back I only found out by stumbling on to their
conversation left on a post by the time I saw it it was already to late everything snowballed the amount of incest connebts and you dad's not your dad etc that's just the being of it .. I wish I
had 9f seen this then / Ian still waiting for their evidence proving their claims / I've lots spark for it / theirs no point I understood the way you presented it way better than some other ones
I've seen so thanks
13. I have a DNA match with someone on Ancestry, where we only have single segment being shared, 106cm, we can dismiss 4 generations of common ancestors, as one family migrated thousands of miles
away in 1912, additionally, the daughter of the match has performed a DNA test on Ancestry, and matches 99cm with me, albeit, now in two segments, 92cm and 7cm. So surely this is sticky? My
question is, if only 7cm are lost in a generation by such a large sized segment, what figures can be extrapolated from your modelling? Are the segments that are unwilling to be recombined?
14. Could you explain why "a 1 cM span of chromosome has a 99% chance of avoiding recombination per generation"? I don't think I understand the math.
1. That's just the definition of a centi-Morgan, and why the unit of measurement has the prefix "centi-" in it. A cM span of a chromosome is a statistical unit specifically defined by having a 1
/100 chance of a recombination event across it in a generation.
15. Have you calculated the odds of segments longer than 40 cM being passed down a given number of generations? That would be of interest to me and the unknown commentator a couple of lines above.
1. I should probably turn the calculation to a tool on the Borland Genetics site if other people are interested in this kind of thing. My next "programming marathon" for the site will begin in
July and I'll put it on my list of ideas for new site content. Thanks!
2. This comment has been removed by the author.
3. Thanks Kevin. That would be great--there's no rush to respond. I tried running some numbers for a 47 cM shared segment (the size my dad shares with someone who I think could be a 4th cousin
twice removed). This matches' ancestor did have 15 or 16 kids that may have had offspring. If I am reading the formula right, the chance of inheriting the 47 cM segment intact is 62.5% for
each generation distant. So the odds of sharing a large segment are very low, but it's also hard to be confident in which generation the lines connect. I'm reading about a 37% chance that it
is at the 4C2R level versus something more distant (adding up the rows below 0.57% until they get close to 0). However, we have other shared segments in the 30-40 cM range with folks who are
5th cousins of this match, so that would seem to push the odds of the closer relationship much higher again. I'm think I'm imagining a single large segment versus of the WATO calculator....
P(Shared 47cm segment) Steps Relationship
62.50% 1 sibling
39.06% 2 sibling 1R
24.41% 3 1C
15.26% 4 1C1R
9.54% 5 2C
5.96% 6 2C1R
3.73% 7 3C
2.33% 8 3C1R
1.46% 9 4C
0.91% 10 4C1R
0.57% 11 5C
0.36% 12 5C1R
0.22% 13 6C
0.14% 14 6C1R
0.09% 15 7C
0.05% 16 7C1R
0.03% 17 8C
0.02% 18 8C1R
0.01% 19 9C
0.01% 20 9C1R
0.01% 21 10C
0.00% 22 10C1R
16. You are going to be getting a ton of questions from me! I'll start with this sentence: "There are exactly three possibilities in our inheritance model: A) This 7 cM segment is passed intact from
parent to child; B) the segment is not passed at all and instead the parent passes genetic material from his or her opposite parent to the child; or C) at least one recombination occurs across
this span and the parent passes pieces of both copies of his or her chromosome across this span (say 4 cM from one grandparent and 3 cM from the other)." You are saying that your computer model
has exactly 3 possibilities not that there are only 3 possibilities in reality right? The other possibilities is that the child receives somewhere between 37.5% and 74.99% of their DNA from each
parent and that might mean getting about half of that 7 cm chunk or it could mean getting none of it or it could mean getting the whole thing so long as the end result is that 37.5% to 74.99% of
their DNA came from that parent. So your computer model is only addressing the segments that are inherited unchanged?
1. You're absolutely right that the overall genetic inheritance from each parent to a child is a complex process with many possible outcomes. However, when we focus on a single genetic segment
and its inheritance through generations, we are indeed limited to three primary outcomes for that specific segment:
The segment is inherited in full, without recombination.
The segment is not inherited at all because another segment from the alternate chromosome is chosen.
The segment undergoes recombination, and only a part (or parts) of it is inherited.
This model allows us to calculate the statistical likelihood of a segment being inherited in a particular way and to estimate its age based on the known rates of recombination, given the
hindsight that the segment exists and is a certain fixed length in a descendant (as evident perhaps by match start/stop coordinates). While multiple recombination events can occur along a
chromosome, our study is concerned with the inheritance of one specific segment and what it can tell us about our ancestry.
For the purposes of this analysis and the genetic tools we use, we're examining the inheritance of this single segment to make conclusions about its age (the most recent estimated date when
that segment was cut to its current size). By doing so, we can infer information about the common ancestor from whom the segment was inherited.
I hope this explanation clarifies the focus of the model and the article. I should also point out that I wish to refine the probabilities by taking into account whether the segment in
question is the ONLY segment shared with a match. Because if so, for example, as a first-order perturbation to our model, we can probably safely remove the <100 years wedge of the pie chart
and renormalize the remaining percentages in the other wedges to add up to 100% since someone that closely related likely shares multiple segments. I say first order, because a higher order
analysis would take into account additional information that might be available to us besides the length of our segment under study, such as the number of and statistical lengths of all
shared segments with a match.
2. Thank you.
17. "Next, let’s move on to 20 cM. Why 20 cM? Because that’s where Ancestry sets the threshold for shared matches. Here, the picture is very different, with quartiles (rounding to the nearest
generation) being 0-25 years, 25-75 years, and 75-175 years, and >175 years. It’s no surprise that Ancestry considers this to be the fourth cousin boundary, since the average fourth cousin shares
an ancestor born about 125 years prior, and since a 20 cM single-segment match is likely to be related at fourth cousin level or closer about 75.5% of the time! Here’s that same pie chart for a
20 cM segment:" Ancestry's white paper does not actually address 4th cousins except tangentially they are not listed in their recombination chart. I have been studying this because Ancestry is
inconsistent in what relationship categories are assigned to people who share between 9 and 18 centimorgans - their main match page frustratingly misleadingly labels these matches as I think 6th
to 8th cousin or 5th to distant cousin, but when you click on the individual match a page opens up with it's long list of category batches which are really just all the categories that belong to
various degrees of relatedness. At the top ranked batch for 9-18 cM is always exactly as it should be the 9th degree of relatedness because there are exactly specifically and pointedly nine times
DNA has to divide on the path between Tester 1 and Tester 2 in all those relationship categories. 4th cousin is in the top ranked batch and I have many known source documented 4th cousin matches
between 9 and 18 centimorgans because that is .20% of their max centimorgans of 6600. 20 cM is an 8th degree relationship where DNA only divides 8 times. The only type of 4th cousin whose DNA
divides 8 times is when they descend from a set of identical twins - basically you have a path that is 9 people long or 9 division instances long but dna does not divide for the twin sibling so
therefore the 4th cousins end up sharing more DNA as if they are 1/2 3rd cousins. A 20 cm amount of shared DNA is not generating Ancestry estimates of 4th cousin (although if they added one more
line to their meiosis chart in the white paper, 20 cM is what you'd get at 9 meiosis - problem with their meiosis chart is that they call people to their siblings 2 meiosis as if they are 25% 2nd
degree relatives and they are not they are 1st degree relatives. Ancestry's white paper and their description of siblings as 2nd degree relatives differs from the definitions of 1st degree
relationships at the Human Genome Project and also with Gina the federal act on genetic privacy. | {"url":"https://borlandgenetics.blogspot.com/2020/06/help-my-segments-areso-sticky-back-in.html","timestamp":"2024-11-02T21:09:11Z","content_type":"text/html","content_length":"435710","record_id":"<urn:uuid:634b1fac-5509-46a8-b535-ab36937a9d30>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00795.warc.gz"} |
Overview of axes_grid1 toolkit
Controlling the layout of plots with the axes_grid toolkit.
What is axes_grid1 toolkit?
axes_grid1 is a collection of helper classes to ease displaying (multiple) images with matplotlib. In matplotlib, the axes location (and size) is specified in the normalized figure coordinates, which
may not be ideal for displaying images that needs to have a given aspect ratio. For example, it helps if you have a colorbar whose height always matches that of the image. ImageGrid, RGB Axes and
AxesDivider are helper classes that deals with adjusting the location of (multiple) Axes. They provides a framework to adjust the position of multiple axes at the drawing time. ParasiteAxes provides
twinx(or twiny)-like features so that you can plot different data (e.g., different y-scale) in a same Axes. AnchoredArtists includes custom artists which are placed at some anchored position, like
the legend.
Demo Axes Grid
A class that creates a grid of Axes. In matplotlib, the axes location (and size) is specified in the normalized figure coordinates. This may not be ideal for images that needs to be displayed with a
given aspect ratio. For example, displaying images of a same size with some fixed padding between them cannot be easily done in matplotlib. ImageGrid is used in such case.
Simple Axesgrid
• The position of each axes is determined at the drawing time (see AxesDivider), so that the size of the entire grid fits in the given rectangle (like the aspect of axes). Note that in this
example, the paddings between axes are fixed even if you changes the figure size.
• axes in the same column has a same axes width (in figure coordinate), and similarly, axes in the same row has a same height. The widths (height) of the axes in the same row (column) are scaled
according to their view limits (xlim or ylim).
Simple Axes Grid
• xaxis are shared among axes in a same column. Similarly, yaxis are shared among axes in a same row. Therefore, changing axis properties (view limits, tick location, etc. either by plot commands
or using your mouse in interactive backends) of one axes will affect all other shared axes.
When initialized, ImageGrid creates given number (ngrids or ncols * nrows if ngrids is None) of Axes instances. A sequence-like interface is provided to access the individual Axes instances (e.g.,
grid[0] is the first Axes in the grid. See below for the order of axes).
ImageGrid takes following arguments,
Name Default Description
nrows_ncols number of rows and cols. e.g., (2,2)
ngrids None number of grids. nrows x ncols if None
direction "row" increasing direction of axes number. [row|column]
axes_pad 0.02 pad between axes in inches
add_all True Add axes to figures if True
share_all False xaxis & yaxis of all axes are shared if True
aspect True aspect of axes
label_mode "L" location of tick labels thaw will be displayed. "1" (only the lower left axes), "L" (left most and bottom most axes), or "all".
cbar_mode None [None|single|each]
cbar_location "right" [right|top]
cbar_pad None pad between image axes and colorbar axes
cbar_size "5%" size of the colorbar
axes_class None
direction of increasing axes number. For "row",
grid[0] grid[1]
grid[2] grid[3]
For "column",
grid[0] grid[2]
grid[1] grid[3]
You can also create a colorbar (or colorbars). You can have colorbar for each axes (cbar_mode="each"), or you can have a single colorbar for the grid (cbar_mode="single"). The colorbar can be placed
on your right, or top. The axes for each colorbar is stored as a cbar_axes attribute.
The examples below show what you can do with ImageGrid.
Demo Axes Grid
AxesDivider Class
Behind the scene, the ImageGrid class and the RGBAxes class utilize the AxesDivider class, whose role is to calculate the location of the axes at drawing time. While a more about the AxesDivider is
(will be) explained in (yet to be written) AxesDividerGuide, direct use of the AxesDivider class will not be necessary for most users. The axes_divider module provides a helper function
make_axes_locatable, which can be useful. It takes a existing axes instance and create a divider for it.
ax = subplot(1,1,1)
divider = make_axes_locatable(ax)
make_axes_locatable returns an instance of the AxesLocator class, derived from the Locator. It provides append_axes method that creates a new axes on the given side of ("top", "right", "bottom" and
"left") of the original axes.
colorbar whose height (or width) in sync with the master axes
Simple Colorbar
scatter_hist.py with AxesDivider
The "scatter_hist.py" example in mpl can be rewritten using make_axes_locatable.
axScatter = subplot(111)
axScatter.scatter(x, y)
# create new axes on the right and on the top of the current axes.
divider = make_axes_locatable(axScatter)
axHistx = divider.append_axes("top", size=1.2, pad=0.1, sharex=axScatter)
axHisty = divider.append_axes("right", size=1.2, pad=0.1, sharey=axScatter)
# the scatter plot:
# histograms
bins = np.arange(-lim, lim + binwidth, binwidth)
axHistx.hist(x, bins=bins)
axHisty.hist(y, bins=bins, orientation='horizontal')
See the full source code below.
Scatter Hist
The scatter_hist using the AxesDivider has some advantage over the original scatter_hist.py in mpl. For example, you can set the aspect ratio of the scatter plot, even with the x-axis or y-axis is
shared accordingly.
The ParasiteAxes is an axes whose location is identical to its host axes. The location is adjusted in the drawing time, thus it works even if the host change its location (e.g., images).
In most cases, you first create a host axes, which provides a few method that can be used to create parasite axes. They are twinx, twiny (which are similar to twinx and twiny in the matplotlib) and
twin. twin takes an arbitrary transformation that maps between the data coordinates of the host axes and the parasite axes. draw method of the parasite axes are never called. Instead, host axes
collects artists in parasite axes and draw them as if they belong to the host axes, i.e., artists in parasite axes are merged to those of the host axes and then drawn according to their zorder. The
host and parasite axes modifies some of the axes behavior. For example, color cycle for plot lines are shared between host and parasites. Also, the legend command in host, creates a legend that
includes lines in the parasite axes. To create a host axes, you may use host_subplot or host_axes command.
Example 1. twinx
Parasite Simple
Example 2. twin
twin without a transform argument assumes that the parasite axes has the same data transform as the host. This can be useful when you want the top(or right)-axis to have different tick-locations,
tick-labels, or tick-formatter for bottom(or left)-axis.
ax2 = ax.twin() # now, ax2 is responsible for "top" axis and "right" axis
ax2.set_xticks([0., .5*np.pi, np.pi, 1.5*np.pi, 2*np.pi])
ax2.set_xticklabels(["0", r"$\frac{1}{2}\pi$",
r"$\pi$", r"$\frac{3}{2}\pi$", r"$2\pi$"])
Simple Axisline4
A more sophisticated example using twin. Note that if you change the x-limit in the host axes, the x-limit of the parasite axes will change accordingly.
Parasite Simple2
It's a collection of artists whose location is anchored to the (axes) bbox, like the legend. It is derived from OffsetBox in mpl, and artist need to be drawn in the canvas coordinate. But, there is a
limited support for an arbitrary transform. For example, the ellipse in the example below will have width and height in the data coordinate.
Simple Anchored Artists
mpl_toolkits.axes_grid1.inset_locator provides helper classes and functions to place your (inset) axes at the anchored position of the parent axes, similarly to AnchoredArtist.
Using mpl_toolkits.axes_grid1.inset_locator.inset_axes(), you can have inset axes whose size is either fixed, or a fixed proportion of the parent axes. For example,:
inset_axes = inset_axes(parent_axes,
width="30%", # width = 30% of parent_bbox
height=1., # height : 1 inch
loc='lower left')
creates an inset axes whose width is 30% of the parent axes and whose height is fixed at 1 inch.
