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15 Best R Courses Online: Training Classes & Certifications – TangoLearn
15 Best R Programming Course Online for Beginners Experts
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According to polls, data mining surveys, and examinations of scientific literature databases, R is quite popular. R has been ranked 14th in the TIOBE index, a measure of programming language
popularity, since August 2021.
The best is to take up a course with R programming training online. We have listed the 15 best R courses online. Make your pick!
15 Best R Programming Course for Beginners
Bonus R Programming Certification Online
15 Best R Courses Online for This Year
This is a true step-by-step course. Every subsequent tutorial builds on what you’ve already learned and takes you one step further.
After each video, you’ll discover a new helpful topic that you can put into practice right away. Of course, the finest thing is that you learn by watching real-life examples.
This best R programming course online is jam-packed with real-world analytical problems that you will learn to answer. Some of these will be solved together, while others will be assigned as
In conclusion, this is one of the best online courses for R programming that has been developed for students of all skill levels, and you will succeed even if you have no programming or statistical
Rating 4.6 based out of 43,700 ratings
Duration 10 hours 30 minutes
Level Beginner level course
Refund Policy 30-day return policy
Certificate Provided Yes
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Course Type Paid
Course Instructor Kirill Eremenko, Ligency Team, Ligency I Team
Scope for Improvement (Cons) It is an amazing beginner friendly course. However, you will need more training to kick start your career after this course.
Topics Covered
• Core programming principles
• Types of variables
• Using variables
• Logical variables and operators
• ‘While’ Loop
• ‘For’ Loop
• ‘If’ Statement
• Fundamentals of R
• Vector
• Creating vectors
• Vectorized operations
• Power of Vectorized operations
• Matrices
• Building the first matrix
• Naming dimensions
• Matrix operations
• Subsetting
• Data frames
• Importing data into R
• Exploring the dataset
• Building data frames
• Merging data frames
• Introduction to qplot
• Advanced visualization with GGPlot2
• Plotting with layers
• What is a factor?
• Histograms and density charts
• Conclusion
Learning Outcomes
• With these best R courses online, you get a good understanding of R programming and R Studio
• In this tutorial, you’ll learn how to make vectors in R.
• Find out how to make variables.
• R’s types include integer, double, logical, character, and others.
• Learn how to use the while() and for() functions in R.
• You’ll learn how to create and use matrices.
• Learn the matrix(), rbind(), and cbind() functions ()
• Learn how to use R to install packages.
• Learn how to personalize R studio to fit your needs.
• Recognize the Rule of Large Numbers
• Learn about the normal distribution.
• Experiment with statistical, financial, and sports data in R.
There is no requirement for prior knowledge or expertise to pursue this best R Programming certification course.
Is it the right course for you?
This is the course for you, if:
• You wish to learn how to program in R.
• You have taken R classes but found them too difficult.
• You are someone who enjoys challenging tasks and wants to learn R by doing.
Review Janith C.
Thank you for this awesome course. I really enjoyed the step by step approach of the concepts. Did really enjoy learning the basics!
This is one of the best R courses online for complete novices with no programming knowledge and experienced developers who want to move into Data Science!
This complete R programming training online is equivalent to a Data Science bootcamp that typically costs thousands of dollars, but you can now study everything for a fraction of that price!
This is one of the most comprehensive data science and machine learning courses on Udemy, with over 100 HD video lectures and complete code notebooks for each lecture!
In this best R programming course online, you will learn how to program in R, generate stunning data visualizations, and utilize R for machine learning.
Rating 4.7 based out of 14,126 ratings
Duration 17 hours 30 minutes course
Level Beginner level course
Refund Policy 30-day return policy
Certificate Provided Yes
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Course Type Paid best online course for R programming
Course Instructor Jose Portilla
Scope for Improvement (Cons) Some learners may struggle with the course teaching pattern. Not everything is spoon-fed in this course.
Topics Covered
• Windows installation setup
• Mac OS installation setup
• Linux Installation
• Development Environment
• More about R basics
• Arithmetic in R
• Variables
• R basic data types
• Vector basics
• Vector operations
• Comparison operators
• Vector indexing and slicing
• Introduction to R Matrices
• Matrix arithmetic
• Matrix operations
• Matrix selection and indexing
• Factor and categorical matrices
• R data frames
• Data frame indexing and selection
• Data input and output with R
• R programming basics
• Advanced R programming
• Data manipulation with R
• Data visualization with R
• Interactive Visualizations with Plotly
• Machine Learning with R
• Linear Regression
• Logistic Regression
• Decision trees and random forests
Learning Outcomes
• Understand R programming
• Analyze and alter data with R
• Do data visualizations
• Learn to use R to manage csv, excel, SQL, and web scraping files.
• Learn machine learning algorithms in R
• Use R as a tool for data science
To take up these best R courses online, you need:
• Access to a computer with the ability to download files.
• Math Fundamentals
Is it the right R programming training online for you?
Anyone interested in pursuing a career as a data scientist can attend this class.
Review Damien O.
After taking the SQL course by Jose, I really enjoyed Jose’s teaching style, and in this course I was certainly not let down. Efficient, effective, and clear, as I hoped.
You’ll begin by working with financial data, cleaning it up, and preparing it for analysis in the first section of this best R Programming certification course.
Next, you will be given the task of creating graphs depicting income, expenses, and profit for various sectors.
In the second segment, you will prepare several data analysis jobs to assist Coal Terminal in determining which equipment is underutilized.
In the third section, you’ll visit the meteorology bureau to evaluate weather forecast patterns.
Rating 4.6 based out of 7492 ratings
Duration 6 hours course
Level Intermediate level course
Refund Policy 30-day return policy
Certificate Provided Yes
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded R programming training online
Course Type Paid
Course Instructor Kirill Eremenko, Ligency I Team, Ligency Team
Scope for Improvement (Cons) Repetition of the same content is found in this course. Also, the course name suggests advanced level but actually it is best for intermediate learners.
Topics Covered
• Data preparation
• Importing data into R
• gsub() and sub()
• Dealing with missing data
• Data filters
• Removing records with missing data
• Resetting the dataframe index
• Lists in R
• Handling Date-Times in R
• Naming components of a list
• Extracting components lists
• Adding and deleting components
• Subsetting a list
• Creating a time series plot
• ‘’Apply” family of functions
• Using lapply()
• Using sapply()
• Nesting apply functions()
• Conclusion
Learning Outcomes
This R programming course online will teach you:
• How is R used to prepare data?
• Recognize missing records in dataframes
• Find missing information in your dataframes.
• To replace lost records, use the Median Imputation technique.
• To replace missing records, use the Factual Analysis approach.
• Learn how to utilize the which() function.
• Understand how to reset the dataframe index.
• Replace strings using the gsub() and sub() methods.
• Explain why NA is a logical constant of the third type.
• Convert dates and times to POSIXct time.
• Lists in R can be created, used, appended, modified, renamed, accessed, and subset.
• When working with Lists, understand when to use [] and when to use [[]] or the $ sign.
• In R, make a time series plot.
• Recognize how the Apply family of functions works.
• Using a for() loop, recreate an apply statement.
• Basic knowledge of R
• Knowledge of the GGPlot2 package is recommended
• Knowledge of dataframes
• Knowledge of vectors and vectorized operations
Is it the best R Programming certification course for you?
• Anyone with a basic understanding of R who wants to advance their expertise or someone who has already finished the R Programming A-Z course is most eligible for this course.
• This course is NOT intended for complete R beginners.
Review Andrew I.
Great course! Great series of courses! Love what they are doing with creating structured pathways of learning for different careers.
R for Statistics and Data Science is one of the best R courses online that will take you from a total beginner in R programming to a professional capable of performing data manipulation on demand.
This is the best R programming course online that provides you with the comprehensive skill set to confidently begin a new data science project and critically evaluate your own and others’ work.
Rating 4.6 based out of 3646 ratings
Duration 6 hours 30 minutes course
Level Beginner level course
Refund Policy 30-day return policy
Certificate Provided Yes
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded best R Programming certification courses
Course Type Paid
Course Provider 365 careers,365 Simona
Scope for Improvement (Cons) The instructor speaks a bit too fast, making the instructions hard to follow in some places.
Topics Covered
• The building blocks of R
• Vectors and vector operations
• Matrices
• Fundamentals of programming with R
• Data frames
• Manipulating data
• Visualizing data
• Exploratory data analysis
• Hypothesis Testing
• Linear Regression Analysis
Learning Outcomes
• In these best R courses online, you will learn programming principles and use R’s conditional expressions, functions, and loops to your advantage.
• Create your own functions in R.
• Transfer your data into and out of R.
• Use R as a great way to learn the fundamental tools of data science.
• Use the Tidyverse ecosystem of packages to manipulate data.
• In R, investigate data systematically.
• Graphics grammar and the ggplot2 package
• Data visualization to draw conclusions
• When and how to transform data: best practices
• Data indexing, slicing, and subsetting
• Learn the principles of statistics and put them to use.
• R hypothesis testing
• Recognize and apply regression analysis in R
• Use dummy variables to your advantage.
• Learn how to create data-driven decisions!
• Have some fun by disassembling Star Wars and Pokemon data, as well as some more serious data sets.
Who should take this R Programming course online?
• Data scientists with aspirations
• Beginners in programming with interest in statistics and data analysis
• Anyone interested in learning to code and putting their knowledge into practice.
Review Jhonatan Antonio M.
Great!… Very clear explanations, the instructor has a fluent english language very easy to understand, Thanks.
R is the most extensively used statistical programming language. It is strong, versatile, and simple to use. As a result, it is the preferred option for thousands of data analysts in businesses and
academics. You will learn this and more with these best R courses online.
This best online course for R programming will teach you the fundamentals of R in a short time, allowing you to take the first step toward becoming a professional R data scientist.
Rating 4.6 based out of 2626 ratings
Duration 9.5 hours course
Level Beginner level best R Programming certification course
Refund Policy 30-day return policy
Certificate Provided Yes
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Course Type Paid
Course Provider Bogdan Anastasiei
Scope for Improvement (Cons) This course has not been updated for long.
Topics Covered
• Getting Started with R
• Vectors
• Matrices and Arrays
• Lists
• Factors
• Data Frames
• Programming Structures
• Working With Strings
• Plotting in Base R
• Download Links
Learning Outcomes
• Use vectors, matrices, and lists.
• Work with variables
• Control data frames
• Create intricate programming structures (loops and conditional statements)
• Create their own binary operations and functions.
• Create charts in R.
• There are no additional requirements; all you need to know is how to use a computer.
Is it the best R programming course online for you?
• Aspiring data scientists
• Researchers in academia
• Doctoral students
• Anyone who wishes to learn R
Review Jennifer.
Great course! Product is exactly as described. Lectures are easy to understand and implement in R workspace. This course has really been a benefit to me. Thank you!
Coursera is a known platform that offers the best R courses online. This course will teach you how to program in R and how to use R for successful data analysis.
You’ll learn how to install and configure the tools required for a statistical programming environment, as well as how to express generic programming language principles as they’re implemented in a
high-level statistical language.
The R Programming course online addresses practical topics in statistical computing such as programming in R, reading data into R, accessing R packages, writing R functions, debugging, profiling R
code, organizing and commenting R code, etc.
Rating 4.5 based out of 21373 ratings
Duration 57 hours
Level Intermediate level course
Refund Policy 7-days free trial
Certificate Provided Yes
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Course Type Paid
Course Provider Roger. D. Peng, Jeff Leek, Brian Caffo
The assignments were too hard and not well linked to the rest of the content of this R programming certification online course.
Scope for Improvement (Cons)
Topics Covered
• How to write codes
• R Console Input and Evaluation
• Data types- R objects and attributes
• Vectors and lists
• Matrices
• Factors
• Missing Values
• Data frames
• Names attribute
• Textual Data formats
• Subsetting basics
• Subsetting lists
• Subsetting matrices
• Subsetting – partial matching
• Vectorized operations
• Programming with R
• R Functions
• Scoping rules
• Coding standards
• Loop functions and Debugging
• Simulation and Profiling
• The str function
Learning Outcomes
• Understand key programming language ideas
• Set up statistical programming software.
• Utilize R loop functions and debugging tools.
• Using R profiler, collect detailed information.
Familiarity with regression is recommended.
Review MR.
Really interesting course. The interactive coding sessions with swirl are especially useful. Would be great, if you provided sample solutions for the programming assignments, in particular for week
This best R Programming certification course will teach you the fundamentals of the R programming language, including data types, manipulation techniques, and how to accomplish basic programming
With the help of the best R courses online, you will begin to comprehend common data structures, programming fundamentals, and how to handle data.
This R Programming course online has a strong emphasis on hands-on and practical learning. You will use RStudio to develop a simple program, manipulate data in a data frame or matrix, and complete a
final project as a data analyst utilizing Watson Studio and Jupyter notebooks to collect and analyze data-driven insights.
Rating 4.6 based out of 85 ratings
Duration 11 hours course
Level Beginner level course
Refund Policy 7-days free trial
Certificate Provided Yes, you get R programming certification online
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Course Type Paid
Course Provider Yan Luo
The concepts are not taught deeply. This R programming training online is just an overview of the topics covered.
Scope for Improvement (Cons)
Topics Covered
• R basics
• Basic data types
• Math, Variables and Strings
• Writing and Running R in Jupyter Notebooks
• Common data structures
• Vectors and factors
• Vector Operations
• Lists
• Arrays and Matrices
• Data frames
• R programming fundamentals
• Conditions and loops
• Functions in R
• String operations in R
• Regular expressions
• Data format in R
• Debugging
• Working with data
• Reading text files in R
• Writing and saving to files
• HTTP request and REST API
• Web Scraping in R
• Project
• Conclusion
Learning Outcomes
These best R courses online, will teach you to:
• Manipulate numeric and textual data types with the R programming language and RStudio or Jupyter Notebooks.
• With this bestR Programming course for beginners, you will also define and modify R data structures such as vectors, factors, lists, and data frames.
No prior knowledge is required to pursue this R programming certification online course.
Review MR.
Exceptional course for beginners in R programming and data science enthusiasts. Highly recommended!
Learn the fundamentals of programming essential for a career in data science. You will be able to use R, SQL, Command-Line, and Git by the end of the best R courses online.
Rating 4.8 based out of 91 ratings
Duration 3 months course
Level Beginner level course
Refund Policy 2-day return policy
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded R Programming course online
Course Type Paid
Course Provider Josh Bernhard, Derek Steer, Juno Lee, Richard Kalehoff and Karl Krueger
Topics Covered
• Introduction to SQL
• Introduction to R Programming
• Introduction to Version Control
Learning Outcomes
• Learn SQL basics, including JOINs, Aggregations, and Subqueries. Discover how to utilize SQL to solve challenging business problems.
• Understand the essentials of R programming, such as data structures, variables, loops, and functions. With these best R courses onlinealso know how to use the popular data visualization library
ggplot2 to visualize data.
• Learn about version control and how to share your work with others in the data science profession.
Best Online Course for R Programming – Bonus Entries!
Learn with online courses and lessons from Harvard, MIT, and other world-class colleges. Understand the R console, the R community, algorithms, and more.
Some of the best R courses offered by this platform are listed below. You may enroll in them for free and upgrade to the paid option if you need a certification:
LinkedIn Learning has some of the best R courses online for novices and experienced learners.
In this R programming training online with professor and data scientist Barton Poulson, you’ll learn the fundamentals of R and get started with finding insights from your own data.
The tutorials in this R programming course online show you how to get started with R, including how to install R, RStudio, and code packages that increase R’s capabilities. You’ll also learn how to
utilize R and RStudio for basic data modeling, visualization, and statistical analysis.
By the end of the course, you’ll have a solid understanding of R’s power and flexibility, as well as how to use it to explore and analyze a wide range of data.
Rating 4.6 based out of 1212 ratings
Duration 2 hours 51 minutes course
Level Beginner and intermediate level course
Certificate Provided Yes
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Course Type Paid
Course Provider Barton Poulson
Scope for Improvement (Cons) Detailed explanations are missing from this course.
Topics Covered
• R for Data Science
• What is R
• R in context
• Installing R
• Environments for R
• Installing RStudio
• Navigating the RStudio Environment
• Data types and structures
• Comments and headers
• Packages for R
• The Tidyverse
• Data Visualization
• Using colors in R
• Creating histograms
• Data Wrangling
• Data Analysis
A thorough understanding of R programming is essential for data analysis. In this R Programming course online, you will learn how to manipulate various objects.
First, you will learn the fundamental syntax of R coding with these best R courses online.
Following that, you will investigate the data types and data structures available in R.
Finally, you will learn to write your functions using control flow statements.
Rating 4 based out of 53 ratings
Duration 2 hours 2 minutes course
Level Beginner level course
Certificate Provided No
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Course Type Paid
Course Provider Mihaela Danci
Topics Covered
• Why R?
• Integrated Development Environment
• Variables and operators
• Data types
• Code style
• Organizing your code
• Exploring vectors and factors
• Data structures
• Creating vectors
• Manipulating vectors
• Sets
• Using matrices, arrays, and lists
• Working with Data Frames
• Discovering data frames
• Creating data frames
• Manipulating data frames
• Working with tidyverse
• Managing control statements
• Conditional statements
• Switch statement
• Loops
• Building function
• Discovering functions
• Function components
Learning Outcomes
This R programming training online will teach you how to use the R programming language along with the following topics:
• What is R, and why should you use it?
• R data types, R variables and operators
• Investigating vectors and factors in R
• R data frames contain matrices, arrays, and lists.
• Control statements
• Creating your first R function
Is it the right R programming course online for you?
This course is excellent for anyone with interest in data analysis. This R course is ideal if you want to learn the fundamental syntax of R programming as well.
This data science with R course prepares you to apply your Data Science skills in a variety of settings, assisting companies in analyzing data and making more informed business decisions.
These best R courses online incorporate cutting-edge curriculum and dedicated mentoring sessions to help you develop job-ready skills.
Rating 4.3 based out of 9457 ratings
Duration 64 hours course
Level Beginner level course
Certificate Provided Yes
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Refund 7-days money back guarantee
Course Type Paid
Course Provider Simplilearn
Topics Covered
• Introduction to business analytics
• Introduction to R Programming
• Data structures
• Data Visualization
• Statistics for data science
• Regression analysis
• Classification
• Clustering
• Association
Learning Outcomes
• Business analytics
• R programming and its packages
• Data structures and data visualization
• Apply functions and DPLYR function
• Graphics in R for data visualization
• Hypothesis testing
• Apriori algorithm
• kmeans and DBSCAN clustering
Is it the right R programming training online for you?
This Data Science with R certification training is beneficial for all ambitious young data scientists, including IT professionals or software developers.
In ‘Introduction to R’ is one of the best R courses online that will help you master the basics of this widely used open-source language, including factors, lists, and data frames.
With the knowledge gained in this R Programming course online, you will be ready to undertake your first very own data analysis.
Rating 4.3 based out of 9457 ratings
Duration 4 hours course
Level Beginner level R programming certification online course
Certificate Provided No
Course Material Provided Yes
Live Classes/Recorded Lessons Recorded lessons
Course Type Free
Course Provider DataCamp
Topics Covered
• Intro to basics
• Vectors
• Matrices
• Factors
• Data Frames
• Lists
The best R courses online on Codecademy will introduce you to essential programming ideas in R. After you’ve mastered the fundamentals, you’ll learn how to organize, edit, and clean data frames, a
valuable data structure in R.
Then you’ll learn how to construct data visualizations to highlight data insights! Finally, to become a data analysis specialist, finish with statistics and hypothesis testing.
Want a free R programming certification online? Great Learning Academy offers one of the most excellent R programming courses, which will lead you through the basics of R programming, such as data
types, data structures, control statements, and so on, as well as hands-on practice for each.
This is a free R programming course online that also bags you a certificate. Here you will learn about R packages, functions, operators, matrix, vectors and commands.
Related read: Difference Between Software Engineer and Software Developer
We at TangoLearn tried to list the best R programming course for beginners and experienced learners. Ss select the most suitable course for yourself and take your career to the next level.
Best R Programming Courses is rated 4.7 and reviewed by 14 R Programming Experts & 30+ R Programming Classes Students
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Microeconomics - Online Tutor, Practice Problems & Exam Prep
So now we're going to learn how to take points on the graph and turn them into a curve, as well as how to shift that curve on the graph. We're also going to learn how to shift curves just visually
with no math. It's a tool we'll use quite often in this class. So let's go ahead and look on the graph here. You'll see I have the points from the previous video where we learned how to put points on
the graph. I've got those points here already. So when we're turning points into a curve, what we do is we start at the leftmost point and we work our way rightwards. Okay. This one seems pretty
simple. It's just going to make a line and, yes, a line is a curve. It's just a straight curve. So this is what this curve may look like right here in green. Alright. I just want to make an example
here. I'm going to do something right here on top. Let's say we had points that looked like this. Ignore the other points right now, but let's say there were points like this. There's a specific way
we want to connect those. Right? We want to start left to right just like I said. You never want to double back and start going back to the left or back to the right. Let me show you an example here.
Right. So these points, we would connect them something like this. Right. And I don't want you to get confused and connect them maybe like this. Right? That's not how we would connect these points.
You start at the leftmost point and you go to the right. Cool?
So now let's talk about shifting this green curve right here. How do we shift it on the graph? Let's say someone told us we had to shift this curve, let's say, 2 units to the right. 2 units to the
right. Okay? So how do we do that? What we're going to do, and the easiest way I find to do it is I pick the leftmost point, so in this case it would be this point that I'm going to circle here in
black, Right? And we're going to move it 2 spaces to the right, so I'm going to count here 2. That's 1 and that's 2 right there. That's going to be our new point that I'll put in blue. Cool? So you
do that with your leftmost point and I like to just go straight to the rightmost point and do the same thing. Grab my pen and I'll pick this rightmost point right down here and I'm going to move it 2
spaces to the right. 1, 2. Right? And I'm going to put my point right there, my new point, and now that I have two points, if you just connect these two points you'll have your new line. So I'll do
that one in green as well. So here we go. Connecting these two points, we've got our shifted curve. So this new green curve right here, it's shifted 2 to the right. Actually, I'm going to do it in
blue so we can see which one's which. So the blue curve has been shifted 2 to the right. Cool? And a lot of times in this class, like I said, we're going to be doing shifting of curves just visually.
We're not going to put any math behind it. We're going to have a reason we're shifting the curve, and then we're going to have to see what happens, after we've shifted the curve. And when I say see
what happens, we're going to see what happened to the new price and the new quantity. But we'll get more into that in the next chapter. So I'm going to draw a couple graphs here just to explain what
I mean by shifting visually. So a lot of times on a test or on a practice problem you're just going to kind of draw a graph kind of willy nilly like this and a lot of them are going to be graphs that
look like this. We're going to have an X and remember I suggested having at least 2 colors, and we're going to use those quite often. So in this case what we're going to do is kind of like we did
above, on the graph. We're going to shift the red line, to the right. So now it's not 2 units, we're just shifting to the right. Cool? So what you do is you start and you're going to pick a point on
the graph. You're going to move it to the right and then you're going to draw a parallel line just like that. Right. So when we do these kinds of shifts what we're doing is looking for these points
of intersection. Where this was the point of intersection originally, now we're at this point of intersection here, right? So we would be judging what happened to the price and what happened to the
quantity after this shift, Right? So we can make, assessments of that just visually without doing any math. But we'll deal more with analyzing it, when the time comes. Now I'm just trying to expose
you to shifting the graphs like that. So let's do a couple more examples here. Now let's shift the red line to the left. So we're going to start with the red line and you can see now if I want to the
left, I kinda end up off the graph here. Right? So maybe I can pick a point like a little further down and go to the left here, and now I can draw a parallel line. Right? So you just want to make
sure you're going the correct direction, and let's see where our new intersection is. We were here before and now we've moved down there, right? So we will be able to make assessments about price and
quantity based on that movement. Cool, a couple more examples here. So now let's move the blue line. Let's see what happens when we move the blue line to the right. So same thing, we're going to pick
a point here, move it directly to the right, and draw a parallel line. Cool? So point of intersection was there and now it's there. Alright, now let me get out of the way. I'm going to do one more
example here in this last corner. So sometimes we actually have to shift both the lines on the same graph, and that's when it starts to get a little difficult, remembering which line was which. So I
like to draw my x so that the original point of intersection is right in the middle. Right? When I'm doing it visually, I just keep that point of intersection in the middle and then I'm going to look
at my new points of intersection. So let's say we had to move, let's just move them both to the right in this example. So first let's move the red line to the right. So I'm going to pick a point
here, move it to the right, and draw my new line. And now I'm going to draw the blue line, we're also going to shift to the right. So pick a point, go to the right, and draw my new line. So you can
see now there's a lot of points of intersection here. It's like which one is the new one, which one's the old one. So you gotta be careful and pick the correct intersection here. So you want to have
good eyes at finding these because this is going to happen quite a bit in this class when we're studying demand and supply. So which one did you pick? This This is going to be our new point of
intersection right here. Let me go yeah I'll leave it in green so we can see that that's the new point of intersection right there. Cool. So this is how we're going to shift curves visually on the
graph and we can also do it mathematically like we did on the left But mostly, we're going to do it like we did on the right in this course. See you in the next video. | {"url":"https://www.pearson.com/channels/microeconomics/learn/brian/reading-and-understanding-graphs/relationships-between-variables?chapterId=49adbb94","timestamp":"2024-11-03T13:33:43Z","content_type":"text/html","content_length":"301702","record_id":"<urn:uuid:8c44ec21-629a-4a3a-8a50-b4c1a5a159fd>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00472.warc.gz"} |
Dekalb Middle School Weekly Lesson Plan s5
DeKalb Middle School Weekly Lesson Plan
Teacher: Birmingham/Jones EXCEL ACADEMY:
Week of: Sept 4-8, 2017 Focus: Computation
Subject: Math Interventions: MobyMax, Drops in the Bucket
Class Hour: 7:30 – 2:42 Harvard Tier 2 & 3
Day/Date /
CC Standard(s) (and/or Big Idea/Goals)
Lesson Activities
/ Assessment
Standard Essential Question
Practice Assignment
*(This is subject to change)
Monday / NO SCHOOL
Tuesday / 7.NS.A.1
7.NS.A.3 / Review, Reinforcement, and/or Enrichment Activities of Adding & Subtracting Integers / informal observation of verbal and written answers, self-evaluation and reflection, peer reviews,
feedback from technological practice, and/or written assessment / How can you use addition and subtraction of integers to solve real-world problems? / Practice solving mathematical and word problems
with same- and different-sign integers using addition and subtraction
Wednesday / 7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. / PPT - Multiplying and Dividing Integers
PPT - Perfection Learning / informal observation of verbal and written answers, self-evaluation and reflection, peer reviews, feedback from technological practice, and/or written assessment / How can
you use multiplication and division of integers to solve real-world problems? / Practice solving problems with positive and negative integers using multiplication
Thursday / 7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. / PPT - Multiplying and Dividing Integers
PPT - Perfection Learning / informal observation of verbal and written answers, self-evaluation and reflection, peer reviews, feedback from technological practice, and/or written assessment / How can
you use multiplication and division of integers to solve real-world problems? / Practice solving problems with positive and negative integers using multiplication
Friday / 7.NS.A.2
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. / PPT - Multiplying and Dividing Integers
PPT - Perfection Learning / informal observation of verbal and written answers, self-evaluation and reflection, peer reviews, feedback from technological practice, and/or written assessment / How can
you use multiplication and division of integers to solve real-world problems? / Practice solving problems with positive and negative integers using division | {"url":"https://docest.com/doc/269183/dekalb-middle-school-weekly-lesson-plan-s5","timestamp":"2024-11-12T23:31:32Z","content_type":"text/html","content_length":"23908","record_id":"<urn:uuid:fc8f3895-1dd6-4f8b-8b19-63beb51f1b3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00647.warc.gz"} |
Fibonacci C code | C, C++ and C#
Aug 4, 2018
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I was given this question in my interview but couldn't crack it.
Please go through it and help me out.
As we know in Fibonacci series every number after the first two is the sum of the two preceding ones.
Instead of adding two preceding numbers, multiply them and print the result modulo 10^9+7.
Since this is easy, let’'s make it bit difficult. Let'’s say there are K numbers to begin with.
You have to find nth number, where nth number will be product of k previous numbers modulo 10^9+7.
Input Format
First line contains T number of test case,
In each test case
First line contains two integers n, k delimited by space
Second line contains k integers delimited by space
T lines, each line contains modified Fibonacci number modulo 109+7
Example 1
4th , 5th , 6th modified Fibonacci numbers are 6 , 36 , 648 respectively
Similarly 10th modified Fibonacci number will be 845114970
My Solution
#define Modulo 1000000007
long long int Num2Mod(long long int A,long long int B);
int K[10],n,k,i,j,t,first,second =1;
long long int product,list[1000000];
scanf("%d %d",&n,&k);
product = 1;
product = Num2Mod(product,list[i+j]);
list[i+k] = product;
long long int Num2Mod(long long int A,long long int B)
return ((A%Modulo*B%Modulo)%Modulo);
Help me out here please.
Thank You.
Last edited by a moderator:
Jun 27, 2018
Reaction score
OK, there are a number of problems there that I can instantly see that would be bad from an interviewers point of view:
1. You have two unused variables in your code called "first" and "second".
Leaving unused variables in code is a little bit sloppy and definitely NOT what you want to be doing in an interview situation!
2. No prompts for the user.
As your code stands, your program will just hang there waiting for input.
A user will not know what kind of input is expected without some kind of prompt. Your interviewers may have wanted to see instructional prompts in your program.
3. Some of the variables are badly named.
Single letter variable names are not very helpful. By using clear, concise variable names, your code will be more readable and understandable.
So for example your variable 't' could have been called something like 'numTests', or 'numberOfTests', or 'testsToPerform'. Likewise with some of your other variables. Giving variables descriptive
names instantly makes the code more meaningful.
4. No comments in the code
Comments are extremely valuable and can help interviewers to gauge a candidates level of understanding of a problem. That way even if the code/logic is slightly wrong - the comments can help to make
programmers intentions/understanding clear to the interviewers (or others viewing the code).
5. No bounds checks:
There are no bounds checks on any of the user-input values. User input should NEVER be trusted and should always be checked.
- You haven't checked the value of t (the number of tests) to ensure it is between 1 and 10. The user could set the program to run for any number of tests. It was clearly specified that the number
should be between 1 and 10, so you should check it!
- You haven't checked the value of k (the number of existing numbers) to ensure it is between 1 and 10 (or 1 and 100) - not sure why you have two limits there?! But either way, neither of the limits
have been enforced.
- You haven't checked the value of n (the number of numbers to calculate up to).
And even if you didn't have time to add bounds checks - you could at least put a comment in the code.
/* TODO - Bounds check n - ensure 1 < n < x */
That shows interviewers that you understand that this would normally need to be bounds checked.
But without a bounds check OR some kind of comment to indicate that you know a bounds-check is appropriate - the interviewer will assume that you hadn't considered bounds checking, which would imply
to the interviewer that you are either a sloppy programmer, or you didn't know any better!
6. Allocating extremely large items on the stack
The biggest problem that I can see in your code is in this line, which immediately set off alarm bells in my head:
long long int product,list[1000000];
The declaration of the array variable called 'list' - where you are trying to create an array on the stack, to hold up to one million long long int's. This would almost certainly require more memory
than the stack would ever be allocated and would probably cause your program to crash as soon as you tried to run it. That is a pretty fatal mistake if you ask me. That alone may have been enough to
blow the entire interview!
The stack size can vary from platform to platform and although it may be possible to use compiler options to customise/increase the stack size for a program - allocating such a large item on the
stack is not a good idea! Anything that large should be declared dynamically, on the heap NOT on the stack.
Instead of declaring 'list' as a huge array of long long int, it would be better and much more efficient to use a long long int pointer instead. In the declaration at the top of the scope - you
should initialise the pointer to NULL, then later on - after you have read the value for n (the number of numbers to find/calculate), you can use malloc to allocate an appropriately sized block of
memory on the heap and assign the returned pointer to list.
//.... variable declarations at the top of the scope ....
long long int* list='\0'; // list initialised to null
// ....
// .... More code ....
// ....
//...after reading the value of n...
// Allocate memory
list = malloc(n*sizeof(long long int));
// Check the pointer is valid:
printf("Failed to allocate space for list - exiting\n");
return EXIT_FAILURE;
// ....
// .... More code ....
// ....
// Don't forget to free the memory when it has been finished with:
list='\0'; // Explicitly reset the pointer to NULL to avoid having a dangling pointer.
That approach is much more efficient. You will only be allocating the memory that you require for the list of values.
And more importantly there is no chance that you will exceed the stack-size and cause a crash.