You may creates your inset whose size is determined so that the data scale of the inset axes to be that of the parent axes multiplied by some factor. For example,
inset_axes = zoomed_inset_axes(ax,
0.5, # zoom = 0.5
loc='upper right')
creates an inset axes whose data scale is half of the parent axes. Here is complete examples.
Inset Locator Demo
For example, zoomed_inset_axes() can be used when you want the inset represents the zoom-up of the small portion in the parent axes. And mpl_toolkits/axes_grid/inset_locator provides a helper
function mark_inset() to mark the location of the area represented by the inset axes.
Inset Locator Demo2
RGB Axes
RGBAxes is a helper class to conveniently show RGB composite images. Like ImageGrid, the location of axes are adjusted so that the area occupied by them fits in a given rectangle. Also, the xaxis and
yaxis of each axes are shared.
from mpl_toolkits.axes_grid1.axes_rgb import RGBAxes
fig = plt.figure()
ax = RGBAxes(fig, [0.1, 0.1, 0.8, 0.8])
r, g, b = get_rgb() # r,g,b are 2-d images
ax.imshow_rgb(r, g, b,
origin="lower", interpolation="nearest")
Simple Rgb
The axes_divider module provides helper classes to adjust the axes positions of a set of images at drawing time.
• axes_size provides a class of units that are used to determine the size of each axes. For example, you can specify a fixed size.
• Divider is the class that calculates the axes position. It divides the given rectangular area into several areas. The divider is initialized by setting the lists of horizontal and vertical sizes
on which the division will be based. Then use new_locator(), which returns a callable object that can be used to set the axes_locator of the axes.
First, initialize the divider by specifying its grids, i.e., horizontal and vertical.
for example,:
rect = [0.2, 0.2, 0.6, 0.6]
horiz=[h0, h1, h2, h3]
vert=[v0, v1, v2]
divider = Divider(fig, rect, horiz, vert)
where, rect is a bounds of the box that will be divided and h0,..h3, v0,..v2 need to be an instance of classes in the axes_size. They have get_size method that returns a tuple of two floats. The
first float is the relative size, and the second float is the absolute size. Consider a following grid.
h0,v2 h1 h2 h3
• v0 => 0, 2
• v1 => 2, 0
• v2 => 3, 0
The height of the bottom row is always 2 (axes_divider internally assumes that the unit is inches). The first and the second rows have a height ratio of 2:3. For example, if the total height of the
grid is 6, then the first and second row will each occupy 2/(2+3) and 3/(2+3) of (6-1) inches. The widths of the horizontal columns will be similarly determined. When the aspect ratio is set, the
total height (or width) will be adjusted accordingly.
The mpl_toolkits.axes_grid1.axes_size contains several classes that can be used to set the horizontal and vertical configurations. For example, for vertical configuration one could use:
from mpl_toolkits.axes_grid1.axes_size import Fixed, Scaled
vert = [Fixed(2), Scaled(2), Scaled(3)]
After you set up the divider object, then you create a locator instance that will be given to the axes object.:
locator = divider.new_locator(nx=0, ny=1)
The return value of the new_locator method is an instance of the AxesLocator class. It is a callable object that returns the location and size of the cell at the first column and the second row. You
may create a locator that spans over multiple cells.:
locator = divider.new_locator(nx=0, nx=2, ny=1)
The above locator, when called, will return the position and size of the cells spanning the first and second column and the first row. In this example, it will return [0:2, 1].
See the example,
Simple Axes Divider2
You can adjust the size of each axes according to its x or y data limits (AxesX and AxesY).
Simple Axes Divider3 | {"url":"https://matplotlib.org.cn/tutorials/toolkits/axes_grid.html","timestamp":"2024-11-13T01:28:01Z","content_type":"text/html","content_length":"130409","record_id":"<urn:uuid:0125850e-4920-42d0-953b-b09c5ccab977>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00153.warc.gz"} |
TGD diary{{post.title}}TGD diary
TGD predicts revolution in quantum theory based on three new principles.
1. ZEO solving the basic paradox of quantum measurement theory. Ordinary ("big") state function reduction involves time reversal forcing a generalization of thermodynamics and leading to a theory of
quantum self-organization and self-organized quantum criticality (homeostasis in living matter).
2. Phases of ordinary matter labelled by effective Planck constant h[eff]=nh[0] identified as dark matter and explaining the coherence of living matter in terms of dark matter at magnetic body
serving as a master, and predicting quantum coherence in all scales at the level of magnetic bodies. h[eff]/h[0]=n has interpretation as the dimension for an extension of rationals and is a
measure of algebraic complexity. Evolution corresponds to the increase of n.
Extensions of rationals are associated with adelic physics providing description of sensory experience in terms of real physics and of cognition in terms of p-adic physics. Central notion is
cognition representation providing unique discretization of X^4 in terms of points with imbedding space coordinates in the extension of rationals considered M^8-H duality realizes the hierarchy
of rational extensions and assigns them to polynomials defining space-time regions at the level of M^8 and mapped to minimal surfaces in H by M8-H duality.
3. The replacement of the unitary S-matrix with the Kähler metric of the Kähler space defined by WCW spinor fields satisfying the analog of unitarity and predicting positive definite transition
probabilities defining matrix in Teichmueller space. Einstein's geometrization of classical physics extends to the level of state space, Equivalence Principle generalizes, and interactions are
coded by the geometry of the state space rather than by an ad hoc unitary matrix. Kähler geometry for the spinor bundle of WCW has Riemann connection only for a maximal group of isometries
identified as super-symplectic transformations (SS). This makes the theory unique and leads to explicit analogs of Feynman rules and to a proof that theory is free of divergences.
In this article the third principle, which is new, is formulated and some of its consequences are discussed. The detailed formulation allows understanding of how normal ordering divergences and other
divergences cancel.
See the article Zero energy ontology, hierarchy of Planck constants, and Kähler metric replacing unitary S-matrix: three pillars of new quantum theory or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.
The understanding of the unitarity of the S-matrix has remained a major challenge of TGD for 4 decades. It has become clear that some basic principle is missing. Assigning S-matrix to a unitary
evolution works in non-relativistic theory but fails already in generic QFT. The solution of the problem turned out to be extremely simple. Einstein's great vision was to geometrize gravitation by
reducing it to the curvature of space-time. Could the same recipe work for quantum theory? Could the replacement of the flat Kähler metric of Hilbert space with a non-flat one allow to identify
unitary S-matrix as a geometric property of Hilbert space?
An amazingly simple argument demonstrates that one can construct scattering probabilities from the matrix elements of Kähler metric and assign to the Kähler metric a unitary S-matrix assuming that
some additional conditions guaranteeing that the probabilities are real and non-negative are satisfied. If the probabilities correspond to the real part of the complex analogs of probabilities, it is
enough to require that they are non-negative: complex analogs of probabilities would define the analog of Teichmueller matrix. Teichmueller space parameterizes the complex structures of space: could
the allowed WCW K\"ahler metrics- or rather the associated complex probability matrices - correspond to complex structures for some space? By the strong from of holography, the most natural candidate
would be Cartesian product of Teichmueller spaces of partonic 2 surfaces with punctures and string world sheets.
Under some additional conditions one can assign to Kähler metric a unitary S-matrix but this does not seem necessary. The experience with loop spaces suggests that for infinite-D Hilbert spaces the
existence of non-flat Kähler metric requires a maximal group of isometries. Hence one expects that the counterpart of S-matrix is highly unique.
In the TGD framework the world of classical worlds (WCW) has Kähler geometry allowing spinor structure. WCW spinors correspond to Fock states for second quantized spinors at space-time surface and
induced from second quantized spinors of the imbedding space. Scattering amplitudes would correspond to the Kähler metric for the Hilbert space bundle of WCW spinor fields realized in zero energy
ontology and satisfying Teichmueller condition guaranteeing non-negative probabilities.
Equivalence Principle generalizes to level of WCW and its spinor bundle. In ZEO one can assign also to the Kähler space of zero energy states spinor structure and this suggests strongly an infinite
hierarchy of second quantizations starting from space-time level, continuing at the level of WCW, and continuing further at the level of the space of zero energy states. This would give an
interpretation for an old idea about infinite primes asan infinite hierarchy of second quantizations of an arithmetic QFT.
See the article The analog of unitary S-matrix from a curved K\"ahler geometry of the space of WCW spinor fields or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD.
The pulsations of Earth - kind of mini earthquakes - occurring with a period of 26 s represent a mysterious phenomenon. In TGD framework the interpretation would rely on the notions of magnetic body
(MB) controlling ordinary matter, and dark matter as phases of ordinary matter labelled by h[eff]=nh[0] and giving rise to quantum coherence at MBs in all scales. The strange findings about
earthquakes suggest that they correspond to macroscopic quantum jumps ("big" state function reductions (BSRs)) changing the arrow of time. Also classically the earthquake corresponds to a
discontinuous process in which tectonic plates slide with respect to each other so that the identification as a macroscopic BSFR is natural in TGD framework. Could the periodic mini earthquakes
correspond to a sequence of BSFRs?
Deep ocean waves hitting the shore should somehow induce this periodic microseism as a sliding of the tectonic plates with respect to each other. If there is a lattice like structure of
incompressible cylindrical plates, the compression by sea waves arriving at shore induces a volume preserving vertical stretching of these cylinders inducing the detected Rayleigh wave.
Cyclotron periods of ions at MB are quantized and 26 s could be understood as a resonance period for the coupling between the tectonic dynamics and that of MB. The problem is that the periods
associated with the deep ocean waves are below 20 s so that a linear coupling preserving frequency does not allow understanding of the 26 s period. However, non-linear coupling allows period doubling
at the limit of chaos. Could 26 s period be seen as 8:th period doubling of
T=.1 s which corresponds to alpha rhythm in EEG and fundamental biorhythm, the secondary p-adic time scale of electron, and the cyclotron frequency of iron ion in the endogenous magnetic field B[end]
=(2/5)B[E] identified as monopole flux part of Earth's magnetic field B[E] and playing a key role in TGD inspired quantum biology?
See the article 26 second pulsation of Earth: analog for 8^th period doubling of EEG alpha rhythm?.
For a summary of earlier postings see Latest progress in TGD.
I attach below the introduction of the article "Homeostasis as self-organized quantum criticality" written together with Reza Rastmanesh. I have dropped references. They can be found from the article
which I shall add to Research Gate soon.
This article started as an attempt to understand the properties of cold shock proteins (CSPs) and heat shock proteins (HSPs) in TGD framework. As a matter of fact , these proteins have great deal of
similarity and have much more general functions, so it is easier to talk about stress proteins (SPs) having two different modes of operation. time
As we proceed, it will be revealed that this issue is only one particular facet of a much bigger problem: how self-organized quantum criticality (SOQC) is possible? Criticality means by definition
instability but SOQC is stable, which seems to be in conflict with the standard thermodynamics. In fact, living systems as a whole seem to be quantum criticalt and manage to stay near criticality,
which means SOQC. Note that the self-organized criticality (SOC) is generalized to SOQC.
Topological Geometrodynamics (TGD) is a 43 year old proposal for a unification of fundamental interactions. Zero energy ontology (ZEO) is basic aspect of quantum TGD and allows to extend quantum
measurement theory to a theory of consciousness and of living systems. ZEO also leads to a quantum theory of self-organization predicting both arrows of time. Could ZEO make SOQC possible as well?
Summary of the basic properties of CSPs and HSPs
Let's consider a summary of CSPs and HSPs or briefly SPs.
1. There is a large variety of cold shock proteins (CSP) and heat shock proteins (HSPs). CSPs and HSPs are essentially the same proteins and labelled by HSPX, where X denotes the molecular weight of
the protein in kDaltons. The value range of X includes the values {22,60,70,90,104,110} and HSPs are classified into 6 families: small HSPs, HSPX, X ∈ {40,60,70,90,110}. At least HSP70 and
HSP90 have ATPase at their end whereas HSP60 has ATP binding site. CSPs and HSPs consist of about 10^3-10^4 amino acids so that X varies by one order of magnitude.
Their lengths in the un-folded active configuration are below 1 micrometer. CSPs/HSPs are expressed when the temperature of the organism is reduced /increased from the physiological temperature.
CSPs possess cold-shock domains consisting of about 70-80 amino-acids thought to be crucial for their function. Part of the domain is similar to the so called RNP-1 RNA-binding motif. In fact, it
has turned that CSP and HSP are essentially the same object and stress protein (SP) is a more appropriate term.
Wikipedia article about cold shock domain mentions Escherichia Coli as an example. When the temperature is reduced from 37 C to 10 C, there is 4-5 hours lag phase after which growth is resumed at
a reduced rate. During lag phase expression of around 13 proteins containing cold shock domains is increased 2-10 fold. CSPs are thought to help the cell to survive in temperatures lower than
optimum growth temperature, by contrast with HSPs, which help the cell to survive in temperatures greater than the optimum, possibly by condensation of the chromosome and organization of the
prokaryotic nucleoid. What is the mechanism behinds SP property is the main question.
2. SPs have a multitude of functions involved with the regulation, maintenance and healing of the system. They appear in stress situations like starvation, exposure to cold or heat or to UV light,
during wound healing or tissue remodeling, and during the development of the embryo. SPs can act as chaperones and as ATPAses.
SPs facilitate translation, and protein folding in these situations, which suggests that they are able to induce local heating/cooling of the molecules involved in these processes. CSPs could be
considered like ovens and HSPs like coolants; systems with very large heat capacity acting as a heat bath and therefore able to perform temperature control. SPs serve as kind of molecular
blacksmiths - or technical staff - stabilizing new proteins to facilitate correct folding and helping to refold damaged proteins. The blacksmith analogy suggests that this involves a local
"melting" of proteins making it possible to modify them.
What "melting" could mean in this context? One can distinguish between denaturation in which the folding ability is not lost and melting in which it is lost. Either local denaturation or even
melting would be involved depending on how large the temperature increase is. In a aqueous environment the melting of water surrounding the protein as splitting of hydrogen bonds is also
involved. One could also speak also about local unfolding of protein.
3. There is evidence for large change Δ C[p] of heat capacity C[p] (C[p]= dE/dT for pressure changing feed of heat energy) for formation ion nucleotide-CSP fusion. This could be due to the high C[p]
of CSP. The value of heat capacity of SPs could be large only in vivo, not in vitro.
4. HSPs can appear even in hyper-thermophiles living in very hot places. This suggests that CSPs and HSPs are basically identical - more or less - but operate in different modes. CSPs must be able
to extract metabolic energy and they indeed act as ATPases. HSPs must be able to extract thermal energy. If they are able to change their arrow of time as ZEO suggests, they can do this by
dissipating with a reversed arrow of time.
To elucidate the topic from other angles, the following key questions should be answered:
1. Are CSPs and HSPs essentially identical?
2. Can one assign to SPs a high heat capacity (HHC) possibly explaining their ability to regulate temperature by acting as a heat bath? One can also ask whether HHC is present only in vivo that is
in a aqueous environment and whether it is present only in the unfolded configuration of HP?