Besides - there is no point allocating space for a million values if the user only wants to calculate up to the 10th.
So for extremely large items, or items where the size could vary at run-time - it is much better to find out how much space you need first and then dynamically allocate memory for it on the heap.
And once 'list' is pointing to the start of a block of memory - I'd leave it pointing there. Don't use pointer arithmetic to iterate through the array. Because if you aren't careful - you might lose
track of where the start of the block is and could be unable to deallocate the memory properly. So to dereference each number in the array, you could simply use array notation (as you are already
doing with your stack-smashing statically allocated array).
So for this problem, I think your interviewers would have expected you to have demonstrated some knowledge of memory management and would have been expecting your list to be a dynamically allocated
-- This next point is to do with posting here:
Finally, for future reference - when posting here - please use code-tags as it preserves the formatting of your code.
Because you failed to use code-blocks in your original post - I'm assuming that this hideously broken code:
Is actually supposed to be like this:
To explain what has happened:
When you posted your code as plain-text, all instances of [i] would have been interpreted by the editor here as markup code for italics. So everything in your original post would have been italicised
from the first instance of [i] and all instances of [i] would have disappeared from your post. And this made your code look like it had extra bugs in it.
Obviously whoever updated your post to use code-tags didn't realise that this was a problem and didn't re-add the missing instances of [i]. But then they might not know C well enough to know that
this could be an issue. So please - always use code-tags!
There a couple of things about the above section of code too:
8. The array 'K' has been hard-coded to a size of 10. But because you haven't range-checked the value of the variable 'k', it is possible that a user could enter a value for 'k' that exceeds 10.
Which would mean that when entering values into 'K' - the bounds of the array could be exceeded. Which is another fairly major, yet preventable bug. Yet another mistake that your interviewers would
hold against you and yet another reason for you to range-check user-entered values!
9. The array called 'K' is completely redundant anyway, you don't need it.
You could just populate the 'list' array directly:
scanf("%lld", &list[i]);
And on a personal note:
I'd explicitly declare Modulo as a const long long int, rather than using a macro.
const long long int Modulo=1000000007;
Because you are using 'long long int' everywhere else and because the value of 'Modulo' is a constant value that you will be using - it makes sense to explicitly declare it as such, rather than using
#define - where it will be defined as an int by the pre-processor. But that's just a stylistic thing and my personal preference!
Other than that, the rest of the algorithm/logic looks good..... I think! I haven't completely desk-checked the numbers, or the algorithm you're using for the final calculations - but from reading
your description and from taking a quick look at this bit - off the top of my head - I think the rest of the logic is more or less OK and should yield the expected answers.
But although your program
like it should be able to produce the correct results - I'd guess that most of the problems that I have pointed out in your code are probably the main reasons that you failed the question.
The biggest one will be the problem with the massive array of long long ints on the stack. If your interviewers tried compiling and running your program, I'm pretty certain it would crash straight
away! But once you have fixed that issue - your program probably will correctly solve the problem. But again, only as long as the user doesn't enter values that exploit any of the other bugs I have
mentioned in your program.
So you have a few things to work on. Hopefully this provides some insight for you!
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A general formulation of Bead Models applied to flexible fibers and active filaments at low Reynolds number PDF
A general formulation of Bead Models applied to flexible fibers and active filaments at low Reynolds number Blaise Delmottea,b, Eric Climenta,b, Franck Plourabou´ea,b,∗ aUniversity of Toulouse INPT-UPS:
Institut de M´ecanique des Fluides, Toulouse, France. bIMFT - CNRS, UMR 5502 1, All´ee du Professeur Camille Soula, 31 400 Toulouse, France. 5 1 0 Abstract 2 This contribution provides a general
framework to use Lagrange multipliers for the simula- n a tion of low Reynolds number fiber dynamics based on Bead Models (BM). This formalism J provides an efficient method to account for kinematic
constraints. We illustrate, with several 3 1 examples, to which extent the proposed formulation offers a flexible and versatile framework for the quantitative modeling of flexible fibers deformation and
rotation in shear flow, the ] n dynamics of actuated filaments and the propulsion of active swimmers. Furthermore, a new y contact model called Gears Model is proposed and successfully tested. It
avoids the use of d - numerical artifices such as repulsive forces between adjacent beads, a source of numerical u difficulties in the temporal integration of previous Bead Models. l f . s Keywords:
Bead Models, fibers dynamics, active filaments, kinematic constraints, Stokes c i flows s y h p [ 1. Introduction 1 v The dynamics of solid-liquid suspensions is a longstanding topic of research while
it 5 combines difficulties arising from the coupling of multi-body interactions in a viscous fluid 3 9 with possible deformations of flexible objects such as fibers. A vast literature exists on the 2
response of suspensions of solid spherical or non-spherical particles due to its ubiquitous 0 . interest in natural and industrial processes. When the objects have the ability to deform 1 0 many
complications arise. The coupling between suspended particles will depend on the 5 positions (possibly orientations) but also on the shape of individuals, introducing intricate 1 effects of the
history of the suspension. : v When the aspect ratio of deformable objects is large, those are generally designated as i X fibers. Manypreviousinvestigationsoffiberdynamics,
havefocusedonthedynamicsofrigid r a fibers or rods [1, 2]. Compared to the very large number of references related to particle ∗Corresponding author: Franck Plourabou´e. Tel.: +33 5 34 32 28 80 Email
addresses: blaise.delmotte@imft.fr (Blaise Delmotte), eric.climent@imft.fr (Eric Climent), franck.plouraboue@imft.fr (Franck Plourabou´e ) Preprint submitted to Journal Computational Physics January
14, 2015 suspensions, lower attention has been paid to the more complicated system of flexible fibers in a fluid. Suspension of flexible fibers are encountered in the study of polymer dynamics [3, 4]
whose rheology depends on the formation of networks and the occurrence of entanglement. The motion of fibers in a viscous fluid has a strong effect on its bulk viscosity, microstruc- ture, drainage
rate, filtration ability, and flocculation properties. The dynamic response of such complex solutions is still an open issue while time-dependent structural changes of the dispersed fibers can
dramatically modify the overall process (such as operation units in wood pulp and paper industry, flow molding techniques of composites, water purification). Biological fibers such as DNA or actin
filaments have also attracted many researches to understand the relation between flexibility and physiological properties [5]. Flexible fibers do not only passively respond to carrying flow gradients but
can also be dynamically activated. Many of single cell micro-organisms that propel themselves in a fluid utilizealongflagellumtailconnectedtothecellbody. Spermatozoa(andmoregenerallyone- armed
swimmers) swim by propagating bending waves along their flagellum tail to generate a net translation using cyclic non-reciprocal strategy at low Reynolds number [6]. These natural swimmers have been
modeled by artificial swimmers (joint microbeads) actuated by an oscillating ambient electric or magnetic field which opens breakthrough technologies for drug on-demand delivery in the human body [7].
Many numerical methods have been proposed to tackle elasto-hydrodynamic coupling between a fluid flow and deformable objects, i.e. the balance between viscous drag and elastic stresses. Among those,
“mesh-oriented” approaches have the ambition of solving a complete continuum mechanics description of the fluid/solid interaction, even though some approximations are mandatory to describe those at
the fluid/solid interface. Without being all-comprehensive, one can cite immerse boundary methods (e.g. [8, 9, 10, 11]), extended finite elements (e.g. [12]), penalty methods [13, 14], particle-mesh
Ewald methods [15], regularized Stokeslets [16, 17], Force Coupling Method [18]. In the specific context of low Reynolds numberelastohydrodynamics [19], difficulties arise when numerically solving the
dynamics of rigid objects since the time scale associated with elastic waves propagation within the solid can be similar to the viscous dissipation time- scale. In the context of self propelled
objects the ratio of these time scales is called “Sperm number”. When the Sperm number is smaller or equal to one, the object temporal response is stiff, and requires small time steps to capture fast
deformation modes. In this regime, fluid/structure interaction effects are difficult to capture. One possible way to circumvent such difficulties is to use the knowledge of hydrodynamic interactions of
simple objects in Stokes flow. This strategy is the one pursued by the Bead Model (BM) whose aim is to describe a complex deformable object by the flexible assembly of simple rigid ones. Such flexible
assemblies are generally composed of beads (spheres or ellipsoids) interacting by some elastic and repulsive forces, as well as with the surrounding fluid. For long elongated structures, alternative
approaches to BM are indeed possible such as slender body approximation [20, 21, 22] or Resistive Force Theory [23, 24, 25]. One important advantage of BM which might explain their use among various
commu- 2 nities (polymer Physics [26, 29, 31, 34], micro-swimmer modeling in bio-fluid mechanics [38, 39, 40, 43], flexible fiber in chemical engineering [46, 48, 50, 52]), relies on their para- metric
versatility, their ubiquitous character and their relative easy implementation. We provide a deeper, comparative and critical discussion about BM in Section 2. However, we would like to stress that
the presented model is more clearly oriented toward micro-swimmer modeling than polymer dynamics. OneshouldalsoaddthatBMcanbecoupledtomesh-orientedapproachesinordertopro- vide accurate description of
hydrodynamic interactions among large collection of deformable objects at moderate numerical cost [43]. Many authors only consider free drain, i.e no Hy- drodynamic Interactions (no HI), [27, 49, 48,
53] or far field interactions associated with the Rotne-Prager-Yamakawa tensor [40, 36, 35, 54]. This is supported by the fact that far-field hydrodynamic interactions already provide accurate
predictions for the dynamics of a single flexible fiber when compared to experimental observations or numerical results. In order to illustrate the method we use, for convenience, the
Rotne-Prager-Yamakawa tensor to model hydrodynamic interactions. We wish to stress here that this is not a limitation of the pre- sented method, since the presented formulation holds for any mobility
problem formulation. However, it turns out that for each configuration we tested, our model gave very good com- parisons with other predictions, including those providing more accurate description of
the hydrodynamic interactions. The paper is organized as follows. First, we give a detailed presentation of the Bead Modelforthesimulationofflexiblefibers. Inthissection,weproposeageneralformulationof
kinematicconstraintsusingtheframeworkofLagrangemultipliers. Thisgeneralformulation is used to present a new Bead Model, namely the Gears Model which surpasses existing models on numerical aspects.
The second part of the paper is devoted to comparisons and validations of Bead Models for different configurations of flexible fibers (experiencing a flow or actuated filaments). Finally, we conclude the
paper by summarizing the achievements we obtain with our model and open new perspectives to this work. 2. The Bead Model 2.1. Detailed Review of previous Bead Models The Bead Model (BM) aims at
discretizing any flexible object with interacting beads. Interactions between beads break down into three categories: hydrodynamic interactions, elastic and kinematic constraint forces. Hydrodynamics
of the whole object result from multibody hydrodynamic interactions between beads. In the context of low Reynolds num- ber, the relationship between stresses and velocities is linear. Thus, the
velocity of the assembly depends linearly on the forces and torques applied on each of its elements. Elastic forces and torques are prescribed according to classical elasticity theory [55] of flexible
mat- ter. Constraint forces ensure that the beads obey any imposed kinematic constraint, e.g. fixed distance between adjacent particles. All of these interactions can be treated separately as long as
they are addressed in a consistent order. The latter is the cornerstone which differentiates previous works in the literature from ours. Numerous strategies have been 3 employed to handle kinematic
constraints. [32, 40, 35, 34] and [50] used a linear spring to model the resistance to stretching and compression without any constraint on the bead rotational motion (Fig. 1). The resulting
stretching force reads: Fs = −k (r −r0 ) (1) s i,i+1 i,i+1 where • k is the spring stiffness, s • r = r − r is the distance vector between two adjacent beads (for simplicity, i,i+1 i+1 i equations and
figures will be presented for beads 1 and 2 and can easily be generalized to beads i and i+1), • r0 is the vector corresponding to equilibrium. 1,2 However, regarding the connectivity constraint, the
spring model is somehow approxi- mate. A linear spring is prone to uncontrolled oscillations and the problem may become unstable. Many other authors, among which [28, 29, 30], thus use non-linear
spring models for a better description of polymer physics. Nevertheless, the repulsive force stiffness has an important numerical cost in time-stepping as will be discussed in section 2.6.3.
Furthermore, unconstrained bead rotational motion leads to spurious hydrodynamic interactions and thus limits the range of applications for these BM. Alternatively, [49, 48, 53, 47] and [46]
constrained the motion of the beads such that the contact point for each pair c remains the same. While more representative of a flexible i object, this approach exhibits two main drawbacks: 1. a gap
between beads is necessary to allow the object to bend (see Fig. 2), 2. it requires an additional center to center repulsive force, and thus more tuning numer- ical parameters to prevent overlapping
between adjacent beads. Consider two adjacent beads, with radius a, linked by a hinge c (typically called ball 1 and socket joint). The gap ε defines the distance between the sphere surfaces and the
joint g (see Fig. 3). Denote p the vector attached to bead i pointing towards the next joint, i.e. i the contact point c . i The connectivity between two contiguous bodies writes: [r +(a+ε )p ]−[r −
(a+ε )p ] = 0 (2) 1 g 1 2 g 2 and its time derivative [r˙ −(a+ε )p ×ω ]−[r˙ +(a+ε )p ×ω ] = 0. (3) 1 g 1 1 2 g 2 2 r˙ and ω are the translational and rotational velocities of bead i. The constraint
forces and i i torques associated to (3) are obtained either by solving a linear system of equations involv- ing beads velocities [53], or by inserting (3) into the equations of motion when
neglecting 4 Figure 1: Spring model: linear spring to keep constant the inter-particle distance. Figure 2: Joint Model: overlapping due to bending if no gap between beads. Figure 3: Joint Model: c is
separated by a gap ε from the beads. 1 g Figure 4: Gears Model: contact velocity must be the same for each bead (no-slip condition). 5 hydrodynamic interactions [49, 48]. Thegapwidth2ε
controlsthemaximumcurvatureκJ allowedwithoutoverlapping. From g max the sine rule, one can derived the simple equation relating ε and κJ g max (cid:114) (cid:16) (cid:17)2 1− a κJ = a+εg (4) max a
Once aware of these limitations, the gap ε , range and strength of the repulsive force should g be prescribed depending on the problem to be addressed. [56] and [43] proposed a more sophisticated
Joint Model than those hitherto cited, using a full description of the links dynamics along the curvilinear abscissa. They derived a subtle constraint formulation which ensures that the tangent
vector to the centerline is continuous and that the length of links remains constant. These two works are worth mentioning since they avoid an empirical tuning of repulsive forces. Yet, [56] computed
the constraint forces and torques with an iterative penalty scheme instead of using an explicit formulation. Finally, it is worth mentioning that the bead model proposed in [31] circumvents the
inextensibility difficulty by imposing constraints on the relative velocities of each successive segments, so that their relative distance is kept constant. Using bending potential, [31] permit overlap
between beads with restoring torque (cf. Fig. 2). A Lagrangian multiplier formulation of tensile forces is also used in [57], which is equivalent to a prescribed equal distance between successive
beads. Again, inextensibility condition does not prevent bead overlapping due to bending in this formulation. The computation of contact forces which is proposed in the following section 2.2
generalizes the Lagrangian multiplier formulation of [31] to generalized forces. Using more complex constraints involving both translational and angular velocities, we show that it is possible to
accommodate both non-overlapping and inextensibility conditions without additional repulsive forces (using the rolling no-slip contact with the gears model detailed in 2.3). This proposed general
formulation is also well suited for any type of kinematic boundary conditions as illustrated in Section 3.4. 2.2. Generalized forces, virtual work principle and Lagrange multipliers The model and
formalism proposed in this article rely on earlier work in Analytical Me- chanics and Robotics [58, 59]. The concept of generalized coordinates and constraints which has proven to be very useful in
these contexts is described here. Generalized coordinates refer to a set of parameters which uniquely describes the configuration of the system relative to somereference parameters(positions,
angles,...). For describingobjectsof complex shape, let us consider the position r of each bead i ∈ {1,N } with associated orientation vector p i b i which is defined by three Euler angles p ≡
(θ,φ,ψ). In the following, any collection of vector population (r ,..r ,..r ) ≡ R will be capitalized, so that R is a vector in R3Nb. Hence the 1 i N b collection of orientation vectors p will be
denoted P, which is a vector of length 3N , the i b collection of velocities dri = r˙ = v , will be denoted V, the collection of angular velocity dt i i p˙ ≡ ω will be Ω, the collection of forces f ,
F, the collection of torques γ , Γ. All V, Ω, F i i i i and Γ are vectors in R3Nb. 6 Let us then define some generalized coordinate q for each bead, which is defined by q ≡ i i (r ,p ) ≡ {r ,r ,r ,θ ,φ
,ψ }sothatthecollectionofgeneralizedpositions(q ,..q ,..q ) ≡ i i 1,i 2,i 3,i i i i 1 i N b Q is a vector in R6Nb. Generalized velocities are then defined by vectors q˙ ≡ (v ,ω ) with i i i associated
generalized collection of velocities Q˙ . Articulated systems are generically submitted to constraints which are either holonomic, non-holonomic or both [33]. Holonomic constraints do not depend on
any kinematic param- eter (i.e any translational or angular velocity) whereas non-holonomic constraints do. In the following we consider non-holonomic linear kinematic constraints associated with
generalized velocities of the form [60] JQ˙ +B = 0, (5) such that J is a 6N ×N matrix and B is a vector of N components. N is the number of b c c c constraints acting on the N beads. B and J might
depend (even non-linearly) on time t b and generalized positions Q, but do not depend on any velocity of vector Q˙ , so that relation (5) is linear in Q˙ . In subsequent sections, we provide specific
examples for which this class of constraints are useful. Here we describe, following [60, 58] how such constraints can be handledthankstosomegeneralizedforcethatcanbedefinedfromLagrangemultipliers.
The idea formulated to include constraints in the dynamics of articulated systems is to search additional forces which could permit to satisfy these constraints. First, one must rely on generalized
forces f ≡ (f ,γ ) which include forces and torques acting on each bead, whose i i i collection (f ,f ,..f ) is denoted F. Generalized forces are defined such that the total work 1 i N b variation δW
is the scalar product between them and the generalized coordinates variations δQ δW = F·δQ = F·δR+Γ·δP, (6) so that, on the right hand side of (6) one also gets the translational and the rotational
components of the work. Then, the idea of virtual work principle is to search some virtual displacement δQ that will generate no work, so that F·δQ = 0. (7) At the same time, by rewriting (5) in
differential form JdQ+Bdt = 0, (8) admissible virtual displacements, i.e those satisfying constraints (8), should satisfy JδQ = 0. (9) Combining the N constraints (9) with (7) is possible using any
linear combination of these c constraints. Such linear combination involves N parameters, the so-called Lagrange multi- c pliers which are the components of a vector λ in RNc. Then from the difference
between (7) and the N linear combination of (9) one gets c (F − λ·J)·δQ = 0. (10) 7 Prescribing an adequate constraint force F = λ·J, (11) c permits to satisfy the required equality for any virtual
displacement. Hence, the con- straints can be handled by forcing the dynamics with additional forces, the amplitude of which are given by Lagrange multipliers, yet to be found. Note also, that this
first result im- plies that both translational forces and rotational torques associated with the N constraints c are both associated with the same Lagrange multipliers. This formalism is particularly
suitable for low Reynolds number flows for which trans- lational and angular velocities are linearly related to forces and torques acting on beads by the mobility matrix M (cid:18) (cid:19) (cid:18)
(cid:19) (cid:18) (cid:19) V F V∞ = M + +C : E∞. (12) Ω Γ Ω∞ V∞ = (cid:0)v∞,...,v∞(cid:1) and Ω∞ = (cid:0)ω∞,...,ω∞(cid:1) correspond to the ambient flow evaluated at 1 N 1 N b b the centers of mass r
. E∞ is the rate of strain 3 × 3 tensor of the ambient flow. C is a i third rank tensor called the shear disturbance tensor, it relates the particles velocities and rotations to E∞ [54]. Matrix M (and
tensor C) can also be re-organized into a generalized mobilitymatrixM (generalizedtensorC resp.) inordertodefinethelinearrelationbetween the previously defined generalized velocity and generalized
force Q˙ = MF+V ∞ +C : E∞, (13) where V ∞ = (cid:0)v∞,ω∞,...,v∞,ω∞(cid:1). The explicit correspondence between the classical ma- 1 1 N N trix M and the hereby proposbed gebneralized coordinate
formulation M is given in Appendix A. Hence, as opposed to the Euler-Lagrange formalism of classical mechanics, the dynam- ics of low Reynolds number flows does not involve any inertial contribution,
and provide a simple linear relationship between forces and motion. In this framework, it is then easy to handle constraints with generalized forces, because the total force will be the sum of the
known hydrodynamic forces F , elastic forces F , inner forces associated to active fibers F h e a and the hereby discussed and yet unknown contact forces F to verify kinematic constraints c F = F
(cid:48) +F , with (14) c F(cid:48) = F +F +F . (15) h e a Hence,ifoneisabletocomputetheLagrangemultipliersλ,thecontactforceswillprovide the total force by linear superposition (14), which gives the
generalized velocities with (13). Now, let us show how to compute the Lagrange multiplier vector. Since the generalized force is decomposed into known forces F(cid:48) and unknown contact forces F =
λ·J, relations c (14) and (13) can be pooled together yielding MF = MλJ = Q˙ −MF(cid:48) −V ∞ −C : E∞. (16) c 8 So that, using (5), JMJTλ = −B−J (MF(cid:48) +V ∞ +C : E∞), (17) one gets a simple
linear system to solve for finding λ, where JT stands for the transposition of matrix J. 2.3. The Gears Model The Euler-Lagrange formalism can be readily applied to any type of non-holonomic con-
straint such as (3). In the following, we propose an alternative model based on no-slip condition between the beads: the Gears Model. This constraint, first introduced in a Bead Model (BM) by [27],
conveniently avoid numerical tricks such as artificial gaps and repulsive forces. However, [27] and [61] relied on to an iterative procedure to meet requirements. Here, we use the Euler-Lagrange
formalism to handle the kinematic constraints associated to the Gears Model. Considering two adjacent beads (Fig. 4), the velocity v at the contact point must be c1 the same for each sphere: v1 −v2 =
0. (18) c1 c1 vL and vR are respectively the rigid body velocity at the contact point on bead 1 and bead c1 c1 2. Denote σ1 the vectorial no-slip constraint. (18) becomes σ1(r˙ ,ω ,r˙ ,ω ) = 0, (19)
1 1 2 2 i.e. [r˙ −ae ×ω ]−[r˙ −ae ×ω ] = 0, (20) 1 12 1 2 21 2 where e is the unit vector connecting the center of bead 1, located at r , to the center 12 1 of bead 2, located at r (e = e − e ). The
orientation p vector attached to bead i, is 2 12 2 1 i not necessary to describe the system. Hence, from (20) one realises that σ1 is linear in translational and rotational velocities. Therefore
equation (19) can be reformulated as (cid:16) (cid:17) σ1 Q˙ = J1Q˙ = 0. (21) where, Q˙ is the collection vector of generalized velocities of the two-bead assembly Q˙ = [r˙ , ω , r˙ , ω ]T , (22) 1 1
2 2 J1 is the Jacobian matrix of σ1: ∂σ1 J1 = k, k = 1,...,3, l = 1,...,12, (23) kl ∂Q l 9 (cid:2) (cid:3) J1 = J1 J1 1 2 (24) = (cid:2) I −ae× −I ae× (cid:3), 3 12 3 21 and 0 −e e 3 2 e× = e3
0 −e1 . (25) −e e 0 2 1 For an assembly of N beads, N −1 no-slip vectorial constraints must be satisfied. The b b Gears Model (GM) total Jacobian matrix JGM is block bi-diagonal and reads J1 J1 1
2 J2 J2 JGM = 2 ..3. ... (26) JNb−1 JNb−1 N −1 N b b where Jα is the 3×6 Jacobian matrix of the vectorial constraint α for the bead β. β The kinematic constraints for the whole assembly
then read JGMQ˙ = 0. (27) The associated generalized forces F are obtained following Section 2.2. c 2.4. Elastic forces and torques We are considering elastohydrodynamics of homogeneous flexible and
inextensible fibers. These objects experience bending torques and elastic forces to recover their equilibrium shape. Bending moments derivation and discretization are provided. Then, the role of
bending moments and constraint forces is addressed in the force and torque balance for the assembly. 2.4.1. Bending moments The bending moment of an elastic beam is provided by the constitutive law
[55, 62] dt m(s) = Kbt× , (28) ds where Kb(s) is the bending rigidity, t is the tangent vector along the beam centerline and s is the curvilinear abscissa. Using the Frenet-Serret formula dt = κn,
(29) ds the bending moment writes m(s) = Kbκb, (30) 10
See more | {"url":"https://www.zlibrary.to/dl/a-general-formulation-of-bead-models-applied-to-flexible-fibers-and-active-filaments-at-low-reynolds-number","timestamp":"2024-11-12T19:56:12Z","content_type":"text/html","content_length":"156348","record_id":"<urn:uuid:f90867cb-0ac3-4e39-81da-e012d00bf719>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00223.warc.gz"} |
This entity is
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weapon_mg42 is a point entity available in Day of Defeat: Source.
The MG 42 is a machine gun used by the Axis machine gunners.
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The name that other entities refer to this entity by, via Inputs/Outputs or other keyvalues (e.g. parentname or target).
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Amount of reserve ammo to be added. Fallback value is 0. !FGD
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This entity's orientation in the world. Pitch is rotation around the Y axis, yaw is the rotation around the Z axis, roll is the rotation around the X axis.
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Prevents the model from moving.
Fires when an NPC picks up this weapon. (!activator is the NPC.)
Fires when the player +uses this weapon. (!activator is the player.)
Fires when a player picks up this weapon. (!activator is the player.)
Fires when the player 'proves' they've found this weapon. Fires on: Player Touch, +USE pickup, Physcannon pickup, Physcannon punt. | {"url":"https://developer.valvesoftware.com/wiki/Weapon_mg42","timestamp":"2024-11-09T07:31:13Z","content_type":"text/html","content_length":"24230","record_id":"<urn:uuid:0f867ff0-12a8-4e9a-908d-bdefca61abf2>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00053.warc.gz"} |
The New Angle On What Range in Math Just Released | Telgesa
16 Oct The New Angle On What Range in Math Just Released
Posted at 18:13h
Uncategorized @lt 0 Comments
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pursue a minor (like business administration), or earn a masters degree in Biology with a particular track (like neuroscience). | {"url":"https://www.telgesa.lt/uncategorized-lt/the-new-angle-on-what-range-in-math-just-released/","timestamp":"2024-11-10T20:33:38Z","content_type":"text/html","content_length":"41100","record_id":"<urn:uuid:f49c6efc-95c1-4138-983e-98ad7b280531>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00163.warc.gz"} |
Stepping Patterns in Ants: I. Influence of Speed and Curvature
The locomotory behaviour of workers of 12 ant species belonging to four different genera (Formicinae: Cataglyphis, Formica, Lasius; Myrmicinae: Myrmica) was studied by filming individuals walking on
smoked-glass plates. Subsequent multivariate analyses of footfall positions and walking kinematics revealed a set of constant features characterizing ant locomotion. The alternating tripod gait
prevails over a wide range of speeds. The temporal rigidity of tripod coordination is paralleled by spatially rigid footfall patterns. Tripod geometry is preserved irrespective of speed and
curvature. When walking around curves, tripods are rotated relative to the walking trajectory. Whereas stride length on the inner side of the curve is shortened, that on the outer side is independent
of curvature.
During recent decades, terrestrial locomotion of insects has become the subject of a steadily growing number of studies focusing on the neural background and the morphological and mechanical
constraints of locomotion. There is now detailed knowledge of how temporal and spatial coordination of the legs is brought about in some insects (Carausius morosus: Bässler, 1985; Cruse, 1985a,b;
Cruse and Schwarze, 1988; Cruse and Knauth, 1989; Dean, 1989; Periplaneta americana: Delcomyn, 1985, 1991a,b) and important insights into the functional relationship between walking mechanics, gait
patterns and body morphology have been obtained (Acheta domestica: Harris and Ghirardella, 1980; Blatella germanica: Franklin, 1985; Carausius morosus: Cruse, 1976; Jander, 1985; Dean, 1991;
Periplaneta: Full and Tu, 1990, 1991; Full et al. 1991;Carabidae: Evans, 1977).
The studies presented here focus on interleg coordination patterns in ants. Ants appear to be especially well suited for comparative studies, as this taxon exhibits a high degree of inter-and
intraspecific variability, with respect to both morphology and behaviour. The analyses shed some light on three different, but mutually connected, aspects of insect locomotion.
Part I of the studies provides a basic description of the constant and variable components of spatiotemporal interleg coordination. Part II (Zollikofer, 1994a) analyzes the role of body morphology
(size, species, caste) and mechanical constraints on walking kinematics, and part III (Zollikofer, 1994b) deals with alterations in the locomotory behaviour observed in ant workers carrying loads.
Colonies of four species of Cataglyphis from Tunisia (C. bicolor, C. bombycina, C. albicans and C. fortis) were held in the laboratory under constant conditions (30°C, 50% relative humidity, 14 h:10
h light:dark cycle). Individuals belonging to the genera Formica (F. pratensis, F. lefrançoisi, F. rufa), Lasius (L. niger, L. fuliginosus, L. flavus) and Myrmica (M. ruginodis) were collected at
different sites and kept in boxes.
Data acquisition and analysis
Data sampling consisted of filming the ants walking on smoked-glass plates. This method, first used by Manton (1952) in semiquantitative analyses of arthropod stepping patterns, still provides an
elegant means of measuring tarsal imprints and body position simultaneously.
A video camera (50 half-frames s^−1) was placed above the walking area (30 cm×30 cm Perspex box). Smoked-glass plates (7 cm×10 cm or 9 cm×12 cm) were calibrated with a reference point grid using an
x,y-plotter and were then positioned beneath the visual field of the camera. During the tests, the ant was allowed to move in any direction and to stop and to resume walking at any point of its path.
Each individual had to perform at least 15 runs, the glass plates being replaced after 1–2 runs. At the end of the test series, the ants were killed and weighed to the nearest 0.1 mg (body mass m).
The position of the centre of mass was determined by suspending the ant on a nylon fibre glued to the thorax and moving the point of attachment until the ant assumed a horizontal position. The
influence of leg positions was accounted for by repeating centre of mass determination with the legs cut off at the coxae. Morphological dimensions (Table 1) were measured to the nearest 0.1 mm.
Video data sampling was performed by copying frame-by-frame the position of the head of the ant and the direction of the longitudinal axis of the body onto a tracing foil overlay placed on the
monitor screen. The reference points imprinted on the glass plates were sampled for later calibration. The successive body positions (frame interval 20 ms) were digitized (MOP digitizer; 0.1 mm
resolution) and transmitted to a Cromemco Minicomputer. Image distortions due to the video system were corrected for by referring the coordinate values of each sampled data point to the nearest
reference grid points. For further analyses, the original data (head position and body axis vector) were replaced by the coordinates of the centre of mass.
The footprints, as well as the calibration marks, were identified on photographic replicas of the glass plates (enlargements 2:1–5:1). The smoke layer put onto the plates was sufficiently fine to be
pierced by the tarsal claws and basal bristles of the distal tarsal segments, yielding individually discernible footprints. Digitizing followed similar procedures to those described for body
positions. Both body and tarsal position data were then match-merged into a common coordinate system by referencing to the calibration grid. The accuracy of the tarsal data was checked by making
direct measurements of the distances between tarsal imprints on the glass plates themselves and comparing the results with data obtained by digitizing. The sampling error of the video data was
estimated by analyzing repeated measurements of a single test run. The tolerance of positional information was ±0.15 mm for tarsal distances, and the reproducibility of velocity values was at least
The geometry of stepping patterns was sampled by a set of 14 variables specifying the distances between footprints as well as their orientation relative to the walking trajectory. Variables were
defined without presuming the existence of any regular coordination (e.g. tripod) pattern, but were systematically grouped and named according to the results of the intercorrelation analysis
presented below (Table 1). It should be pointed out that stride lengths are defined on purely geometric criteria, i.e. as the distance between successive imprints of a given tarsus. Stride length s
(the mean value of the stride lengths of the legs that act in phase) measures the distance travelled in a full stride (Alexander, 1977).
The movement of the centre of mass was described by speed (v, determined from two consecutive positions of the centre of mass) and the local radius of curvature (r, determined from three consecutive
positions). Average values of v and r were attributed to each stepping cycle. Calculations of averages were based on the consecutive body positions (time intervals 20 ms) situated within the polygon
outlined by the tarsal positions belonging to one cycle.
Further data handling and statistical analyses were performed with SAS (Statistical Analysis Software) utilities and procedures. The test runs of each animal were analyzed with SAS procedure VARCLUS.