The notion of quantum criticality
The basic postulate of quantum TGD is that the TGD Universe is quantum critical. There is only a single parameter, Kähler coupling strength α[K] mathematically analogous to a temperature and theory
is unique by requiring that it is analogous to critical temperature. Kähler coupling strength has discrete spectrum labelled by the parameters of the extensions of rationals. Discrete p-adic coupling
constant evolution replacing continuous coupling constant evolution is one aspect of quantum criticality.
What does quantum criticality mean?
1. Quite generally, critical states define higher-dimensional surfaces in the space of states labelled for instance by thermo-dynamical parameters like temperature, pressure, volume, and chemical
potentials. Critical lines in the (P,T) plane is one example. Bringing in more variables one gets critical 2-surfaces, 3-surfaces, etc. For instance, in Thom's catastrophe theory cusp catastrophe
corresponds to a V-shaped line, whose vertex is a critical point whereas butterflly catasrophe to 2-D critical surface. In thermodynamics the presence of additional thermodynamical variables like
magnetization besides P and T leads to higher-dimensional critical surfaces.
2. There is a hierarchy of criticalities: there are criticalities inside criticalities. Critical point is the highest form of criticality for finite-D systems. Triple point, for instance, for water
in which one cannot tell whether the phase is solid, liquid or gas. This applies completely generally irrespective of whether the system is a thermo-dynamical or quantal system. Also the
catastrophe theory of Thom gives the same picture. The catastrophe graphs available in the Wikipedia article illustrate the situation for lower-dimensional catastrophes.
3. In TGD framework finite measurement resolution implies that the number of degrees of freedom (DFs) is effectively finite. Quantum criticality with finite measurement resolution is realized as an
infinite number of hierarchies of inclusions of extensions of rationals. They correspond to inclusion hierarchies of hyperfinite factors of type II[1] (HFFs). The included HFF defines the DFs
remaining below measurement resolution and it is possible to assign to the detected DFs dynamical symmetry groups, which are finite-dimensional. The symmetry group in never reachable ideal
measurement resolution is infinite-D super-symplectic group of isometries of "world of classical worlds" (WCW) consisting of preferred extremals of Kähler action as analogs of Bohr orbits.
Super-symplectic group extends the symmetries of superstring models.
4. Criticality in living systems is a special case of criticality - and as the work of Kauffman suggests - of quantum crticality as well. Living matter as we know, it most probably corresponds to
extremely high level of criticality so that very many variables are nearly critical, not only temperature but also pressure. This relates directly to the high value of h[eff] serving as IQ. The
higher the value of h[eff], the higher the complexity of the system, and the larger the fluctuations and the scale of quantum coherence. There is a fractal hierarchy of increasingly quantum
critical systems labelled by a hierarchy of increasing scales (also time scales).
In ZEO classical physics is an exact part of quantum physics and quantum physics prevails in all scales. ZEO makes discontinuous macroscopic BSFRs to look like smooth deterministic time
evolutions for the external observer with opposite arrow of time so that the illusion that physics is classical in long length scales is created.
Number theoretical physics or adelic physics is the cornerstone of TGD inspired theory of cognition and living matter and makes powerful predictions.
p-Adic length scale hypothesis deserves to be mentioned as an example of prediction since it has direct relevance for SPs.
1. p-Adic length scale hypothesis predicts that preferred p-adic length scales correspond to primes p≈ 2^k: L(k)= 2^(k-151)/2L(151), L(151)≈ 10 nm, thickness of neuronal membrane and a scale often
appearing molecular biology.
2. TGD predicts 4 especially interesting p-adic length scales in the range 10 nm- 25 μ. One could speak of a number theoretical miracle. They correspond to Gaussian Mersenne primes M[G,k] = (1+i)^
k-1 with prime k ∈{151,157,163,167} and could define fundamental scales related with DNA coiling for instance.
3. The p-adic length scale L(k=167)= 2^(167-151)/2L(151)= 2.5 μ m so that SPs could correspond to k∈{165,167,169} . L(167) corresponds to the largest Gaussian Mersenne in the above series of 4
Gaussian Mersennes and to the size of cell nucleus. The size scale of a cold shock domain in turn corresponds to L(157), also associated with Gaussian Mersenne. Note that the wavelength defined
by L(167) corresponds rather precisely to the metabolic currency .5 eV.
4. HSPX, X∈ {60,70,90} corresponds to a mass of X kDaltons (Dalton corresponds to proton mass). From the average mass 110 Dalton of amino acid and length of 1 nm one deduces that the straight
HSP60, HSP70, and HSP90 have lengths about .55 μm, .64 μ, and .8 μm. The proportionality of the protein mass to length suggests that the energy scale assignable to HSPX is proportional to X.
(HSP60, HSP70, HSP90) would have energy scales (2.27, 1.95,1.5 eV) for h[eff]=h naturally assignable to biomolecules. The lower boundary of visible photon energies is a 1.7 eV.
Remark: One has h= h[eff]=nh[0] for n=6. What if one assumes n=2 giving h[eff]=h/3 for which the observations of Randel Mills give support? This scales down the energy scales by factor 1/3 to
(.77,.65,0.5) eV not far from the nominal value of metabolic energy currency of about .5 eV.
There are strong motivations to assign to HSPs the thermal energy E=T=.031 eV at physiological temperature: this is not the energy E[max]= .084 eV at the maximum of the energy distribution, which
is by a factor 2.82 higher than E. The energies above are however larger by more than one order of magnitude. This scale should be assigned with the MBs of SPs.
5. The wavelengths assignable to HSPs correspond to the "notes" represented by dark photon frequencies. There is an amusing co-incidence suggesting a connection with the model of bio-harmony: the
ratios of energy scales of HSP60 and HSP70 to the HSP90 energy are 3/2 and 1.3, respectively. If HSP90 corresponds to note C, HSP60 corresponds to G and HSP70 to note E with ratio 1.33. This
gives C major chord in a reasonable approximation! Probably this is an accident. Note also that the weights X of HSPXs are only nominal values.
Hagedorn temperature, HHC, and self-organized quantum criticality (SOC)
Self-organized criticality (SOC) is an empirically verified notion. For instance, sand piles are SOQC systems. The paradoxical property of SOQC is that although criticality suggests instability,
these systems stay around criticality. In standard physics SOQC is not well-understood. TGD based model for SOQC involves two basic elements: ZEO and Hagedorn temperature.
1. ZEO predicts that quantum coherence is possible in all scales due to the hierarchy of effective Planck constants predicted by adelic physics. "Big" (ordinary) state function reductions (BSFRs)
change the arrow of time. Dissipation in reversed arrow of time looks like generation of order and structures instead of their decay - that is self-organization. Hence SOQC could be made possible
by the instability of quantum critical systems in non-standard time direction. The system paradoxically attracted by the critical manifold in standard time direction would be repelled from it in
an opposite time direction as criticality indeed requires.
2. Surfaces are systems with infinite number of DFs. Strings satisfy this condition as also magnetic flux tubes idealizable as strings in reasonable approximation. The number of DFs is infinite and
this implies that when one heats this kind of system, the temperature grows slowly since heat energy excites new DFs. The system's maximum temperature is known as Hagedorn temperature and it
depends on string tension for strings.
In the TGD framework, magnetic flux tubes can be approximated as strings characterized by a string tension decreasing in long p-adic length scales. This implies a very high value of heat capacity
since very small change of temperature implies very large flow of energy between the system and environment.
T[H] could be a general property of MB in all scales (this does not yet imply SOQC property). An entire hierarchy of Hagedorn temperatures determined by the string tension of the flux tube, and
naturally identifiable as critical temperatures is predicted. The temperature is equal to the thermal energy of massless excitations such as photons emitted by the flux tube modellable as a black
Remark: If the condition h[eff]=h[gr] , where h[gr] is gravitational Planck constant introduced originally by Nottale, holds true, the cyclotron energies of the dark photons do not depend on h
[eff], which makes them an ideal tool of quantum control.
Hagedorn temperature would make them SOQC systems by temperature regulation if CSP type systems are present they can serve as ovens by liberating heat energy and force the local temperature of
environment to their own temperature near T[H]. Their own temperature is reduced very little in the process. These systems can also act as HSP/CSP type systems by extracting heat energy from/
providing it to the environment and in this manner reduce/increase the local temperature. System would be able to regulate its temperature.
A natural hypothesis is that T[H] corresponds to quantum critical temperature and in living matter to the physiological temperature. The ability to regulate the local temperature so that it stays
near T[H] has interpretation as self-organized (quantum) criticality (SOC). In the TGD framework these notions are more or less equivalent since classical physics is an exact part of quantum physics
and BSFRs create the illusion that the Universe is classical in long (actually all!) scales.
Homeostasis is a basic aspect of living systems. System tends to preserve its flow equilibrium and opposes the attempts to modify it. Homeostasis involves complex many-levels field back circuits
involving excitatory and inhibitory elements. If living systems are indeed quantum critical systems, homeostasis could more or less reduce to SOQC as a basic property of the TGD Universe.
I will add the article "Homeostasis and self-organized quantum criticality" to Research Gate.
See either the article Homeostasis as self-organized quantum criticality or the chapter with the same title.
For a summary of earlier postings see Latest progress in TGD. | {"url":"https://matpitka.blogspot.com/2020/11/","timestamp":"2024-11-06T04:43:00Z","content_type":"application/xhtml+xml","content_length":"164137","record_id":"<urn:uuid:15e48e3d-e311-4064-a160-d188585886f1>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00595.warc.gz"} |
Lund University Publications
Essays on Financial Models
(2000) In Lund Economic Studies 92.
This thesis consists of five essays exploring the validity of some extensively used financial models with a focus on the Swedish equity and derivative markets. The essays are of both an empirical
and a theoretical nature. In the first paper, The Search for Chaos and Nonlinearities in Swedish Stock Index Returns, an investigation of the presence of nonlinearities in general and chaos in
particular on the Swedish stock market is performed. Some properties of stock returns are hard to grasp with linear models. Nonlinearities must be introduced and can be of both a stochastic and a
deterministic nature. In the former case, the movements are generated by external shocks, although these shocks may exhibit complicated interdependences. In the... (More)
This thesis consists of five essays exploring the validity of some extensively used financial models with a focus on the Swedish equity and derivative markets. The essays are of both an empirical
and a theoretical nature. In the first paper, The Search for Chaos and Nonlinearities in Swedish Stock Index Returns, an investigation of the presence of nonlinearities in general and chaos in
particular on the Swedish stock market is performed. Some properties of stock returns are hard to grasp with linear models. Nonlinearities must be introduced and can be of both a stochastic and a
deterministic nature. In the former case, the movements are generated by external shocks, although these shocks may exhibit complicated interdependences. In the latter case, the movements are
self-generating due to the nonlinear dynamics of the system, but still behave in a way seemingly indistinguishable from pure randomness. This is called chaotic motion, or chaos. Since the stock
market crash of 1987 in particular, much effort has been made, trying to uncover the existence of different types of nonlinearities in financial and economic time series. Using the BDS test, we
examine whether the rejection of the null hypothesis of IID stock returns arises from nonlinear or linear dependences in the conditional mean process, chaos, nonstationarities, or autoregressive
conditional heteroscedasticity (ARCH). The results indicate that ARCH-effects are responsible for the rejections of IID. The second paper, The Compass Rose Pattern of the Stock Market: How Does
it Affect Parameter Estimates, Forecasts, and Statistical Tests?, deals with the discrete nature of stock returns, imposed by the fact that stock prices move in discrete steps, or ticks.
Recently, a geometrical pattern in a scatter plot of stock returns versus lagged stock returns, has been found. We believe that the effects of discreteness need a closer examination and, in this
paper, we do Monte Carlo simulations on artificial stock prices with different degrees of rounding. We find AR-GARCH parameter estimates to be affected by the discreteness imposed by rounding. On
basis of the compass rose and the discreteness we investigate different possibilities of improving predictions of stock returns, theoretically and empirically. The distributions of some
correlation integral statistics, that is, the BDS test and Savit and Green's dependability index, are also influenced by the compass rose pattern. However, throughout the paper, we must impose
heavy rounding of the stock prices to find significant effects on our estimates, forecasts, and statistical tests. Discreteness in stock returns is also the issue in the third paper, GARCH
Estimation and Discrete Stock Prices. The results from the previous paper indicate the break-down of statistical models and tests based on state-continuity as the tick size to price ratio
increases. Still, modeling such low-price stocks might be desirable in many situations. The continuous-state GARCH model is often used in modeling financial asset returns, but is misspecified if
applied to returns calculated from discrete price series. I propose a modification of the above model for handling such cases, by modeling the dependent variable as an unobserved stochastic
variable. The focus is on the GARCH framework, but the same ideas could also be used for other stochastic processes. Using Swedish stock price data and a stochastic optimization algorithm, that
is, simulated annealing, I compare the parameter estimates and asymptotic standard errors from the approximative and the extended model. I find small deviations between the two models for longer
time series, but larger differences for shorter series, mainly in the conditional variance parameters. None of the models provide continuous residuals. By constructing generalized residuals, I
show how valid residual diagnostic and specification tests can be performed. The fourth paper, A Neural Network Versus Black-Scholes: A Comparison of Pricing and Hedging Performances, studies
option pricing. The Black-Scholes formula is a well-known model for pricing and hedging derivative securities. It relies, however, on several highly questionable assumptions. This paper examines
whether a neural network (MLP) can be used to find a call option pricing formula better corresponding to market prices and the properties of the underlying asset than the Black-Scholes formula.
The neural network model is applied to out-of-sample pricing and delta-hedging of daily Swedish stock index call options from the period 1997-1999. The relevance of a hedge-analysis stressed in
this paper. The Black-Scholes model with historical and implicit volatility estimates is used as benchmarks. Comparisons reveal that the neural network models outperform the benchmarks both in
pricing and hedging performances. The moving block bootstrap procedure is used to test the statistical significance of the results. Although the neural networks are superior, the results are
often insignificant at the 5% level. In the fifth paper, Comparison of Mean-Variance and Exact Utility Maximization in Stock Portfolio Selection, portfolio optimization is considered. The
mean-variance approximation to expected utility maximization has been subject to much controversy ever since introduced by Markowitz. Given different correlated assets, how shall an investor
create a portfolio maximizing his expected utility? The validity of the mean-variance approximation has been verified, but only in the limited case of choosing among 10-20 securities. This paper
examines how well the approximation works in a larger allocation problem. The effects of limited short selling of the risky assets, as well as including synthetic options, that is, assets with
high levels of skewness and kurtosis, in the security set is also explored. The results show that the mean-variance approximative portfolios have less skewness than the exact solution portfolios,
but welfare losses, measured as the reduction in the certainty equivalent, are still small. (Less)
Abstract (Swedish)
Popular Abstract in Swedish
Alla finansiella modeller bygger på antaganden, approximationer och förenklingar. Syftet med avhandlingen är att undersöka giltigheten och rimligheten i ett antal välkända och populära
finansiella och ekonometriska modeller, med en tonvikt på de svenska aktie- och optionsmarknaderna. De ekonomiska och statistiska effekterna av alltför restriktiva modellantaganden undersöks, och
i de fall modellerna fungerar illa, presenteras alternativa modeller eller utvidgningar av existerande modeller. Avhandlingen består av fem essäer, och är både av empirisk och teoretisk natur.
Områden som behandlas är bl.a. prissättning och "hedgning" av optioner, portföljoptimering och kaosteori.
□ Professor Engsted, Tom, Department of Finance, The Aarhus School of Business, Aarhus, Denmark
publishing date
publication status
options, neural networks, hedging, portfolio optimization, econometrics, Economics, ekonomisk teori, ekonomiska system, ekonomisk politik, ekonometri, generalized residuals, discreteness, GARCH,
compass rose, nonlinearities, Chaos, economic theory, economic systems, Nationalekonomi, economic policy
Lund Economic Studies
116 pages
Department of Economics, Lund University
defense location
EC3:211, Holger Crafoords Ekonomicentrum III
defense date
2000-10-09 15:00:00
LU publication?
8878698d-30e1-4062-92f2-942c610cea30 (old id 40823)
date added to LUP
2016-04-01 15:46:44
date last changed
2019-05-21 16:26:39
abstract = {{This thesis consists of five essays exploring the validity of some extensively used financial models with a focus on the Swedish equity and derivative markets. The essays are of both an empirical and a theoretical nature. In the first paper, The Search for Chaos and Nonlinearities in Swedish Stock Index Returns, an investigation of the presence of nonlinearities in general and chaos in particular on the Swedish stock market is performed. Some properties of stock returns are hard to grasp with linear models. Nonlinearities must be introduced and can be of both a stochastic and a deterministic nature. In the former case, the movements are generated by external shocks, although these shocks may exhibit complicated interdependences. In the latter case, the movements are self-generating due to the nonlinear dynamics of the system, but still behave in a way seemingly indistinguishable from pure randomness. This is called chaotic motion, or chaos. Since the stock market crash of 1987 in particular, much effort has been made, trying to uncover the existence of different types of nonlinearities in financial and economic time series. Using the BDS test, we examine whether the rejection of the null hypothesis of IID stock returns arises from nonlinear or linear dependences in the conditional mean process, chaos, nonstationarities, or autoregressive conditional heteroscedasticity (ARCH). The results indicate that ARCH-effects are responsible for the rejections of IID. The second paper, The Compass Rose Pattern of the Stock Market: How Does it Affect Parameter Estimates, Forecasts, and Statistical Tests?, deals with the discrete nature of stock returns, imposed by the fact that stock prices move in discrete steps, or ticks. Recently, a geometrical pattern in a scatter plot of stock returns versus lagged stock returns, has been found. We believe that the effects of discreteness need a closer examination and, in this paper, we do Monte Carlo simulations on artificial stock prices with different degrees of rounding. We find AR-GARCH parameter estimates to be affected by the discreteness imposed by rounding. On basis of the compass rose and the discreteness we investigate different possibilities of improving predictions of stock returns, theoretically and empirically. The distributions of some correlation integral statistics, that is, the BDS test and Savit and Green's dependability index, are also influenced by the compass rose pattern. However, throughout the paper, we must impose heavy rounding of the stock prices to find significant effects on our estimates, forecasts, and statistical tests. Discreteness in stock returns is also the issue in the third paper, GARCH Estimation and Discrete Stock Prices. The results from the previous paper indicate the break-down of statistical models and tests based on state-continuity as the tick size to price ratio increases. Still, modeling such low-price stocks might be desirable in many situations. The continuous-state GARCH model is often used in modeling financial asset returns, but is misspecified if applied to returns calculated from discrete price series. I propose a modification of the above model for handling such cases, by modeling the dependent variable as an unobserved stochastic variable. The focus is on the GARCH framework, but the same ideas could also be used for other stochastic processes. Using Swedish stock price data and a stochastic optimization algorithm, that is, simulated annealing, I compare the parameter estimates and asymptotic standard errors from the approximative and the extended model. I find small deviations between the two models for longer time series, but larger differences for shorter series, mainly in the conditional variance parameters. None of the models provide continuous residuals. By constructing generalized residuals, I show how valid residual diagnostic and specification tests can be performed. The fourth paper, A Neural Network Versus Black-Scholes: A Comparison of Pricing and Hedging Performances, studies option pricing. The Black-Scholes formula is a well-known model for pricing and hedging derivative securities. It relies, however, on several highly questionable assumptions. This paper examines whether a neural network (MLP) can be used to find a call option pricing formula better corresponding to market prices and the properties of the underlying asset than the Black-Scholes formula. The neural network model is applied to out-of-sample pricing and delta-hedging of daily Swedish stock index call options from the period 1997-1999. The relevance of a hedge-analysis stressed in this paper. The Black-Scholes model with historical and implicit volatility estimates is used as benchmarks. Comparisons reveal that the neural network models outperform the benchmarks both in pricing and hedging performances. The moving block bootstrap procedure is used to test the statistical significance of the results. Although the neural networks are superior, the results are often insignificant at the 5% level. In the fifth paper, Comparison of Mean-Variance and Exact Utility Maximization in Stock Portfolio Selection, portfolio optimization is considered. The mean-variance approximation to expected utility maximization has been subject to much controversy ever since introduced by Markowitz. Given different correlated assets, how shall an investor create a portfolio maximizing his expected utility? The validity of the mean-variance approximation has been verified, but only in the limited case of choosing among 10-20 securities. This paper examines how well the approximation works in a larger allocation problem. The effects of limited short selling of the risky assets, as well as including synthetic options, that is, assets with high levels of skewness and kurtosis, in the security set is also explored. The results show that the mean-variance approximative portfolios have less skewness than the exact solution portfolios, but welfare losses, measured as the reduction in the certainty equivalent, are still small.}},
author = {{Amilon, Henrik}},
issn = {{0460-0029}},
keywords = {{options; neural networks; hedging; portfolio optimization; econometrics; Economics; ekonomisk teori; ekonomiska system; ekonomisk politik; ekonometri; generalized residuals; discreteness; GARCH; compass rose; nonlinearities; Chaos; economic theory; economic systems; Nationalekonomi; economic policy}},
language = {{eng}},
publisher = {{Department of Economics, Lund University}},
school = {{Lund University}},
series = {{Lund Economic Studies}},
title = {{Essays on Financial Models}},
volume = {{92}},
year = {{2000}}, | {"url":"https://lup.lub.lu.se/search/publication/40823","timestamp":"2024-11-02T02:09:58Z","content_type":"text/html","content_length":"61934","record_id":"<urn:uuid:5b4d1ad5-9485-44df-b3a7-32bc27491a86>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00319.warc.gz"} |
Slingshot: Trajectory Inference for Single-Cell Data
This vignette will demonstrate a full single-cell lineage analysis workflow, with particular emphasis on the processes of lineage reconstruction and pseudotime inference. We will make use of the
slingshot package proposed in (Street et al. 2017) and show how it may be applied in a broad range of settings.
The goal of slingshot is to use clusters of cells to uncover global structure and convert this structure into smooth lineages represented by one-dimensional variables, called “pseudotime.” We provide
tools for learning cluster relationships in an unsupervised or semi-supervised manner and constructing smooth curves representing each lineage, along with visualization methods for each step.
The minimal input to slingshot is a matrix representing the cells in a reduced-dimensional space and a vector of cluster labels. With these two inputs, we then:
• Identify the global lineage structure by constructing an minimum spanning tree (MST) on the clusters, with the getLineages function.
• Construct smooth lineages and infer pseudotime variables by fitting simultaneous principal curves with the getCurves function.
• Assess the output of each step with built-in visualization tools.
We will work with two simulated datasets in this vignette. The first (referred to as the “single-trajectory” dataset) is generated below and designed to represent a single lineage in which one third
of the genes are associated with the transition. This dataset will be contained in a SingleCellExperiment object (Lun and Risso 2017) and will be used to demonstrate a full “start-to-finish”
# generate synthetic count data representing a single lineage
means <- rbind(
# non-DE genes
matrix(rep(rep(c(0.1,0.5,1,2,3), each = 300),100),
ncol = 300, byrow = TRUE),
# early deactivation
matrix(rep(exp(atan( ((300:1)-200)/50 )),50), ncol = 300, byrow = TRUE),
# late deactivation
matrix(rep(exp(atan( ((300:1)-100)/50 )),50), ncol = 300, byrow = TRUE),
# early activation
matrix(rep(exp(atan( ((1:300)-100)/50 )),50), ncol = 300, byrow = TRUE),
# late activation
matrix(rep(exp(atan( ((1:300)-200)/50 )),50), ncol = 300, byrow = TRUE),
# transient
matrix(rep(exp(atan( c((1:100)/33, rep(3,100), (100:1)/33) )),50),
ncol = 300, byrow = TRUE)
counts <- apply(means,2,function(cell_means){
total <- rnbinom(1, mu = 7500, size = 4)
rmultinom(1, total, cell_means)
rownames(counts) <- paste0('G',1:750)
colnames(counts) <- paste0('c',1:300)
sce <- SingleCellExperiment(assays = List(counts = counts))
The second dataset (the “bifurcating” dataset) consists of a matrix of coordinates (as if obtained by PCA, ICA, diffusion maps, etc.) along with cluster labels generated by \(k\)-means clustering.
This dataset represents a bifurcating trajectory and it will allow us to demonstrate some of the additional functionality offered by slingshot.
library(slingshot, quietly = FALSE)
rd <- slingshotExample$rd
cl <- slingshotExample$cl
dim(rd) # data representing cells in a reduced dimensional space
## [1] 140 2
length(cl) # vector of cluster labels
## [1] 140
Upstream Analysis
Gene Filtering
To begin our analysis of the single lineage dataset, we need to reduce the dimensionality of our data and filtering out uninformative genes is a typical first step. This will greatly improve the
speed of downstream analyses, while keeping the loss of information to a minimum.
For the gene filtering step, we retained any genes robustly expressed in at least enough cells to constitute a cluster, making them potentially interesting cell-type marker genes. We set this minimum
cluster size to 10 cells and define a gene as being “robustly expressed” if it has a simulated count of at least 3 reads.
# filter genes down to potential cell-type markers
# at least M (15) reads in at least N (15) cells
geneFilter <- apply(assays(sce)$counts,1,function(x){
sum(x >= 3) >= 10
sce <- sce[geneFilter, ]
Another important early step in most RNA-Seq analysis pipelines is the choice of normalization method. This allows us to remove unwanted technical or biological artifacts from the data, such as
batch, sequencing depth, cell cycle effects, etc. In practice, it is valuable to compare a variety of normalization techniques and compare them along different evaluation criteria, for which we
recommend the scone package (Cole and Risso 2018). We also note that the order of these steps may change depending on the choice of method. ZINB-WaVE (Risso et al. 2018) performs dimensionality
reduction while accounting for technical variables and MNN (Haghverdi et al. 2018) corrects for batch effects after dimensionality reduction.
Since we are working with simulated data, we do not need to worry about batch effects or other potential confounders. Hence, we will proceed with full quantile normalization, a well-established
method which forces each cell to have the same distribution of expression values.
FQnorm <- function(counts){
rk <- apply(counts,2,rank,ties.method='min')
counts.sort <- apply(counts,2,sort)
refdist <- apply(counts.sort,1,median)
norm <- apply(rk,2,function(r){ refdist[r] })
rownames(norm) <- rownames(counts)
assays(sce)$norm <- FQnorm(assays(sce)$counts)
Dimensionality Reduction
The fundamental assumption of slingshot is that cells which are transcriptionally similar will be close to each other in some reduced-dimensional space. Since we use Euclidean distances in
constructing lineages and measuring pseudotime, it is important to have a low-dimensional representation of the data.
There are many methods available for this task and we will intentionally avoid the issue of determining which is the “best” method, as this likely depends on the type of data, method of collection,
upstream computational choices, and many other factors. We will demonstrate two methods of dimensionality reduction: principal components analysis (PCA) and uniform manifold approximation and
projection (UMAP, via the uwot package).
When performing PCA, we do not scale the genes by their variance because we do not believe that all genes are equally informative. We want to find signal in the robustly expressed, highly variable
genes, not dampen this signal by forcing equal variance across genes. When plotting, we make sure to set the aspect ratio, so as not to distort the perceived distances.
pca <- prcomp(t(log1p(assays(sce)$norm)), scale. = FALSE)
rd1 <- pca$x[,1:2]
plot(rd1, col = rgb(0,0,0,.5), pch=16, asp = 1)
## Loading required package: Matrix
## Attaching package: 'Matrix'
## The following object is masked from 'package:S4Vectors':
## expand
rd2 <- uwot::umap(t(log1p(assays(sce)$norm)))
colnames(rd2) <- c('UMAP1', 'UMAP2')
plot(rd2, col = rgb(0,0,0,.5), pch=16, asp = 1)
We will add both dimensionality reductions to the SingleCellExperiment object, but continue our analysis focusing on the PCA results.
reducedDims(sce) <- SimpleList(PCA = rd1, UMAP = rd2)
Clustering Cells
The final input to slingshot is a vector of cluster labels for the cells. If this is not provided, slingshot will treat the data as a single cluster and fit a standard principal curve. However, we
recommend clustering the cells even in datasets where only a single lineage is expected, as it allows for the potential discovery of novel branching events.
The clusters identified in this step will be used to determine the global structure of the underlying lineages (that is, their number, when they branch off from one another, and the approximate
locations of those branching events). This is different than the typical goal of clustering single-cell data, which is to identify all biologically relevant cell types present in the dataset. For
example, when determining global lineage structure, there is no need to distinguish between immature and mature neurons since both cell types will, presumably, fall along the same segment of a
For our analysis, we implement two clustering methods which similarly assume that Euclidean distance in a low-dimensional space reflect biological differences between cells: Gaussian mixture modeling
and \(k\)-means. The former is implemented in the mclust package (Scrucca et al. 2016) and features an automated method for determining the number of clusters based on the Bayesian information
criterion (BIC).
library(mclust, quietly = TRUE)
## Package 'mclust' version 5.4.7
## Type 'citation("mclust")' for citing this R package in publications.
## Attaching package: 'mclust'
## The following object is masked from 'package:mgcv':
## mvn
cl1 <- Mclust(rd1)$classification
colData(sce)$GMM <- cl1
plot(rd1, col = brewer.pal(9,"Set1")[cl1], pch=16, asp = 1)
While \(k\)-means does not have a similar functionality, we have shown in (Street et al. 2017) that simultaneous principal curves are quite robust to the choice of \(k\), so we select a \(k\) of 4
somewhat arbitrarily. If this is too low, we may miss a true branching event and if it is too high or there is an abundance of small clusters, we may begin to see spurious branching events.
cl2 <- kmeans(rd1, centers = 4)$cluster
colData(sce)$kmeans <- cl2
plot(rd1, col = brewer.pal(9,"Set1")[cl2], pch=16, asp = 1)
Using Slingshot
At this point, we have everything we need to run slingshot on our simulated dataset. This is a two-step process composed of identifying the global lineage structure with a cluster-based minimum
spanning tree (MST) and fitting simultaneous principal curves to describe each lineage.