This procedure performs a covariance analysis on the correlation matrix of all variables (in this case a 14X14 matrix) and subsequently cluster-analyzes the results to specify groups of highly
intercorrelated variables.
The fact that stride length increases with increasing speed suggests that speed may act as a strong correlator on intertarsal geometry. In order to eliminate speed effects, the analyses are based on
partial correlations, i.e. correlations between any variables, but keeping constant speed.
The analysis of interleg coordination follows a three-step procedure. First, the temporal correlation of footfalls is investigated. Second, the spatial correlations between footfall positions are
screened for constant patterns. Third, the influence of walking speed and curvature on spatial patterns is examined.
Temporal coordination of the legs
The temporal pattern of interleg coordination of worker ants is simple: at speeds ranging from slow walking to high-velocity running, ants exhibit a fairly strict alternating tripod pattern (Fig. 1)
(Hughes, 1952). The fore-and hindlegs of one body side together with the contralateral midleg move in phase relative to each other and in antiphase relative to the opposite legs, yielding the
footfall pattern L1R2L3 alternating with R1L2R3, where L is the left and R is the right side and 1, 2 and 3 represent the fore-, mid-and hindlegs, respectively.
At very low speed, when resuming locomotion after a stop or when walking on strongly bent trajectories, the tripod gait is replaced by metachronal coordination.
Spatial coordination of the legs
A graphic examination of the stepping pattern geometry (Fig. 2) reveals that the spatial arrangement of the legs is highly regular and reflects the rigidity of the temporal patterns. The analysis of
intercorrelations between the variables describing footfall geometry yielded tree diagrams of the correlative fit for each individual ant. As no substantial differences between individuals or between
species could be revealed, the following results apply to worker ants in general. An overall outcome of the analysis is that geometric tripods between legs L1R2L3 and/or R1L2R3 are shown to be
spatially constant entities. Furthermore, the size and the shape of tripods depend neither on the distance between them nor on their orientation relative to the walking direction.
In quantitative terms, these results are expressed as follows. Each individual tree diagram consists of four subsets comprising variables which are highly intercorrelated to each other (r^2>0.6; P
<0.02). Each subset of variables characterizes a distinct geometric property of the footfall patterns (see Table 1). (1) Subset A, all variables describing the size and shape of a tripod (L1R2L3 and
R1L2R3); (2) subset B, all variables describing the position of a tripod relative to the trajectory (inclination and lateral shift); (3) subset C, distances from a midleg tarsus (R2 or L2) of a given
tripod to tarsal positions of the subsequent tripods; (4) subset D, distances from foreleg and hindleg tarsi (L1 and L3, or R1 and R3) to subsequent tripods.
Whereas subsets A and B describe intratripod geometry, subsets C and D represent distances between subsequent tripods. Moreover, subsets C and D correspond to distances measured from opposite body
sides of a given tripod. This indicates decoupling of right and left leg movements, a point that will be discussed under curve-walking.
Effects of speed and curvature on stepping pattern geometry
This analysis tests the correlation of footfall geometry with speed and curvature, based on linear regression models. Each individual was analyzed separately. The results presented here (Table 2)
refer to workers of all species, as scale effects or species-specific differences could not be detected.
Speed is positively correlated with all variables indicating distances between subsequent tripods (subsets C and D of the above analysis; Fig. 3A). However, there is no correlation between speed and
any of the variables describing tripod shape and position (subsets A and B; Fig. 3B). Thus, speed has no influence on the spatial relationship between the legs acting together as a tripod (Table 2,
tripod shape and position). With increasing speed, the tripods are simply placed further apart from each other (Table 2, stride lengths; Fig. 2A,B), without any shape alteration.
When walking along curved paths (Fig. 2C), the tripod L1R2L3 supporting the body during a left turn is geometrically similar to R1L2R3 in a right turn, as in both cases the tarsus of the midleg is
placed on the concave (inner) side of the curve. Correspondingly, tripod L1R2L3 in right turns is equivalent to R1L2R3 in left turns. Data have been categorized according to this criterion.
Correlations of curvature (inverse of the radius of curvature) with the tarsal constellation were studied after speed effects had been eliminated by calculating partial correlation coefficients (
Table 2).
The size and shape of the tripods are not altered with respect to curvature. Conversely, intertripod distances as well as tripod positioning vary with curvature; stride length of the legs acting on
the inner side of the curve is shortened, whereas stride length on the outer side does not depend on curvature. This finding confirms the decoupling of the left and right body sides, as stated
earlier. With increasing curvature, the forelegs on the concave side of the curve are placed closer to the body axis and the hindlegs are placed farther from it. The opposite situation is found on
the convex side (Table 2, lateral distances of the tarsi l1, l2, l3; Figs 2C, 3C).
These findings demonstrate that, depending on the curvature, footfall positions change relative to the longitudinal axis of the body, yet the spatial arrangement of the legs belonging to one tripod
is always held constant.
Gait patterns
The alternating tripod gait has been thoroughly studied in a variety of insect species (Coleoptera, Dermaptera, Hemiptera, Blattariae, Orthoptera: Hughes, 1952; Delcomyn, 1971; Graham, 1972; Manton,
1972; Burns, 1973; Evans, 1977; Kozacik, 1981). The alternating tripod gait is a widespread interleg coordination pattern for walking at moderate to high speed. Tripod coordination is generally
lacking during slow walking (Periplaneta americana, Spirito and Mushrush, 1979; Carausius morosus, Graham, 1972; Neoconocephalus robustus, Graham, 1978) or, obviously, in species using less than six
legs in locomotion (Mantis religiosa, Roeder, 1937; Romalea microptera, Graham, 1972). As has been demonstrated in this paper, tripod coordination in ants predominated over a wide range of speed and
curvature. Moreover, tripods proved to be highly constant spatial entities. Given the above evidence, it is assumed that the spatiotemporal constancy of this gait pattern may be a general feature of
fast-running insects (Periplaneta americana, Delcomyn, 1971; Carabidae: Evans, 1977).
In vertebrates, Taylor et al. (1980) observed that the specific cost of locomotion increases with decreasing body mass. This is mainly due to the higher number of stepping cycles a small animal has
to perform in order to cover a given distance. Following this argument, increasing levels of energy consumption result because muscular efficiency is inversely proportional to contraction speed.
Hence, in order to minimize the cost of locomotion, an animal should minimize the number of stepping cycles and maximize stride length.
For insects, from a geometric point of view, there are two strategies for maximizing stride length. First, to extend the ranges of action of the legs simultaneously (from anterior to posterior
extreme position); second, to extend the spans between temporally successive legs. Maximum spans between successive legs are attained if the temporal onset of retraction of a given leg is maximally
shifted relative to the onset of retraction of a neighbouring leg. This is the case when adjacent legs are moving in antiphase. From a static point of view, stability must be maintained by supporting
the body with at least three legs.
The alternating tripod gait represents an optimal pattern with respect to both geometric and static demands. The antiphase relationship between contralateral as well as adjacent legs yields longer
strides than any other coordination pattern. At the same time, three-point supports are established. More generally, every antiphase relationship between adjacent legs will yield maximum stride
length, e.g. alternating tetrapods in Arachnidae (Wilson, 1967; Ward and Humphreys, 1981; Land, 1972) and Scorpionidae (Bowerman, 1975) and even in sideways-walking crabs (Uca pugnax, Barnes, 1975;
Ocypode ceratophthalma, Burrows and Hoyle, 1973). The above argument suggests that the prevalence of symmetrical gait patterns may reflect kinetic rather than neuronal constraints.
Apart from the constant phase relationships, the results presented here show that the legs belonging to one tripod build up a spatially constant entity. This implies that, depending on speed and
curvature, the tarsal positions of a tripod may vary considerably relative to the body axis and to the position of the succeeding tripod while remaining constant within tripods. Except for some
qualitative descriptions given by Manton (1972), spatial constancy of tarsal positions has not yet been described in insects, although it may be a common feature of the terrestrial locomotion of
What is the neural basis of spatiotemporal constancy? The emergence of gait patterns in walking insects is attributed to the action of central pattern generators (CPGs) and/or to the influence of
sensory input (Cruse, 1985a; Delcomyn, 1985). In Carausius morosus (Cruse, 1985a,b; Cruse and Schwarze, 1988; Cruse and Knauth, 1989), interleg coordination patterns can be explained by a series of
well-defined coupling mechanisms that control the timing of protraction of adjacent and of opposite legs, and there is no evidence for the action of CPGs in this slow-walking species (Cruse, 1985a).
In fast-running insects, however, CPGs may play an important role. In Periplaneta americana (Delcomyn, 1985, 1991a,b), the amputation of a leg altered the phase relationship and the consistency of
the bursting activity of motor neurones. These effects were drastic at low speed but disappeared at high walking speeds. Thus, while sensory feedback seems to be essential for interleg coordination
during slow walking, CPGs become increasingly important with increasing speed (Delcomyn, 1991b). Following this argument, the rigidity of tripod gait patterns observed in ants indicates that CPGs are
dominant over sensory input at medium to high walking speeds. Tripods have been shown to be spatiotemporal entities that are invariant over a wide range of speeds and curvatures. Thus, CPGs appear to
generate both the patterns of temporal coordination and of spatial arrangement of the legs belonging to a tripod.
Walking around curves requires path differences between the legs of the left and the right body sides. To achieve this, insects use different strategies. In Apis mellifera (Zolotov et al. 1975), in
Geotrupes stercorosus and in G. stercorarius (Frantsevich and Mokrushov, 1980), the legs on the inner side walk backwards, whereas in Blatella germanica a slightly modified tripod pattern is used (
Franklin et al. 1981). In both Apis mellifera (Zolotov et al. 1975) and Carausius morosus (Jander, 1985), the stride frequency is lowered on the inner side, resulting in the uncoupling of the
stepping rhythms on each side of the body. Stride length reduction on the inner side is reported for every species cited above.
During curve-walking, ants use comparatively conservative strategies. Distances between successive footfalls on the inner side are shortened, whereas on the outer side they remain unchanged.
Uncoupling of the two body sides has been demonstrated by the loose correlative fit between intertripod distances on the two body sides. The spatial tripod pattern, however, is maintained even when
ants walk around narrow curves. Thus, the footfall positions of legs belonging to one tripod are held constant relative to each other, although they may vary relative to body position.
The results presented here were part of a PhD thesis. I would like to thank Professor Rüdiger Wehner for his constant support and innumerable discussions and Dr Rima Huston for proof-reading the
final version. I am much indebted to Dr Reinhard Blickhan and to two anonymous referees for many suggestions on earlier versions of the manuscript.
R. McN
. (
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J. exp. Biol.
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The control of walking in Orthoptera. I. Leg movements during normal walking
J. exp. Biol.
The mechanism of rapid running in the ghost crab, Ocypode ceratophthalma
J. exp. Biol.
The function of the legs in the free walking stick insect, Carausius morosus
J. comp. Physiol.
Which parameters control the leg movement of a walking insect?
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Leg coordination in the stick insect Carausius morosus: effects of cutting thoracic connectives
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Effects of load on leg movements and step coordination of the stick insect Carausius morosus
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The locomotion of the cockroach Periplaneta americana
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Sense organs and the pattern of motor activity during walking in the american cockroach
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Perturbation of the motor system in freely walking cockroaches. II. The timing of motor activity in leg muscles after amputation of a middle leg
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Locomotion in the Coleoptera Adephaga, especially Carabidae
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Basics In Statistics Question And Answers - Class Notes
Basics In Statistics Question And Answers
Basics In Statistics Definitions
The sample is part of a population called the universe, reference, or parent population
Biostatistics is that branch of statistics concerned with mathematical facts and data related to biological events
The variable is a state, condition, concept, or event whose value is free to vary within the population
Read And Learn More: Percentive Communitive Dentistry Question And Answers
Basics In Statistics Important Notes
1. Measures of central tendency
• Arithmetic mean
□ Simplest measure
□ Obtained by summing up all the observations divided by the number of observations
□ The arithmetic mean is very sensitive to extreme scores.
• Median
□ Median is the simplest division of the set of measurements into two halves
□ When the distribution has odd numbers, the middle value is the median, when the distribution has an even number of elements, the average of two middle scores is the median
□ The median is insensitive to small numbers.
• Mode
• Mode is the most frequently occurring value in a set of observations
2. Sampling
• Simple random sampling
□ Used when the population is small and homogenous.
• Systemic sampling stratified sampling
□ Used when the population is large, non-homogenous, and scattered
• Multistage sampling
□ Employed in large country surveys
□ Carried out in several stages
• Multiphase sampling
□ Here sampling is done in different phases
• Cluster sampling
□ Involves grouping the population and then surveying
• Stratified sampling
□ Used when the population is large, nonhomogenous
3.Properties of the normal curve
• Bell-shaped
• Symmetrical
• The height of the curve is maximum at the mean
• Mean = median = mode
• The area under the curve between any two points can be found in terms of the relationship between mean and standard deviation.
Mean ± 1 SD = 68.3% of observation
Mean ± 2 SD = 95.4% of observation
Mean + 3 SD = 99.7% of observation
4. Classification of data
• Qualitative data
□ Qualitative data is data with frequency but no magnitude
□ Nonparametric tests are used for it
• Quantitative data
□ Quantitative data is data with a magnitude
□ Parametric tests are used for it
5. The Chi-square test is used
• To test the association between the cause and effect
• To find the goodness of fit
• To test the differences between two/more proportions
6. Tests
Basics In Statistics Long Essays
Question 1. Define sample. What are the ideal requisites of sampling, describe different sampling methods.
Sample: is part of a population called the universe, reference, or parent population
Sample Ideal Requisites:
• Efficiency
• Representativeness
• Measurability
• Size-large
• Adequate coverage
• Goal orientation
• Feasibility
• Economic
Sample Sampling Methods:
Sample Probability Sampling:
• Simple Random Sampling
□ Each member of the population has an equal chance of being included in the sample
□ The member is determined by chance only © Methods of random selection are
□ Lottery method
□ Table of random numbers
• Systematic
• Systematic is obtained by selecting one unit at random and then selecting additional units at evenly spaced intervals till an adequate sample size is obtained
• Systematic can be adopted as long as there is no periodicity of occurrence of any particular event in the population
• Stratified Random
• The population to be sampled is subdivided into strata
• A simple random sample is then chosen from it
• Used for a heterogeneous population
• Systematic ensures more representativeness, provides greater accuracy, and can concentrate over a wider area
• Systematic eliminates sampling variation
Sample Cluster Sampling:
• Useful when a population forms natural groups
• First, a sample of the clusters is selected and then all units in clusters are surveyed
Sample Advantage:
Sample Disadvantage:
Cannot be generalized
Sample Non-Probability Sampling:
Sample Accidental Sampling:
• Sampling is a matter of taking what you can get
• Sampling is not randomly obtained
Sample Advantage:
Sample Accidental Sampling is inexpensive and less time-consuming
Sample Purposive Sampling:
• Sample Purposive Sampling is a nonrepresentative subset of some larger population
• A sample is achieved by asking a participant to suggest someone else willing for the study
1. Quota Sampling:
Quota Sampling involves the selection of proportional samples of subgroups within a target population to ensure generalization
2. Dimensional Sampling:
A small sample is selected then each selected case is examined in detail
3. Mixed Sampling:
Constitute a combination of both probability and nonprobability sampling
Question 2. Define biostatistics. Write in detail the uses of biostatistics in dental public health.
• Biostatistics is that branch of statistics concerned with mathematical facts and data related to biological events
• Biostatistics deals with the statistical methodologies involved in biological sciences
Biostatistics Uses:
• Measure the state of health of the community
• Identify the health problems
• Compare the health status of one country with another and past status with the present
• Predict health trends
• Plan and administer dental health services
• Evaluate the achievement of public health program
• Fix priorities in public health program
• Evaluate the efficacy of vaccines, sera, etc
• Measure mortality and morbidity
• Test whether the difference between 2 populations is real or a chance occurrence
• Study correlation between attributes in the same population
• Promote health legislation
• Help the dentist to think quantitatively
Question 3. Define sampling. Classify sampling. Enumerate any one sampling.
Sampling is the process or technique of selecting a sample of appropriate characteristics and adequate size
Probability Sampling:
Simple Random Sampling:
• Each member of the population has an equal chance of being included in the sample
• The member is determined by chance only
• Methods of random selection are
□ Lottery method
□ Table of random numbers
Sampling Systematic:
• Sampling Systematic is obtained by selecting one unit at random and then selecting additional units at evenly spaced intervals till an adequate sample size is obtained
• Sampling Systematic can be adopted as long as there is no periodicity of occurrence of any particular event in the population
1. Stratified Random:
• The population to be sampled is subdivided into strata
• A simple random sample is then chosen from it
• Used for a heterogeneous population
• It ensures more representativeness, provides greater accuracy and can concentrate over a wider area
• It eliminates sampling variation
2. Cluster Sampling:
• Useful when a population forms natural groups
• First, a sample of the clusters is selected and then all units in clusters are surveyed
Sampling Advantage:
Sampling Disadvantage:
Cannot be generalized
Question 4. Enumerate various measures of dispersion and describe in detail the test of significance.
Measures Of Dispersion:
• Range
□ The range of the difference between the smallest and largest results in a set of data
• Mean deviation
□ Mean deviation is the average of the deviation from the arithmetic mean
• Standard deviation
Measures Of Dispersion Test Of Significance:
Measures Of Dispersion Test Of Significance deals with the techniques to know how far the differences between the estimates of different samples are due to sampling variations
1. Standard Error of Mean (SE):
Gives the standard deviation of the mean of several samples from the same population
= standard deviation / √n
2. Standard Error of Proportion:
= p and q = proportion of occurrence of an event
in 2 groups
n = sample size
Measures Of Dispersion Standard Error Of Difference Between Two Means:
Indicates whether the samples represent two different universe
Measures Of Dispersion Standard Error Of Difference Between Proportion:
Indicate whether the difference is significant or has occurred by chance
Measures Of Dispersion Chi-Square Test:
Measures Of Dispersion Uses:
• Test whether the difference in the distribution of attributes in different groups is due to sampling variation or not
• Test the significance of the difference between 2 proportion
• Used when there are more than 2 groups to be compared
Measures Of Dispersion Z Test:
• Test the significance of differences in means for large samples
• ‘t’ Test
Measures Of Dispersion Synonym:
Student’s t-test
Measures Of Dispersion Uses:
• Used when the sample size is small
• Used to test the hypothesis
• Find the significance of the difference between the 2 proportions
Measures Of Dispersion Types:
• Unpaired’t’ test
• Applied to unpaired data made on individuals of 2 different sample
• Test if the difference between the means is real or not
Measures Of Dispersion Paired’t test:
Applied to paired data obtained from one sample only
Question 5. Define biostatistics. Describe in detail the normal curve. Write a note on measures of central tendency.
(or) Normal distribution/ Properties of normal curve/ Gaussian curve.
(or) Mean, Median, Mode.
(or) Measures of central tendency.
• Biostatistics is that branch of statistics concerned with mathematical facts and data related to biological events
• Biostatistics deals with the statistical methodologies involved in biological sciences
Biostatistics Normal Curve:
• Biostatistics A Normal Curve is a pattern followed by very many sets of continuous measurements.
• Biostatistics A Normal Curve is characterized by a symmetric, bell-shaped curve
• In a normal curve
□ The area between one standard deviation on either side of the mean will include approximately 68% of the values
□ The area between two standard deviations on either side of the mean will include approximately 95% of the values
□ The area between three standard deviations on either side of the mean will include approximately 99.5% of the values
Biostatistics Characteristics:
• Biostatistics Characteristics is smooth, symmetrical bell-shaped
• The maximum number of observations is at the center and gradually decreases at the extremities
• The total area is 1, the mean is 0 and standard deviation is 1
• Mean, median and mode coincide at center
Basics In Statistics Short Essays
Question 1. Presentation of statistical data.
(or) Pie Chart
(or) Histogram
(or) Pictogram
(or) Uses of biostatistics
Presentation of statistical data Tabulation
• Tables are simple devices used for data presentation
• Prepared manually or mechanically
Presentation of statistical data Types:
1. Simple Table:
Way table containing one characteristic of data only
Presentation of statistical data Master Table:
Contains all the data obtained from a survey
Presentation of statistical data Frequency Distribution Table: Two-column table
• 1st column: lists classes of data
• 2nd column: lists the frequency of each class
Charts/ Diagrams:
1. BarCharts:
• BarCharts is a diagram of columns/ bars
□ The height of the bars determines the value of the particular data
□ The width of the bar remains the same
□ The bars are separated by spaces
□ The bars can be either vertical/ horizontal
Presentation of statistical data Types:
• Simple bar chart
• Represents only one variable
Presentation of statistical data Multiple bar chart
Consist of a set of bars of the same width corresponding to the different sections without any gap in between
• Component bar chart
□ Individual bars are divided into 2 or more parts
□ Used to compare the sub-groups
2. Pie Chart:
• The entire graph looks like a pie and its components are represented by its slices
□ The Pie Chart is divided into different sectors corresponding to the frequencies of the variables
□ The segments are then shaded/ colored
3. Histogram:
• A histogramis a pictorial presentation of data
• Class intervals are presented on the X-axis and frequencies on the Y axis
• No space occurs between the cells
4. Pictogram:
They are small pictures used for data presentation USA
5. Line Diagram:
• Used for continuous variable
• Time is represented on the X-axis and value on the Y-axis
6. Statistical Maps:
• Refer to the geographic area
• Dot/ point is used to represent the area
Question 2. Types of diagram.
1. Bar Charts:
• Bar Charts is a diagram of columns/ bars
□ The height of the bars determines the value of the particular data
□ The width of the bar remains the same
□ The bars are separated by spaces
□ The bars can be either vertical/ horizontal
Bar Charts Types:
• Simple bar chart
□ Represents only one variable
2. Multiple bar chart:
Consist of a set of bars of the same width corresponding to the different sections without any gap in between
3. Component bar chart:
• Individual bars are divided into 2 or more parts
• Used to compare the sub-groups
4. Pie Chart:
• The entire graph looks like a pie and its components are represented by its slices
• The Pie Chart is divided into different sectors corresponding to the frequencies of the variables
• The segments are then shaded/ colored
5. Histogram:
• Histogram is a pictorial presentation of data
• Class intervals are presented on the X-axis and frequencies on the Y-axis
• No space occurs between the cells
6. Pictogram:
They are small pictures used for data presentation
Question 3. Types of samples/ Probability sampling methods/ Sampling methods.
(or) Cluster sampling
Probability Sampling
Simple Random Sampling:
• Each member of the population has an equal chance of being included in the sample
• The member is determined by chance only
• Methods of the random selection are e
□ Lottery method
□ Table of random numbers
Probability Sampling Systematic:
• Probability Sampling Systematic is obtained by selecting one unit at random and then selecting additional units at evenly spaced intervals till an adequate sample size is obtained
• It can be adopted as long as there is no periodicity of occurrence of any particular event in the population
Probability Sampling Stratified Random:
• The population to be sampled is subdivided into strata
• A simple random sample is then chosen from it
• Used for a heterogeneous population
• Stratified Random ensures more representativeness, provides greater accuracy and can concentrate over a wider area
• Stratified Random eliminates sampling variation
Probability Sampling Cluster Sampling:
• Useful when a population forms natural groups
• First, a sample of the clusters is selected and then all units in clusters are surveyed
Probability Sampling Advantage:
Probability Sampling Disadvantage:
Cannot be generalized
Probability Sampling Non-Probability Sampling:
Probability Sampling Accidental Sampling:
• Accidental Sampling is a matter of taking what you can get
• Accidental Sampling is not randomly obtained
Probability Sampling Advantage:
Probability Sampling is inexpensive and less time-consuming
Probability Sampling Purposive Sampling:
• Purposive Sampling is a nonrepresentative subset of some larger population
• A sample is achieved by asking a participant to suggest someone else willing for the study
Probability Sampling Quota Sampling:
Quota Sampling involves the selection of proportional samples of subgroups within a target population to ensure generalization
Probability Sampling Dimensional Sampling:
A small sample is selected then each selected case is examined in detail
Probability Sampling Mixed Sampling:
Constitute a combination of both probability and nonprobability sampling
Question 4. Simple random sampling.
Simple random sampling
• Each member of the population has an equal chance of being included in the sample
• The member is determined by chance only
• Methods of random selection are the
□ Lottery method
□ Table of random numbers
Question 5. Multistage sample.
Multistage sample
Multistage sample is a sampling procedure often used when the sampling units can be defined in a hierarchical manner
Multistage sample Steps:
• Select the groups/cluster
• Then subsamples are taken in subsequent stages
□ 1st stage: choice of states within countries
□ 2nd stage: choice of towns within each state
□ 3rd stage, choice of neighborhoods in each town
Question 6. Tests of significance.
(or)’t’ test.
Tests of significance
Tests of significance deal with the techniques to know how far the differences between the estimates of different samples is due to sampling variations
Tests of significance Standard Error Of Mean(Se):
Gives the standard deviation of the mean of several samples from the same population
Tests of significance Standard Error Of Proportion:
\(=\sqrt{\frac{p q}{n}} \mathrm{p} \and \mathrm{q}=\) proportion of occurrence of an event
in 2 groups n= sample size
Tests of significance Standard Error Of Difference Between Two Means
Indicates whether the samples represent two different universe
Tests of significance Standard Error Of Difference Between Proportion
Indicate whether the difference is significant or has occurred by chance
Tests of significance Chi-Square Test
Tests of significance Uses:
• Test whether the difference in the distribution of attributes in different groups is due to sampling variation or not
• Test the significance of the difference between 2 proportion
• Used when there are more than 2 groups to be compared
Tests of significance Z Test:
Test the significance of differences in means for large samples
Tests of significance ‘t’ Test:
Tests of significance Synonym:
Student’s t-test
Tests of significance Uses:
• Used when the sample size is small
• Used to test the hypothesis
• Find the significance of the difference between the 2 proportions
Tests of significance Types:
Tests of significance Unpaired’t’ test:
• Applied to unpaired data made on individuals of 2 different sample
• Test if the difference between the means is real or not
• Paired’t’ test
• Applied to paired data obtained from one sample only
Question 7. Statistical analysis.
Statistical analysis
• Statistical analysis is based on
• Population
□ Statistical analysis is the collection of units of observations that are of interest and is the target of the investigation
□ Statistical analysis is essential to identify the population clearly and precisely
□ The success of the investigation will depend on the identification of the population
• Variable
□ The variableis a state, condition, concept/ event whose value is free to vary within the population
Classification of Statistical Analysis:
• Independent
• Manipulated/ treated in a study
• The result of the independent variable
• Confounding
□ Confound the effect of the independent variable on the dependent
• Background
□ Considered for possible inclusion in the study
• Probability distribution
□ The probability distribution is a link between population and its characteristics
□ A probability distribution is a way to enumerate the different values the variable can have and how frequently each value appears in the population
□ A probability distribution is characterized by parameters i.e. quantities
Question 8. Standard deviation.
Standard deviation
• Standard deviation is the square root of the mean of the squared deviations from arithmetic
• Standard deviation is the most commonly used measure of dispersion
Standard deviation Synonym
Root Mean Square Deviation
Standard deviation Calculation
• Calculate the mean of the series, X
• Take the deviation mean X- X,
• Square these deviations and add them up 5^ 2
• Divide the result by the total number of observation
• Obtain the square root of it (Standard deviation)
Standard deviation Significance:
• The greater the standard deviation, the greater the magnitude of dispersion
• The lesser the standard deviation, the higher the degree of uniformity of observation
Question 9. Bar diagram/ charts.
Bar diagram
• Bar diagram is a diagram of columns/ bars
• The height of the bars determines the value of the particular data
• The width of the bar remains the same
• The bars are separated by spaces
• The bars can be either vertical/ horizontal
Bar diagram Types:
1. Bar diagram Simple bar chart
Represents only one variable
2. Bar diagram Multiple bar chart
Consist of a set of bars of the same width corresponding to the different sections without any gap in between
3. Bar diagram Component bar chart
Individual bars are divided into 2 or more parts Used to compare the sub-groups
Basics In Statistics Short Question And Answers
Question 1. Primary and secondary data.
secondary data Primary Data:
• Obtained directly from an individual
• secondary data Primary Data is first-hand information
secondary data Advantage:
• Precise information
• Reliable
secondary data Disadvantages:
secondary data Methods:
• Direct personal interviews
• Oral health examination
• Questionnaire
secondary data Secondary Data:
• Obtained from outside sources
□ Used to serve the purpose of the objective of the study
□ Example: Hospital records
Question 2. Frequency polygon.
Frequency polygon
Pictorial presentation of data
Frequency polygon Method:
• Obtained from histogram
• Mark the midpoint over histogram bars
• Next, connect these points in a straight line
• Example. Agewise prevalence of dental caries
Question 3. Stratified random sampling.
Stratified random sampling
• The population to be sampled is subdivided into strata
• A simple random sample is then chosen from it
• Used for a heterogeneous population
• Stratified random sampling ensures more representativeness, provides greater accuracy and can concentrate over a wider area
Question 4. Mode.
Mode is a value occurring with the greatest frequency
Mode Advantage:
• Eliminates extreme variation
• Easily located
• Easy to understand
Mode Disadvantage:
• Uncertain location e Not exactly defined
• Not useful in a small number of cases
Question 5. Null hypothesis.
Null hypothesis
• The null hypothesis asserts that there is no real difference between the two groups under consideration and the difference found is accidental and arises out of sampling variation
• The null hypothesis is the first step in the testing of the hypothesis
Question 6. Variable.
Variable is a state, condition, concept, or event whose value is free to vary within the population
Classification of Variable:
• Independent
□ Manipulated/ treated in a study
• Dependent:
□ The result of an independent variable
• Confounding
□ Confound the effect of the independent variable on the dependent
• Background
□ Considered for possible inclusion in the study
Question 7. Qualitative data.
Qualitative data
When data is collected on the basis of attributes/ qualities like sex, it is called qualitative data
Question 8. Chi-square test.
Chi-square test Uses:
• Test whether the difference in the distribution of attributes in different groups is due to sampling variation or not
• Test the significance of the difference between 2 proportion
• Used when there are more than 2 groups to be compared
Basics In Statistics Viva Voce
1. Mean, median and mode are measures of central tendency
2. Range, standard deviation, and coefficient of variation are measures of dispersion
3. The range is the difference between the smallest item and the value of the largest item
4. A census is a collection of information from all the individuals in a population
5. Sampling is the collection of information from representative units in a sample
6. Standard deviation is the most important and widely used measure of studying dispersion
7. A bar diagram is used to represent qualitative data
8. Histogram used to depict quantitative data
9. A frequency polygon is used to represent the frequency distribution of quantitative data
10. A pie diagram is used to show percentage breakdowns for qualitative data
11. A line diagram is useful to study the changes in values in the variable over time
12. Pictogram is the method to impress the frequency of occurrence of events to the common man
13. The chi-square test is a non-parametric test for qualitative data
14. For large samples, z test is preferred
15. For small samples, a t-test is preferred
16. The value of the mean in a normal distribution is zero
17. Standard deviation is also called root mean square deviation
18. The median is also called the 50th percentile
19. The standard error of the mean depicts the deviation | {"url":"https://classnotes.guru/basics-in-statistics-question-and-answers/","timestamp":"2024-11-04T14:04:20Z","content_type":"text/html","content_length":"126079","record_id":"<urn:uuid:5a5d0a8c-64c6-43f3-99c3-f9f0dc532a10>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00710.warc.gz"} |
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The Global Shortest Path Visualization Approach with Obstructions
Guan-Qiang Dong, Zong-Xiao Yang, Lei Song, Kun Ye, and Gen-Sheng Li
Institute of Systems Science and Engineering, Henan Engineering Laboratory of Wind Power Systems,
Henan University of Science and Technology
Luoyang 471003, China
April 28, 2015
August 31, 2015
October 20, 2015
NP hard problem, geometry-experiment approach (GEA), steiner minimal tree, obstacle-avoiding Steiner minimal tree (OASMT)
Shortest path experiment device
The avoidance obstacle path planning problem is stated in an obstacle environment. The minimum Steiner tree theory is the basis of the global shortest path. It is one of the classic NP-hard
problem in nonlinear combinatorial optimization. A visualization experiment approach has been used to find Steiner point and system’s shortest path is called Steiner minimum tree. However,
obstacles must be considered in some problems. An Obstacle Avoiding Steiner Minimal Tree (OASMT) connects some points and avoids running through any obstacle when constructing a tree with a
minimal total length. We used a geometry experiment approach (GEA) to solve OASMT by using the visualization experiment device discussed below. A GEA for some systems with obstacles is used to
receive approximate optimizing results. We proved the validity of the GEA for the OASMT by solving problems in which the global shortest path is obtained successfully by using the GEA.