These two steps can be run separately with the getLineages and getCurves functions, or together with the wrapper function, slingshot (recommended). We will use the wrapper function for the analysis
of the single-trajectory dataset, but demonstrate the usage of the individual functions later, on the bifurcating dataset.
The slingshot wrapper function performs both steps of trajectory inference in a single call. The necessary inputs are a reduced dimensional matrix of coordinates and a set of cluster labels. These
can be separate objects or, in the case of the single-trajectory data, elements contained in a SingleCellExperiment object.
To run slingshot with the dimensionality reduction produced by PCA and cluster labels identified by Gaussian mixutre modeling, we would do the following:
sce <- slingshot(sce, clusterLabels = 'GMM', reducedDim = 'PCA')
As noted above, if no clustering results are provided, it is assumed that all cells are part of the same cluster and a single curve will be constructed. If no dimensionality reduction is provided,
slingshot will use the first element of the list returned by reducedDims.
The output is a SingleCellExperiment object with slingshot results incorporated. All of the results are stored in a PseudotimeOrdering object, which is added to the colData of the original object and
can be accessed via colData(sce)$slingshot. Additionally, all inferred pseudotime variables (one per lineage) are added to the colData, individually. To extract all slingshot results in a single
object, we can use either the as.PseudotimeOrdering or as.SlingshotDataSet functions, depending on the form in which we want it. PseudotimeOrdering objects are an extension of SummarizedExperiment
objects, which are flexible containers that will be useful for most purposes. SlingshotDataSet objects are primarily used for visualization, as several plotting methods are included with the package.
Below, we visuzalize the inferred lineage for the single-trajectory data with points colored by pseudotime.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.000 8.631 21.121 21.414 34.363 43.185
colors <- colorRampPalette(brewer.pal(11,'Spectral')[-6])(100)
plotcol <- colors[cut(sce$slingPseudotime_1, breaks=100)]
plot(reducedDims(sce)$PCA, col = plotcol, pch=16, asp = 1)
lines(SlingshotDataSet(sce), lwd=2, col='black')
We can also see how the lineage structure was intially estimated by the cluster-based minimum spanning tree by using the type argument.
plot(reducedDims(sce)$PCA, col = brewer.pal(9,'Set1')[sce$GMM], pch=16, asp = 1)
lines(SlingshotDataSet(sce), lwd=2, type = 'lineages', col = 'black')
Downstream Analysis
Identifying temporally dynamic genes
After running slingshot, we are often interested in finding genes that change their expression over the course of development. We will demonstrate this type of analysis using the tradeSeq package
(Van den Berge et al. 2020).
For each gene, we will fit a general additive model (GAM) using a negative binomial noise distribution to model the (potentially nonlinear) relationshipships between gene expression and pseudotime.
We will then test for significant associations between expression and pseudotime using the associationTest.
# fit negative binomial GAM
sce <- fitGAM(sce)
# test for dynamic expression
ATres <- associationTest(sce)
We can then pick out the top genes based on p-values and visualize their expression over developmental time with a heatmap. Here we use the top 250 most dynamically expressed genes.
topgenes <- rownames(ATres[order(ATres$pvalue), ])[1:250]
pst.ord <- order(sce$slingPseudotime_1, na.last = NA)
heatdata <- assays(sce)$counts[topgenes, pst.ord]
heatclus <- sce$GMM[pst.ord]
heatmap(log1p(heatdata), Colv = NA,
ColSideColors = brewer.pal(9,"Set1")[heatclus])
Detailed Slingshot Functionality
Here, we provide further details and highlight some additional functionality of the slingshot package. We will use the included slingshotExample dataset for illustrative purposes. This dataset was
designed to represent cells in a low dimensional space and comes with a set of cluster labels generated by \(k\)-means clustering. Rather than constructing a full SingleCellExperiment object, which
requires gene-level data, we will use the low-dimensional matrix of coordinates directly and provide the cluster labels as an additional argument.
Identifying global lineage structure
The getLineages function takes as input an n \(\times\) p matrix and a vector of clustering results of length n. It maps connections between adjacent clusters using a minimum spanning tree (MST) and
identifies paths through these connections that represent lineages. The output of this function is a PseudotimeOrdering containing the inputs as well as the inferred MST (represented by an igraph
object) and lineages (ordered vectors of cluster names).
This analysis can be performed in an entirely unsupervised manner or in a semi-supervised manner by specifying known initial and terminal point clusters. If we do not specify a starting point,
slingshot selects one based on parsimony, maximizing the number of clusters shared between lineages before a split. If there are no splits or multiple clusters produce the same parsimony score, the
starting cluster is chosen arbitrarily.
In the case of our simulated data, slingshot selects Cluster 1 as the starting cluster. However, we generally recommend the specification of an initial cluster based on prior knowledge (either time
of sample collection or established gene markers). This specification will have no effect on how the MST is constructed, but it will impact how branching curves are constructed.
lin1 <- getLineages(rd, cl, start.clus = '1')
## class: PseudotimeOrdering
## dim: 140 2
## metadata(3): lineages mst slingParams
## pathStats(2): pseudotime weights
## cellnames(140): cell-1 cell-2 ... cell-139 cell-140
## cellData names(2): reducedDim clusterLabels
## pathnames(2): Lineage1 Lineage2
## pathData names(0):
plot(rd, col = brewer.pal(9,"Set1")[cl], asp = 1, pch = 16)
lines(SlingshotDataSet(lin1), lwd = 3, col = 'black') | {"url":"https://bioconductor.statistik.tu-dortmund.de/packages/3.13/bioc/vignettes/slingshot/inst/doc/vignette.html","timestamp":"2024-11-05T13:56:56Z","content_type":"text/html","content_length":"1048939","record_id":"<urn:uuid:aba2d307-e682-4c5a-8a3c-d28b14d897e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00851.warc.gz"} |
Example 7.7 Iterative Outlier Detection
This example illustrates the iterative nature of the outlier detection process. This is done by using a simple test example where an additive outlier at observation number 50 and a level shift at
observation number 100 are artificially introduced in the international airline passenger data used in Example 7.2. The following DATA step shows the modifications introduced in the data set:
data airline;
set sashelp.air;
logair = log(air);
if _n_ = 50 then logair = logair - 0.25;
if _n_ >= 100 then logair = logair + 0.5;
In Example 7.2 the airline model, ARIMA, was seen to be a good fit to the unmodified log-transformed airline passenger series. The preliminary identification steps (not shown) again suggest the
airline model as a suitable initial model for the modified data. The following statements specify the airline model and request an outlier search.
/*-- Outlier Detection --*/
proc arima data=airline;
identify var=logair( 1, 12 ) noprint;
estimate q= (1)(12) noint method= ml;
outlier maxnum=3 alpha=0.01;
The outlier detection output is shown in Output 7.7.1.
Output 7.7.1: Initial Model
Shift 0.49325 199.36 <.0001
Additive -0.27508 104.78 <.0001
Additive -0.10488 13.08 0.0003
Clearly the level shift at observation number 100 and the additive outlier at observation number 50 are the dominant outliers. Moreover, the corresponding regression coefficients seem to correctly
estimate the size and sign of the change. You can augment the airline data with these two regressors, as follows:
data airline;
set airline;
if _n_ = 50 then AO = 1;
else AO = 0.0;
if _n_ >= 100 then LS = 1;
else LS = 0.0;
You can now refine the previous model by including these regressors, as follows. Note that the differencing order of the dependent series is matched to the differencing orders of the outlier
regressors to get the correct “effective” outlier signatures.
/*-- Airline Model with Outliers --*/
proc arima data=airline;
identify var=logair(1, 12)
crosscorr=( AO(1, 12) LS(1, 12) )
estimate q= (1)(12) noint
input=( AO LS )
method=ml plot;
outlier maxnum=3 alpha=0.01;
The outlier detection results are shown in Output 7.7.2.
Output 7.7.2: Airline Model with Outliers
Additive -0.10310 12.63 0.0004
Additive -0.08872 12.33 0.0004
Additive 0.08686 11.66 0.0006
The output shows that a few outliers still remain to be accounted for and that the model could be refined further. | {"url":"http://support.sas.com/documentation/cdl/en/etsug/65545/HTML/default/etsug_arima_examples07.htm","timestamp":"2024-11-06T10:14:18Z","content_type":"application/xhtml+xml","content_length":"28108","record_id":"<urn:uuid:09fdeaed-e291-47aa-b4b8-c3bb4c968919>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00050.warc.gz"} |
The Building Blocks of Machine Learning
Tags: : Machine Learning
One can easily be confused by the sea of methods and terms in machine learning. I find the endless terminology confusing and counter productive. One might have a perfect understanding of a method
“A”, but is unaware that the new state of the art algorithm, “B++”, is merely a small twist to his familiar “A”. I have spent hours trying to disambiguate terms just to realize that a simple idea was
occluded by terminology.
In this post, I try to collect the fundamental ideas underlying machine-learning. Most algorithms out there can be shown to be some compounding of these ideas:
\[\newcommand{\loss}{l} % A loss function \newcommand{\risk}{R} % The risk function \newcommand{\riskn}{\mathbb{R}} % The empirical risk \newcommand{\argmin}[2]{\mathop{argmin} _{#1}\set{#2}} % The
argmin operator \newcommand{\set}[1]{\left\{ #1 \right\}} % A set \newcommand{\rv}[1]{\mathbf{#1}} % A random variable \newcommand{\x}{\rv x} % The random variable x \newcommand{\y}{\rv y} % The
random variable x \newcommand{\X}{\rv X} % The random variable x \newcommand{\Y}{\rv Y} % The random variable y \newcommand{\X}{\rv z} % The random variable x \newcommand{\Y}{\rv Z} % The random
variable y \newcommand{\estim}[1]{\widehat{#1}} % An estimator\]
Risk Minimization
This is possibly the most fundamental concept in machine learning. The idea stems from (statistical) decision theory, and consists of defining what is the loss incurred by making an error, \(\loss(Z;
\theta)\). Once defined, one would naturally seek to make predictions, \(\theta^*\), that are accurate, i.e., minimize the average loss known as the risk: \(\risk(\theta):= \int \loss(Z;\theta) dZ\),
and \(\theta^*:=\argmin{\theta}{\risk(\theta)}\).
As the whole population is typically inaccessible, we cannot compute the risk. Instead, we have access to a sample, and so we instead minimize the empirical risk \(\riskn(\theta):= \frac 1n \sum \
loss(Z_i;\theta)\), and \(\estim{\theta^*}:= \argmin{\theta}{\riskn(\theta)}\).
The vast majority of supervised and unsupervised learning algorithms are merely empirical risk minimizers (ERM). Some examples include Ordinary Least Squares, Maximum Likelihood estimation, PCA.
Typical examples of algorithms that cannot be cast as pure ERM problems are non-parametric methods like k-nearest neighbour and kernel methods. This is because optimization in an infinite dimension
is typically impossible (with exceptions, see the kernel trick).
Inductive Bias
Inductive bias is the idea that without constraining the class of prediction function (“hypothesis” in the learning literature), there is non point in learning. Indeed, our predictions and
approximation will be overfitted to the sample, and perform poorly on new data.
Inductive bias can be introduced in several ways:
• By restricting the functional form of your predictors (the “hypothesis class”).
• By preferring simple solutions, i.e., adding regularization.
Ridge regression demonstrates these two forms of inductive bias: we constrain the predictor to be linear in the features, and also add \(l_2\) regularization.
Risk Estimation
Having defined a loss function, we obviously seek for predictors and estimates that minimize the risk. We may think that choosing the model with the smallest empirical risk, \(\riskn{\theta)}\), may
be a good way to compare and choose models. This, however, is a poor strategy that will certainly lead to overfitting. This is because \(\riskn(\estim{\theta^*})\) is a biased estimate of \(\risk(\
estim{\theta^*})\). The “richer” the hypothesis class, the larger the bias, leading us to prefer more complicated models.
To choose the best model, we would like some unbiased estimate of \(\risk(\estim{\theta^*})\). Notable methods that aim at estimating the risk of the selected model, a.k.a. the generalization error,
or test error are:
Dimensionality Reduction
The fact that a scientists/machine collected a set of \(p\) variables, does clearly not mean that we need them all, or that we need them in the original scale. Indeed, variables may include redundant
information. It is thus of great interest, both for interpretability and for computational speedups, to remove the redundancy in the data by reducing its dimension.
Some examples include:
Basis Augmentation
Basis augmentation consists of generating new variables from non linear combinations of existing ones. This is motivated by the observation that relation between variables may be linear in some
non-linear transformation of these. Examples include:
The Kernel Trick
As previously stated, it is typically impossible so solve some risk minimization problem over an infinite dimensional function space. Exceptions arise, however, if using a particular type of
regularization. Indeed, it was Grace Whaba’s observation, when studying smoothing splines, that with the right regularization, the solution to a risk minimization problem in an infinite dimensional
function space, is contained within a finite dimensional simple space (that of cubic splines). This observation was later generalized to what is currently known as the Kernel Trick.
Informally speaking, the kernel trick states that by choosing an appropriate regularization, we can constrain the solution of an infinite dimensional ERM problem to a simple functional subspace,
known as a Reproducing Kernel Hilbert Space. The reason that RKHS spaces appear in this context is that functions in RKHS can be represented as linear combinations of distances between points in the
space, which is a far- from-trivial property of a function.
Generative Models
A data scientist trained as a Statistician will first think of a sampling distributions, a.k.a., a generative model. This may be an overkill for the simple purpose of discriminative analysis,
dimensionality reduction and clustering. If, however, a generative model can be assumed, then it immediately lends itself to learning using likelihood principals.
Assuming the generative model has a latent variable, allows the design of algorithms that pool information from different samples in a manner that no algorithm designer could have though of. Examples
Written on June 12, 2015 | {"url":"http://www.john-ros.com/building-blocks-of-machine-learning/","timestamp":"2024-11-11T14:37:31Z","content_type":"text/html","content_length":"18384","record_id":"<urn:uuid:b84e6818-0097-4b2f-a34d-bbf267c297e9>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00195.warc.gz"} |
How do you calculate acceleration of gravity? | HIX Tutor
How do you calculate acceleration of gravity?
Answer 1
You can use a pendulum to calculate the acceleration due to gravity, $\text{g}$. The equations to use are the following:
$\text{T}$ is period in seconds, $\text{L}$ is length of pendulum in meters, and $\text{g}$ is the acceleration due to gravity in ${\text{m/s}}^{2}$.
A pendulum with a length of $\text{1.15 m}$ has a period of $\text{2.151 s}$.
$\text{g}$ = ${\text{4pi"^2"L/T}}^{2}$ = ${\text{(4)(3.1415)"^2"(1.15 m)/(2.151 s)}}^{2}$ = ${\text{9.81 m/s}}^{2}$
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer 2
To calculate the acceleration of gravity, you can use the formula:
[a = \frac{F}{m}]
where (a) is the acceleration, (F) is the force of gravity acting on the object, and (m) is the mass of the object. The force of gravity can be calculated using the formula:
[F = mg]
where (g) is the acceleration due to gravity (approximately (9.81 , \text{m/s}^2) on the surface of Earth).
Sign up to view the whole answer
By signing up, you agree to our Terms of Service and Privacy Policy
Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
Not the question you need?
HIX Tutor
Solve ANY homework problem with a smart AI
• 98% accuracy study help
• Covers math, physics, chemistry, biology, and more
• Step-by-step, in-depth guides
• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-calculate-acceleration-of-gravity-1-8f9af8b766","timestamp":"2024-11-04T18:27:22Z","content_type":"text/html","content_length":"572305","record_id":"<urn:uuid:85964167-b875-4c6b-bacd-a54e091a24d5>","cc-path":"CC-MAIN-2024-46/segments/1730477027838.15/warc/CC-MAIN-20241104163253-20241104193253-00510.warc.gz"} |
Characteristics of coaxial waveguide
Rectangular and circular waveguides are used as emitters in phased arrays [1]. We are interested in arrays made with open endings of coaxial waveguides excited by a mode
Methods for mathematical modeling of waveguide phased arrays are well developed [1]. There are a lot of works devoted to studying arrays with emitters in the form of open endings of circular and
rectangular waveguides and also waveguides with a more complex form of the cross-section. Nevertheless, there is almost no works where you could find the results of a numeric study of characteristics
of phased arrays with coaxial waveguide emitters.
The aim of the articles is to study scanning characteristics, possibilities of matching the mentioned emitters on the basis of design features and the traditional methods, for example, with the help
of electrical insertions.
When designing the mathematical model of phased arrays with coaxial waveguide emitters, we used the technique described in [1]. The mentioned mathematical model is realized in the form of a computer
program GuidesArray Coaxial. The results of numerical experiment are given below.
In fig.1 you can see the set of curves representing the dependence of the reflection coefficient at the entry point of an emitter on the radius Т,
In fig.2 you can see partial diagrams of the directivity of coaxial waveguide emitters in Е- and H-planes (curves 1 and 2 respectively) in an array with the triangular grid of the location of
emitting elements. The array spacing is d=0,7142Т, H is shifted relative to the above-mentioned direction towards the normal. The numerical experiment shows that this shift is determined by the
excitation in the area of the aperture of the waveguide for the
In fig.3 you can see graphs showing how the reflection coefficient depends on the scanning angle in Е- and H-planes (curves 1 and 2 respectively) when matching emitters with the help of dielectrical
insertions towards the normal at the angle of 30° relative to the normal (curve 3), and also the dependence of the amplitude of Т-mode appearing at the aperture of the waveguide, related to the
amplitude of an incident wave (curve 4). Curves 1 and 2 represent the insertions with the thickness t=0,142l=0,2121|R|=0,622. Curve 3 represents the insertion with the parameters t=0,1185l=0,2166|R|=
0,721. The above-mentioned curves correspond to the triangular array of placing emitters with the array spacing d=0,7142T, H-plane was almost absent and the dazzling of the array occurs when the
direction of diffractional maximum coincided with the border between actual and imaginary angles. The graphs in the fig.3 show that dielectric insertions ensure effective matching, as it is the case
with arrays having circular and rectangular waveguides. The difference is that in our case the insertions can be used as constructive elements supporting the central conductor of the coaxial
In the fig. 4 you can see the graphs showing the dependence between the modules of wave reflection coefficients d=0,7142Т, Е- and H- planes of the emitter, excited by waves, the reflection
coefficients of which are
The results of the numerical experiment confirm the possibility of the effective use of coaxial waveguide emitters as emitting elements of phased arrays. | {"url":"http://www.eds-soft.com/en/publications/articles/2003/index.php?ID=21","timestamp":"2024-11-10T04:58:09Z","content_type":"application/xhtml+xml","content_length":"40583","record_id":"<urn:uuid:a09af7fd-522d-4a43-8942-65022cc6eec4>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00519.warc.gz"} |
How to Use StandardScaler and MinMaxScaler Transforms in Python
Coding is a skill that is now essential in every industry, even apart from IT, since it is the base of machine learning and AI, which are the two cornerstones of future innovations. Python, one of
the most used programming languages in the world, has a vast library and is used in almost all sectors unanimously because of its ease and flexibility. Given below is a short account of how to
perform two important algorithms, and why they’re necessary.
What is StandardScaler Transform?
In simplest terms, StandardScaler is used so that the data (distribution) is arranged so as to have a standard deviation 1 and mean value 0. For data with multiple variables, this is done
independently for each variable, or each column in the table.
How can it be implemented?
This algorithm uses the strictest form of standardization to transform data, and uses the following formula:
a_scaled = (a-m) / d
where m is the mean and d is the standard deviation.
First, define a StandardScaler instance with default hyperparameters. After that, a fit_transform() function can be called to pass this instance to a dataset, thus creating a new dataset, which is
basically just a transformed version of the previous one.
What is MinMaxScaler Transform?
This transform algorithm uses a given range to scale and transform each feature. The feature_range parameter is specified with default at (0,1), and works better for cases with non-Gaussian
distribution or with small standard deviation, and best for data with no outliers.
a_scaled = (a – min(a)) / (max(a) – min(a))
Importing and usage of the MinMaxScaler is exactly the same as of StandardScaler, with only a few parameters being different on a new instance initiation.
A MinMaxScaler instance has to be defined by default hyperparameters. Then, the fit_transform() function can be called to pass it to the main dataset, which now becomes a transformed version of | {"url":"https://deeptechknowledge.com/how-to-use-standardscaler-and-minmaxscaler-transforms-in-python/","timestamp":"2024-11-10T08:14:03Z","content_type":"text/html","content_length":"60005","record_id":"<urn:uuid:cce4707b-777c-4e85-98c9-3fea8b037614>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00357.warc.gz"} |
Discrete-Time Dynamical Systems
What is a discrete-time dynamical system, and how can we represent one using functions?
How can we make predictions about a discrete-time dynamical system?
We had to be careful with units. The initial value was given in meters, while the constant amount of change each year was given in centimeters. We converted to a consistent unit of measure before
writing the updating function.
Write the updating function and initial value for this DTDS. Focus on using correct notation.
\(t\) \(h_t\) \(h_{t+1}\)
\(0\) \(2\) \(3.5\)
\(1\) \(3.5\) \(5\)
\(2\) \(5\) \(6.5\)
\(3\) \(6.5\) \(8\)
\begin{equation*} h_{50} = 1.5(50)+ 2 = 77 \text{ m} \end{equation*}
Finding the explicit solution function from a recursive updating function, however, can be a challenging (and sometimes impossible) task. In the following examples, we’ll look at a few cases in which
we can determine the solution function explicitly.
In order to do so, it is important to recognize the information that an updating function is actually giving us. For example, if you re-write the updating function
\begin{align*} \dfrac{AROC_{[t+1,t+2]}}{AROC_{[t,t+1]}} \amp= \dfrac{b_{t+2} - b_{t+1}}{b_{t+1} - b_t} \\ \amp= \dfrac{b_{t+1}}{b_t}\\ \amp= \dfrac{2b_t}{b_t}\\ \amp= 2 \end{align*}
We can see once again that the solution function cannot be linear, so we will check the ratio of consecutive average rates of change:
\begin{align*} \dfrac{AROC_{[t+1,t+2]}}{AROC_{[t,t+1]}} \amp= \dfrac{p_{t+2} - p_{t+1}}{p_{t+1} - p_t} \\ \amp= \dfrac{(0.5p_{t+1} +1) - (0.5p_t +1)}{p_{t+1} - p_t}\\ \amp= \dfrac{0.5p_{t+1} -
0.5p_t}{p_{t+1} - p_t}\\ \amp= \dfrac{0.5(p_{t+1} - p_t)}{p_{t+1} - p_t}\\ \amp=0.5 \end{align*}
\(t\) \(p_t\) \(p_{t+1}\) \(10 \cdot 0.5^t\)
\(0\) \(10\) \(6\) \(10\)
\(1\) \(6\) \(4\) \(5\)
\(2\) \(4\) \(3\) \(2.5\)
\(3\) \(3\) \(2.5\) \(1.25\)
\(4\) \(2.5\) \(2.25\) \(0.625\)
\(5\) \(2.25\) \(2.125\) \(0.3125\)
\begin{align*} 10 \amp= v \cdot 0.5^0+2\\ 10 \amp= v +2\\ 8 \amp= v \end{align*}
We can summarize our findings from the previous examples with the following:
Determine the solution function associated with each updating function and initial value.
A discrete-time dynamical system describes a sequence of measurements made at equally spaced intervals. It is common to represent a DTDS using an updating function (a recursive function) and an
initial value.
We can make predictions about a DTDS by describing its solution function. We can do this by creating a table of several values, plotting several points on a graph, and in some cases, finding the
equation of the solution function explicitly.
What is the updating function and initial condition describing this DTDS? Graph the updating function, labeling axes appropriately.
Find the equation of the solution function for this DTDS explicitly. | {"url":"https://www.math.colostate.edu/~shriner/sec-1-7.html","timestamp":"2024-11-05T04:13:41Z","content_type":"text/html","content_length":"82909","record_id":"<urn:uuid:2cc52fb1-83e5-4e98-b23d-ef3178b5ef8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00612.warc.gz"} |
Big Slot Machine Winners | Delta Downs Racetrack Casino Hotel
You could be the next big winner! Join us in our congratulations to our lucky players, and come down to Delta Downs to see if you can hit the jackpot.
Crystal Star Deluxe
Delta Downs
James H.
Double Gold
Delta Downs
Hilario C.
Shamrock Fortunes
Delta Downs
Carlos R.
Double Double Gold
Delta Downs
Christopher H.
Blazing Triple Double
Delta Downs
Margaret O.
Black Diamond Deluxe
Delta Downs
Patricia T.
Ultra Reels Golden Steed
Delta Downs
Luis L.
Dragon Link
Delta Downs
Cindy C.
Fortune Rooster
Delta Downs
Helen H.
Lightning Link
Delta Downs
Felipe P.
Jewel of the Dragon
Delta Downs
Gale E.
Wheel of Fortune
Delta Downs
Jacob M.
Lightning Link
Delta Downs
Elizabeth T.
Kitty Wheels
Delta Downs
Edward A.
Kitty Wheels
Delta Downs
Maryjane G.
Genie's Wheel
Delta Downs
Bonniebelle N.
Mo' Mummy
Delta Downs
Claudia T.
Dragon Cash
Delta Downs
Anne C.
Lightning Link
Delta Downs
James H.
Texas Tea
Delta Downs
Lauren S.
Buffalo Link
Delta Downs
Carlos G.
Lock it Link
Delta Downs
Jeremy W.
Ultimate Fire Link
Delta Downs
Jesse J.
Lightning Link
Delta Downs
Terri A.
Double Diamonds
Delta Downs
Guillermo A.
Happy Prosperous
Delta Downs
Annmarie G.
Timber Wolf
Delta Downs
Ruby H.
Dragon Link
Delta Downs
Deomontina E.
Dragon Link
Delta Downs
Nora R.
Top Dollar
Delta Downs
Eileen P.
Lightning Link
Delta Downs
Jimmie G.
Rakin' Bacon
Delta Downs
Cole J.
Triple Double Diamond
Delta Downs
Guillermo A.
Lightning Link
Delta Downs
Rachel K.
Blazing Triple Double Jackpot Wild
Delta Downs
Erika P.
Cai Shen's Dragon
Delta Downs
Venancio S.
Ultimate Fire Link
Delta Downs
Mercelyn M.
Dragon Link
Delta Downs
John C.
Wonder 4 Boost
Delta Downs
Jose J.
Spin It Grand
Delta Downs
Lasonya B.
Triple Double Cash
Delta Downs
Joseph M.
Wonder 4 Boost
Delta Downs
Timothy C.
Jade Gems
Delta Downs
Cynthia P.
Dragon Link
Delta Downs
Keith E.
Buffalo Link
Delta Downs
Rosalind B.
Buffalo Link
Delta Downs
Dylan R.
Dragon Light gold
Delta Downs
Ernesto C.
Lightning Cash
Delta Downs
Kristin G.
Black & White Wild
Delta Downs
Nancy F.
Triple 7
Delta Downs
Mary H.
Double Gold
Delta Downs
Thomas R.
Cash Fortune Deluxe
Delta Downs
Delia L.
Hot Peppers
Delta Downs
Janet J.
Dragon Cash
Delta Downs
Billy B.
Coin Trio
Delta Downs
Brandi J.
Blazing 7
Delta Downs
Conitra I.
Dragon Cash
Delta Downs
Frances D.
Dragon Cash
Delta Downs
Kira B.
Super Link 888
Delta Downs
Monica D.
Delta Downs
Norma N.
Buffalo Link
Delta Downs
Roy D.
Rising Fortunes
Delta Downs
Amber T.
Cash Falls
Delta Downs
Ceasar T.
Lucky 7's
Delta Downs
Jasmine J.
Dragon Cash
Delta Downs
Jimmie P.
Ten Times Pay
Delta Downs
Jose L.
Double Diamond
Delta Downs
Julia C.
Best Things in Life
Delta Downs
Kenneth S.
Delta Downs
Kevin E.
Ultimate Fire Link
Delta Downs
Lisa F.
Great Fortune Deluxe
Delta Downs
Lisa G.
Delta Downs
Michael G.
Triple Double Jackpot Wild
Delta Downs
Ruby D.
Delta Downs
Shirley R.
Hot Peppers
Delta Downs
Tony N.
Dragon Cash
Delta Downs
Annie D.
Double Black Jack
Delta Downs
Gloria E.
Delta Downs
Hilario C.
Triple Gold Bars
Delta Downs
James M.
Buffalo Chief
Delta Downs
Juan R.
Hot Peppers
Delta Downs
Larry H.
Double Gold
Delta Downs
Larry H.
Mo' Money
Delta Downs
Ngun H.
Mr. Cashman
Delta Downs
Rodriques P.
Dragon Link
Delta Downs
Shahid M.
Dragon Link
Delta Downs
Brenda D. Orange, TX
Ultimate Fire Link
Delta Downs
Aniceto S. Port Author, TX
Lightning Link
Delta Downs
Christi R. Spring, TX
Buffalo Boost Gold
Delta Downs
Penny B. Baytown, TX
Fortune Bags
Delta Downs
Victor H. Cypress, TX
Blazing Hot Double Jackpot
Delta Downs
Shelli L. Beaumont, TX
Double Diamond
Delta Downs
Jacquelyn G. Hillister, TX
Jewel of the Dragon
Delta Downs
Pedro R. Pasadena, TX
Fire Link
Delta Downs
Lea F. Spring, TX
Red Festival
Delta Downs
Carolyn F. Livington, TX
Dragon Link
Delta Downs
Jose G. Auburn, WA
Dragon Cash
Delta Downs
Rigoberto S. Houston, TX
Lightning Dollar Link
Delta Downs
Sandra Y. Humble, Tx
Buffalo Gold
Delta Downs
Melony T. Silsbee, TX
Piggy Bankin'
Delta Downs
Draykedria B. Beaumont, TX
Dancing Drums
Delta Downs
Maria S. Houston, TX
Double Diamond
Delta Downs
Ruben N. Houston, TX
Big Hot Flaming Pots
Delta Downs
Edward S. — Dequincy, LA
Dragon Cash
Delta Downs
Lauren M. — Brookshire, TX
Dancing Drums
Delta Downs
Amanda K. — Beaumont, TX
777 Red Hot
Delta Downs
Herman S. — Oyster Creek, TX
Buffalo Chiefs
Delta Downs
Lillie B. — Beaumont, TX
Cash Fortunes Deluxe
Delta Downs
Delores M. — Lafayette, LA
Dragon Cash
Delta Downs
Candace G. Huffman, TX
Triple Diamond Strike
Delta Downs
Cari C. Bryan, TX
Ultimate Fire Link
Delta Downs
Debra J. Orange, TX
Supercharged Cash
Delta Downs
Henry W.