Cite this article as:
G. Dong, Z. Yang, L. Song, K. Ye, and G. Li, “The Global Shortest Path Visualization Approach with Obstructions,” J. Robot. Mechatron., Vol.27 No.5, pp. 579-585, 2015.
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2083-2087, 2008.
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EvoTRANSLOG, pp. 42-51, 2007.
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15. [15] Z. Feng, Y. Hu, T. Jing, X. Hong, X. Hu, and G. Yan, “An O(nlogn) algorithm for obstacle-avoiding routing tree construction in the lambda-geometry plane,” Proc. ISPD, pp. 48-55, 2006.
16. [16] Y. Hu, Z. Feng, T. Jing, X. Hong, Y. Yang, G. Yu, X. Hu, and G. Yan, “FORst: a 3-step heuristic for obstacle-avoiding rectilinear Steiner minimal tree construction,” J. of Information
and Computational Science, pp. 107-116, 2004.
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Decision Tree
Decision Tree Summary
Decision Trees are a supervised learning method, used most often for classification tasks, but can also be used for regression tasks. The goal of the decision tree algorithm is to create a model,
that predicts the value of the target variable by learning simple decision rules inferred from the data features, based on divide and conquer.
During the training of the decision tree algorithm for a classification task, the dataset is split into subsets on the basis of features.
• If one subset is pure regarding the labels of the dataset, splitting this branch is stopped.
• If the subset is not pure, the splitting is continued.
After the training, the overall importance of a feature in a decision tree can be computed in the following way:
1) Go through all the splits for which the feature was used and measure how much it has reduced the impurity compared to the parent node.
2) The sum of all importance is scaled to 100. This means that each importance can be interpreted as a share of the overall model importance.
You can also plot the finished decision tree in your notebook.
If you do not know how, no problem, I created two end-to-end examples with Google Colab.
Table of Contents
Definitions of Decision Trees
Before we look at the decision tree in more detail, we need to get to know its individual elements of it. There are already names of the nodes from the following example listed.
• Root Node: represents the whole sample and gets divided into sub-trees.
• Splitting: the process of dividing a node into two or more sub-nodes.
• Decision Node: node, where a sub-node splits into further sub-nodes → Wind
• Leaf: nodes that can not split anymore are called terminal nodes (pure) → Overcast
• Sub-Tree: sub-section of the entire tree
• Parent and Child Node: A node, which is divided into sub-nodes is called the parent node of sub-nodes where as sub-nodes are the child of the parent’s node. → Weak and Strong are child nodes of
parent Wind.
The following table shows a dataset with 14 samples, 3 features, and the label “Play” that we will use as an example to train a decision tree classifier by hand.
The following decision tree shows what the final decision tree looks like. The tree has a depth of 2 and at the end all nodes are pure. You see that in the first step, the dataset is divided by the
feature “Outlook”. In the second step the previous divided feature “Sunny” is divided by “Humidity” and “Rain” is divided by “Wind”.
You might ask yourself how we know which feature we should use to divide the dataset. We will answer this question in the last section of this article.
Decision Tree Advantages & Disadvantages
Decision Tree Advantages
• The main advantage of decision trees is, that they can be visualized and therefore are simple to understand and interpret.
□ Therefore visualize the decision tree as you are training by using the export function (see the Google Colab examples). Use max_depth=3 as an initial depth, because the tree is easy to
visualize and overview. The objective is to get a feeling for how well the tree is fitting your data.
• Requires little data preparation, because decision trees do not need feature scaling, encoding of categorical values, or imputation of missing values. However, if you are using sklearn, you have
to input all missing values, because the current implementation does not support missing values. Also if you want that the decision tree works with categorical values, you have to use the sklearn
implementation of HistGradientBoostingRegressor and HistGradientBoostingClassifier.
• Decision tree predictions are very fast on large datasets, because the cost of using the tree (predicting data) is logarithmic in the number of data points used to train the tree. Note that only
the prediction is very fast, not the training of a decision tree.
• Decision trees are able to handle multi-output classification problems.
Decision Tree Disadvantages
• Decision Trees have a tendency to overfit the data and create an over-complex solution that does not generalize well.
□ How to avoid overfitting is described in detail in the “Avoid Overfitting of the Decision Tree” section
• Decision trees can be unstable because small variations in the data might result in a completely different tree being generated.
□ Use ensemble techniques, either stacking with other machine learning algorithms or bagging techniques like Random Forst to avoid this problem.
• There are concepts that are hard to learn for decision trees because trees do not express them easily, such as XOR, parity, or multiplexer problems.
• If your dataset is unbalanced, decision trees will likely create biased trees.
□ You can balance your dataset before training by balancing techniques like SMOTE or preferably by normalizing the sum of the sample weights sample_weights for each class to the same value.
□ If the samples are weighted, it will be easier to optimize the tree structure using the weight-based pre-pruning criterion min_weight_fraction_leaf, which ensures that leaf nodes contain at
least a fraction of the overall sum of sample weights.
Avoid Overfitting of the Decision Tree
Before Training the Decision Tree
We know that decision trees tend to overfit data with a large number of features. Getting the right ratio of samples to the number of features is important since a tree with few samples in high
dimensional space is very likely to overfit. Therefore we can use different dimensionality reduction techniques beforehand to limit the number of features the decision tree is able to be trained on.
During Training the Decision Tree
To avoid overfitting the training data, we need to restrict the size of the Decision Tree during training. The following chapter shows the parameter overview and how to prevent overfitting during the
After Training the Decision Tree
After training the decision tree, we can prune the tree. The following steps are done when pruning is performed.
1. Train the decision tree to a large depth
2. Start at the bottom and remove leaves that are given negative returns when compared to the top.
You can use the Minimal Cost-Complexity Pruning technique in sklearn with the parameter ccp_alpha to perform pruning of regression and classification trees.
Decision Tree Parameter Overview
The following list gives you an overview of the main parameters of the decision tree, how to use these parameters, and how you can use the parameter against overfitting.
• criterion: The function that is used to measure the quality of a split
□ Classification: can be Gini, Entropy, or Log Loss
□ Regression: can be Squared Error, Friedman MSE, Absolute Error, or Poisson
• max_depth: The maximum depth of the tree
□ Initial search space: int 2…4
□ The number of samples required to populate the tree doubles for each additional level the tree grows to.
□ Overfitting: when the algorithm is overfitting, reduce max_depth, because a higher depth will allow the model to learn relations very specific to a particular sample.
• min_samples_split: The minimum number of samples required to split an internal node
□ Initial search space: float 0.1…0.4
□ Overfitting: when the algorithm is overfitting, increase min_samples_split, because higher values prevent a model from learning relations that might be highly specific to the particular
sample selected for a tree.
• min_samples_leaf: The minimum number of samples required to be at a leaf node. A split point at any depth will only be considered if it leaves at least min_samples_leaf training samples in each
of the left and right branches.
□ Initial search space: float 0.1…0.4
□ Generally, lower values should be chosen for imbalanced class problems because the regions in which the minority class will be in majority will be very small.
□ For classification with few classes, min_samples_leaf=1 is often the best choice.
□ Overfitting: when the algorithm is overfitting, increase min_samples_leaf
• max_leaf_nodes: The maximum number of terminal nodes or leaves in a tree.
□ Can be defined in place of max_depth. Since binary trees are created, a depth of n would produce a maximum of 2^n leaves.
• max_features: The number of features to consider while searching for the best split.
□ The features will be randomly selected.
□ As a thumb-rule, the square root of the total number of features works great but we should check up to 30-40$% of the total number of features.
□ Overfitting: Higher values can lead to overfitting.
• min_impurity_decrease: If the weighted impurity decrease is greater than the min_impurity_decrease threshold, the node is split.
□ Overfitting: when the algorithm is overfitting, increase min_impurity_decrease
The following Google Colab shows the differences of different parameters on the trained decision tree as well as the performance of the model for a classification task.
How to use Decision Trees for Regression
In most cases, decision trees are used for classification, but it is also possible to use decision trees for regression. The main difference is that instead of predicting a class in each node, it
predicts a value. You can find an end-to-end example of how to use decision trees for regression in the following Google Colab. Also, all charts from this section are in the notebook.
The following picture shows how the dataset was divided by the feature “bmi”. The value of the target variable is the median of all values in the sub-nodes.
You see that when the prediction would only be done by this one feature “bmi” there would be only 5 different values for the output. With a deeper decision tree, you have more values for the output
but you see the general problem with decision trees in regression tasks. Therefore if you have a regression task my suggestion is not to use decision trees in the first place, but to focus on other
machine learning algorithms, because they likely will have a better performance.
Just like for classification tasks, Decision Trees are prone to overfitting when dealing with regression tasks. Without any regularization (i.e., using the default hyperparameters), you get the
predictions in the following chart, which is trained on the same dataset as the decision tree from the previous decision tree. The only difference is that the decision tree that is overfitting was
trained with standard parameters and the decision tree from the previous scatter-plot was limited to max_depth=3 and min_samples_leaf=30.
How to Select the Feature for Splitting?
The decision of which features are used for splitting heavily affects the accuracy of the model. Decision trees use multiple algorithms to decide to split a node into two or more sub-nodes. The
creation of sub-nodes increases the homogeneity of the resulting sub-nodes. The decision tree splits the nodes on all available variables and then selects the split which results in the most
homogeneous sub-nodes and therefore reduces the impurity. The decision criteria are different for classification and regression trees. The following are the most used algorithms for splitting
decision trees:
Split on Outlook
Split on Humidity
Gini Index
The Gini coefficient is a measure of statistical dispersion and is the most commonly used measure of inequality. The Gini coefficient measures the inequality among values of a frequency distribution.
A Gini coefficient of zero expresses perfect equality, where all values are the same, whereas a Gini coefficient of 1 (100%) expresses maximal inequality among values.
The following lines show the calculation of the Gini index when we split for “Outlook” and “Humidity” in the first decision node.
Split on Outlook
G(sunny) = (\frac{2}{5})^2 + (\frac{3}{5})^2 = 0.52 \\
G(overcast) = (\frac{4}{4})^2 + (\frac{0}{4})^2 = 1 \\
G(rain) = (\frac{3}{5})^2 + (\frac{2}{5})^2 = 0.52 \\
G(Outlook) = \frac{5*0.52 + 4*1 + 5*0.52}{14} = 0.657
Split on Humidity
G(high) = (\frac{3}{7})^2 + (\frac{4}{7})^2 = 0.51 \\
G(normal) = (\frac{6}{7})^2 + (\frac{1}{7})^2 = 0.75
G(Humidity) = \frac{7*0.51 + 7*0.75}{14} = 0.63
We see that the Fini coefficient for “Outlook” with 0.657 is higher compared to “Humidity” with 0.63. Therefore when we would use the Gini Index as a criterion to divide the dataset, the first node
would be split the dataset by “Outlook”.
Entropy is the expected value (mean) of the information of an event. The objective of entropy is to define information as a negative of the logarithm of the probability distribution of possible
events or messages. As a result of the Shannon entropy, the decision tree algorithm tries to reduce the entropy with every split as much as possible. Therefore the feature for splitting is selected
which reduces the entropy most until an entropy of 0 is achieved and the category of the feature is pure.
E(parent node) = -(\frac{9}{14})*\log_2(\frac{9}{14})-(\frac{5}{14})*\log_2(\frac{5}{14})=0.94
Split on Outlook
E(sunny) = -(\frac{2}{5})*\log_2(\frac{2}{5})-(\frac{3}{5})*\log_2(\frac{3}{5})=0.97 \\
E(overcast) = -(\frac{4}{4})*\log_2(\frac{4}{4})-(\frac{0}{4})*\log_2(\frac{0}{4})=0 \\
E(rain) = -(\frac{3}{5})*\log_2(\frac{3}{5})-(\frac{2}{5})*\log_2(\frac{2}{5})=0.97
E(Outlook) = \frac{5}{14}*0.97+\frac{4}{14}*0+\frac{5}{14}*0.97=0.69
Split on Humidity
E(high) = -(\frac{3}{7})*\log_2(\frac{3}{7})-(\frac{4}{7})*\log_2(\frac{4}{7})=0.99 \\
E(normal) = -(\frac{6}{7})*\log_2(\frac{6}{7})-(\frac{1}{7})*\log_2(\frac{1}{7})=0.59
E(Humidity) = \frac{7}{14}*0.99+\frac{7}{14}*0.59=0.79
From the example, you see that we would divide the dataset by “Outlook” again because the entropy is reduced the most to 0.69 from 0.94.
Reduction in variance is an algorithm used for continuous target variables (regression problems). This algorithm uses the standard formula of variance to choose the best split. The split with lower
variance is selected as the criteria to split the samples. Assumption: numerical value 1 for playing and 0 for not playing.
\text{parent node mean} = \frac{9*1+5*0}{14} = 0.64 \Rightarrow
V(\text{parent node}) = \frac{9(1-0.64)^2+5(0-0.64)^2}{14} = 0.97
Split on Outlook
\text{sunny mean} = \frac{2*1+3*0}{5} = 0.4 \Rightarrow
V(\text{sunny}) = \frac{2(1-0.4)^2+3(0-0.4)^2}{5} = 0.24 \\
\text{overcast mean} = \frac{4*1+0*0}{4} = 1 \Rightarrow
V(\text{overcast}) = \frac{4(1-1)^2+0(0-1)^2}{4} = 0 \\
\text{rain mean} = \frac{3*1+2*0}{5} = 0.6 \Rightarrow
V(\text{rain}) = \frac{3(1-0.6)^2+2(0-0.6)^2}{5} = 0.24
V(outlook) = \frac{5}{14}*0.24 + \frac{4}{14}*0 + \frac{5}{14}*0.24 = 0.17
Split on Humidity
\text{high mean} = \frac{3*1+4*0}{7} = 0.43 \Rightarrow
V(\text{high}) = \frac{3(1-0.43)^2+4(0-0.43)^2}{7} = 0.24 \\
\text{normal mean} = \frac{6*1+1*0}{7} = 0.86 \Rightarrow
V(\text{high}) = \frac{6(1-0.86)^2+1(0-0.86)^2}{7} = 0.12
V(humidity) = \frac{7}{14}*0.24 + \frac{7}{14}*0.12 = 0.18
Because the variance of “Outlook” is lower compared to “Humidity” (0.17 vs. 0.18), the dataset would be divided by “Outlook”.
Read my latest articles:
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Soil Terminologies | Relationships Between Soil Properties
Soil terminologies/properties cover the properties of soil that are essential in predicting the engineering behavior of soils. These properties include void ratio, water content, air content, etc. It
also covers important interrelationships between these properties. Both the properties and relationship form an important part of the GATE Civil Engineering exam.
Soil Phase Diagram
Soil is a three-phase system in general. It contains air, water, and solids as a part of the system. Soil can also exist as a two-phase system depending on the field conditions. The two-phase systems
that could exist are listed below.
1. Dry soil system - solids and air
2. Saturated soil system - solids and water
Before getting into various terminologies it is important to understand certain terms related to weights and volumes of phase as listed below.
• Va - Volume of air
• Vw - Volume of water
• Vv - Volume of voids
• Vs - Volume of solids
• V - Total volume of the soil mass
• Wa - Weight of air (zero)
• Ww - Weight of water
• Wv - Weight of material occupying voids (neglected)
• Ws - Weight of solids
• W - Total weight of the soil mass
Soil Terminologies
Basic Relations
There are six basic relations using weights and volumes of phases as discussed below.
Porosity (n)
The porosity of a soil mass is the ratio of the volume of voids to the total volume of the soil mass. It is commonly expressed as a percentage and ranges from 0 to 100%.
Void Ratio (e)
The void ratio of a soil mass is the ratio of the volume of voids to the volume of solids in the soil mass. Its value is always greater than zero.
Void ratio is more commonly used than porosity as the volume of solids (Vs) remains constant upon application of pressure.
Degree of Saturation (S)
The degree of saturation of a soil mass is the ratio of the volume of water in the voids to the volume of voids. It is commonly expressed as a percentage and ranges from 0 to 100%.
For fully saturated soil mass Vv = Vw and the degree of saturation becomes 100%. For a dry soil mass, Vw = 0, and hence the degree of saturation becomes 0.
Percent of Air Voids (na)
The percentage of air voids of a soil mass is the ratio of the volume of air to the total volume of soil mass. It is commonly expressed as a percentage and ranges from 0 to 100%.
Air Content (ac)
The air content of a soil mass is the ratio of the volume of air to the total volume of voids. It is commonly expressed as a percentage and ranges from 0 to 100%.
Water content (w)
The water content of a soil mass is the ratio of the weight of water to the weight of solids of the soil mass. It is commonly expressed as a percentage and ranges from 0 to 100%.
Unit Weights of Soil
There are six unit weights that are required to be understood all of which are discussed further.
Bulk Unit Weight (γ)
The bulk unit weight of a soil mass is the weight per unit volume of the soil mass. It is also called the 'mass unit weight'.
Unit Weight of Solids (γs)
The unit weight of solids is the weight of soil solids per unit volume of solids alone. It is also called the 'absolute unit weight'.
Unit Weight of Water (γw)
The unit weight of water is the weight per unit volume of water. The unit weight of water is 9.81 kN/m^3 at 4°C which is commonly used as the standard.
Saturated Unit Weight (γsat)
The saturated unit weight is nothing but the bulk unit weight in a saturated condition.
Submerged Unit Weight (γ')
Submerged unit weight is its unit weight in submerged condition.
Dry Unit Weight (γd)
The dry unit weight of a soil mass is the soil solids per unit of total volume.
Specific Gravity of Soil
There are two specific gravity terms relating to the soil as discussed further.
Mass Specific Gravity (Gm)
The mass specific gravity of a soil is the ratio of mass unit weight of the soil to the unit weight of water. This is also called the 'bulk specific gravity or 'apparent specific gravity.
Specific Gravity of Solids (Gs)
The specific gravity of solids is the ratio of the unit weight of solids to the unit weight of water. It is also called the 'absolute specific gravity or 'grain specific gravity. This term is
relatively constant as it is based on the unit weight of solids (γs) and hence used in almost all relations.
Important Relations Between Soil Terminologies
Below mentioned are some of the important relations between the above-discussed terminologies. Note that G in the below relations refers to grain specific gravity i.e., Gs.
Void ratio and porosity
Void ratio, degree of saturation, and water content
Air content and degree of saturation
Percent of air void, porosity, and air content
Unit weight, void ratio, grain specific gravity, and degree of saturation
γ = ((G + Se)*γw) / (1 + e)
γsat = ((G + e)*γw) / (1 + e)
γsub = (G - 1)*γw / (1 + e)
Dry unit weight, water content, percent of air void
γd = (1 - na) * ((G*γw) / (1+e))
For your easy reference, all the relations are attached as an image below which could be downloaded and used for preparation.
Relations between Soil Properties
Example Problem
Question: If the porosity of the soil sample is 20%, the void ratio is? (GATE: 1997)
Therefore, the void ratio of the soil sample is 0.25
Practise Problem
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ManPag.es -
slasd5.f −
subroutine SLASD5 (I, D, Z, DELTA, RHO, DSIGMA, WORK)
SLASD5 computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by sbdsdc.
Function/Subroutine Documentation
subroutine SLASD5 (integerI, real, dimension( 2 )D, real, dimension( 2 )Z, real, dimension( 2 )DELTA, realRHO, realDSIGMA, real, dimension( 2 )WORK)
SLASD5 computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of a 2-by-2 diagonal matrix. Used by sbdsdc.
This subroutine computes the square root of the I-th eigenvalue
of a positive symmetric rank-one modification of a 2-by-2 diagonal
diag( D ) * diag( D ) + RHO * Z * transpose(Z) .
The diagonal entries in the array D are assumed to satisfy
0 <= D(i) < D(j) for i < j .
We also assume RHO > 0 and that the Euclidean norm of the vector
Z is one.
I is INTEGER
The index of the eigenvalue to be computed. I = 1 or I = 2.
D is REAL array, dimension (2)
The original eigenvalues. We assume 0 <= D(1) < D(2).
Z is REAL array, dimension (2)
The components of the updating vector.
DELTA is REAL array, dimension (2)
Contains (D(j) - sigma_I) in its j-th component.
The vector DELTA contains the information necessary
to construct the eigenvectors.
RHO is REAL
The scalar in the symmetric updating formula.
DSIGMA is REAL
The computed sigma_I, the I-th updated eigenvalue.
WORK is REAL array, dimension (2)
WORK contains (D(j) + sigma_I) in its j-th component.
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
September 2012
Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA
Definition at line 117 of file slasd5.f.
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Cracking the Code: Mastering Poker Math for Texas Hold'em Wins - Texas Hold'em Kingdom
Poker is a game that combines skill, strategy, and luck. While many players rely on their intuition and gut feelings to make decisions at the table, there is another aspect of the game that should
not be overlooked: poker math. Understanding and applying mathematical concepts can greatly enhance your performance in Texas Hold’em, giving you an edge over your opponents.
The Importance of Poker Math in Texas Hold’em: Enhance Your Game with Numbers
One of the most important aspects of poker math is calculating odds. Knowing the odds of certain hands appearing or improving can help you make informed decisions about whether to call, raise, or
fold. For example, if you have a flush draw with two cards to come, you can calculate the probability of hitting your flush by multiplying the number of outs (cards that will complete your hand) by 2
and adding 1. This gives you an approximate percentage chance of completing your hand.
Furthermore, understanding pot odds is crucial in determining whether a particular bet is worth making. Pot odds compare the current size of the pot to the cost of a contemplated call. By comparing
these two numbers, you can determine if the potential payout justifies the risk. If the pot odds are higher than the odds of completing your hand, it may be a profitable decision to call.
Another important concept in poker math is expected value (EV). EV measures the average amount of money you can expect to win or lose from a particular play over the long run. By calculating the EV
of different actions, you can make more informed decisions about which plays are likely to be profitable in the long term. For example, if you have a 50% chance of winning $100 and a 50% chance of
losing $50, the EV of that play would be positive ($25), indicating that it is a good decision to make.
In addition to odds, pot odds, and EV, understanding implied odds can also give you an advantage at the poker table. Implied odds take into account potential future bets that can be won if you hit
your hand. For example, if there is a large stack behind you who is likely to call a big bet if you hit your draw, the implied odds of making that call may outweigh the immediate pot odds.
It’s worth noting that poker math is not an exact science and should not be relied upon as the sole basis for decision-making. It is merely a tool that can help guide your choices and give you a
statistical advantage over time. Poker is still a game of skill, strategy, and adaptability, so it’s important to consider other factors such as player tendencies, table dynamics, and position when
making decisions.
In conclusion, mastering poker math is a key component of becoming a successful Texas Hold’em player. By understanding and applying concepts such as odds, pot odds, EV, and implied odds, you can make
more informed decisions at the table and increase your chances of winning in the long run. While poker math is not a guarantee of success, it can certainly give you an edge over your opponents and
enhance your overall gameplay. So, next time you sit down at the poker table, don’t forget to bring your calculator along with your poker face!
Mastering Pot Odds and Expected Value: Key Concepts for Poker Success
In the world of Texas Hold’em, where skill and strategy are essential for success, understanding poker math is crucial. Being able to calculate pot odds and expected value can give players a
significant advantage at the table. These key concepts allow players to make informed decisions and maximize their chances of winning.
Pot odds, in simple terms, refer to the ratio between the current size of the pot and the cost of a contemplated call. By calculating pot odds, players can determine whether it is profitable to
continue playing a hand or fold. This calculation involves comparing the number of outs (cards that can improve a player’s hand) with the size of the bet.
For example, if a player has four cards to a flush after the flop and there is $100 in the pot, they would need to call a $20 bet. In this scenario, the pot odds would be 5:1 ($100/$20). If the
player believes they have a better than 20% chance of hitting their flush on the next card, it would be a profitable call.
Understanding pot odds alone is not enough; players must also consider expected value (EV). Expected value takes into account both the probability of winning a hand and the potential payout. It helps
determine whether a particular play will yield long-term profits or losses.
To calculate expected value, players multiply the probability of winning by the amount that can be won and subtract the probability of losing multiplied by the amount that will be lost. A positive
expected value indicates a potentially profitable play, while a negative expected value suggests a losing proposition.
For instance, suppose a player has a pair of Kings, and they estimate that their opponent has a 30% chance of having a better hand. The pot currently stands at $200, and their opponent bets $50. To
calculate the expected value, the player multiplies the probability of winning (70%) by the amount that can be won ($250, including their opponent’s bet). They then subtract the probability of losing
(30%) multiplied by the amount that will be lost ($50).
In this case, the expected value would be $145 (($250 x 0.7) – ($50 x 0.3)). With a positive expected value, it would be a profitable decision to call the $50 bet.
Mastering pot odds and expected value requires practice and familiarity with poker math. However, once players become comfortable with these concepts, they can use them to make more informed
decisions at the table.
Additionally, understanding pot odds and expected value allows players to assess risk versus reward accurately. By comparing the potential payout to the likelihood of winning a hand, players can
determine whether a particular play is worth pursuing.
Furthermore, mastering these key concepts can help players avoid common mistakes, such as chasing draws without proper odds or calling bets when the expected value is negative. It provides a solid
foundation for making rational decisions based on mathematical calculations rather than emotions or gut feelings.
Moreover, pot odds and expected value are not only applicable in specific scenarios but can also be used as general guidelines during gameplay. By constantly assessing the current pot size, the cost
of calls, and the potential payout, players can adjust their strategy accordingly.
In conclusion, mastering pot odds and expected value is essential for success in Texas Hold’em. These key concepts allow players to make informed decisions based on mathematical calculations rather
than relying solely on intuition. By understanding the relationship between the size of the pot, the cost of calls, and the potential payout, players can maximize their chances of winning and
minimize losses. Practice and familiarity with poker math are necessary to effectively utilize these concepts, but the effort pays off in improved decision-making skills and overall poker success.
Calculating Hand Equity: How to Make Informed Decisions at the Poker Table
Poker is a game of skill and strategy, where players must make calculated decisions based on the information available. One crucial aspect of poker that often separates the pros from the amateurs is
understanding and utilizing poker math. By mastering poker math, players can gain an edge over their opponents and make more informed decisions at the poker table.
One fundamental concept in poker math is calculating hand equity. Hand equity refers to the percentage chance of winning a hand at any given point in the game. It is essential to have a solid
understanding of hand equity to make accurate decisions about whether to fold, call, or raise.
To calculate hand equity, you need to consider two factors: your hole cards and the community cards. Your hole cards are the two private cards dealt to you at the beginning of the hand. The community
cards are the five cards dealt face-up on the table for all players to use.
Let’s say you have been dealt two hearts โ the Ace of hearts and the King of hearts. After the flop, three community cards are revealed โ the Queen of spades, the Ten of hearts, and the Four of
clubs. To determine your hand equity, you need to evaluate how likely it is for your hand to improve and beat your opponents’ hands.
There are various methods to calculate hand equity, but one commonly used approach is using outs. Outs are the number of cards left in the deck that could improve your hand. In this example, let’s
say you want to hit a flush (five cards of the same suit). There are nine remaining hearts in the deck, so you have nine outs.
To calculate your hand equity, divide the number of outs by the number of unseen cards. Since there are 52 cards in a standard deck, and we know five community cards (the flop), there are 47 unseen
cards. Dividing nine by 47 gives you approximately 19% hand equity.
Knowing your hand equity allows you to make informed decisions about whether to continue playing the hand or fold. In this scenario, if the pot odds (the ratio of the current pot size to the cost of
a contemplated call) are greater than your hand equity percentage, it may be profitable to call or even raise. If the pot odds are lower, folding would likely be the best decision.
Additionally, understanding hand equity can help you determine the strength of your opponents’ hands. By comparing their actions with the potential range of hands they could have based on the
community cards, you can estimate their hand equity and adjust your strategy accordingly.
Calculating hand equity is not limited to flush draws; it applies to various scenarios in poker. Whether you’re calculating your chances of hitting a straight, a full house, or even determining the
probability of your opponent holding a specific hand, mastering poker math will undoubtedly enhance your gameplay.
In conclusion, poker math is a vital tool for any serious player looking to improve their game. Understanding hand equity and how to calculate it accurately empowers players to make more informed
decisions at the poker table. By considering factors such as outs, pot odds, and opponents’ actions, players can gain an edge and increase their chances of success in Texas Hold’em and other poker
variants. So, embrace the numbers, practice your calculations, and get ready to crack the code of poker math for those coveted wins!
Advanced Poker Math Techniques: Gaining an Edge over Your Opponents
When it comes to playing poker, many people focus solely on their intuition and reading their opponents. However, mastering poker math can give you a significant advantage at the table. Understanding
the underlying mathematics of the game allows you to make more informed decisions and increase your chances of winning.
One fundamental concept in poker math is probability. Knowing the odds of certain events occurring can help you determine whether a particular move is likely to be profitable or not. For example, if
you have a flush draw with four cards to a suit after the flop, you can calculate the probability of hitting your flush by multiplying the number of outs (cards that will complete your hand) by 2 and
then adding 1. This calculation gives you an approximate percentage chance of completing your flush by the river.
Another crucial aspect of poker math is pot odds. Pot odds compare the current size of the pot to the cost of making a particular bet. By calculating the ratio between these two numbers, you can
determine whether calling or folding is the more profitable decision. If the pot odds are higher than the odds of completing your hand, it may be worth making the call.
Furthermore, understanding implied odds can take your poker game to the next level. Implied odds refer to the potential future bets you can win if you hit your hand. While pot odds only consider the
current pot size, implied odds factor in the additional chips you can expect to win from your opponents in later betting rounds. By considering both pot odds and implied odds, you can make better
decisions about staying in a hand even when the immediate odds may not seem favorable.
Equity is another critical mathematical concept in poker. Equity represents your share of the pot based on the strength of your hand compared to your opponents’ hands. By calculating your equity, you
can determine whether it is profitable to make a particular bet or raise. For example, if you have a 50% chance of winning the pot, and the current pot size is $100, then your equity is $50.
Understanding poker math also involves knowing how to calculate expected value (EV). EV takes into account both the probability of winning a hand and the potential payoff. By multiplying the
probability of winning by the amount you stand to win and subtracting the probability of losing multiplied by the amount you stand to lose, you can determine the expected value of a particular
decision. Positive expected value decisions are typically profitable in the long run, while negative expected value decisions should be avoided.
Lastly, being able to quickly perform calculations at the table is crucial for applying poker math effectively. Practice mental math exercises and familiarize yourself with common probabilities and
odds to improve your speed and accuracy. Developing this skill will allow you to make more informed decisions in real-time without giving away any tells to your opponents.
In conclusion, mastering poker math is an essential skill for any serious Texas Hold’em player looking to gain an edge over their opponents. Understanding concepts such as probability, pot odds,
implied odds, equity, and expected value can help guide your decision-making process and increase your chances of success. So, take the time to study and practice these advanced poker math
techniques, and watch your game soar to new heights.
Applying Probability Theory to Texas Hold’em: Maximizing Your Chances of Winning
When it comes to playing poker, many players rely solely on their intuition and gut feeling. While these can be valuable tools in certain situations, mastering the art of poker math can significantly
increase your chances of winning in Texas Hold’em.
Probability theory is a branch of mathematics that deals with analyzing and quantifying uncertain events. By applying probability theory to poker, you can gain insights into the likelihood of
different outcomes and make more informed decisions at the table.
One fundamental concept in poker math is understanding the odds of hitting specific hands. For example, knowing the probability of flopping a flush draw or making a straight can help you decide
whether to stay in a hand or fold. By calculating these odds, you can determine if the potential reward outweighs the risk.
To calculate probabilities in poker, you need to consider two key factors: the number of outs and the number of unseen cards. Outs are cards that will improve your hand, while unseen cards refer to
the remaining deck. By dividing the number of outs by the number of unseen cards, you can estimate your chances of hitting a particular hand.
Let’s say you have two hearts in your hand, and there are two more hearts on the flop. With nine hearts left in the deck (13 total minus your two plus the four on the board), you have nine outs.
Since there are 47 unseen cards (52 total minus your two and the five on the board), your probability of hitting a heart on the turn is approximately 19%.
Understanding pot odds is another crucial aspect of poker math. Pot odds compare the size of the current pot to the cost of a contemplated call. By comparing these two values, you can determine
whether calling a bet is financially justified based on the probability of improving your hand.
Suppose the pot is $100, and your opponent bets $20. To calculate the pot odds, you divide the size of the pot ($100) by the cost of the call ($20), resulting in a ratio of 5:1. If your probability
of improving your hand is greater than 1 in 5 (or 20%), calling would be a profitable decision in the long run.
Additionally, understanding implied odds can give you an edge in poker. Implied odds take into account potential future bets that you may win if you hit your desired hand. For example, if you have a
flush draw on the turn, and your opponent has a large stack, there’s a higher chance they’ll continue betting on the river if you hit your flush. This additional potential profit should be factored
into your decision-making process.