Supercharged Cash
Delta Downs
Joyce J.
Quick Hit
Delta Downs
Kor C. Houston, TX
Dragon Link
Delta Downs
Maricruz E. Houston, TX
Supercharged Cash
Delta Downs
Olympia B.
Supercharged Cash
Delta Downs
Paola W.
Rich Little Piggies
Delta Downs
Patricia F. Port Arthur, TX
Supercharged Cash
Delta Downs
Roderic M.
Supercharged Cash
Delta Downs
Sebastian H.
Supercharged Cash
Delta Downs
Shannon H.
Supercharged Cash
Delta Downs
Susan C.
Double Gold
Delta Downs
Virginia L. Orange, TX
Lightning Link
Delta Downs
Mario S. — Houston, TX
Delta Downs
Santa E. — Houston, TX
Black Diamond
Delta Downs
Timothy M. — Galveston, TX
Mega Diamond
Delta Downs
Jaclynn F. — San Antonio, TX
Supercharged Cash
Delta Downs
Katherine C. — Deyton, TX
Supercharged Cash
Delta Downs
Linda M. — Segur, TX
Triple Red Hot
Delta Downs
Lillian P. — Port Arthur, TX
Hot Peppers
Delta Downs
Jonathan S. — Port Neches, TX
Delta Downs
John M. — Washington, TX
Mighty Cash
Delta Downs
Nga V.
Double Diamond
Delta Downs
George G.
Heat Em Up
Delta Downs
Joshua T.
Lock It Link
Delta Downs
Ricardo S.
Double Diamond
Delta Downs
George G.
Dragon Link
Delta Downs
David G.
Triple Diamond Strike
Delta Downs
Larry H.
Twelve Times
Delta Downs
Lillie B.
Dragon Link
Delta Downs
Liza V.
Hot Peppers
Delta Downs
Larry H.
Dragon Link
Delta Downs
Sangitaben K.
Dancing Drums
Delta Downs
Norma M.
Dancing Drums
Delta Downs
Alicia H.
Dragon Link
Delta Downs
Jacqueline B.
Top Dollar
Delta Downs
Christine S.
Triple Double Diamond
Delta Downs
Susan W.
Dragon Link
Delta Downs
Darlene B.
Dragon Link
Delta Downs
Gloria P.
Delta Downs
Dorothy P.
Blazing Triple 7
Delta Downs
Marcia R.
Quick Hits
Delta Downs
Diana T.
Dragon Link
Delta Downs
Martin Z.
Dancing Drums Prosperity
Delta Downs
Cody M.
Double Pepper
Delta Downs
Andre B.
Dragon Link
Delta Downs
Tony B.
Coin Trio
Delta Downs
Judy L.
Buffalo Gold Revolution
Delta Downs
Kevin S.
Double Money Link
Delta Downs
Bradley H.
Imperial Wealth
Delta Downs
Mario S.
Dancing Drums Prosperity
Delta Downs
Morris M.
Blue Festival
Delta Downs
Yadira C.
Buffalo Chief
Delta Downs
Matthew M.
Triple Double Red Hot Strike
Delta Downs
Patsy L.
Red, White & Blue
Delta Downs
Martin Z.
Dragon Cash
Delta Downs
Martin Z.
Buffalo Link
Delta Downs
Kenneth W.
Dragon Link
Delta Downs
Jose P.
Delta Downs
Hector Z.
Gran Coronas
Delta Downs
Elizabeth N.
Dragon Link
Delta Downs
Sharon P.
Wonder 4 Boost
Delta Downs
Severiano R.
Jackpot Gems Deluxe
Delta Downs
Michael S.
Dragon Link
Delta Downs
Michael B.
Double Top Dollar
Delta Downs
Michael H.
Wheel of Fortune
Delta Downs
Larry H.
Quick Hit
Delta Downs
Judith L.
Lightning Cash
Delta Downs
Juan T.
Wonder 4
Delta Downs
James S.
Double Gold
Delta Downs
Kimberly P.
Dollar Storm
Delta Downs
Harold M.
Triple Double Diamond
Delta Downs
George G.
Quick Hit
Delta Downs
Brandon S.
Ultimate Fire Link
Delta Downs
Ana Ruth S.
Triple Diamond Strike
Delta Downs
Allison R.
Double Jackpot
Delta Downs
Stacy K.
Double Double Gold
Delta Downs
Richard H.
Dragon Link
Delta Downs
Renee L.
Triple Diamond
Delta Downs
Pauline A.
Jinse Dao Phoenix
Delta Downs
Oswaldo F.
Triple Double Diamond
Delta Downs
Mickey H.
Dragon Link
Delta Downs
Michael B.
Lightning Link
Delta Downs
Lauren M.
Buffalo Link
Delta Downs
LaTosha P.
Lightning Link
Delta Downs
Kirk V.
Delta Downs
Katrina C.
Dragon Link
Delta Downs
Ivan O.
Dollar Storm
Delta Downs
Gloria B.
Ultimate Fire Link
Delta Downs
Crystal D.
Lock It Link
Delta Downs
Charles M.
Delta Downs
Carmen M.
Triple Gems Jackpot
Delta Downs
Ammy N.
Real Riches Dragon's Wealth
Delta Downs
Gwendolyn P.
Triple Diamond Deluxe
Delta Downs
Leroy B.
12x Pay
Delta Downs
Constance H.
Delta Downs
Craig T.
Double Top Dollar
Delta Downs
Tammy H.
Dancing Drums
Delta Downs
Natalia C.
Dragon Link
Delta Downs
Jose G.
Choy's Kingdom
Delta Downs
Alejandrino T.
Buffalo Chief
Delta Downs
Cynthia C.
Dancing Drums
Delta Downs
Lynn F.
Dragon Cash
Delta Downs
Eugene D.
Dragon Cash
Delta Downs
Maria C.
Tiki Fire
Delta Downs
Maria C.
Lightning Link
Delta Downs
Michelle B.
Delta Downs
Tara T.
Dancing Drums
Delta Downs
Stephanie W.
Double Top Dollar
Delta Downs
Shirley P.
Triple Red Hot
Delta Downs
Regina G.
Dragon Link - Golden Century
Delta Downs
Randi J.
Buffalo Chief
Delta Downs
Molly B.
5 Treasures
Delta Downs
Michelle L.
Double Double Gold
Delta Downs
Michelle M.
Triple Diamond Strike
Delta Downs
Margarita C.
Wild Wild Gems
Delta Downs
Lillian K.
Double Gold
Delta Downs
Joy S.
Triple Double Strike
Delta Downs
Joy S.
Long Teng Hu Xiao - Mighty Cash
Delta Downs
Joey W.
Lightning Link
Delta Downs
Jennifer R.
Lightning Link
Delta Downs
Embrick H.
Buffalo Chief
Delta Downs
Cynthia C.
Dragon Link
Delta Downs
Cody P.
2x 3x 4x 5x Red Hot
Delta Downs
Carolyn P.
Dancing Drums
Delta Downs
Valerie R.
Coin Collector
Delta Downs
Sara L.
Hot Hit
Delta Downs
Robert H.
Wild Gems
Delta Downs
Paula H.
Quick Hits
Delta Downs
Max R.
Delta Downs
Laurie B.
Delta Downs
Laura S.
Double Top Dollar
Delta Downs
Kevin P.
Dancing Drums
Delta Downs
Katrina R.
Treasure Ball
Delta Downs
Jennifer B.
Delta Downs
James B.
Lock It Link
Delta Downs
Gloria G.
Delta Downs
Elsa R.
Double Pepper
Delta Downs
David G.
Fire Link
Delta Downs
Brittanie H.
Delta Downs
Andrew L. | {"url":"https://deltadowns.boydgaming.com/play/winners","timestamp":"2024-11-09T15:21:45Z","content_type":"text/html","content_length":"866257","record_id":"<urn:uuid:e550ef57-f141-496f-b402-93802103cc28>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00453.warc.gz"} |
CSE 6140 / CX 4140 Assignment 1 Solved - Ideal coders
Help Protect the City
1 Background
Let G = (V,E) be a simple, unweighted, undirected graph. The distance between two vertices u,v ∈ V , given by d(u,v), is defined as the length of the shortest path between u and v in the graph G.
Note that d(u,u) = 0.
The farness of a vertex u in a graph G is defined as:
far(u) = X d(u,v)
The closeness centrality of u is then defined as:
1 CC(u) = far(u)
2 Problem
Your TAs have been working hard helping Batman connect his cell phone audio player to bluetooth on his batmobile, so we are turning to you for help with this problem.
Your job is to design, analyze, and implement an algorithm to efficiently compute the heist-closeness centrality for each vertex in a graph.
Concretely, you need to do the following:
1. Design an algorithm to compute the heist-closeness centrality, as defined by (1). Write thehigh-level idea, then concrete details and pseudo-code for the algorithm.
2. Prove the feasibility of the algorithm. That is, prove that the algorithm will correctly computethe heist-closeness centrality for each vertex.
3. Compute and prove the run-time and space complexity of the algorithm. Identify if differentdata structures will impact the complexity.
4. Using one of the provided templates, implement your new algorithm and ensure it runscorrectly on the sample graphs provided.
3 I/O File Formats
There are three file formats used for this assignment which are described in this section.
3.1 Graph File Format
The input graph files are in a format that is easy to construct a CSR from. The file extension used is .graph. The format is as follows:
n m s1 d1 s2 d2
… sm dm
The edges must be sorted numerically by source.
3.2 Heist Nodes File Format
The file extension .h is used for heist nodes. The heist nodes are provided in the following format:
k v1 v2 … vk
Here, k is the number of heist nodes, or k = |H|. Each following vi is a given vertex ID that is a heist node. The values should be formatted as ASCII integers.
3.3 Output File Format
The output file format has an extension of .hc and is as follows:
n v1-hcc v2-hcc … vn-hcc
Here, n is n = |V |, as in the input graph format. Each following line corresponds to the respective vertex’s computed CHC value. The values should be formatted as an ASCII integer and doubles
Please follow these instructions carefully.
• You should submit two files:
1. A PDF containing your typed written assignment: the algorithm, the feasibility proof,the run-time and space complexity analysis and proof, and the implementation report and evaluation.
2. A single source code file (C++, Python, or Java) that implements your algorithm andcompiles or runs in the templates provided at https://github.gatech.edu/ucatalyurek7/cse6140-Fall18.git. This
file will be used to replace hcc.cpp, HCC.java, or hcc.py appropriately when testing
There are no reviews yet.
Be the first to review “CSE 6140 / CX 4140 Assignment 1 Solved” | {"url":"https://idealcoders.com/product/cse-6140-cx-4140-assignment-1-solved/","timestamp":"2024-11-09T03:43:02Z","content_type":"text/html","content_length":"167092","record_id":"<urn:uuid:3c2d6ac6-7f8b-49f5-9490-e7bfe6ddc365>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00511.warc.gz"} |
Understanding the Birthday Paradox: Probability Demystified
Written on
Chapter 1: The Birthday Paradox Unveiled
The concept of the birthday paradox is a captivating exploration of probability that reveals how likely it is for individuals in a small group to share the same birthday. Surprisingly, in a group of
just 23 people, there is a 50% chance that at least two individuals have their birthdays on the same day, a fact that may seem counterintuitive given the 365 days in a year.
In this article, we will derive a formula that estimates the likelihood of shared birthdays across groups of different sizes. Let's dive in!
To start, we will skip the general formula for now and look at a specific example involving thirty individuals. First, we consider one person's birthday. When we introduce another person, the chances
that they share a birthday with the first is 1 out of 365. As we add more people, the calculations become increasingly complex since we must evaluate each new person's birthday against those already
considered. The most effective way to determine the probability of a shared birthday is by utilizing the classic method of probability calculation:
Written by me
In this context, P(Event) signifies the probability of a specific event occurring. The beauty of probability is that it can be boiled down to either happening or not happening. For instance, if
there’s a 5% chance of your train being late, this translates to a 95% chance of it being on time. As we delve deeper into probability theory, let’s express these percentages as decimal
probabilities: 5% becomes 0.05, and 95% converts to 0.95.
It’s far simpler to calculate the probability that no one in our group of thirty shares a birthday, meaning we need to find the likelihood that each individual has a unique birthday. The first person
has no restrictions — they can celebrate on any day. Thus, P(No shared birthday) = 1. The second person now has one day less available to avoid a shared birthday:
Written by me
The third individual now has two fewer options out of the total 365 days:
Written by me
Are you spotting the trend? Each successive individual has one less birthday option than the previous one, so by the time we reach the thirtieth person, they have 365–29 days available to choose
from. Let's calculate the probability for thirty individuals:
Written by me
The capital Pi notation indicates a product of all integer values of i ranging from 0 to 29. By applying our traditional method, we find that in a group of thirty:
Written by me
The resulting probability is approximately 0.7, which translates to more than a 2/3 chance that at least two individuals in a group of thirty will share a birthday!
Armed with this understanding, we can now calculate the probability of shared birthdays for any size group. If we denote the number of people as n, we can adapt our previous calculation. For the
thirty-person example, we calculated the product of thirty fractions, each with a denominator of 365 and a numerator decreasing by one for each new person. To generalize this for n individuals, we
simply replace the 29 with n - 1:
Written by me
Once again, we use our traditional method:
Written by me
This is the elegant formula we sought! Note that it only applies for integer values of n ranging from 1 to 366. Let’s plug in some numbers to see what we uncover.
For a group of twenty people, the chance of a shared birthday is around 41%. In a group of fifty individuals, the probability skyrockets to 97%. Fifty is significantly fewer than the total number of
days in a year!
What if we flip the question and ask how many individuals are needed to reach a 99% probability of a shared birthday? The answer is just 56 people.
For a 99.9% likelihood? Only 70 individuals are required. And for a staggering 99.99% chance, it takes 80 people.
Even with these calculations, this phenomenon can still feel unbelievable, which is why it earns the title of paradox.
Challenge: Utilize the formula to determine how many individuals are required for a 75% chance of a shared birthday. WolframAlpha can assist with calculations involving products. Simply navigate to
the ‘math input’ section, found under calculus.