While mastering poker math can undoubtedly enhance your game, it’s essential to remember that it’s just one piece of the puzzle. Combining mathematical analysis with reading opponents’ behavior,
managing your bankroll effectively, and making strategic decisions based on the overall context of the game will ultimately lead to success at the poker table.
In conclusion, applying probability theory to Texas Hold’em can significantly improve your chances of winning. By calculating the odds of hitting specific hands, understanding pot and implied odds,
and integrating these calculations into your decision-making process, you can make more informed choices and increase your profitability in the long run. So, start studying poker math, sharpen your
analytical skills, and crack the code to master Texas Hold’em. | {"url":"https://texasholdemkingdom.com/cracking-the-code-mastering-poker-math-for-texas-holdem-wins/","timestamp":"2024-11-09T00:57:00Z","content_type":"text/html","content_length":"86183","record_id":"<urn:uuid:2c0aaa70-c3da-48b5-95ad-17ca0d9e1c67>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00578.warc.gz"} |
Geoscientific Data Adds Value in Unconventional Reservoirs: Statistically Quantified
A dataset of 25 metrics collected from the annual investment reports of 9 public operators in the Bakken Formation is used for statistical analysis of the value of geoscientific data. According to
results of this analysis, for each million dollars invested in the geosciences, P90 reserves increase, on average, by 91 +/- 22 mboe after five years for these 9 public operators. Assuming a profit
margin of $15/bbl. returns on investment in geoscientific data average 6% for these 9 operators in the Bakken. The Bakken was chosen for this article because it extends into Canada. Operators in
other basins see even larger returns (Alvarado, 2016 & 2017).
Under the assumption that through exploration expenses (EXPEX) the cost of geoscientific information can be measured, fixed effects panel regressions with robust standard errors are constructed.
These regressions use proven reserves (P90, defined as resources attainable with 90% certainty) as the dependent variable, EXPEX as the variable of interest, and a series of controls accounting for
other line items expensed in the EXPEX account. Using the coefficients of these regressions, the ceteris paribus (everything else held constant) effect of investments in geoscientific information on
P90 is quantified.
In 2014, approximately 12% of the 8.7 mmbpd produced in the US came from the tight oil Bakken Formation. In the current macroeconomic scenario, where excess in supply and decrease in demand are
driving oil prices down, several oil and gas operators are reducing their exploration budgets and workforce, and taking a conservative approach to research and development. However, several industry
experts point out that a key component in the economic development of unconventional resources is the understanding of the technical drivers of fluid flow, which can only be achieved by the
application of new technologies (Moniz et al., 2011). In this sense, using low commodity prices to rationalize a reduction of capital allocated to exploration and research for unconventional
resources may be detrimental for a firm’s long-term value. This paper is motivated to help decision makers quantify the financial returns of investments in geoscientific information for
unconventional resources independently of oil prices. The objective is to provide a deterministic economic model using real data that quantifies the average financial return of investing in new
In this paper, geoscientific information is defined as geological, geophysical, and technical data describing the in situ conditions of a reservoir. Some examples of geoscientific information are
seismic imaging, petroelastic inversions, microseismic surveys, FMIs, geomechanical models, acid jobs, and tracers, to name a few. The value of the geoscientific information depends on the cost to
acquire this information and its derived benefits. For example, monitoring hydraulic stimulations with multicomponent seismic permits a clear image of the fracture network, thus providing a better
understanding of the petroelastic properties of the rock (Barkved, 2004). In this example, the investment is made in a geoscientific project (the acquisition of seismic data) from which geoscientific
information (subsurface images) is obtained. Using this information, operators make better decisions that optimize operations (well planning, directionality, completion depth), leading to higher
financial returns.
Managers in the industry use Value of Information (VOI) exercises as the standard procedure to evaluate the need of technical information (Bailey et al., 2011; Borison, 2005; Strunk, 2006). VOI
exercises are deterministic calculations from the real options theory where, in its simplest terms, information is valued as the difference between the value of an asset with current and future
information. However, VOI exercises cannot be used to quantify the value of geosciences on average for public shale operators. This is so because VOI exercises are subjective, they depend on the risk
aversion of the decision maker (Eeckhoudt and Godfroid, 2000), they require a very specific definition of the desired information (Bailey et al., 2011), and furthermore these exercises require
defining uncertainty through probabilistic functions backed by historical data (Strunk, 2006). In this paper, returns from investments in geoscientific information are quantified on average for
public operators with key assets in the Bakken. Instead of making use of the real option theory and VOI exercises, this work uses econometrics and statistical regressions on a dataset containing
several corporate metrics describing the past 20 years of exploration activity.
The findings presented in this paper are the result of equity research and statistical analysis. First, in the equity research part, the annual investment reports of the top public producers in the
Bakken Formation as of 2014 were investigated (Figure 1). 25 metrics were collected for each of these operators from 1995 to 2014, which include the 1998 and 2008 downturns, using the investment
reports available online to the public through the Securities and Exchange Commission (SEC). Some of the metrics in the dataset capture changes in exploration and development strategies among
operators (exploration expenses, proven reserves, acreage, drilled wells, etc.), other metrics describe their business and financial position (enterprise multiples, reserve life indexes, price per
flowing barrel, etc.), and some metrics capture macroeconomic scenarios (oil prices, interest rates, time trends). A detailed list of the metrics is provided in Appendix A. The metric of interest in
this paper is exploration expenses (EXPEX) because investments in the acquisition of technical information are expensed in this account.
Figure 1. Top public operators in terms of daily production in the Bakken Formation in 2014. These operators accounted for 44% of the whole Bakken production in that same year.
Now, let’s describe the statistical analysis used in this paper. The metrics gathered from the investment reports were organized as a panel. In statistics, a panel is a bi-dimensional array that
measures the changes in metrics (X) across companies (i) through time (t). The panel contains 148 observations from 9 public operators. The average EXPEX in the panel is $630 M with a minimum of $4
M, a maximum of $2144 M, and a standard deviation of $572 M. Analogously, P90 has an average of 7318 mmboe with a minimum of 122 mmboe, a maximum of 55946 mmboe, and a standard deviation of 12215
mmboe. Due to the large range in the values of EXPEX and P90, histograms showing the distribution of both metrics in natural logarithmic form are given in Figure 2. Figure 2 also shows a relationship
plot showing a clear positive relation between EXPEX and P90. Furthermore, the correlation between P90 and EXPEX for this panel of operators in the Bakken is 0.65, which means that up to 43% of the
variations in P90 could be explained by changes in EXPEX (Baltagi, 2011).
Figure 2. Histograms describing the dataset and a relation plot showing a clear positive relation between EXPEX and P90.
A type of multivariate regression called fixed-effects panel regression with robust standard errors is constructed to quantify the effects of EXPEX on Proven Reserves (P90). This type of statistical
regression permits quantifying the interaction between variables across companies, addressing any unobserved differences among them (Woolridge 2012; Baltagi, 2011; or Greene, 2012). In short, this
type of regression allows us to quantify the relationship between metrics among companies with different budgets, market capitalization, enterprise value or any other intrinsic characteristic. Since
each metric is a time series, the metrics are highly correlated and autocorrelated (e.g. the reserves of EOG in 2007 are likely to depend in some degree on its reserves, production, and acreage in
2006). The problem with constructing statistical regressions with autocorrelated variables is that the resulting model tends to underestimate standard errors (Greene, 2012). Hence, a method called
the Arellano procedures was used to construct robust standard errors and 95% confidence intervals.
To account for the cumulative effect of investments in the geosciences, finite distributed lag models (FDLM) were used. FDLM are multiple regressions where the lagged values of a variable of interest
are also used in the regression equation (Baltagi, 2011; Woolridge, 2012). As an example, it takes time for a pharmaceutical company to find a new cure so the effect of investments in R&D on profits
will show up with a lag and will be significant for many years afterwards. Similarly, it can take years for investments in geoscientific information to pay off, so the effects of exploration on other
corporate metrics are likely to show up with lags and be significant for several periods afterwards. However, due to limitations in the dataset, a maximum of 5 lags of EXPEX was considered in each
Finally, once the relationship between EXPEX and P90 is quantified using the regression coefficients, this information is used in a discounted cash flow model where ROIs are quantified as a
continuously compounded interest rate using the following formula (Equation 1):
Where FV represents future value, PV present value, i the interest rate (ROI in this case), and t the number of periods (years).
Table 1 shows the regression coefficient of the variable of interest EXPEX (highlighted in yellow) and selected controls obtained from the fixed effects panel regression of P90 vs. EXPEX for the top
operators in the Bakken Formation. Under the regression coefficients, the reader will find the standard errors and symbols illustrating statistical significance. The result of this regression can be
interpreted as follows: a million dollars invested in the geosciences will increase the amount of proven reserves by 91 +/- 22 (95% confidence interval) thousands of barrels of oil equivalent after 5
Table 1. Regression coefficients from the P90 versus EXPEX regression in the Bakken Panel. Numbers in parenthesis are standard errors. The stars below the regression coefficient depict their
significance levels (*** for 99.9% statistical significance, ** for 95%, * for 90%). R2 coefficient indicates the goodness of fit.
Using this information, a discounted cash flow model is constructed to estimate expected returns on investment. On Table 2, this procedure is illustrated in a practical way. The first row represents
the time after investment (in years). The regression models quantified an increment of 91 thousand barrels of oil equivalent with a lower confidence level (LCL) of 69 and an upper confidence level
(UCL) of 113 as illustrated on the second row of Table 2. The third row assumes a fixed price of $40/bbl. The fourth row assumes a total cost per barrel of $25/ bbl, leaving a profit margin of $15/
bbl. Under these assumptions, geoscientific projects have an average ROI of 6% for these 9 operators.
Table 2. Discounted cash flow model illustrating findings of regressions and estimating ROIs.
In this paper, the financial returns from geoscientific information are quantified using panel data econometrics. Specifically, results indicate that $1 MM invested in geosciences increases the
amount of proven reserves (P90) by 91 +/- 22 (two standard deviations) mboe on average 5 years after the geoscience investment for the top public operators in the Bakken Formation. Assuming a profit
margin of $15/bbl, these increments in P90 attributed to EXPEX can have ROIs as high as 10% under the assumptions stated in this paper.
Given access to time series describing the changes in acreage, production, management, reserves, costs, and type of technologies at a field or basin level, which are often available in annual and/or
government reports, econometrics could then be used to quantify the returns from specific exploration technologies (microseismic, tracer data, seismic surveys) across different basins independently
of commodity prices.
The Reservoir Characterization Project (RCP) from Colorado School of Mines. Tom Davis, Graham Davis, Peter Maniloff, Hortense Viallard.
About the Author(s)
Fernando Alvarado Blohm is a recent graduate from Colorado School of Mines whose research focused on oil and gas economics. His background is in geophysics and he works in strategic asset allocation
and risk management in Houston, Texas.
Alvarado Blohm, F., 2016, Quantifying the value in geoscientific information using panel data econometrics, M.S. thesis, Colorado School of Mines.
Alvarado Blohm, F., 2017, Quantifying the value in geoscientific information using panel data econometrics, SPE Annual Technical Conference and Exhibition.
Bailey, W. J., B. Couet, and M. Prange, 2001, Forecast optimization and value of information under uncertainty in Y. Z. Ma and P. R. La Pointe, eds., Uncertainty analysis and reservoir modeling: AAPG
Memoir 96, 217–233.
Baltagi, B. H., 2011, Econometrics, 3rd edition: Springer-Verlag Berlin Heidelberg.
Barkved, O., B. Bartman, B. Compani, J. Gaiser, T. Johns, P. Kristiansen, T. Probert, M. Thompson, and R. Van Dok, 2004, The many faces of multicomponent seismic data: Oilfield Review, 16, no. 2.
Borison, A., 2005, Real options analysis: where are the emperor’s clothes?: Journal of Applied Corporate Finance, 17, no. 2, 17-31.
Demirmen, F. 2007. Reserves Estimation: The Challenge for the Industry. Journal of Petroleum Technology, 59, no. 5: 80–89. SPE-103434-PA.
Eeckhoudt, L., and P. Godfroid, 2000, Risk aversion and the value of information: The Journal of Economic Education, 31, 382–388.
Etherington, J., T. Pollen, and L. Zuccolo, 2005. Comparison of selected reserves and resource classifications and associated definitions: SPE, Oil and Gas Reserves Committee, Mapping Subcommittee
final report.
Greene, W., 2012. Econometric Analysis, 7th edition: Pearson Education Inc.
Moniz et al., 2011.The future of natural gas: MIT Energy Initiative, http://energy.mit.edu/publication/future-natural-gas/.
Pickering, E., and S. Bickel, 2006. The value of seismic information: Oil and Gas Financial Journal, 3, no. 5.
PRMS, 2007, Petroleum resources management system: SPE, AAPG, World Petroleum Council, Society of Petroleum Evaluation Engineers.
PRMS, 2011, Guidelines for application of the petroleum resources management system: SPE, AAPG, World Petroleum Council, Society of Petroleum Evaluation Engineers.
Strunk, A., 2006, Decision frameworks inc – value of information training course: slides from lectures (8 sessions).
Woolridge, J., 2012, Introductory Econometrics: A Modern Approach, 5th edition: South-Western.
Appendix A
List of metrics used in the regression analysis presented in this paper:
1. Exploration Expenses (EXPEX): Recorded in $M. Exploration expenses is the variable of interest in the dataset as costs related to geology and geophysics (G&G) are expensed in this account.
Specifically, investments in exploration technologies and the acquisition of geoscientific data are recorded here. However, other line items not related to exploration per se are also expensed in
this account such as leasehold impairments, dry well costs, the cost of land, and sometimes even development costs.
2. Proven Reserves (P90): Recorded in mmboe. Proven reserves are technically defined as the volume of hydrocarbons sized by a reliable technology that can be recovered in the current infrastructure
with the simple average annual crude price (PRMS, 2007). The acronym, P90, comes from the probabilistic definition used when the range of uncertainty is represented by a probability distribution.
In this context, proven reserves correspond to the lowest 10th percentile of a probability density function meaning that “P90” barrels or more can be recovered with 90% probability (PRMS, 2011;
Etherington et al., 2005; Demirmen, 2007).
3. Total Production (Q): Recorded in mbpd (thousands of barrels of oil equivalent per day).
4. Production Costs (C): Recorded in $/bbl. This metric describes the costs of getting a barrel of crude to the surface.
5. Net Acreage Developed Total (NADT): Recorded in thousands of acres. Net acreage is calculated by multiplying gross acreage by the operator’s working interest. The word developed means that this
acreage is spaced, or assignable, to productive wells.
6. Net Acreage Undeveloped Total (NAUT): Recorded in thousands of acres. Similar to NADT but with the difference that undeveloped acreage is not assignable to a producing well. This acreage is
commonly held to keep potential prospects within range, maintain mineral rights, and prevent competitors from developing nearby.
7. Development Wells Producing (WDP): These are the net number of wells for development that were producing (not dry).
8. Development Wells Dry (WDD): These are the net number of wells for development that resulted in dry holes.
9. Exploratory Wells Producing (WEP): These are wildcat or exploratory wells that were drilled to confirm a prospect and resulted in being productive wells. This metric is reported in the same
section as development wells.
10. Exploratory Wells Dry (WED): These are wildcat or exploratory wells that resulted in dry holes.
11. Price of Oil Brent (POB): Recorded in $/bbl. According to microeconomic theory, any price represents the equilibrium between supply and demand for a product. Applied to POB, this metric
represents the global supply and demand balance from 1995 to 2014.
12. Price of Oil West Texas Intermediate (POWTI): Recorded in $/bbl and gathered from the EIA. Same as POB, POWTI is a crude benchmark for light crude but more sensitive to US production.
13. Nominal 5 Year Constant Maturity Treasury Notes (I5N): This metric is unitless and is taken from the Federal Reserve System. It measures the risk free rates for investments with a five-year pay
14. Basin Maturity (Period): This is a variable that counts the years from 1995 to 2014 as periods from 1 to 20. The reason for including this variable is to control for time trends among variables.
In the context of this paper, this variable addresses the “drilling frenzy” in unconventional resources happening across formations and basins during the 2000s as a result of high oil prices.
15. CEOs (CEO): These are dummy variables counting for changes in CEOs among operators for the past 20 years. This metric addresses management changes that could affect the decision making process
within companies through time.
16. Earnings Before Interests, Taxes, Depreciation, and Amortization (EBITDA): Recorded in $M. EBITDA is sometimes published within the summary of financial position or debt balance in 10-Ks. When
not reported, EBITDAX is estimated from the income statement by locating the earnings before interests and taxes and adding exploration expenses and depreciation, amortization, and depletion
allowances. Since EBITDAX measures earnings before fiscal obligations and income-sheltering allowances, this metric represents earnings generated by current management and quantifies the added
value of managers.
17. Reserve Life Index (RLI): Calculated by dividing proven reserves by annual production and thus has units of years. This metric can be interpreted as the years of operations any oil and gas
operator has left given its current reserves and current production.
18. Price Per Flowing Barrel (PFB): Calculated as Enterprise Value (EV) divided by production (Q) and thus has units of $/bbl. This metric is used to compare the market value of a barrel of crude
coming from a specific operator. EV is calculated as market capitalization plus debt minus cash and thus EV assigns a dollar value to the whole company.
19. Enterprise Multiple (EM): Unitless. This metric is calculated by dividing EV by EBITDAX and is a commonly used financial ratio used by analysts in different industries because is unaffected by
capital structure. In contrast with RLI or PFB, this metric by itself says very little about the financial position of a company or the state of its current operations, nevertheless when used
along with other ratios is useful in comparing business performance beyond company size. | {"url":"https://csegrecorder.com/articles/view/geoscientific-data-adds-value-in-unconventional-reservoirs","timestamp":"2024-11-10T02:22:47Z","content_type":"text/html","content_length":"45629","record_id":"<urn:uuid:49f0c8ef-52e7-4f73-b8bf-ecc71509fce3>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00471.warc.gz"} |
An open-ended implentation of artificial neural networks in Julia.
Some neat features include:
• Poised to deliver cutting-edge synergy for your business or housecat in real-time!
• Twitter-ready out of the box!
• Both HAL9000 and Skynet proof!
• Low calorie, 100% vegan, and homeopathic friendly!
• Excellent source of vitamin Q!
Some less exciting features:
• Flexible network topology with any combination of activation function/layer number.
• Support for a number of common node activation functions in addition to support for arbitrary activation functions with the use of automatic differentiation.
• A broad range of training algorithms to chose from.
Over time we hope to develop this library to encompass more modern types of neural networks, namely deep belief networks.
Currently we only have support for multi-layer perceptrons, these are instantiated by using the MLP(genf,layer_sizes,act) constructor to describe the network topology and initialisation procedure as
• genf::Function is the function we use to initialise the weights (commonly rand or randn);
• layer_sizes::Vector{Int} is a vector whose first element is the number of input nodes, and the last element is the number of output nodes, intermediary elements are the numbers of hidden nodes
per layer;
• act::Vector{Function} is the vector of activation functions corresponding to each layer.
• actd::Vector{Function} is the vector corresponding to the derivatives of the respective functions in the act vector. All of the activation functions provided by NeuralNets have derivatives (which
can be seen in the dictionary NeuralNets.derivs)
For example, MLP(randn, [4,8,8,2], [relu,logis,ident], [relud,logisd,identd]) returns a 3-layer network with 4 input nodes, 2 output nodes, and two hidden layers comprised of 8 nodes each. The first
hidden layer uses a relu activation function, the second uses logis. The output nodes lack any activation function and so we specify them with the ident 'function'—but this could just as easily be
another logis to ensure good convergence behaviour on a 1-of-k target vector like you might use with a classification problem.
Once your neural network is initialised (and trained), predictions are made with the prop(mlp::MLP,x) command, where x is a column vector of the node inputs. Of course prop() is also defined on
arrays, so inputting a k by n array of data points returns a j by n array of predictions, where k is the number of input nodes, and j is the number of output nodes.
There is 'native' support for the following activation functions. If you define an arbitrary activation function its derivative is calculated automatically using the ForwardDiff.jl package. The
natively supported activation derivatives are a bit over twice as fast to evaluate compared with derivatives calculated using ForwardDiff.jl.
• ident the identify function, f(x) = x.
• logis the logistic sigmoid, f(x) = 1 ./(1 .+ exp(-x)).
• logissafe the logistic sigmoid with a 'safe' derivative which doesn't collapse when evaluating large values of x.
• relu rectified linear units , f(x) = log(1 .+ exp(x)).
• tanh hyperbolic tangent as it is already defined in Julia.
Once the MLP type is constructed we train it using one of several provided training functions.
• train(nn, trainx, valx, traint, valt): This training method relies on calling the external Optim.jl package. By default it uses the gradient_descent algorithm. However, by setting the
train_method parameter, the following algorithms can also be selected: levenberg_marquardt, momentum_gradient_descent, or nelder_mead. The function accepts two data sets: the training data set
(inputs and outputs given with trainx and traint) and the validation set (valx, valt). Input data must be a matrix with each data point occuring as a column of the matrix. Optional parameters
□ maxiter (default: 100): Number of iterations before giving up.
□ tol (default: 1e-5): Convergence threshold. Does not affect levenberg_marquard.
□ ep_iterl (default: 5): Performance is evaluated on the validation set every ep_iter iterations. A smaller number gives slightly better convergence but each iteration takes a slightly longer
□ verbose (default: true): Whether or not to print out information on the training state of the network.
• gdmtrain(nn, x, t): This is a natively-implemented gradient descent training algorithm with momentum. Returns (N, L), where N is the trained network and L is the (optional) list of training
losses over time. Optional parameters include:
□ batch_size (default: n): Randomly selected subset of x to use when training extremely large data sets. Use this feature for 'stochastic' gradient descent.
□ maxiter (default: 1000): Number of iterations before giving up.
□ tol (default: 1e-5): Convergence threshold.
□ learning_rate (default: .3): Learning rate of gradient descent. While larger values may converge faster, using values that are too large may result in lack of convergence (you can typically
see this happening with weights going to infinity and getting lots of NaNs). It's suggested to start from a small value and increase if it improves learning.
□ momentum_rate (default: .6): Amount of momentum to apply. Try 0 for no momentum.
□ eval (default: 10): The network is evaluated for convergence every eval iterations. A smaller number gives slightly better convergence but each iteration takes a slightly longer time.
□ store_trace (default: false): Whether or not to store information on the training state of the network. This information is returned as a list of calculated losses on the entire data set.
□ show_trace (default: false): Whether or not to print out information on the training state of the network.
• adatrain
• lmtrain | {"url":"https://juliapackages.com/p/neuralnets","timestamp":"2024-11-10T08:10:22Z","content_type":"text/html","content_length":"39618","record_id":"<urn:uuid:ffc5a97a-5ba9-4520-b176-938ffcbd8b08>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00406.warc.gz"} |
Electric Definition and 1000 Threads
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the
phenomenon of electromagnetism, as described by Maxwell's equations. Various common phenomena are related to electricity, including lightning, static electricity, electric heating, electric
discharges and many others.
The presence of an electric charge, which can be either positive or negative, produces an electric field. The movement of electric charges is an electric current and produces a magnetic field.
When a charge is placed in a location with a non-zero electric field, a force will act on it. The magnitude of this force is given by Coulomb's law. If the charge moves, the electric field would be
doing work on the electric charge. Thus we can speak of electric potential at a certain point in space, which is equal to the work done by an external agent in carrying a unit of positive charge from
an arbitrarily chosen reference point to that point without any acceleration and is typically measured in volts.
Electricity is at the heart of many modern technologies, being used for:
Electric power where electric current is used to energise equipment;
Electronics which deals with electrical circuits that involve active electrical components such as vacuum tubes, transistors, diodes and integrated circuits, and associated passive interconnection
technologies.Electrical phenomena have been studied since antiquity, though progress in theoretical understanding remained slow until the seventeenth and eighteenth centuries. The theory of
electromagnetism was developed in the 19th century, and by the end of that century electricity was being put to industrial and residential use by electrical engineers. The rapid expansion in
electrical technology at this time transformed industry and society, becoming a driving force for the Second Industrial Revolution. Electricity's extraordinary versatility means it can be put to an
almost limitless set of applications which include transport, heating, lighting, communications, and computation. Electrical power is now the backbone of modern industrial society.
View More On Wikipedia.org
1. L
Hi, I don't know if I have calculated the electric field correctly in task a, because I get different values for the Poisson equation from task b The flow of the electric field only passes
through the lateral surface, so ##A=2\pi \varrho L## I calculated the enclosed charge as follows...
2. H
Suppose two orthogonal neighbouring orbitals ##|\phi _1 \rangle## and ##|\phi _2 \rangle## so that ##\langle \phi_1|\phi _2 \rangle =0##. Applying an electric field adds a new term ##u (c_1^{\
dagger}c_1-c_2^{\dagger}c_2)## to the Hamiltonian which u is a constant potential. Obviously, we still...
4. M
Hi, I am reading Griffiths Introduction to electrodynamics. Currently I am solving problem 2.11 which asks to find an electric field inside and outside a spherical shell of radius R. Inside: $$\
int{E \cdot da} = \frac{Q}{e_0} = |E|4\pi r^2 = \frac{Q}{e_0} = 0$$ The result is $$0$$ because we...
this is the field I was provided and this is the charge density that I have reached I tried to use this yet the output was different I also used Cartesian it gave me the same output as the
spherical ones
6. K
Here's what I've tried. First of all, I assume that q is positive. For particle A, then, I can write $$q E -k {\left( x _{A }-x _{B }\right) }=m \ddot{x }_{A }, $$ where ##x _{A } ## and ##x _{B
} ## are the coordinates of the particles relative to their equilibrium positions from the point of...
Homework Statement: circuits - terms Relevant Equations: - How exactly can the electric potential be constant between two points in a wire; (assuming that it is electron current); if the electron
is moving from a region of high electric potential to a low electric potential because of the...
I did a thought experiment and I can't figure out what the mistake is. There is a system of 2 electric motors weighing 1 kg each with batteries in the Earth's orbit. The motors are rigidly
connected by a 1-meter-long bar. If one motor starts rotating in one direction on a signal, the entire...
It seems that electric mining equipment is all the craze right now: Liebherr electric excavator Caterpillar 240-ton electric haul truck Liebherr and Fortescue partner on world’s first autonomous
electric haul truck EPCA plans to convert 50-70 mining trucks to electric power annually Liebherr -...
Hello to everyone. I have some doubts about one problem of quantum mechanics. My attempt. I need to calculate the coefficient ##W_{ij}=<\psi_i | H' |\psi_j>## where ##H' = -eE(t)z## is a
perturbation term in the hamiltonian and ##|\psi_i> = |\psi_{nlm}>##. We have four states and sixteen...
Suppose there is very long current carrying wire. A charged particle is present somewhere around it. The current in the wire varies with time, thus by biot-savart's law there should be time
varying magnetic field. I want to know that will this time varying magnetic field produce electric field...
12. C
Hi, I wonder if someone can help with the following problem? We have a sealed box in space and inside the box is an electric motor with the stator attached to the box. The rotor arm is attached
to the inner race of a bearing and the outer race of the bearing is also attached to the box. There...
My understanding of this question is that, if you have a proton standing against a positive electric field, and move it in the opposite direction of the field, you're putting in work and
therefore should have greater electric potential energy. But that idea breaks down when you consider a...
I would like to discuss a few ways to apply derivatives in physics (I don't understand it fully). I don't need a full solution, I only need to understand how to successfully apply the derivatives
First example, Thin insulating ring of mass M, uniformly charged by charge ##+q## has a small cut...
surfafce area = 0.502 E = -q/A2(en) = 3800 -q = 3800*(A2(en)) -q = 1.68*10^(-8) -q = 3.37*10^(-8) V = kq/r V = (9.0*10^9)(-3.37*10^(-8))/0.2 V = -1519 V
I did make the problem simpler by looking at the the part from d/2 down the upper plate here are my initial parameters I am making my size step be h since lowering it may make calculating harder
I am especially getting weird results for the field and capacitance R = 0.1; % Radius of the...
Let's assume that we have a hollow sphere with holes at opposite ends of the diameter. What would be the field inside the hollow sphere? I know that we can look at this as the superposition of
the hollow sphere without holes and 2 patches with opposite surface charge density. For some reason, in...
18. S
here is my attempted solution. ## d^2 = z^2 + \frac {L^2} {3} ## ## C ## is coulomb constant since the point is symmetric, only the vertical component of the electric field remains. So, $$ E = 3
E_y =3 \frac {C Q cos \theta} {d^2} $$ $$ E= 3 \frac {C Q z} {d^3} $$ thus part (a) is done ( i...
Today, I watched a video about electric field created by an infinite plate by Khan Academy. They were talking about the clever application of the Gauss's law in this case (the cylinder method),
so I wondered if I could apply the same thing but to 2 plates. For example, let's say that the plates...
We take out "formulas" for electric potential from the relation $$V=\int E.dx$$ Some general formulas are : For a hollow sphere : ##\frac{Q} {4π\epsilon_0 x}## when x>R, x =distance of that point
from the center And the problem is we just input the distance in sums to calculate absolute...
I understand the following .a conductor is made of atoms and atoms always strive to be at equilibrium and that's why the electric field inside a conductor is zero because the electros distribute
themselves in such a way so that they are in equilibrium , yet they do produce an electric field...
q encolsed =0 Second case q enclosed q by gauss law At 2r
25. P
I have having trouble understanding Maxwell's Equations. Can anyone recommend some good book or website that can help me to understand these Equations? How can electric and magnetic fields travel
perpendicular to each other? What causes electromagnetic waves to first radiate from its source? I...
The most common explanation I know is that anomaly cancelation implies the sum of electric charges of each particle must cancel generation-wise, so 3 Q(Up) + 3 Q(Down) + Q(electron) = 0, and
electroweak doublets imply Q(Up) - Q(Down) = Q(neutrino) - Q(electron), so with Q(neutrino) = 0 it solves...
I tried resolving the semi infinite rods into arcs of 90 degree each placed on the three axes but that doesnt take me anywhere.... Alternatively I tried finding out the field at the point due to
each rod but im unable to find the perpendicular distance from the point to the rod...I dont think...
Lambda = charge density I tried first taking out the field due to the circular arc and I got $$ (lambda / 4π (epsilon knot) ) (2 sin (theta)) $$ For reference this is the arc that was provided in
the question of angle 2(theta) and the tangent What I dont understand is how can the fields be...
This is the general suggested approach given in a textbook. My question is why can I not directly write it in vector form? E1 vector + E2 vector =0 should be valid no? Why are they choosing to
write E1 mag + E2 mag=0 Then find a vector form Then convert the magnitude equation into a vector...
31. B
Please help me with this homework! I haven't had any solutions since it is all unclear.
Here is the exercise: And these are my attempts: This is for the first question about the electric field. (I know I'm missing the drawing, which is a drawing of the plane layer of thickness 2e
with a cylinder on it as a GAUSS SURFACE ). As for the second question, I'm not sure about it, so I...
33. P
So for this problem I think I am doing something weird with the trig and/or vector components. I calculated the problem like this: First drew a picture, q1 and q2 on the x axis. q3 located
equidistant between them but negative .300m in the y direction. First finding magnitude of Electric...
Using either H&R's Chapter 27 Example 3 or Problem 590 of the ##\mathbf{Physics Problem Solver}##, I've been unable to get the component ##E_x## or ##E_y##. There are now different angles at the
charges. My thanks to berkeman for LaTeX advice, but any errors are of course my own. Thanks in...
36. L
The first image is for a conducting sheet (part of it anyway), the second is for a nonconducting sheet. Gauss' law seems to tell me that the electric field strength are different - they differ by
a factor of two. Is this true? The charge enclosed in both of them are the same, and my intuition...
37. L
Consider a negatively charged spherical conductor. On the surface of it, what is the direction of its electric field? Well, the definition of the direction of an electric field is the direction a
positive test charge would go if placed at that point. But... it wouldn't move anywhere! So is the...
38. J
I have a student trying to build a simple solar powered vehicle for a high school design thinking class. He solar panel produces about 3.1 V as measured on a multimeter, but will not power the
electric motor she had chosen. She tested the motor with a pair of AA batteries (2.9 V on multimeter)...
How can I find the sustaining time of an electric guitar? The influence of other components besides the strings can be neglected. I need it for my term paper.
40. A
Suppose there is an electric charge of 350 micro coulombs in space. The electric field at a distance of less than one meter will be more than 3,000,000 volts/meter considering that this field is
greater than the electric breakdown of air and the charge has no place to discharge, what happens...