Chapter 2: Exploring the Birthday Paradox Further
This video titled "The Birthday Paradox" delves into the intriguing mathematical principles behind shared birthdays, providing a visual explanation that enhances understanding of the topic.
In this video, "The Birthday Paradox - Explained," the presenter further clarifies the concept and its implications, making the paradox accessible to all viewers. | {"url":"https://acelerap.com/understanding-birthday-paradox.html","timestamp":"2024-11-13T18:37:09Z","content_type":"text/html","content_length":"14247","record_id":"<urn:uuid:ab499bad-e639-4c81-a47a-f902aaf04092>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00680.warc.gz"} |
Ruckig: Background Information on Intermediate Waypoints
In comparison to the Community Version, Ruckig Pro allows to generate more complex trajectories defined by intermediate waypoints. Therefore, Ruckig takes a list of positions as input and calculates
a trajectory reaching them successively before moving to the target state. As this is a much harder problem than solving state-to-state motions (as the Community Version does), Ruckig Pro is not able
to guarantee for a time-optimal trajectory. In fact, trajectory planning with intermediate waypoints is a non-convex problem and therefore NP-hard. However, Ruckig calculates much faster trajectories
than other approaches for a wide range of trajectory types, primarily due to the joint calculation of the path and its time parametrization. On top, Ruckig is real-time capable and considers
In this regard, following information help to obtain robust and ideal optimization results:
1. Ruckig guarantees to output trajectories faster than a trajectory stopping at each intermediate waypoint (with zero velocity and acceleration).
2. Ruckig prefers as few waypoints as possible, both improving the (relative) trajectory duration as well as the computational performance. Note that the complexity of the calculation increases with
the number of waypoints, in contrast to many other (time-parametrization) approaches that scale with the trajectory duration or path length. While Ruckig is tested with 50+ waypoints, we strongly
recommend to limit the number of waypoints to a lower number.
3. Ruckig includes public hyperparameters for tuning the waypoint calculation, in particular Ruckig.calculator.waypoints_calculator.number_global_steps = 96 and
Ruckig.calculator.waypoints_calculator.number_local_steps = 16 with their respective default values. For a higher number of waypoints, we recommend to reduce the number of global steps and
increase the number of local steps, e.g. to global_steps=16 and local_steps=256. Additionally, there is the Ruckig.calculator.waypoints_calculator.number_smoothing_steps = 0 parameter for
smoothing the trajectory. It is turned off by default, and a higher number (e.g. in the range 0 - 64) will improve the smoothing.
4. Ruckig prefers waypoints that are far away from each other (relative to the given velocity limit) and not positioned in a straight line. When using waypoints from a motion planning algorithm
(such as RRT), we strongly recommend to pre-process your list by a waypoint filter (such as the integrated Ruckig::filter_intermediate_positions method) first. Fortunately, a straight line of
dozens of waypoints is not a typical use-case seeking for near time-optimal trajectories. For a single DoF, Ruckig automatically filters waypoints without loss of generality. | {"url":"https://docs.ruckig.com/md_pages_2__intermediate__waypoints.html","timestamp":"2024-11-13T21:04:45Z","content_type":"application/xhtml+xml","content_length":"7895","record_id":"<urn:uuid:43056cd9-e05a-4979-ba5c-691bfef44342>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00355.warc.gz"} |
Pearson BTEC Level 3 Unit 8 Mechanical Principles of Engineering Systems : Statics, Assignment, UK
Vocational Scenario or Context
The use and application of mechanical systems is an essential part of modern life. The design, manufacture and maintenance of these systems are the concern of engineers and technicians who must be
able to apply a blend of practical and theoretical knowledge to ensure that systems work safely and efficiently. Science underpins all aspects of engineering and a sound understanding of its
principles is essential for anyone seeking to become an engineer. Having completed learning outcome LO8.1 of Mechanical Principles of Engineering Systems in your course at Newham College, you are
required by your employer to demonstrate your understanding in effects of loading in static engineering systems particularly effects of mechanical force on solids. To do this complete the tasks
Task 1
Calculate the magnitude, direction and position of the resultant force and turning moment about point for the system of forces acting on the bridge support plate shown in figure 1. What would be the
equilibrant force and its direction? P1
Figure 1 Bridge support plate
Task 2
Simply supported beam Calculate the support reactions for the roof beam shown figure 2. The uniformly distributed load (UDL) is across the whole length of the beam. P2 Figure 2 Roof beam
Figure 2 Roof beam
Task 3
Component subjected to direct uniaxial loading
The rectangular steel bar, figure 3 is part of a static test rig and extends under a tensile load of kN. If the elastic modulus for the steel material is 208 GN/m2, calculate:
the stress in the bar,
the resulting strain, and
the extension of the bar. P3
Component subjected to shear loading
A rivet which has a diameter D mm is being used to hold together the sides of a storage vessel and is subject to the loadings shown in the figure 4. Calculate:
the resulting shear stress
the resulting shear strain if the shear modulus for the rivet material is 79 GN/m2. P3
Figure 4 Loads on a rivet holding together a storage vessel
The diagram in figure 5 shows the section through part of a car trailer and the corresponding loading. The failure criteria for the diameter high tensile bolt shown is:
Calculate the maximum permissible factor of safety for the working conditions detailed. M1
P3 calculate the induced direct stress, strain and dimensional change in a component subjected to direct uniaxial loading and the shear stress and strain in a component subjected to shear loading
M1 calculate the factor of safety in operation for a component subjected to combined direct and shear loading against given failure criteria
Are You Looking for Answer of This Assignment or Essay
The post Pearson BTEC Level 3 Unit 8 Mechanical Principles of Engineering Systems : Statics, Assignment, UK appeared first on Students Assignment Help UK. | {"url":"https://adeptance.com/pearson-btec-level-3-unit-8-mechanical-principles-of-engineering-systems-statics-assignment-uk/","timestamp":"2024-11-14T02:01:45Z","content_type":"text/html","content_length":"26282","record_id":"<urn:uuid:0bc79432-9d1b-493a-a5b5-a5ee172e4ca8>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00148.warc.gz"} |
row-col-notation: Row and column notation in RCLabels: Manipulate Matrix Row and Column Labels with Ease
It is often convenient to represent matrix row and column names with notation that includes a prefix and a suffix, with corresponding separators or start-end string sequences. There are several
functions to generate specialized versions or otherwise manipulate row and column names on their own or as row or column names.
flip_pref_suff() Switches the location of prefix and suffix, such that the prefix becomes the suffix, and the suffix becomes the prefix. E.g., "a -> b" becomes "b -> a" or "a [b]" becomes "b [a]".
get_pref_suff() Selects only prefix or suffix, discarding notational elements and the rejected part. Internally, this function calls split_pref_suff() and selects only the desired portion.
notation_vec() Builds a vector of notation symbols in a standard format. By default, it builds a list of notation symbols that provides an arrow separator (" -> ") between prefix and suffix.
paste_pref_suff() paste0's prefixes and suffixes, the inverse of split_pref_suff(). Always returns a character vector.
preposition_notation() Builds a list of notation symbols that provides (by default) square brackets around the suffix with a preposition ("prefix [preposition suffix]").
split_pref_suff() Splits prefixes from suffixes, returning each in a list with names pref and suff. If no prefix or suffix delimiters are found, x is returned in the pref item, unmodified, and the
suff item is returned as "" (an empty string). If there is no prefix, and empty string is returned for the pref item. If there is no suffix, and empty string is returned for the suff item.
switch_notation() Switches from one type of notation to another based on the from and to arguments. Optionally, prefix and suffix can be flipped.
Parts of a notation vector are "pref_start", "pref_end", "suff_start", and "suff_end". None of the strings in a notation vector are considered part of the prefix or suffix. E.g., "a -> b" in arrow
notation means that "a" is the prefix and "b" is the suffix. If sep only is specified for notation_vec() (default is " -> "), pref_start, pref_end, suff_start, and suff_end are set appropriately.
For functions where the notation argument is used to identify portions of the row or column label (such as split_pref_suff(), get_pref_suff(), and the from argument to switch_notation()), (Note:
flip_pref_suff() cannot infer notation, because it switches prefix and suffix in a known, single notation.) if notation is a list, it is treated as a store from which the most appropriate notation is
inferred by infer_notation(choose_most_specific = TRUE). Because default is RCLabels::notations_list, notation is inferred by default. The argument choose_most_specific tells what to do when two
notations match a label: if TRUE (the default), the notation with most characters is selected. If FALSE, the first matching notation in notation will be selected. See details at infer_notation().
If specifying more than one notation, be sure the notations are in a list. notation = c(RCLabels::bracket_notation, RCLabels::arrow_notation) is unlikely to produce the desired result, because the
notations are concatenated together to form a long string vector. Rather say notation = list(RCLabels::bracket_notation, RCLabels::arrow_notation).
For functions that construct labels (such as paste_pref_suff()), notation can be a list of notations over which the paste tasks is mapped. If notation is a list, it must have as many items as there
are prefix/suffix pairs to be pasted.
If either pref or suff are a zero-length character vector (essentially an empty character vector such as obtained from character()) input to paste_pref_suff(), an error is thrown. Instead, use an
empty character string (such as obtained from "").
notation_vec( sep = " -> ", pref_start = "", pref_end = "", suff_start = "", suff_end = "" ) preposition_notation(preposition, suff_start = " [", suff_end = "]") split_pref_suff( x, transpose =
FALSE, inf_notation = TRUE, notation = RCLabels::notations_list, choose_most_specific = TRUE ) paste_pref_suff( ps = list(pref = pref, suff = suff), pref = NULL, suff = NULL, notation =
RCLabels::arrow_notation, squish = TRUE ) flip_pref_suff( x, notation = RCLabels::notations_list, inf_notation = TRUE, choose_most_specific = TRUE ) get_pref_suff( x, which = c("pref", "suff"),
inf_notation = TRUE, notation = RCLabels::notations_list, choose_most_specific = TRUE ) switch_notation( x, from = RCLabels::notations_list, to, flip = FALSE, inf_notation = TRUE )
sep A string separator between prefix and suffix. Default is " -> ".
pref_start A string indicating the start of a prefix. Default is NULL.
pref_end A string indicating the end of a prefix. Default is the value of sep.
suff_start A string indicating the start of a suffix. Default is the value of sep.
suff_end A string indicating the end of a suffix. Default is NULL.
preposition A string used to indicate position for energy flows, typically "from" or "to" in different notations.
x A string or vector of strings to be operated upon.
A boolean that tells whether to purr::transpose() the result. Set transpose = TRUE when using split_pref_suff() in a dplyr::mutate() call in the context of a data frame. Default
transpose is FALSE.
inf_notation A boolean that tells whether to infer notation for x. Default is TRUE. See infer_notation() for details.
notation A notation vector generated by one of the *_notation() functions, such as notation_vec(), arrow_notation, or bracket_notation.
choose_most_specific A boolean that tells whether to choose the most specific notation from the notation argument when the notation argument is a list.
ps A list of prefixes and suffixes in which each item of the list is itself a list with two items named pref and suff.
pref A string or list of strings that are prefixes. Default is NULL.
suff A string of list of strings that are suffixes. Default is NULL.
squish A boolean that tells whether to remove extra spaces in the output of paste_*() functions. Default is TRUE.
which Tells which to keep, the prefix ("pref") or the suffix ("suff").
from The notation to switch away from.
to The notation to switch to.
flip A boolean that tells whether to also flip the notation. Default is FALSE.
A string separator between prefix and suffix. Default is " -> ".
A string indicating the start of a prefix. Default is NULL.
A string indicating the end of a prefix. Default is the value of sep.
A string indicating the start of a suffix. Default is the value of sep.
A string indicating the end of a suffix. Default is NULL.
A string used to indicate position for energy flows, typically "from" or "to" in different notations.
A string or vector of strings to be operated upon.
A boolean that tells whether to purr::transpose() the result. Set transpose = TRUE when using split_pref_suff() in a dplyr::mutate() call in the context of a data frame. Default is FALSE.
A boolean that tells whether to infer notation for x. Default is TRUE. See infer_notation() for details.
A notation vector generated by one of the *_notation() functions, such as notation_vec(), arrow_notation, or bracket_notation.
A boolean that tells whether to choose the most specific notation from the notation argument when the notation argument is a list.
A list of prefixes and suffixes in which each item of the list is itself a list with two items named pref and suff.
A string or list of strings that are prefixes. Default is NULL.
A string of list of strings that are suffixes. Default is NULL.
A boolean that tells whether to remove extra spaces in the output of paste_*() functions. Default is TRUE.
Tells which to keep, the prefix ("pref") or the suffix ("suff").
A boolean that tells whether to also flip the notation. Default is FALSE.
For notation_vec(), arrow_notation, and bracket_notation, a string vector with named items pref_start, pref_end, suff_start, and suff_end; For split_pref_suff(), a string list with named items pref
and suff. For paste_pref_suff(), split_pref_suff(), and switch_notation(), a string list in notation format specified by various notation arguments, including from, and to. For keep_pref_suff, one of
the prefix or suffix or a list of prefixes or suffixes.
notation_vec() arrow_notation bracket_notation split_pref_suff("a -> b", notation = arrow_notation) # Or infer the notation (by default from notations_list) split_pref_suff("a -> b") split_pref_suff
(c("a -> b", "c -> d", "e -> f")) split_pref_suff(c("a -> b", "c -> d", "e -> f"), transpose = TRUE) flip_pref_suff("a [b]", notation = bracket_notation) # Infer notation flip_pref_suff("a [b]")
get_pref_suff("a -> b", which = "suff") switch_notation("a -> b", from = arrow_notation, to = bracket_notation) # Infer notation and flip prefix and suffix switch_notation("a -> b", to =
bracket_notation, flip = TRUE) # Also works for vectors switch_notation(c("a -> b", "c -> d"), from = arrow_notation, to = bracket_notation) # Functions can infer the correct notation and return
multiple matches infer_notation("a [to b]", allow_multiple = TRUE, choose_most_specific = FALSE) # Or choose the most specific notation infer_notation("a [to b]", allow_multiple = TRUE,
choose_most_specific = TRUE) # When setting the from notation, only that type of notation will be switched switch_notation(c("a -> b", "c [to d]"), from = arrow_notation, to = bracket_notation) # But
if notations are inferred, all notations can be switched switch_notation(c("a -> b", "c [to d]"), to = bracket_notation) # A double-switch can be accomplished. # In this first example,
`RCLabels::first_dot_notation` is inferred. switch_notation("a.b.c", to = arrow_notation) # In this second example, # it is easier to specify the `from` and `to` notations. switch_notation("a.b.c",
to = arrow_notation) %>% switch_notation(from = first_dot_notation, to = arrow_notation) # "" can be used as an input paste_pref_suff(pref = "a", suff = "", notation = RCLabels::from_notation)
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/RCLabels/man/row-col-notation.html","timestamp":"2024-11-09T20:49:32Z","content_type":"text/html","content_length":"38088","record_id":"<urn:uuid:ce2e67d9-0e39-4c49-b133-f207b022f5f7>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00625.warc.gz"} |