A recent article published in the Proceedings of the National Academy of Sciences (PNAS) describes a large electric capacitors based on carbon black and concrete. The device would be used for
electric power storage - often in proximity to the electric power demand, for example, a home.
I am using an old monitor (MITSUBISHI RDT27IWLM). The power consumption changes when the screen is white or black, but does the frequency of the weak electromagnetic waves emitted from the
monitor change? Or is the frequency the same, only the output is stronger/weaker?
There are two identical spheres with the same charge that are the vertices of an equilateral triangle. ##+3 \mu C## will exert an outward electric field, which is drawn in the FBD below (see the
attached pic), Since the horizontal force components (1x and 2x) are equal and opposite at point P...
There are three charges with +1 μC and −1 μC, are placed at the opposite corners of a cube with edges of length 1 cm, and the distance from P to B is 1cm 2. I labeled them as A, P, and B, which
is shown in the diagram below. Since we need to find the magnitude of the charge at point P and the...
46. R
How does an electric field of a moving charge, for example a moving electron, inside a wire looks like? Does it looks like this with distorted circular radial lines?
47. M
My question is specifically with calculating the intensity. The book solution is I=P/(4*pi*r^2) but would this not give me a weaker electrical amplitude in the final calculation after plugging it
in to I=(1/2)*√(ε0/μ0)*(E02) ?
Hi I want to know the power efficiency of linear piezo electric motors in percentile.
49. L
Hi, unfortunately, I am not sure if I have calculated the task correctly The electric field of a point charge looks like this ##\vec{E}(\vec{r})=\frac{Q}{4 \pi \epsilon_0}\frac{\vec{r}}{|\vec{r}|
^3}## I have now simply divided the electric field into its components i.e. #E_x , E-y, E_z#...
50. N
Dear Experts, When a thin conducting sheet with no charge on is placed at a certain distance from a point charge, does it shield the electric field caused due to the point charge from reaching
the other side of the sheet. As an extension of that idea, when a conducting sheet or slab is placed... | {"url":"https://www.physicsforums.com/tags/electric/","timestamp":"2024-11-06T05:04:48Z","content_type":"text/html","content_length":"171277","record_id":"<urn:uuid:e618e3e6-32d6-4e3f-b367-3b4c9d1e13c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00374.warc.gz"} |
Babylonian Mathematics
Around 2000 B.C., a people called the Amorites invaded Sumer and captured its cities. These people became known as the Babylonians, whose civilisation lasted for a millennium and a half, until the
capture of Babylon by the Persians in 538 B.C.
The Babylonians made significant advances in mathematics over previous civilisations. While retaining much of Sumerian mathematics, as well as most of the Sumerian number system, they then did
something unique in the ancient world: They invented a positional number system. The Babylonians dropped most of the Sumerian symbols that were used to write numbers, and kept only two: The "wedge",
which represented 1, and the "hook", which represented ten.
The Hindu-Arabic number system that we use today is also a positional system. In a positional (or place value) number system, the position of the number indicates the value attached to it. For
example, the value of the "4" in 43 is 40 because it appears in the tens place. On the other hand, the value of "4" in 34 is 4, because it is in the ones place.
Our number system is a base ten system. The Babylonians used a base 60 system. Here is a brief overview of how they formed numbers: 1 was represented as a wedge, 2 as two wedges, and so on up to 9 as
nine wedges, 10 as a hook, 11 as a hook and a wedge, and so on up to 59 which was represented as five hooks and nine wedges. To represent 60, a wedge was placed in the sixties place.
Babylonian Numerals
Babylonian figures for the numbers from one to ten as they appear on the ancient clay tablets
This system wasn't perfect. For example, there was no zero to use as a placeholder. Therefore,a number like 61 (1 × 60 + 1 × 1) would look very similar to 3601 (1 × 3600 + 1 × 1) because in the
latter number, the 60's place was left blank. Around the time of Alexander the Great (more than 200 years after Babylon was captured by the Persians) the Babylonians fixed this problem by using two
oblique wedges as a placeholder. Another problem was that there was no decimal point. This would make a number such as ^1/[2] (30 × ^1/[60]) look the same as 30 (30 × 1). The representation of many
numbers was often ambiguous, so scribes had to use the numbers' context to determine their value. Still, the invention of a positional system was a great achievement in mathematics, considering that
millennia later the Greeks still used cumbersome number systems like the Attic and Ionic numerals. In Europe Hindu-Arabic numerals did not catch on until about 1500 A.D., more than three millennia
after the Babylonians adopted their place value system.
The Babylonians made other significant advances in other areas of mathematics, such as fractions, algebra, and geometry. There are tiny bits about them in my pages on the Pythagorean theorem and .
See also: Mathematics history | {"url":"http://mathlair.allfunandgames.ca/babylonian.php","timestamp":"2024-11-13T08:44:39Z","content_type":"text/html","content_length":"5636","record_id":"<urn:uuid:d6b9c87f-1c8e-4999-9f40-d4e06cb0fc69>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00281.warc.gz"} |
Primary Mathematics Tutor 4A - Revised Edition - Giftedthinkers
Mathematics Tutor 4A is the first of a two-book series specially written to serve as your mathematics companion for primary 4 classes. This new 9 chapter volume, which follow closely to the newly
implemented mathematics syllabus by the Ministry of Education, is comprehensive as it is suitable for the student who self-studies at home. It is dedicated to helping you revise in an effective and
efficient way, as well as to help you prepare confidently for your examinations.
Each topic in Mathematics Tutor 4A contains the following features:
1. Targets to help you focus on the relevant concepts during your revision.
2. Comprehensive Notes and Worked Examples where the important concepts are emphasised. The worked examples will help you learn how to solve similar types of questions.
3. Tutorials by Topics found at the end of each chapter supplement classroom exercises.
4. Thinking skills corner is where you can find more challenging and thought provoking questions. Here you will be able to train your problem-solving skills by applying the concepts learned in the
chapters. Revision exercises are found after every few chapters to help you refresh and consolidate all the concepts learned in these chapters.
ISBN: 9789814564854
Weight: 1,123.00g | {"url":"https://giftedthinkers.net/product/primary-mathematics-tutor-4a-revised-edition/","timestamp":"2024-11-14T23:28:56Z","content_type":"text/html","content_length":"216099","record_id":"<urn:uuid:34964e99-296a-426f-9284-4b6fe89e353b>","cc-path":"CC-MAIN-2024-46/segments/1730477397531.96/warc/CC-MAIN-20241114225955-20241115015955-00388.warc.gz"} |
Top 30 John von Neumann Quotes (2024 Update) - QuoteFancy
We hope you enjoyed our collection of 30 John von Neumann Quotes.
All the images on this page were created with QuoteFancy Studio.
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Learn more
We hope you enjoyed our collection of 30 John von Neumann Quotes.
All the images on this page were created with QuoteFancy Studio.
Use QuoteFancy Studio to create high-quality images for your desktop backgrounds, blog posts, presentations, social media, videos, posters, and more. | {"url":"https://quotefancy.com/john-von-neumann-quotes","timestamp":"2024-11-10T04:40:52Z","content_type":"text/html","content_length":"120632","record_id":"<urn:uuid:f9922aa4-0263-4a7f-8614-91cbef322d97>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00401.warc.gz"} |
[Solved] Consider an individual whose preferences | SolutionInn
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Consider an individual whose preferences are defined over bundles of non-negative amounts of each of two commodities. Suppose that this individual's preferences can be
Consider an individual whose preferences are defined over bundles of non-negative amounts of each of two commodities. Suppose that this individual's preferences can be represented by a utility
function U: R2 R of the form U (x, x) = ln (x + 1) + 2x2, where x denotes the individual's consumption of commodity one, and x2 denotes the individual's consumption of commodity two. This individual
is a price taker in both commodity mar- kets. The price of commodity one is p > 0, and the price of commodity two is p2 > 0. This individual is endowed with an income of y > 0. 1. Does this
individual have quasi-linear preferences? Justify your answer. (3 marks.) 2. Are this individual's preferences locally non-satiated? Justify your answer. marks.) 3. What is this individual's
budget-constrained utility maximisation problem? (2 marks.) 4. Suppose that the individual will optimally consume strictly positive amounts of both commodities. What is the individual's optimal
consumption bundle in this case? Under what circumstances, if any, will this case occur? (7 marks.) 5. Can it ever be optimal for this individual to choose to consume zero units of com- modity one?
If so, what would be his or her optimal consumption of commodity two? Under what circumstances, if any, will this case occur? (3 marks.) 6. Can it ever be optimal for this individual to choose to
consume zero units of com- modity two? If so, what would be his or her optimal consumption of commodity one? Under what circumstances, if any, will this case occur? (3 marks.) 7. What are the
ordinary demand functions (or possibly correspondences) for com- modity one and commodity two for this individual? (2 marks.) 8. What is this individual's indirect utility function? (2 marks.)
There are 3 Steps involved in it
Step: 1
1 INTRODUCTION Quasilinear preferences are a type of utility function where utility can be separated into a linear function of income and a nonlinear ...
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Relative Risk Charts–Hospitalizations
Dave’s more accurate version of the DOH relative risk charts.
1. The Minnesota Department of Health (MDH) publishes graphics showing the relative risk for testing positive, being hospitalized, or dying with Covid, for the unvaccinated compared to the
vaccinated, and for the unvaccinated compared to the boosted. This information in on the Vaccine Breakthrough web page here: https://www.health.state.mn.us/diseases/coronavirus/stats/vbt.html. In
this post we will examine the second graphic from the top on this web page, Adult Cases, Hospitalizations, and Deaths by Age Group, looking only at hospitalization data, with death data to follow
2. We have recently posted several times on the ramifications of MDH’s continued use of the U.S. Census Bureau 2019 American Community Survey (ACS) 5-Year population estimate for Minnesota,
especially for the 65+ age group. The 65+ age group has been increasing by an average of 25,928 people each year, according to the ACS 1-Year population estimates from 2010 through 2021. In
addition to using an older population estimate, MDH is using the 2019 5-Year estimate, which is an average of the years 2015 through 2019. Because the 65+ age group population is increasing
linearly MDH is effectively using a 2017 65+ population to calculate the 65+ unvaccinated population in 2021 and 2022. Note that MDH calculates the unvaccinated population by starting with the
estimated age group population and then subtracting the vaccinated and the boosted populations. Any discrepancy made in the initial population assumption then directly causes the same size
discrepancy in the size of the unvaccinated population. By using too small a 65+ population estimate, MDH therefore calculates a smaller 65+ unvaccinated population, which then leads to higher
rates of cases, hospital admissions, and deaths per 100k than is found using correct population estimates. In addition to using the ACS 2021 1-Year population as our baseline, we are also
extrapolating the 65+ population for 2022 by adding the annual trend of 25,928. Background on the US Census issue can be found here: https://healthy-skeptic.com/2022/12/02/census-estimate-
background/ . We also discuss the impact on the 65+ age group here: https://healthy-skeptic.com/2022/12/06/the-real-vax-effectiveness-rates/.
3. Please note that we are not accusing MDH of intentionally distorting or misrepresenting the data by continuing to use the ACS 2019 5-Year Population Estimate. We have sent repeated emails to MDH
as we have investigated this issue, but have yet to receive any acknowledgement or response, despite a record of over 2 years of continual correspondence with them. The effect of prior infection
is also not taken into account in this analysis. We have repeatedly requested data matching breakthrough events with prior infections, but MDH has stated such data is unavailable.
4. The Adult Cases, Hospitalizations, and Deaths by Age Group graphic on the Vaccine Breakthrough has a pull down menu, allowing us to select between cases, hospitalizations, or deaths. There is
also a pull down menu allowing us to select one of four time periods: 1). All 2). Pre-Omicron, which is up to and including 12/12/2021, per correspondence with MDH. 3). Omicron Period, since 12
/19/2021, and 4). Last 60 days, which is really the last 9 weeks, per correspondence with MDH. In today’s post we will examine the impact that the use of the 2019 ACS 5-Year population
estimate, vs the 2021 ACS 1-Year population estimate for the 18-49 and 50-64 age groups, and the 2022 extrapolated population estimate for the 65+ age group, has on hospitalization rates for
these four time periods. We will also present a 4 week rolling average charts for the relative risks for each age group.
5. Fig. 1A, MDH Relative Risk Graphic, for Hospitalizations, for All time period: This graphic was downloaded 12/12/2022, which represents the hospitalization rate data as published by MDH on 12/08/
2022 using the 2019 ACS 5-Year population estimate. The left side of the graphic presents the number of hospital admissions per 100k for the three adult age groups, when considering all of the
breakthrough data available, which is from 5/02/2021 through 11/13/2022. All three population statuses are presented for each age group. Most important to note is that for each age group the
unvaccinated category has the highest rate of hospitalizations per 100k over the entire pandemic, while the rates for the vaccinated and boosted are fairly close together for all three age
groups. The right side of the graphic presents the relative risk of the unvaccinated to the vaccinated, and the unvaccinated to the boosted. The risk ratios are calculated by taking the ratio of
the hospitalization rates per 100k shown on the left side of the graphic.
6. Fig. 1B, Replicated Relative Risk Graphic for Hospitalizations, for All time period, using 2021 ACS 1-Year population estimate and 2022 extrapolation for 65+ age group: In this graphic we
replicate MDH’s graphic, but recalculating all the rates and risk ratios using the 2019 ACS 1-Year population estimate for the 18-49 and 50-64 age groups, and the 2022 extrapolated population
estimate for the 65+ age group. Note that on the left side we calculate identical hospitalization rates per 100k for the vaccinated and boosted categories of each age group, compared to the MDH
graphic in Fig. 1A. We calculate slightly lower hospitalization rates per 100k for the unvaccinated 18-49 age group, slightly higher hospitalization rates for the unvaccinated 50-64 age group,
and greatly lower hospitalization rates for the unvaccinated 65+ age group. These changes in unvaccinated hospitalization rates are driven solely by MDH’s use on the 2019 ACS 5-Year population
estimate, vs the 2021 ACS 1-Year and 2022 extrapolated population. When we look at the risk ratios on the right side of the graphic, the biggest changes are for the 65+ age group. We calculate
that the unvaccinated 65+ age group is 2.1 times more likely to be a Covid hospitalization than the vaccinated, compared to MDH’s ratio of 5.1 times. We calculated that the 65+ unvaccinated age
group are 2.6 times more likely to be a Covid hospitalization than the boosted, compared to MDH’s ratio of 6 times. There are minimal changes to the risk ratios for the 18-49 and 50-64 age
7. Fig. 2A, MDH Relative Risk Graphic, for Hospitalizations, for Pre-Omicron time period: This is MDH’s graphic for the hospitalization rates for the Pre-Omicron time period, as downloaded on 12/12/
2022. Compared to the All Time period in Fig. 1A, the unvaccinated have higher hospitalization rates for the Pre-Omicron time period than for the All Time period, and the relative risk ratios are
significantly larger than those in Fig. 1A.
8. Fig. 2B, Replicated Relative Risk Graphic for Hospitalizations, for Pre-Omicron time period, using 2021 ACS 1-Year population estimate and 2022 extrapolation for 65+ age group: The same pattern
of impacts for using the 2021 ACS 1-Year population estimate as seen in Fig. 1B can be seen here. The biggest changes are for the 65+ unvaccinated age group, which has its hospitalization rate
per 100k cut approximately in half. This then causes the risk ratios for the 65+ age group to also be cut approximately in half, compared to Fig. 2A
9. Fig. 3A, MDH Relative Risk Graphic, for Hospitalizations, Omicron time period, since 12/19/2021: This is MDH’s graphic for the hospitalization rates for the Omicron time period, as downloaded on
12/12/2022. The hospitalization rates for the 65+ age group stand out as being much larger than for the 18-49 and 50-64 age groups, for all vaccination statuses.
10. Fig. 3B, Replicated Relative Risk Graphic for Hospitalizations, for Omicron time period, using 2021 ACS 1-Year population estimate and 2022 extrapolation for 65+ age group: The same pattern of
impacts using the 2021 ACS 1-Year population estimate as seen in Fig. 1B can be seen here. However, when looking at this time period, the relative risk for the unvaccinated 65+ age group compared
to the vaccinated is reduced by a factor of nearly three, from 1.9 times more likely to be a Covid hospitalization to 0.7 times more likely to be a Covid case (therefore less likely to be a case
than the vaccinated). Similarly, the relative risk for the unvaccinated 65+ age group compared to the boosted, is also reduced by a factor of nearly three, from 5.2 times more likely to be
hospitalized down to 1.9 times more likely to be hospitalized (identical risk, in other words). Our calculations have a greater reduction in risk for 2022 because we are adding in another assumed
population growth of 25,928. In addition, as more people got vaccinated in 2022, the unvaccinated 65+ people used by MDH in its calculations got even smaller, meaning that the extra population
added by using the 2021 ACS 1-Year population estimate extrapolated to 2022 made a greater proportional difference than in 2021.
11. Fig. 4A, MDH Relative Risk Graphic, for Hospitalizations, Last 60 days’ time period: This is MDH’s graphic for the hospitalization rates for the 60 days (really the last 9 weeks of data per MDH
communication), as downloaded on 12/12/2022. Hospitalization rates for this time period are the lowest of the 4 time periods analyzed by MDH, and the relative risk for the unvaccinated are also
the lowest of the 4 time periods.
12. Fig. 4B, Replicated Relative Risk Graphic for Hospitalizations, Last 60 days’ time period, using 2021 ACS 1-Year population estimate and 2022 extrapolation for 65+ age group: We see again that
the 18-49 and 50-64 age group rates and relative risks are very minimally affected by the change in population baseline. As before the 65+ unvaccinated hospitalization rates are massively
affected, as well of the relative risk ratios. The unvaccinated 65+ age group is only 0.5 times as likely to be hospitalized compared to the vaccinated, compared to 1.4 times as reported by MDH
in Fig. 4A. The unvaccinated 65+ age group is only 1.1 times as likely to be a Covid hospitalization as the boosted, compared to 3.3 times as reported by MDH in Fig. 4A.
13. Fig. 5: Covid Hospitalization Relative Risk Ratios, 18-49 Age Group: This is a new graph we have developed, presenting the relative risk ratios for hospitalization for Covid of the unvaccinated
to the vaccinated, and of the unvaccinated to the boosted, for the 18-49 age group. These ratios are calculated as 4 week rolling averages. We display the results using MDH’s 2019 ACS 5-Year
population baseline as solid curves, and using the 2021 ACS 1-Year population estimate as dotted curves. The impact on the risk ratios of the change to the 2021 ACS 1-Year population estimate is
minimal. There are quite large risk ratios in the early time period for both the unvaccinated to vaccinated, and unvaccinated to boosted, which is striking along with the rapid decline to very
low levels in early 2022.
14. We have highlighted the horizontal line at a Relative Risk Ratio of 1. At a ratio of one the unvaccinated and vaccinated, or vaccinated and boosted, have identical hospitalization rates and
therefore vaccination or boosting provides no measurable benefit. At Relative Risk Ratios less than one the unvaccinated have lower hospitalization rates than the vaccinated or boosted. Since
January 2022 the Relative Risk Ratios are close to 1, so we would conclude that vaccination or boosting provides only minimal benefit for this age group for hospitalizations since January 2022.
15. Fig. 6: Covid Hospitalization Relative Risk Ratios, 50-64 Age Group: This graph presents the relative risk ratios of the unvaccinated to the vaccinated, and of the unvaccinated to the boosted for
the 50-64 age group. These ratios are calculated as 4 week rolling averages. We display the results using MDH’s 2019 ACS 5-Year population baseline as solid curves, and using the 2021 ACS 1-Year
population estimate as dotted curves. The impact on the risk ratios of the change to the 2021 ACS 1-Year population estimate is minimal, but increases the relative risk ratios slightly. Similar
to the 18-49 age group risk ratios in Fig. 5, the risk ratios for both the unvaccinated to vaccinated, and the unvaccinated to boosted, start out very high with a rapid decline to very low levels
in early 2022. Since April 2022 the Relative Risk Ratios are almost exactly equal to 1, meaning hospitalization rates are not affected by vaccination status.
16. Fig. 7: Covid Hospitalization Relative Risk Ratios, 65+ Age Group: This graph presents the Relative Risk Ratios of the unvaccinated to the vaccinated, and of the unvaccinated to the boosted for
the 65+ age group. These ratios are calculated as 4 week rolling averages. We display the results using MDH’s 2019 ACS 5-Year population baseline as solid curves, and using the 2021 ACS 1-Year
population estimate (and 2022 Relative Risk Ratios calculated using the extrapolated 2022 population estimate) as dotted curves. The impact on the risk ratios of the change to the 2021 ACS 1-Year
population estimate is very large, reducing the relative risk ratios by a factor or 2 to 3 from the 2019 ACS 1-Year population as used by MDH. Similar to the other age groups, the relative risk
ratios for both the unvaccinated to vaccinated, and the unvaccinated to boosted, start out very high with a rapid decline to very low levels in early 2022. The underlying case rates for this
group in Fig. 2 here https://healthy-skeptic.com/2022/12/06/the-real-vax-effectiveness-rates/, shows that the vaccinated and the boosted really did have hospitalization rates per 100k greatly
lower than for the unvaccinated in much of 2021, although the unvaccinated fared better than MDH calculates. Since March 2022 the relative risk ratios are hovering around 1 (considering the
2021-2022 population estimate shown by the dotted curves), meaning that vaccination or boosting did not provide any meaningful reduction in hospitalizations per 100k.
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This election isn't likely to end the intense political division largely caused by Dems and… | {"url":"https://healthy-skeptic.com/2022/12/14/relative-risk-charts-hospitalizations/","timestamp":"2024-11-11T05:08:31Z","content_type":"text/html","content_length":"113544","record_id":"<urn:uuid:4e86678a-854a-4317-a8d4-ad8470e0fbba>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00533.warc.gz"} |
a walk through combinatorics 4th edition pdf
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Numerical integration of data imported for Excel
A few comments which I think may help. The first one is that your expression
delta x = .12
should be
deltax = .12
I.e. no space between the delta and the x. (I realize that this is not used in your notebook to the point you showed, but you would probably use it in the next step.)
The second one is that your data1 parameter is a list that contains a list with the data you are interested in. I.e. it has the form
{number, number, number....}
And you want to interpolate, not data1, but First[data1]. (The reason it is a list inside of a list is that when Importing form Excel, each sheet in the original spreadsheet it placed in a separate
list so you can access each sheet as needed for a multiple sheet spreadsheet.)
Thus your integration expression should be
Interpolation[First[data1], InterpolationOrder -> 2][x],
{x, 1, 256}]
Note that I also changed the integration range. Remember in Mathematica that array indices start at 1, not 0. If you take a look at the interpolation expression's output by evaluating the following
on its own
Interpolation[First[data1], InterpolationOrder -> 2]
you will see the assumed range of the interpolation in the formatted output.
So your integration ultimately will be,
deltax (Integrate[
Interpolation[First[data1], InterpolationOrder -> 2][x],
{x, 1, 256}])
I hope this helps... | {"url":"https://community.wolfram.com/groups/-/m/t/386126","timestamp":"2024-11-06T08:59:53Z","content_type":"text/html","content_length":"164116","record_id":"<urn:uuid:86b5fc96-7712-4ea9-a225-d64ed4505295>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00537.warc.gz"} |
Time and Quantum Mechanics accepted at IARD conference
The physics paper I’ve been working on for several years, Time & Quantum Mechanics, has been accepted for presentation at a plenary session of the 2018 meeting of the IARD — The International
Association for Relativistic Dynamics. I’m very much looking forward to this: the paper should be a good fit to the IARD’s program.
In quantum mechanics the time dimension is treated as a parameter, while the three space dimensions are treated as observables. This assumption is both untested and inconsistent with relativity.
From dimensional analysis, we expect quantum effects along the time axis to be of order an attosecond. Such effects are not ruled out by current experiments. But they are large enough to be
detected with current technology, if sufficiently specific predictions can be made.
To supply such we use path integrals. The only change required is to generalize the usual three dimensional paths to four. We treat the single particle case first, then extend to quantum
We predict a large variety of testable effects. The principal effects are additional dispersion in time and full equivalence of the time/energy uncertainty principle to the space/momentum one.
Additional effects include interference, diffraction, resonance in time, and so on.
Further the usual problems with ultraviolet divergences in QED disappear. We can recover them by letting the dispersion in time go to zero. As it does, the uncertainty in energy becomes
infinite — and this in turn makes the loop integrals diverge. It appears it is precisely the assumption that quantum mechanics does not apply along the time dimension that creates the
ultraviolet divergences.
The approach here has no free parameters; it is therefore falsifiable. As it treats time and space with complete symmetry and does not suffer from the ultraviolet divergences, it may provide a
useful starting point for attacks on quantum gravity. | {"url":"https://timeandquantummechanics.com/2018/04/08/time-and-quantum-mechanics-accepted-at-iard-conference/","timestamp":"2024-11-09T17:07:53Z","content_type":"application/xhtml+xml","content_length":"50873","record_id":"<urn:uuid:de55870a-0c2d-4295-9e45-2fbd1022a018>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00026.warc.gz"} |
IGFAE experiments with quantum teleportation of information to open new doors to future cryptography
The paradox of quantum technologies as the greatest threat and, at the same time, the only solution to guarantee the security of communications in the future is a huge and exciting field of research
for the scientific community. Teleportation or quantum teleportation, with which the IGFAE is experimenting, is revealed as a tool with great potential to contribute to the construction of highly
secure communication infrastructures such as the one projected by the Complementary Quantum Communications Plan (PCCC)
On September 8, 1966, the first episode of Star Trek was broadcast, featuring its famous transporter, a technology that allowed its characters to teleport instantly from one place to another. We
don’t know if the screenwriters of the original version of this iconic series had any idea of quantum physics, but what we do know is that their teleportation fantasies were the utmost representation
of science fiction for more than one generation.
Juan Santos
Who knows what will happen in the future though at least according to the current laws of physics almost six decades later, this continues to be science fiction. It is not possible to do it with
people or objects, but the novelty is that it is feasible when it comes to information, thanks to quantum mechanics. “Instead of transferring the object itself, what is transmitted is its
information, using quantum entangled particles. Although it is a real process, it is not instantaneous and still limited by the speed of light,” explains Juan Santos, a researcher at the Galician
Institute of High Energy Physics (IGFAE) of the University of Santiago de Compostela (USC).
It is about quantum teleportation (a common denomination in the academic environment), a complex phenomenon that is being explored by the scientific community with all the limitations implied by
being a recent field of knowledge. The first research dates back to the 1990s, although the relevance of its potential to contribute decisively to the advancement of quantum technologies is already
At the Department of Particle Physics in the Area of Theoretical Physics of the IGFAE, Juan Santos seeks to do his part in this field by opening new paradigms from the knowledge bases. “It is
important to point out that this work is not experimental, but theoretical. It is inspired by conjectures and not yet fully understood relationships between gravity and quantum physics,” he
specifies. “In particular, we rely on holography, a theoretical proposal that suggests that gravity in curved spaces is related to quantum systems in fewer dimensions. These ideas, while speculative,
offer promising clues as to how information could be efficiently propagated by taking advantage of quantum chaos.”
Chaos behind order
In quantum physics, chaos refers to the rate at which a perturbation in a particular part of a many-body system is distributed throughout it. “In quantum teleportation, this chaos is responsible for
dissolving the signal that is to be sent through the sender system. Although it seems that the signal has been lost because it is scattered throughout the system, in reality, it is still there, only
in a coded form,” explains Santos.
To understand this, reference should be made to one of the key properties of quantum mechanics: entanglement, which appears in quantum systems with more than one particle. In essence, it is a
phenomenon whereby two or more quantum particles become interdependent, so that the state of one is directly related to that of the other or others, regardless of the distance separating them.
“The quantum rules allow for stronger correlations between particles than in classical physics. When it comes to teleportation, a highly entangled type of quantum state is used, shared between the
sender and receiver. Once the two parties are ready, the signal to be sent is introduced into the sender system and then dissolves in it due to its chaotic dynamics,” explains the IGFAE researcher.
This means that, temporarily, the information cannot be recovered without access to the entire system. “It can be understood as a kind of encryption that appears naturally due to the chaos.”
This occurs because the signal is inaccessible until the moment it reappears. The key to recovering the information lies in the fact that entanglement will make the receiver system act as a mirror of
the sender “in which things happen in reverse, as if rewinding a film,” illustrates Santos, explaining how the information reappears in the receiver system.
The fact that chaos can be used as an encryption tool anticipates potential applications for protecting the security of large-scale communications. “In particular, we could think of a quantum
internet in which a set of connected devices is the system with chaotic dynamics. Thus, based on such kind of protocols, perhaps information could be sent securely since it would only be accessible
to the sender and receiver,” describes the researcher. Therefore, the results of this project could contribute to solving the great paradox that implies that quantum technologies are, at the same
time, the greatest threat yet the solution to the security of our communications.
Quantum simulation
The experiments that are part of this research are only theoretical because it is still impossible to physically realize the processes in the laboratory, mostly because chaos-mediated quantum
teleportation requires very precise control of the largest possible quantum systems. Until that point is reached, the solution is to simulate these phenomena on a computer. And to simulate quantum
processes, you must use quantum computers.
“In a classical simulation, a conventional computer performs the mathematical calculations needed to solve a physical problem. What happens is that when simulating quantum systems, the computational
resources required increase exponentially, making it impossible to study sufficiently large systems,” explains Santos. The alternative is quantum simulation, which is capable of recreating these
systems, precisely by applying the laws of quantum physics.
In any case, being something so new and complex, the researcher adds that “creativity is needed when designing the experiment, it is not always obvious what kind of processes must be carried out to
draw any significant conclusions.”
IGFAE is already cooperating in this project with the Galicia Supercomputing Center (CESGA) and the University of Grenoble, and hopes to increase the number of collaborators in the future. There is a
lot of work ahead for the research that is being developed within the framework of the Complementary Quantum Communications Plan (PCCC) from its foundations. “The main objective of the PCCC is to
boost the quantum communications industry in Spain to create a highly secure infrastructure. Our research does not address the deployment of this network in the short or medium term but seeks rather
to find the physical phenomena on which it can be sustained in the longer term,” Santos explains.
The quantum future
Although the rules of quantum mechanics have been established for almost a century, the phase of understanding how to use them to process information is very recent. “This makes everything very
volatile and forces us to study the contributions of other scientists on a daily basis, even if they don’t do exactly the same as we do, and to look for ways to incorporate their ideas or techniques
into our work,” says Juan Santos.
This scientific effervescence explains why, in a field as dynamic as quantum technologies, even the most incipient and theoretical fields of knowledge have very promising prospects for evolution.
According to the researcher, “chaos-mediated quantum teleportation is a relatively new field of research and it has only been in the last two or three years that we began to understand its
theoretical foundations. In the next five years, I expect this understanding to advance significantly, which could allow the first experiments to demonstrate its operation in practice.”
Santos hopes that the contributions being made by the IGFAE team to prepare the initial state for teleportation and understanding the role of chaotic dynamics will help put the center on the map. “In
Santiago, we are very new to this field, so we are at the point of positioning ourselves and starting to become known in the community.”
Contact: juansantos.suarez@usc.es ; tf.pena@usc.es
Faílde, D., Santos-Suárez, J., Herrera-Martí, D. A., & Mas, J. (2023). Hamiltonian Forging of a Thermofield Double. arXiv preprint arXiv:2311.10566. https://arxiv.org/abs/2311.10566
Berenguer, M., Dey, A., Mas J, Santos-Suárez, J., Ramallo, A.V. (2024). Floquet SYK wormholes. arXiv:2404.08394. https://arxiv.org/abs/2404.08394 | {"url":"https://quantum.cesga.es/2024/10/17/igfae-experiments-with-quantum-teleportation-of-information-to-open-new-doors-to-future-cryptography/","timestamp":"2024-11-04T12:24:40Z","content_type":"text/html","content_length":"206130","record_id":"<urn:uuid:e3608695-95dd-4945-ae04-8eef6ef61b06>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00705.warc.gz"} |
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Self-adjoint operator - (Operator Theory) - Vocab, Definition, Explanations | Fiveable
Self-adjoint operator
from class:
Operator Theory
A self-adjoint operator is a linear operator on a Hilbert space that is equal to its own adjoint. This property ensures that the operator has real eigenvalues and allows for various important results
in functional analysis and quantum mechanics. Self-adjoint operators have deep connections with spectral theory, stability, and physical observables.
congrats on reading the definition of self-adjoint operator. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Self-adjoint operators can be defined on both finite-dimensional and infinite-dimensional spaces, making them versatile in various applications.
2. The spectral theorem states that every self-adjoint operator can be diagonalized by an orthonormal basis of eigenvectors corresponding to its real eigenvalues.
3. Self-adjoint operators are crucial in quantum mechanics, where physical observables are represented by such operators to ensure measurements yield real values.
4. In the context of unbounded operators, a symmetric operator is not necessarily self-adjoint; it requires specific conditions on the domain for it to be classified as self-adjoint.
5. Positive self-adjoint operators have non-negative eigenvalues, and this leads to important implications regarding their square roots and functional calculus.
Review Questions
• How does the property of being self-adjoint relate to the concepts of eigenvalues and eigenvectors?
□ Self-adjoint operators guarantee that all their eigenvalues are real numbers, which is crucial when studying the spectral properties of these operators. This characteristic allows for the
construction of orthonormal bases made up of the corresponding eigenvectors. As a result, understanding how self-adjoint operators function helps predict the behavior of systems modeled in
quantum mechanics and other fields.
• Discuss the implications of the spectral theorem specifically for compact self-adjoint operators and how it aids in analyzing their spectra.
□ The spectral theorem states that compact self-adjoint operators can be expressed in terms of their eigenvalues and eigenvectors, allowing one to represent these operators in a diagonal form.
This representation simplifies many problems in analysis and enables a thorough understanding of their spectral properties. It also reveals that the spectrum consists of countably many
eigenvalues, which can be essential for solving differential equations and other applications.
• Evaluate the differences between symmetric and self-adjoint unbounded operators, focusing on their domains and functional implications.
□ Symmetric operators may not have a complete set of eigenvalues or may not map densely into their ranges, meaning they can have issues with defining adjoints. In contrast, self-adjoint
unbounded operators have well-defined adjoints and proper domains, making them more stable for functional analysis. The distinction is significant because self-adjointness ensures that you
can apply various results from spectral theory effectively, allowing for better analysis in applications like quantum mechanics where unbounded operators frequently arise.
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The Landweber exact-functor theorem - good fibrations
The Landweber exact-functor theorem
This post assumes familiarity with formal group laws, the definition of exact sequences, the motivation of the Landweber-Ravenel-Stong construction, that the exactness axioms are one of the
generalized Eilenberg-Steenrod axioms, and the fact that formal group laws over \(R\) are represented by maps from the Lazard ring to \(R\).
Recall the the Landweber-Ravenel-Stong Construction: \(MU^*(X) \otimes_{L} R \simeq E^*(X)\), where \(MU^* \simeq L\) and \(R \simeq E^*(pt)\).
We know that in general, tensoring with abelian groups does not preserve exact sequences (e.g., applying \(-\otimes_{\mathbb{Z}} \mathbb{Z}/2\) to \(0 \to \mathbb{Z} \xrightarrow{\times p} \mathbb{Z}
\to \mathbb{Z}/p \to 0\)).
So, when does the functor \(-\otimes_L R: MU^*(X) \to E^*(X)\) preserve exact sequences?
An object M in an abelian tensor category \(C\) is ‘flat’ if for all \(X \in Obj(C) \), the functor \(X \to X \otimes M\) preserves exact sequences.
Because arbitrary \(MU_*\)-modules do not occur as the \(MU\)-homology of spaces, the requirement of flatness over all of \(MU_*\) can be relaxed.
It is Landweber-exact if it preserves exactness when applied to things in the range of the homology theory, though it doesn’t have to preserve exactness on things that aren’t in the range of the
homology theory. (summarization by Alex Mennen)
Sidenote: All we require is that the map \(\text{Spec }R \to M_{FG}\) is flat. Why to \(M_{FG}\) and not to \(M_{FGL}\)? Every (complex orientable) cohomology theory corresponds to a formal group;
picking a complex orientation corresponds to a choice of coordinate of our formal group law.
I say \(MU_*\) instead of \(MU^*\) for technical reasons, namely, colimits don’t behave nicely in cohomology. But I don’t understand the implications of that well enough to talk about it coherently.
You may ask why we’re only checking for exactness! Since we’re dealing only with CW-complexes: the excision axiom automatically holds, the homotopy equivalence axiom automatically holds because we’re
applying a functor to \(MU\) (functors preserve isomorphisms), and additivity always holds. All we really need to check is exactness.
Flatness and torsion are intimately related. It’s a theorem of Lazard that an object is flat iff it’s a filtered colimit of free modules.
Let’s say we have a map from the Lazard ring \(L \xrightarrow{F} R\) representing our formal group law \(F\). What is this exactness condition, precisely?
Let’s back up a bit and look at what it might mean for a formal group law to be “flat.”
We wish to “add” a point to itself via the formal group law \(p\) times and look at its general form (this is called the \(p\)-series of our formal group law). This allows us to detect stuff like
points of order \(2\) in elliptic curves, that is, the points that when added to themselves give us the origin. Note that \(p\) is a prime in \(\mathbb{Z}\), not necessarily a prime in \(R\).
In general, we can talk about adding a point \(x\) to itself \(n\) times using the following recursive definition:
It doesn’t seem like it at first glance, p-series, or “multiplication by p” map, is EXTREMELY IMPORTANT — it allows us to find periodic phenomena. Let’s look at the map
The kernel of \(\lambda_n\) will be the roots of unity in \(C\).
Similarly, let’s look at the group \(G\).
The kernel of this map will be the points in \(G\) with an order that divides \(p\). If \(p\) is a prime, it’s just the points of order \(p\).
Examples of \(p\)-series: Additive formal group law \(F(x, y) = x + y\)
Multiplicative formal group law \(F(x, y) = x + y + cxy\), where \(c = 1\) we’ll be working with \(c = -1\) at some point later in the post, sorry for the inconsistency!
The \(p\)-series of \(F\) will always be of the form:
\\([p](x) = px + … + v_1x^{p^1} + … + v_nx^{p^n} + … \\)
where \(v_k\) is simply the name we give the coefficient of the expression \(x^{p^k}\).
We’re interested in the tuple of coefficients relevant to the powers of \(p\), that is \((p, v_1, …, v_k, …)\). Let’s mod out the \(p\)-series by \(p\), then by \(v_1\), etc. until we get to \(0\).
We are trying to check that \(v_n\) acts injectively in \(R/(v_0, …, v_{n-1})\) for all \(n\).
Note that \(p\) is a prime in \(\mathbb{Z}\) and \(v_1, …, v_n \in\) the image of \(MU^*\) \in \(R\). The condition of “regular” wards away zero divisors because they are nasty.
For example, our tuple for the multiplicative formal group law is \((p, 1)\), since \(v_1 = 1\), and the rest of the coefficients are 0, so we have:
Tada! It’s Landweber exact so you can bet your muffins that \(MU^*(X) \otimes_L \mathbb{Z}\) is a cohomology theory, in fact, it’s iso to \(K^*(X)\).
That’s a lot to take in, I know, so let’s back up a bit and examine the map \\(Vect^1(X) \xrightarrow{c_1} K^2(X; \mathbb{Z})\\) where the group operators are the tensor product of line bundles and
the tensor product of virtual line bundles.
Note that the dimensions multiply when you do the tensor product, so \(L_1 \otimes L_2\) is still a line bundle.
Let’s say that \(c_1(L) = 1-L\), then we’d expect \(c_1(L_1 \otimes L_2) = 1 – L_1 \otimes L_2\). So, how do we express \(c_1(L_1 \otimes L_2)\) in terms of a formal group law \(F(x,y)\) where \(x =
c_1(L_1)\), and \(y = c_1(L_2))\)?
In other words, using the slightly clearer notation \(F(x,y) \equiv x +_F y\)…for what \(F\) does \(x +_F y = 1-xy\)?
Lost in the algebra? Worry not, I have something that may appeal to your geometric taste: the striking perspective of Morava, in Forms of K-theory.
We can use the properties of the parameterized space below (the set of W-valued genera with the Zariski topology, a point of this space is a formal group law of height 1 over W) to prove things about
the stalks above (K-theories associated to those genera). It’s a pretty beautiful proof method, once we establish the following two claims…
1. moduli space below has a transitive group action, f \\(F(X, Y) \mapsto f^{-1}F(f(X), F(Y))\\)
2. we know at least one of the stalks is exact (the topological K-theory we know and love).
…we know that the rest of the stalks are exact.
Now that I’ve gotten you riled up, let’s back away from \(K\)-theory.
I wrote this post because I was really excited about the following: What if we wish to look at the cohomology theory associated to a supersingular elliptic curve in characteristic \(p\)?
We can’t have torsion in the Landweber-Stong-Ravenel construction, however we can consider the elliptic curve over p-adic completion of \(\mathbb{Z}\), and adjoin \(v_1\), such that \((\mathbb{Z}_p
[[v_1]]/p)/v_1 = \mathbb{Z}/p\).
WOO! Actually, this isn’t too surprising. Recall that the p-adics are the limit of \(\mathbb{Z}/p^n\), thus by definition it comes with maps to all the guys in the limit.
It might be tempting to think that all even periodic cohomology theories (that is, all cohomology theories \(E\) satisfying \(E^{n}(*) \otimes_{E^*(*)} E^2(*) \simeq E^{n+2}(*)\), and \(E^n(*) = 0\)
if n is odd) are associated to the multiplicative formal group law. This is not the case, for example, elliptic cohomology is even periodic (note that even periodic \(\subset\) complex orientable).
Elliptic cohomology is not naturally even periodic. For a given cohomology theory \(E\), even periodicity comes when we must force an ungraded ring into the world of graded rings. One way of doing
this is the following:
\(E^n(pt) = \begin{cases} R & \mbox{if } n \mbox{ is even} \\
0 & \mbox{if } n \mbox{ is odd}. \end{cases} \pmod{2}\)
When we look at \(E^*(pt)\) as a whole, instead of just \(E^n(pt)\) individually, we see that \(E^*(pt) \simeq R[[v_1, v_1^{-1}]]\) is an iso of graded rings.)
I’d also like to mention something slightly more obvious (than the trick with getting supersingular curves in char p through the Landweber-exactness condition) but still awesome:
Tensoring with \(\mathbb{Q}\) gets rid of torsion. Recall that every formal group law over \(\mathbb{Q}\) is isomorphic to the additive formal group law. The difference between cohomology theories
arises due to torsion! \(E \otimes \mathbb{Q}\) simply gives us singular cohomology with some coefficient ring!
This is true more generally: rational spectra (i.e., all the homotopy groups are \(\mathbb{Q}\) – vector spaces) are always determined by their homotopy groups. More precisely: there is a functor
from rational spectra to graded \(\mathbb{Q}\) – vector spaces given by taking homotopy groups, and it’s an equivalence. I’m not yet sure why this is.
I’ve read that an elliptic curve formal group law must be of either height 0, 1, or 2 (1 if singular, 2 if super singular). Why can’t it be of higher height?
Thank you to Akhil Mathew for kindly answering my questions on Landweber-exactness, Achim Krause for talking with me about rational spectra, and the lovely people of Math Overflow for answering my
question on Landweber-exactness.
2 thoughts on “The Landweber exact-functor theorem”
1. power the message house a bit, however instead of that, this is magnificent blog.
1. Pardon, what do you mean by the “message house”? | {"url":"https://catherine.cloud/landweber-exactness/","timestamp":"2024-11-13T21:21:39Z","content_type":"text/html","content_length":"56522","record_id":"<urn:uuid:afad1858-0b4b-4d0e-88fd-912e065ed6bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00387.warc.gz"} |
Live Online classes for kids from 1-10 | Upfunda Academy
What is Multiplication?
Multiplication is a fun and important math skill that helps you find out how many of something there are in total. It helps you count faster! Let's use some examples to help explain what
multiplication equation is and how it works.
A multiplication equation sentence has three parts: the multiplicand, multiplier, and product. The multiplicand is the first number, the multiplier is the second number, and the product is the
Imagine you have 5 apples and you want to know how many apples you would have if you had two sets of those 5 apples. That's where multiplication equation comes in! To multiply, you use a special
symbol called the multiplication equation sign, which looks like this: “x”. So, you would write 5 x 2 to find out how many apples you have in total. And the answer is 10! That means you have 10
apples in total.
There are lots of multiplication shortcut available where another way to think about multiplication equation is that it's just a faster way of adding a number over and over again. For example, if you
want to find out how many apples you would have if you had 3 sets of those 5 apples, you could add 5 apples + 5 apples + 5 apples = 15 apples.
Here are a few fun facts about multiplication equation :
1. multiplication equation is often represented using the symbol "x". For example, 5 x 3 = 15.
2. Multiplication is the repeated addition of the same number.
3. The order of numbers in multiplication equation does not matter. For example, 5 x 3 = 3 x 5. This is called the commutative property of multiplication.
4. Multiplying by zero gives you zero. For example, 5 x 0 = 0.
5. Multiplying by one keeps the number the same. For example, 5 x 1 = 5.
6. Multiplication equation is used in many real-life situations, such as finding the number of minutes in an hourNo of minutes in an hour is 1x60 = 60 minutes. Number of seconds iqqn a day is 24 x
60 x 60 = 86400.
7. In ancient times, people used to use their fingers to do multiplication and keep track of numbers. They would count on their fingers to find the answer.
In conclusion, multiplication equation is a fun and important skill that will help you in many areas of life. Keep practicing and have fun!
Division definition in math
Division is a fun and important math skill that helps you find out how many parts something can be divided into. Let's use some examples to help explain what division is and how it works.
There are importantly three main parts to a division problem: the dividend, the divisor, and the quotient. Imagine you have 10 candies and you want to share them with your 5 friends. How many candies
will each of your friends get? To find out, you can use division. To divide, you use a special symbol called the division sign, which looks like this “÷”. So, you would write 10 ÷ 5 to find out how
many candies each of your friends will get. And the answer is 2! That means each of your friends will get 2 candies.
Another way to think about division is that it's just a way of finding out how many times one number can fit into another number. For example, if you have 15 apples and you want to know how many sets
of 5 apples you can make, you can divide 15 ÷ 5 = 3. That means you can make 3 sets of 5 apples.
So, division definition in math helps us find out how many times one number can fit into another number. It's like sharing something into equal parts!
Multiplication and Division facts!
1. Division was one of the first arithmetic operations used by humans, dating back to ancient civilizations like the Babylonians and Egyptians.
2. In Division, you cannot divide a number by zero, as it would result in a mathematical impossibility. This is due to the fact that any number divided by zero would be equal to infinity, which is
not a real number.
3. In ancient times, people used sticks or stones to do division and keep track of numbers. They would use the sticks or stones to represent each number and find the answer.
4. In the Middle Ages, division was often performed using a method called long division, which involved writing out the problem and dividing it step by step.
5. Division can be used to solve problems related to ratios, rates, and proportions. For example, if you want to find out how many miles per gallon your car gets, you can divide the number of miles
driven by the number of gallons of gas used.
6. Division is related to the inverse operation of multiplication. For example, if you know that 5 x 4 = 20, you can use division to find out that 20 ÷ 5 = 4.
7. Division is an important concept in cryptography, as it is used to find the prime factors of a number, which is an essential step in many encryption algorithms.
8. Division can also be used in data analysis, as it can help you find the average of a set of numbers. For example, if you want to find the average height of a group of people, you can add up all
the heights and then divide by the number of people.
9. Division is also used for finding out how many times one number is bigger or smaller than another number.
Another way to think about division is that it's just a way of finding out how many times one number can fit into another number. In conclusion, the division is a fun and important skill that will
help you in many areas of life. Keep practicing and have fun!
💡 Multiplication Division sums!
1. My rabbit eats only cabbage and carrots. Last week he ate either 10 carrots or 2 heads of cabbage each day. If he ate a total of 6 heads of cabbage last week. how many carrots did he eat?
2. On the first day, a tourist walked 33 kilometers. On the second day, he walked three times as far as he did the first day, and then 5 kilometers more. How many kilometers did he walk on the
second day?
3. My dogs have 18 more legs than noses. How many dogs do I have?
4. Peter has ten balls, numbered from 0 to 9. He distributed these balls among three friends: John got three balls, George four and Ann three. Then he asked each of his friends to multiply the
numbers on the balls they got and the results were: 0 for john, 72 for George and 90 for Ann. What is the sum of the numbers on the balls that John received?
5. In a soccer game the winner gains 3 points, while the loser gains 0 points. If the game is a draw, then the two teams gain 1 point each. A team has played 38 games gaining 80 points. Find the
greatest possible number of games that the team lost.
Answer Key
1. C) 40
2. C) 104
3. C) 15
4. C)10
5. C) 6 | {"url":"https://upfunda.academy/blog/eab8b6a8-e228-4c87-b539-4a5c1ee9189c","timestamp":"2024-11-08T23:18:57Z","content_type":"text/html","content_length":"37070","record_id":"<urn:uuid:608f70d2-356e-46fc-a9c7-7adb29118741>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00484.warc.gz"} |
How to Find the Power of a Statistical Test
When a researcher designs a study to test a hypothesis, he/she should compute the power of the test (i.e., the likelihood of avoiding a Type II error).
How to Compute the Power of a Hypothesis Test
To compute the power of a hypothesis test, use the following three-step procedure.
• Define the region of acceptance. Previously, we showed how to compute the region of acceptance for a hypothesis test.
• Specify the critical parameter value. The critical parameter value is an alternative to the value specified in the null hypothesis. The difference between the critical parameter value and the
value from the null hypothesis is called the effect size. That is, the effect size is equal to the critical parameter value minus the value from the null hypothesis.
• Compute power. Assume that the true population parameter is equal to the critical parameter value, rather than the value specified in the null hypothesis. Based on that assumption, compute the
probability that the sample estimate of the population parameter will fall outside the region of acceptance. That probability is the power of the test.
The following examples illustrate how this works. The first example involves a mean score; and the second example, a proportion.
Sample Size Calculator
The steps required to compute the power of a hypothesis test can be time-consuming and complex. Stat Trek's Sample Size Calculator does this work for you - quickly and accurately. The calculator is
easy to use, and it is free. You can find the Sample Size Calculator in Stat Trek's main menu under the Stat Tools tab. Or you can tap the button below.
Sample Size Calculator
Example 1: Power of the Hypothesis Test of a Mean Score
Two inventors have developed a new, energy-efficient lawn mower engine. One inventor says that the engine will run continuously for 5 hours (300 minutes) on a single ounce of regular gasoline.
Suppose a random sample of 50 engines is tested. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. The inventor tests the null hypothesis that the mean run time
is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes, using a 0.05 level of significance.
The other inventor says that the new engine will run continuously for only 290 minutes on a ounce of gasoline. Find the power of the test to reject the null hypothesis, if the second inventor is
Solution: The steps required to compute power are presented below.
• Define the region of acceptance. In a previous lesson, we showed that the region of acceptance for this problem consists of the values between 294.46 and 305.54 (see previous lesson).
• Specify the critical parameter value. The null hypothesis tests the hypothesis that the run time of the engine is 300 minutes. We are interested in determining the probability that the hypothesis
test will reject the null hypothesis, if the true run time is actually 290 minutes. Therefore, the critical parameter value is 290. (Another way to express the critical parameter value is through
effect size. The effect size is equal to the critical parameter value minus the hypothesized value. Thus, effect size is equal to 290 - 300 or -10.)
• Compute power. The power of the test is the probability of rejecting the null hypothesis, assuming that the true population mean is equal to the critical parameter value. Since the region of
acceptance is 294.46 to 305.54, the null hypothesis will be rejected when the sampled run time is less than 294.46 or greater than 305.54.
Therefore, we need to compute the probability that the sampled run time will be less than 294.46 or greater than 305.54. To do this, we make the following assumptions:
□ The sampling distribution of the mean is normally distributed. (Because the sample size is relatively large, this assumption can be justified by the central limit theorem.)
□ The mean of the sampling distribution is the critical parameter value, 290.
□ The standard error of the sampling distribution is 2.83. The standard error of the sampling distribution was computed in a previous lesson (see previous lesson).
Given these assumptions, we first assess the probability that the sample run time will be less than 294.46. This is easy to do, using the Normal Calculator. We enter the following values into the
calculator: normal random variable = 294.46; mean = 290; and standard deviation = 2.83. Given these inputs, we find that the cumulative probability is 0.942. This means the probability that the
sample mean will be less than 294.46 is 0.942.
Next, we assess the probability that the sample mean is greater than 305.54. Again, we use the Normal Calculator. We enter the following values into the calculator: normal random variable =
305.54; mean = 290; and standard deviation = 2.83. Given these inputs, we find that the probability that the sample mean is less than 305.54 (i.e., the cumulative probability) is 1.0. Thus, the
probability that the sample mean is greater than 305.54 is 1 - 1.0 or 0.0.
The power of the test is the sum of these probabilities: 0.942 + 0.0 = 0.942. This means that if the true average run time of the new engine were 290 minutes, we would correctly reject the
hypothesis that the run time was 300 minutes 94.2 percent of the time. Hence, the probability of a Type II error would be very small. Specifically, it would be 1 minus 0.942 or 0.058.
Example 2: Power of the Hypothesis Test of a Proportion
A major corporation offers a large bonus to all of its employees if at least 80 percent of the corporation's 1,000,000 customers are very satisfied. The company conducts a survey of 100 randomly
sampled customers to determine whether or not to pay the bonus. The null hypothesis states that the proportion of very satisfied customers is at least 0.80. If the null hypothesis cannot be rejected,
given a significance level of 0.05, the company pays the bonus.
Suppose the true proportion of satisfied customers is 0.75. Find the power of the test to reject the null hypothesis.
Solution: The steps required to compute power are presented below.
• Define the region of acceptance. In a previous lesson, we showed that the region of acceptance for this problem consists of the values between 0.734 and 1.00. (see previous lesson).
• Specify the critical parameter value. The null hypothesis tests the hypothesis that the proportion of very satisfied customers is 0.80. We are interested in determining the probability that the
hypothesis test will reject the null hypothesis, if the true satisfaction level is 0.75. Therefore, the critical parameter value is 0.75. (Another way to express the critical parameter value is
through effect size. The effect size is equal to the critical parameter value minus the hypothesized value. Thus, effect size is equal to [0.75 - 0.80] or - 0.05.)
• Compute power. The power of the test is the probability of rejecting the null hypothesis, assuming that the true population proportion is equal to the critical parameter value. Since the region
of acceptance is 0.734 to 1.00, the null hypothesis will be rejected when the sample proportion is less than 0.734.
Therefore, we need to compute the probability that the sample proportion will be less than 0.734. To do this, we take the following steps:
□ Assume that the sampling distribution of the mean is normally distributed. (Because the sample size is relatively large, this assumption can be justified by the central limit theorem.)
□ Assume that the mean of the sampling distribution is the critical parameter value, 0.75. (This assumption is justified because, for the purpose of calculating power, we assume that the true
population proportion is equal to the critical parameter value. And the mean of all possible sample proportions is equal to the population proportion. Hence, the mean of the sampling
distribution is equal to the critical parameter value.)
□ Compute the standard error of the sampling distribution. In a previous lesson, we showed that the standard error of the sample estimate of a proportion σ[P] is:
σ[P] = sqrt[ P * ( 1 - P ) / n ]
where P is the true population proportion and n is the sample size. Therefore,
σ[P] = sqrt[ ( 0.75 * 0.25 ) / 100 ] = 0.0433
Following these steps, we can assess the probability that the sample proportion will be less than 0.734. This is easy to do, using the Normal Calculator. We enter the following values into the
calculator: normal random variable = 0.734; mean = 0.75; and standard deviation = 0.0433. Given these inputs, we find that the cumulative probability is 0.356. This means that if the true
population proportion is 0.75, then the probability that the sample proportion will be less than 0.734 is 0.356. Thus, the power of the test is 0.356, which means that the probability of making a
Type II error is 1 - 0.356, which equals 0.644. | {"url":"https://stattrek.com/hypothesis-test/statistical-power","timestamp":"2024-11-06T12:05:39Z","content_type":"text/html","content_length":"54504","record_id":"<urn:uuid:d4054e2c-d50d-4ce8-a3c5-0ff9f6ef117b>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00411.warc.gz"} |
Real Majorana wavefunction / field: What is the big deal?
It is known that there is a set of gamma matrices that can be purely imaginary (called Majorana basis), thus one can solve the 1st quantized Majorana wave function in terms of real wave function.
However, I am confused by the implications for this real Majorana wave function.
1. Isn't that the wave function should still be complex under time evolution? If so, what is the big deal to call this real Majorana wave function?
2. What is the meaning for this set of real Majorana wave function, when we go from 1st quantized to 2nd quantization language?
p.s. One should clarify 1st quantized and 2nd quantization languages. The Ref below seems to mix two up.
Below from Wilczek on Majorana returns:
This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user annie marie heart
Neat question. The short answer is no, the real spinor does not turn complex under time evolution. That's at the very heart of the "big deal".
To see this explicitly, go to the rest frame, as you may always Lorentz-transform yourself back. So, the Dirac equation reduces to just $$(i\tilde \gamma ^0 \partial_t - m)\psi =0,$$ hence $$(\
partial_t +im\tilde \gamma^0)\psi=0 .$$ Note in this representation $$i\tilde \gamma ^0$$ is real and antisymmetric, and thus antihermitean, as it should be!
Since it does not mix up the real and imaginary parts of the Dirac spinor $$\psi$$, we may consistently take the imaginary part to vanish, so the spinor is real: A Majorana spinor in this, Majorana,
representation. (In the somewhat off-mainstream basis you display, you may take $$C=-i\tilde \gamma^0=-C^T=-C^\dagger=-C^{-1}$$.)
The evident solution, then, is $$\psi (t) = \exp (-itm\tilde \gamma ^0 ) ~~\psi(0)=\exp (-itm ~\tilde \sigma_2\otimes\sigma_1 ) ~~\psi(0)~~,$$ where, as seen, the exponential is real, so the
time-evolved spinor is real, as well, forever and ever. Thus, $$\psi (t) = (\cos (tm)-i\tilde \gamma^0 \sin (tm) ) ~\psi(0) .$$
Now, thruth be told, this is an "existence proof" of the consistency of the split. Few of my friends actually use the Majorana representation in their daily lives. It just reminds you that a Dirac
spinor is resolvable to a Majorana spinor plus i times another Majorana spinor.
The properties of the Dirac equation are the same in both first and second quantization, so all the moves and points made also hold for field theory as well, without departure from the standard
textbook transition of the Dirac equation.
• Edit in response to @annie marie hart 's question.
In effect, the complex nature of Schroedinger's equation is replicated with real matrix quantities, given $$\Gamma\equiv i\tilde \gamma^0 ~\leadsto ~\Gamma^2=-I$$. As a result, all complex unitary
propagation features of Schroedinger's wavefunctions are paralleled here by the real unitary matrices acting on spinor wave functions, normalized in the technically analogous sense.
This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user Cosmas Zachos
thanks very much for the nice answer. vote up
This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user annie marie heart
Another way of saying this could be to think of the free Hamiltonian as a matrix $\chi^T H \chi$, where $\chi$ are Majorana fields. If this is to be nonzero then by fermionic statistics it must be
antisymmetric, and since $H$ is Hermitian we can conclude that the Hamiltonian for a collection of Majoranas must be purely imaginary. This means that the time evolution $e^{-iHt}$ acts as a real
This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user user3521569
How do we appreciate better that the $\exp(-iEt)$ type of time evolution does not get involved? In the naive picture we have $H |\psi_E> = i \hbar \partial_t |\psi_E>$ with a textbook solution $ |\
psi_E>= \exp(-iEt) |\psi_0>$ and $H |\psi_0>=E |\psi_E>$. But $\exp(-iEt)$ complexes the solution...
This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user annie marie heart
Also are all the real solutions, (such as $\psi (t) = (\cos (tm)-i\tilde \gamma^0 \sin (tm) ) ~\psi(0) $) normalizable as in the Hilbert space requires $\int |\psi (t)|^2 d^3x=1$?
This post imported from StackExchange Physics at 2020-11-06 18:50 (UTC), posted by SE-user annie marie heart | {"url":"https://www.physicsoverflow.org/43521/real-majorana-wavefunction-field-what-is-the-big-deal","timestamp":"2024-11-04T12:03:40Z","content_type":"text/html","content_length":"134562","record_id":"<urn:uuid:1b31d8a3-4a96-4d8b-a152-e9e808f472bd>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00382.warc.gz"} |
FREE Tracing Circles Worksheets for Preschool [PDFs] Brighterly
Tracing Circles Worksheets
When kids learn shapes, they come across 2D forms like circles. The learning materials teach them that a circle is a curved line with a flat, smooth, spherical shape. Some resources go the extra mile
and make lessons more interesting. Keep reading to see how circle worksheet for preschool kids can improve the quality of your child’s learning.
How to Use Circle Worksheets for Preschool
Worksheets have become a valuable resource in helping students understand math better. The tutors from Brighterly introduce students to the tracing circles worksheet so they can properly grasp how to
draw a circle. A tracing circle worksheet can introduce students to a circle’s integral components. This circle tracing worksheet allows kids to notice circles around them in their alphabets,
buttons, cookie shapes, door knobs, etc.
In the first stage of teaching pupils about circles, students must learn the definitions of a circle’s components. The parts of a circle are radius, circumference, chord, diameter, sector, segment,
arc, and tangent. Students just beginning to study circles can benefit significantly from the preceding explanations of key terms and ideas in the circle worksheets for preschool.
Learning with Circle Worksheets for Preschool Is Fun!
Circle shape circle worksheets for preschool offer fun and exciting activities to students. It’s no wonder the tutors at Brighterly use the circle worksheets for preschool to encourage learning. The
worksheets’ creators understand that kids need something fun and colorful to catch their attention at all times. So, they provide excellent and interactive materials to capture kids’ attention for
long periods. Students can have a fun time learning without even realizing it.
Worksheets topics
Worksheet #1
Worksheet #2
Worksheet #3
Worksheet #4
Order of Operations with Exponents
More Geometry Worksheets | {"url":"https://brighterly.com/worksheets/tracing-circles-worksheets/","timestamp":"2024-11-06T07:48:34Z","content_type":"text/html","content_length":"95494","record_id":"<urn:uuid:ba4b98e1-e8e8-4540-a294-e786fbb946eb>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00823.warc.gz"} |
Dedekind's Pigeons
10 pigeons in 9 pigeonholes.
Photo credit: Wikipedia
The Pigeonhole Principle states that if you have more pigeons than pigeonholes, then at least two pigeons will end up in the same hole (see photo, 2 pigeons in top left corner).
More generally, if a finite number of objects are put into a smaller number of categories, then there would have to be least 2 of those objects in the same category.
In a formal proof using the Pigeonhole Principle, we would have to be explicit about what you meant by "a finite number of objects" and by "a smaller number of categories." In most textbooks,
these concepts are based on the numerical size of a set (the number of objects in the set). In a formal proof based on this notion, all of the machinery of natural number arithmetic would have to
be constructed and the results included in the proof as well.
Alternatively, we can can use Richard Dedekind's elegant definition of the infinite:
A set X is infinite if and only if there exists a proper subset Y of X and a bijection f mapping X to Y.
The negation of infinite defines finiteness:
A set X is finite if and only if there exists no proper subset Y of X and bijection f mapping X to Y.
We have the following formalisms in DC Proof notation:
The definition of infinite:
ALL(a):[Set(a) => [Infinite(a)
<=> EXIST(b):[Set(b)
& ALL(c):[c e b => c e a]
& EXIST(c):[c e a & ~c e b]
& Bijection(a,b)]]]
The definition of bijection:
ALL(a):ALL(b):[Set(a) & Set(b) => [Bijection(a,b)
<=> EXIST(f):[ALL(c):[c e a => f(c) e b]
& [ALL(c):ALL(d):[c e a & d e a => [f(c)=f(d) => c=d]]
& ALL(c):[c e b => EXIST(d):[d e a & f(d)=c]]]]]]
The definition of finite:
ALL(a):[Set(a) => [Finite(a) <=> ~Infinite(a)]]
The Pigeonhole Principle:
& Set(b)
& Finite(a)
& ALL(c):[c e a => f(c) e b]
& ALL(c):[c e b => g(c) e a]
& ALL(c):ALL(d):[c e b & d e b => [g(c)=g(d) => c=d]]
& EXIST(c):[c e a & ALL(d):[d e b => ~g(d)=c]]
=> EXIST(c):EXIST(d):[c e a & d e a & [f(c)=f(d) & ~c=d]]]
Set(x) means x is a set
Infinite(x) means x is infinite
Bijection(x,y) means there exists a bijective function mapping set x to set y. Note: Bijection is an equivalence relation. It is reflexive, symmetry and transitive.
Finite(x)means x is finite
ALL(c):[c e a => f(c) e b] means f is a function mapping set a to set b. In the language of pigeons and pigeonholes, a is the set of pigeons, and b is the set of pigeonholes. The function f
assigns each pigeon to a unique hole.
ALL(c):ALL(d):[c e b & d e b => [g(c)=g(d) => c=d]] means g is a injective (one-to-one) function.
EXIST(c):[c e a & ALL(d):[d e b => ~g(d)=c]] means g is not a surjective (onto) function and that set a is larger than set b. In the language of pigeons and pigeonholes, g effectively names each
hole after a pigeon. In a sense, it converts the set of holes into a set of pigeons. This is necessary if we want to apply the definition of Dedekind-finiteness. Since there are more pigeons than
holes, at least one pigeon will not have a hole named after it. Note that any pigeon may or may not be assigned to the hole named after it.
EXIST(c):EXIST(d):[c e a & d e a & [f(c)=f(d) & ~c=d]]] means there exists at least two distinct elements of set a which have the same image under function f. In the language of pigeons and
pigeonholes, there exists at least two pigeons that will be assigned to same hole.
For a formal development of the pigeonhole principle using the definition of Dedekind-infinity, see pigeonhole.htm | {"url":"http://www.dcproof.com/DedekindsPigeons.htm","timestamp":"2024-11-05T16:40:11Z","content_type":"text/html","content_length":"11312","record_id":"<urn:uuid:99f4b674-676f-4c89-88b8-f19f8d240c12>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00350.warc.gz"} |
How does grading work with co-teachers? | Writable Help Center
From the teacher's perspective
In a co-taught class, both the primary teacher and the co-teacher will be able to grade student work.
❗️ Note: you will only be able to see the other teacher's review only after you have submitted scores for the student.
In the image above, Viv Tran graded the student first and gave them a score of 9/15. The second teacher then gave the same student a score of 12/15. The student's overall score will be the average of
the two scores, which is would 10.5/15.
There are two places to view student scores in the Student Dashboard.
• Under the Analysis tab: The score you see under the Analysis tab is the score that you gave the student.
• Under the Feedback tab: The score you see under the Feedback tab is the average score given by co-teachers. If only one teacher has graded the assignment, the displayed score reflects the grade
given by that particular teacher.
💡Tip: If you're looking to split grading between co-teachers, check the feedback tab to see which students have been graded already.
From the student's perspective
Students will see an average of both teachers' grades as well as comments from both teachers when reviewing their scores. In the image below, the final score of the assignment is a 10.5, which is the
average of the two scores given by the two teachers (12 and 9).
Related Articles | {"url":"https://intercom.help/writable/en/articles/9262448-how-does-grading-work-with-co-teachers","timestamp":"2024-11-06T02:32:27Z","content_type":"text/html","content_length":"67049","record_id":"<urn:uuid:c0074d56-4fbe-445c-b30b-320614fba876>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00771.warc.gz"} |
hftrialdatagen: Gene intensities simulator and DDHFm tester in DDHFm: Variance Stabilization by Data-Driven Haar-Fisz (for Microarrays)
hftrialdatagen(nreps = 4, nps = 128, plot.it = FALSE, uvp = 0.8)
nreps Number of replicates
nps Number of genes
plot.it Takes TRUE to activate the command of the respective plot and FALSE to deactivate it
uvp a parameter for the denoising
Takes TRUE to activate the command of the respective plot and FALSE to deactivate it
First, genesimulator is called to obtain a vector of mean gene intensities (for a number of genes and a number of replicates for each gene.
Then link{simdurbin2} simulates a series of gene intensities using the (log-normal type) model as described in Durbin and Rocke (2001,2002).
Then for each gene the mean of replicates for that gene is computed.
Optionally, if plot.it is TRUE then the mean is plotted against its standard deviation (over replicates).
Then the intensities are sorted according to increasing replicate mean.
Optionally, if plot.it is TRUE then a plot of the intensities values as a vector (sorted according to increasing replicate mean) is plotted in black, and then the true mean plotted in colour 2 (on my
screen this is red) and the computed replicate mean plotted in green.
Optionally, if plot.it is TRUE then a plot of the transformed means versus the transformed standard deviations is plotted. Followed by a time series plot of the transformed sorted intensities. These
can be studied to see how well DDHF has done the transformation.
Then two smoothing methods are applied the the DDHF transformed data. One method is translation invariant, Haar wavelet universal thresholding. The other method is the classical smoothing spline. If
plot.it is TRUE then these smoothed estimates are plotted in different colours.
Then the mean estimated intensity for each gene is computed and this is returned as the first column of a two-column matrix (ansm). The second column is the true underlying mean. The object hftssq
contains a measure of error between the estimated and true gene means.
ansm Two column matrix containing the estimated gene intensities and the true ones
hftssq Sum of squares between estimated means and true means
yhf Simulated gene intensities
Two column matrix containing the estimated gene intensities and the true ones
# # First run hftrialdatagen # ## Not run: v <- hftrialdatagen() # # Now plot the Haar-Fisz transformed intensities. # ## Not run: ts.plot(v$yhf) # # Now plot the denoised intensities # # Note that
above we have 128 genes and 4 replicates and so there are # 4*128 = 512 intensities to plot. # # However, there are only 128 gene intensities, and estimates. So, for this # plot we choose to plot the
noisy intensities and then for each replicate # group (which are colocated on the plot) plot the (necessarily constant) # true and estimated intensities (ie we plot each true/estimated intensity # 4
times, once for each replicate). # # First estimates... # ## Not run: lines(1:512, rep(v$ansm[,1], rep(4,128)), col=2) # # Now plot the truth # ## Not run: lines(1:512, rep(v$ansm[,2], rep(4,128)),
For more information on customizing the embed code, read Embedding Snippets. | {"url":"https://rdrr.io/cran/DDHFm/man/hftrialdatagen.html","timestamp":"2024-11-01T22:50:57Z","content_type":"text/html","content_length":"27447","record_id":"<urn:uuid:3dbf1706-c87e-408a-8544-905856fb36cb>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00388.warc.gz"} |
Adding and Subtracting Fractions and Mixed Numbers
This page lists the Learning Objectives for all lessons in Unit 16.
Adding Fractions with Like Denominators
The student will be able to:
• Define units, common denominator, simplify, and lowest terms
• Describe the procedure for adding fractions with like denominators.
• Determine the sum of two or more fractions with like denominators by applying the procedure above.
• Simplify the result when necessary.
• Recognize that only the numerators should be added, not the denominators.
• Apply addition procedures to complete five interactive exercises.
Subtracting Fractions with Like Denominators
The student will be able to:
• Define fraction bar.
• State the Multiplication Property of Zero.
• State the additive inverse property of zero.
• Describe the procedure for subtracting fractions with like denominators.
• Find the difference of two fractions with like denominators by applying the procedure above.
• Simplify the result when necessary.
• Recognize that a fraction bar indicates that a division of the numerator by the denominator will be performed
• Recognize that division by zero is undefined.
• Recognize that any fraction with zero in the numerator and a nonzero number in the denominator equals zero.
• Apply subtraction procedures to complete five interactive exercises.
Adding and Subtracting Fractions with Unlike Denominators
The student will be able to:
• Define least common denominator (LCD), least common multiple, and equivalent fractions.
• Find a common denominator by multiplying the denominators together.
• Rename fractions using their least common denominator.
• Identify the LCD of two or more fractions.
• Add fractions with unlike denominators by converting to equivalent fractions with a LCD.
• Simplify the result when necessary.
• Recognize that the numerator and the denominator of a fraction must be multiplied by the same nonzero whole number in order to have equivalent fractions.
• Recognize that there are two methods for adding and subtracting fractions with unlike denominators: common denominators and LCD.
• Describe the procedure for each method of adding and subtracting fractions with unlike denominators.
• Add and subtract fractions with unlike denominators by applying one the procedures above.
• Apply procedures to complete five interactive exercises.
Adding Mixed Numbers
The student will be able to:
• Define mixed numbers.
• Describe the procedure for adding mixed numbers.
• Recognize that a mixed number consists of a whole-number part and a fractional part.
• Recognize that it is easier to arrange the work vertically in order to add mixed numbers.
• Find the sum of two mixed numbers by applying the procedure for adding mixed numbers.
• Find the sum of a mixed number and a whole number by applying the procedure for adding mixed numbers.
• Simplify the result when necessary.
• Apply the procedure to complete five interactive exercises.
Subtracting Mixed Numbers
The student will be able to:
• Identify the whole-number part and the fractional part of a mixed number.
• Describe the procedure for subtracting mixed numbers with like denominators.
• Borrow a mixed number by converting it to an improper fraction.
• Describe the procedure for subtracting mixed numbers with unlike denominators.
• Subtract mixed numbers with like and unlike denominators by following the procedures presented.
• Subtract a mixed number from a whole number by following the procedures presented.
• Subtract a whole number from a mixed number by following the procedures presented.
• Recognize that there is another way to subtract mixed numbers is to convert each mixed number into an improper fraction.
• Recognize subtract mixed numbers by convert each mixed number into an improper fraction can lead to careless errors.
• Solve a real-world problem by subtracting mixed numbers.
• Apply procedures to complete five interactive exercises.
Solving Word Problems
The student will be able to:
• Examine problems involving addition and subtraction of fractions and mixed numbers.
• Identify strategies for solving each problem.
• Apply strategies for solving each problem.
• Connect addition and subtraction of fractions and mixed numbers with the real word.
• Apply all concepts and procedure to complete five interactive exercises with real-world problems.
Practice Exercises
The student will be able to:
• Examine ten interactive exercises for all topics in this unit.
• Identify the concepts and procedures needed to complete each practice exercise.
• Compute all answers and solve all problems by applying appropriate concepts and procedures.
• Self-assess knowledge and skills acquired from the instruction provided in this unit.
Challenge Exercises
The student will be able to:
• Evaluate ten challenging exercises for all topics in this unit.
• Analyze each problem to identify the given information.
• Formulate a strategy for solving each problem.
• Apply strategies to solve problems and write answers.
• Synthesize all information presented in this unit.
• Develop strong problem-solving skills and the ability to handle challenging problems.
The student will be able to:
• Examine the solution for each exercise presented in this unit.
• Compare solutions to completed exercises.
• Identify which solutions need to be reviewed.
• Identify and evaluate incorrect answers to exercises from this unit.
• Amend and label original answers.
• Identify areas of strength and weakness.
• Decide which concepts and procedures need to be reviewed from this unit. | {"url":"https://mathgoodies.com/objectives_unit16/","timestamp":"2024-11-05T16:04:44Z","content_type":"text/html","content_length":"39879","record_id":"<urn:uuid:c07d616a-d59f-41d9-b6d3-ae686c25c5c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00847.warc.gz"} |
Generalized Linear Mixed Models - Data Science Wiki
Generalized Linear Mixed Models :
Generalized linear mixed models (GLMMs) are a type of
regression analysis
that allows for the modeling of both fixed and random effects. This is useful in many research settings, where there may be both individual-level factors (fixed effects) and group-level factors
(random effects) that impact the
of interest.
An example of a GLMM might be a study looking at the relationship between income and happiness. In this study, researchers could include individual-level factors such as age and education as fixed
effects, and group-level factors such as state and region as random effects. This would allow the researchers to account for the potential
of both individual and group-level factors on happiness.
Another example of a GLMM might be a study examining the
of a medical treatment on blood pressure. In this study, researchers could include individual-level factors such as age and gender as fixed effects, and group-level factors such as clinic location
and doctor as random effects. This would allow the researchers to account for the potential influence of both individual and group-level factors on blood pressure.
In both of these examples, the use of GLMMs allows for the inclusion of both fixed and random effects in the analysis, which can provide a more nuanced understanding of the relationship between the
outcome and predictor variables. This is particularly useful when there are both individual-level and group-level factors that may impact the outcome of interest.
Overall, GLMMs are a useful tool for researchers looking to account for both fixed and random effects in their analysis. By including both types of effects, researchers can better understand the
complex relationships between predictor and outcome variables, and provide more accurate and comprehensive results. | {"url":"https://datasciencewiki.net/generalized-linear-mixed-models/","timestamp":"2024-11-13T15:24:34Z","content_type":"text/html","content_length":"41573","record_id":"<urn:uuid:ae900ab4-5a1c-4e1b-9b2e-23cbb5be45e5>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00758.warc.gz"} |
QCM: When is the Sauerbrey equation valid? - BioLogic Learning CenterQCM: When is the Sauerbrey equation valid? - BioLogic Learning Center
Topic 10 min read
QCM: When is the Sauerbrey equation valid?
Latest updated: October 8, 2024
Check out our partners to perform coupled measurements: https://biologic.net/partners
As previously introduced [1], in 1959, Sauerbrey [2] was the first to establish a relationship between mass change and the resonant frequency change:
$$\Delta f_n=-n\frac{2f_{0,n}^2}{\sqrt{\mu_\mathrm{q} \rho_\mathrm{q}}}\Delta m_\mathrm{a} \tag{1}\label{eq1}$$
With $\Delta f_n$, the change of resonant frequency at the $n^{\mathrm{th}}$ harmonic in $\mathrm{Hz}$, $n$ the harmonic order, $f_{0,n}$ the resonant frequency at the $n^{\mathrm{th}}$ harmonic in $
\mathrm{Hz}$, $\mu_\mathrm{q}$ the shear elastic modulus of the quartz in $\mathrm{kg\,m^{-1}\,s^{-1}}$ or $\mathrm{Pa\,s}$, $\rho_\mathrm{q}$ the quartz density in $\mathrm{kg\,m^{-3}}$, and $\Delta
m _\mathrm{a}$ the areal mass of the film in $\mathrm{kg\,m^{-2}}$.
At first, used to monitor mass or thickness of films deposited in vacuum, this relationship can also be used when the quartz and the electrodes are exposed to a solution.
Equation $\eqref{eq1}$ is called the Sauerbrey equation is only valid if the film being dissolved or deposited is considered rigid and thin. Such a film is called a Sauerbrey film.
• “Rigid” means that the acoustic wave will propagate elastically in the film, without any energy loss.
• “Thin” means that the film’s acoustic properties (shear wave modulus and density) can be approximated by the quartz crystal properties. Consequently, the wave velocity in the film is the same as
in the crystal.
• A “Thick” film means that its properties have to be accounted for and that the velocity of the wave is different in the film compared to that of the crystal.
Please note that the Sauerbrey equation is also valid to study tightly adsorbed nanoparticles. The Sauerbrey equation is valid as long the sample of interest is negligibly deformed. Check out our
partners to performed coupled measurements.
How to know if the film is thin or thick ?
If the frequency shift $\Delta f_n$ is over 2% of the initial resonant frequency $f_{0,n}$, the film should be considered thick. The Sauerbrey relationship cannot be used anymore as the film
properties need to be accounted for.
In this case a more complicated relationship needs to be used, that involves a different wave velocity in the quartz and in the film [3]:
$$\Delta m_\mathrm{a}=\frac{\rho_\mathrm{film} v_\mathrm{film}}{2\pi f}\text{arctan}\left(\frac{\rho_\mathrm{q} v_\mathrm{q}}{\rho_\mathrm{film} v_\mathrm{film}} \text{tan}\left(\pi \frac {f_0-f}
{f_0}\right) \right) \tag{2}\label{eq2}$$
With $\Delta m_\mathrm{a}$ the areal mass of the film in $\mathrm{kg\,m^{-2}}$, $\rho_\mathrm{film}$ and $\rho_\mathrm{q}$ the density of the film and the quartz, respectively in $\mathrm{kg\,m^{-3}}
$, $v_\mathrm{film}$ and $v_\mathrm{q}$ the wave propagation velocity in the film and the quartz, respectively in $\mathrm{m\,s^{-1}}$, $f_0$ and $f$ the resonant frequency of the quartz and the
quartz+film composite resonator in $\mathrm{Hz}$.
How to know if the film is rigid or not?
Dissipation measurement
To evaluate the rigidity or elasticity of the film, one should look at the change of half bandwidth shift $\Delta \Gamma$ in $\mathrm{Hz}$ between a clean and a coated sensor as shown in Figure 2 of
the topic Quartz crystal Microbalance: Measurement principles [4]. A bandwidth shift is considered small when it is smaller than the resonant frequency shift $\Delta f$ as it is the case in Figure 2
of the topic Quartz Crystal Microbalance: Measurement principles [4].
Instead of the half bandwidth change, the dissipation factor change $\Delta D$, expressed as a ratio and not a frequency, is measured using:
$$\Delta D=\frac{2\Delta \Gamma}{f_{01}}\tag{3}\label{eq3}$$
With $f_{01}$ the initial fundamental frequency in $\mathrm{Hz}$ as shown in Figure 1 and Figure 2 in the topic Quartz crystal Microbalance: Measurement principles [4].
In the case of dissipation measurement the criterion for rigidity is:
$$\frac{\Delta D}{\Delta f}≪\frac{1}{f_{01}}\tag{4}\label{eq4}$$
Measurements at harmonics/overtones
It is also possible to measure the resonant frequencies at higher harmonics rather than the fundamental one. In the field of acoustic waves, only odd harmonics are measured.
Measuring at harmonics give another way of ensuring that the film coating the bare electrode is rigid. If the value $\Delta f_n/n$ is constant for each harmonic, the film can be considered rigid.
More information on overtones measurements and their use and interest are given in the topic Quartz Crystal Microbalance: Why measuring at overtones? [5].
[1] QCM topics: QCM principles and history
[2] G. Sauerbrey Z. Phys. 155 (1959) 206
[3] T. Pauporté, D. Lincot, in : Microbalance à cristal de quartz, Techniques de l’Ingénieur, (2006) P 2 220.
[4] QCM topics: Measurement principles
[5] QCM topics: Why measure at overtones?
overtones dissipation Quartz Crystal Microbalance Sauerbrey equation gravimetry | {"url":"https://www.biologic.net/topics/bluqcm-when-is-the-sauerbrey-equation-valid/","timestamp":"2024-11-13T15:11:16Z","content_type":"text/html","content_length":"121308","record_id":"<urn:uuid:3461d5c3-c146-443c-8713-ad18d0a93607>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00102.warc.gz"} |
AIcrowd | IITM RL Final Project | Challenges
π Getting Started Code with Random Predictions
BSuite Benchmark for Reinforcement Learning
This notebook uses an open-source reinforcement learning benchmark known as bsuite.
BSuite is a collection of carefully-designed experiments that investigate core capabilities of a reinforcement learning agent.
Your task is to use any reinforcement learning techniques at your disposal to get high scores on the environments specified.
Note: Since the course is on Reinforcement Learning, please limit yourself to using traditional Reinforcement Learning algorithms.
Do not use deep reinforcement learning.
You will be implementing a traditional RL algorithm to solve 3 environments.
Environment 1: CATCH
In this environment , the agent must move a paddle to intercept falling balls. Falling balls only move downwards on the column they are in.
The observation is an array shape (rows, columns), with binary values: 0 if a space is empty; 1 if it contains the paddle or a ball.
The actions 3 discrete actions possible: ['stay', 'left', 'right'].
The episode terminates when the ball reaches the bottom of the screen.
Environment 2: CARTPOLE
This environment implements a version of the classic Cartpole task, where the cart has to counter the movements of the pole to prevent it from falling over.
The observation is a vector representing: (x, x_dot, sin(theta), cos(theta), theta_dot, time_elapsed)
The actions are discrete and there are 3 of them available: ['left', 'stay', 'right'].
Episodes start with the pole close to upright. Episodes end when the pole falls, the cart falls off the table, or the max_time is reached.
Environment 3: MOUNTAIN CAR
This environment implements a version of the classic Mountain Car problem where an underpowered car must power up a hill.
The observation is a vector representing: (x, x_dot, time_elapsed)
There are 3 discrete actions available: ['push left', 'no push', 'push right']
Episodes start with the car at the bottom of the hill with no velocity. An episode ends when you reach position x=0.5, or if 1000 steps have been completed.
Each environment has a NOISE variant which adds a scaled random noise to the received rewards. More details in the BSuite Paper.
π Submission
Before submitting, make sure to accept the rules.
Go to the starter kit notebook and follow the instructions to implement your agent in the notebook.
π ―Scoring
We use BSuite's scoring system to determine score for each environment. The final score is the sum of all the test environments' scores. | {"url":"https://www.aicrowd.com/challenges/iitm-rl-final-project","timestamp":"2024-11-14T17:45:05Z","content_type":"text/html","content_length":"242136","record_id":"<urn:uuid:cb8494bb-a13b-455f-b653-de4ca7071b10>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00406.warc.gz"} |
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Efficient Model Predictive Control for parabolic PDEs with goal oriented error estimation
Title data
Grüne, Lars ; Schaller, Manuel ; Schiela, Anton:
Efficient Model Predictive Control for parabolic PDEs with goal oriented error estimation.
In: SIAM Journal on Scientific Computing. Vol. 44 (2022) Issue 1 . - A471-A500.
ISSN 1095-7197
DOI: https://doi.org/10.1137/20M1356324
This is the latest version of this item.
Project information
Project title: Project's official title
Project's id
Specialized Adaptive Algorithms for Model Predictive Control of PDEs
GR 1569/17-1
Specialized Adaptive Algorithms for Model Predictive Control of PDEs
SCHI 1379/5-1
Project financing: Deutsche Forschungsgemeinschaft
Abstract in another language
We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a Model Predictive Control (MPC) algorithm. In MPC, only an initial part of a
computed solution is implemented as a feedback, which motivates grid refinement particularly tailored to this context. To this end, we present a truncated cost functional as objective for goal
oriented adaptivity and prove under stabilizability assumptions that error indicators decay exponentially outside the support of this quantity. This leads to very efficient time and space
discretizations for MPC, which we will illustrate by means of various numerical examples.
Further data
Available Versions of this Item | {"url":"https://eref.uni-bayreuth.de/id/eprint/68749/","timestamp":"2024-11-09T03:18:22Z","content_type":"application/xhtml+xml","content_length":"27126","record_id":"<urn:uuid:7e5082d8-93a3-4cee-adb1-4d95f6b2aa80>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00721.warc.gz"} |
You will explore methods for evaluating the difference between variables. Comparing two groups of data using a range of t-tests will also be explored. Integration of Fai - Education
You will explore methods for evaluating the difference between variables. Comparing two groups of data using a range of t-tests will also be explored. Integration of Fai
7 Introduction
You will explore methods for evaluating the difference between variables. Comparing two groups of data using a range of t-tests will also be explored. Integration of Faith and Learning will be
presented in a way that examines quantitative analysis in light of God’s plan for how you should handle challenging work.
Learning Outcomes
Upon successful completion of this module, you will be able to:
• Analyze the appropriate inferential statistic to compare groups.
• Evaluate and interpret a one-sample t-test to compare one group or sample to a hypothesized population mean.
• Understand the assumptions and conditions for use of the independent samples t test.
• Analyze and interpret a paired samples t test to check the reliability of a repeated measure.
• Understand your view of God’s perspective on the work involved in conducting meaningful and accurate quantitative analysis.
Discussion Thread: Comparing Groups
Respond to the following short answer questions from Chapter 9 of the Morgan, Leech, Gloeckner, & Barrett textbook:
D7.9.1 (a) Under what conditions would you use a one-sample t test? (b) Provide another possible example of its use from the HSB data.
D7.9.2 In Output 9.2: (a) Are the variances equal or significantly different for the three dependent variables? (b) List the appropriate t, df, and p (significance level) for each t test as you would
in an article. (c) Which t tests are statistically significant? (d) Write sentences interpreting the academic track difference between the means of grades in high school and also visualization. (e)
Interpret the 95% confidence interval for these two variables. (f) Comment on the effect sizes.
D7.9.3 (a) Compare the results of Outputs 9.2 and 9.3. (b) When would you use the Mann–Whitney U test?
D7.9.4 In Output 9.4: (a) What does the paired samples correlation for mother’s and father’s education mean? (b) Interpret/explain the results for the t test. (c) Explain how the correlation and the
t test differ in what information they provide. (d) Describe the results if the r was .90 and the t was zero. (e) What if r was zero and t was 5.0?
D7.9.5 (a) Compare the results of Output 9.4 with Output 9.5. (b) When would you use the Wilcoxon test?
The student will complete 8 short-answer discussions in this course and 1 long-answer
Integrating Faith and Learning discussion. In the thread for each short-answer discussion the
student will post short answers to the prompted questions. The answers must demonstrate
course-related knowledge and support their assertions with scholarly citations in the latest APA
format. Minimum word count for all short answers cumulatively is 200 words. The minimum
word count for Integrating Faith and Learning discussion is 600 words. For each thread the
student must include a title block with your name, class title, date, and the discussion forum
number; write the question number and the question title as a level one heading (e.g. D1.1
Variables) and then provide your response; use Level Two headings for multi part questions (e.g.
D1.1 & D1.1.a, D1.1.b, etc.), and include a reference section.
The student must then post 1 reply to another student’s post. The reply must summarize the
student’s findings and indicate areas of agreement, disagreement, and improvement. It must be
supported with scholarly citations in the latest APA format and corresponding list of references.
The minimum word count for Integrating Faith and Learning discussion reply is 250 words.
You will explore methods for evaluating the difference between variables. Comparing two groups of data using a range of t-tests will also be explored. Integration of Fai
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Place Value Worksheet
A place value is the value of a digit that is located within a number. For whole numbers the place values start with the ones value, then tens, then hundreds, then thousands value. For decimals the
place values have no ones place, it starts with the tenths place. From there they have very similar names with a -th- attached to it such as tenths, hundredths, thousandths. Understanding place
values is very important since it allows us to understand the meaning of a number and perform operations smoothly. We use something called the base ten numbering method where every integer value at
each place can range between zero and nine. Each number position is ten times greater than the number to the right of it. Which is why we call it base ten. If you have a value that goes above nine or
below zero while performing operations, the value will be pushed or carried to the next subsequent place (position).
You find that we have seven categories of related topics available at the top of this page. For just basic place value worksheets, just scroll down. We have lessons that cover the ten-thousandths to
billions place, so you sure to know your stuff. These worksheets help students practice the names of place values and converting groups of numbers into words and numerals. Students will also convert
groups of numbers into words and numerals. Many sheets contain larger numbers.
Get Free Worksheets In Your Inbox! | {"url":"https://www.easyteacherworksheets.com/math/placevalue.html","timestamp":"2024-11-10T01:21:14Z","content_type":"text/html","content_length":"60388","record_id":"<urn:uuid:32c4c5c8-03bd-4c13-8f9f-92ed593ef1e3>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.3/warc/CC-MAIN-20241110005602-20241110035602-00706.warc.gz"} |
Physics in LEGO Mindstorms: Energy Accumulation and Conservation. Part 2. Moment of Intertia
Introducing three main concepts - Energy, Inertia Moment and Angular Velocity. We describe what is the moment of Inertia, how do we calculate it and how do we measure it?
The Moment of Inertia gives you an understanding of how difficult it is to stop an object once it is moving/rotating or to start moving/rotating an object once it has stopped. You can give it a value
and this value depends on the weight and the speed of the object.
Now let's get into the theory. We have a cylinder and this cylinder is rotating on an axle and is rotating in a certain direction. Because of the rotation this cylinder has energy and energy is
marked with the letter E and it has kinetic energy. Now there are different types of cylinders and different ways to calculate the kinetic energy of this cylinder and they all depend on something
called the inertia moment and the the speed of rotation which is called omega. We have these 3 values, the kinetic energy the inertia moment and the speed of rotation of this cylinder. The connection
between these values is the following. E is equal to a half of the inertia moment multiplied by omega squared. Omega marks the speed of rotation of this cylinder. This is the kinetic energy of our
cylinder. For the inertia moment, it depends on the mass and radius of this cylinder. The inertia moment is equal to a half of the mass of the cylinder multiplied by the radius of the cylinder
squared. The radius of the cylinder is right here and the larger the cylinder, the more the inertia moment, the larger the mass, the more the inertia moment and actually the inertia moment shows how
difficult it is once this cylinder is rotating to stop it from rotating or once this cylinder is stopped to start it rotating. That's actually the notion behind the inertia moment.
This formula right here applies only for a solid cylinder. If you look at the LEGO wheel that we are currently using it's a cylinder but it little bit different because the mass of this cylinder is
not equally distributed. Our LEGO wheel has a tire and a rim. If you measure the tire and we've done it it's about 25 grams and the rim is about 15 grams and the whole wheel is about 40 grams. So the
mass is not equally distributed and because of that it get very complicated and we'll use a very simple formula, it won't be very accurate, but it will be quite accurate for the principle that we
would like to show. When you have a cylinder that has most of the mass here between the 2 radii. We have an outer circle and for it we have r outer and for the inner circle we have r inner.
We have the inertia moment for the cylinder that has an outer circle and an inner circle that is equal to a half of the mass multiplied by the radius of the outer circle squared plus radius inner
squared. That's the formula that we are going to use for finding the energy, the inertia moment and from there the energy of our LEGO wheel that is rotating. We have the 2 radii and the mass. Last
but not least we must find the speed of rotation. The speed of rotation of this wheel and the angular velocity is marked with omega. We measured the angular velocity last time.
We measured last time that our motor was rotating with 860 degrees per second. That was the angular velocity of our motor when it was reaching the power of a 100%.
If we know that one circle has 360 degrees how many rotations per second are we doing. If we divide 860 by 360
will give us the number of rotations that our motor is doing for 1 second. This here is equal approximately to 2.39. We have our motor doing 2.39 rotations per second.
Because we are working in the system international units we must have all the units in the system so that we can get the energy. For the energy we would like to get it in joules, that means that we
must have the mass in kg and the mass of our wheel is 0.04 kg, the radius then must be in meters, the larger radius is 0.07 m, that's the radius of the outer circle.
We need the radius of the inner circle is equal to 0.049 m, and we must also have the omega, we must have it in radians per second. What does that means? We have a number of rotations per second
and we multiply this by 2 pi, because n 1 circle we have 2 pi radians. Below the video we'll give you more links for resources that explain exactly the conversion between the rotations per second and
the radians per second If we multiply 2.39 by 2 pi we'll get something like this. So we have 2.39 multiplied by 2*3.14 and the result is about 15 radians per second. Our wheel is doing 15 radians per
Omega is actually 15 radians per second. Because the video is getting too long in the next video we'll continue substituting these values in the formula and getting the result in joules. | {"url":"https://www.fllcasts.com/tutorials/115-physics-in-lego-mindstorms-energy-accumulation-and-conservation-part-2-theory","timestamp":"2024-11-08T10:52:05Z","content_type":"text/html","content_length":"67915","record_id":"<urn:uuid:c5a7819b-d526-4ed1-98e3-f9bf02a81b11>","cc-path":"CC-MAIN-2024-46/segments/1730477028059.90/warc/CC-MAIN-20241108101914-20241108131914-00693.warc.gz"} |
What is the Highest Spacetime Curvature Near Us?
(Image credit: BBC)
In principle, we could unravel exotic physics by irritating spacetime and inducing curvature in it.
What is the highest spacetime curvature accessible to us in our cosmic neighborhood? To answer this question, we should first define a measure of that curvature scale.
In his General Theory of Relativity, Albert Einstein wrote equations that relate the curvature of spacetime to the matter-energy density. As the physicist John Wheeler noted: “Matter tells spacetime
how to curve, and spacetime tells matter how to move.” Gravity is not a force but rather spacetime curvature that affects the motion of matter. A marble with a proper speed will move on a circle on
the flexible surface of a trampoline, which is curved by a bowling ball at its center. When the bowling ball is removed, the rubber surface would turn flat and the marble would move away in a
straight line, just the way that Earth would fly out of the Solar system if the Sun dispersed.
Einstein was inspired by the equivalence principle, whereby all test objects follow the same universal motion under gravity, irrespective of their material composition. Galileo allegedly tested this
principle from the top of the Leaning Tower of Pisa in Italy, and so did the astronaut David Scott during the Apollo 15 mission in 1971 by dropping a hammer and a feather in vacuum and verifying they
both reached the surface of the Moon at the same time. In 2022, the MICROSCOPE satellite confirmed that two masses of titanium and platinum aboard a satellite orbiting Earth fall exactly in the same
way to a precision of one part in quadrillion (10 to the power of -15).
The curvature in the surface of a beach ball is defined by its radius. The larger the radius is, the flatter is the surface from the vantage point of an ant walking on it. The local curvature scale
of spacetime, R, is given by the speed of light divided by the square-root of Newton’s constant times the local mass density, R~[c/sqrt(G*rho)]. The denser the matter is, the shorter R is.
At the nuclear density of a proton, the curvature scale R is about 45 kilometers, comparable to the size of a large city. It is a few times larger than the size of a neutron star with the same mass
density. Nuclear density is nearly a quadrillion times larger than the density of water, because the size of an atom is ~100,000 times bigger than the size of a proton and mass density scales
inversely with size cubed.
Coincidentally, the average density of water is close to the average density of the Sun or of Jupiter. At this matter density, the curvature scale is about 8 times the Earth-Sun separation. The
spacetime curvature scale R produced by the denser rock of planet Earth is half that value.
The highest spacetime curvature near astrophysical objects is just outside the event horizon of black holes. The lowest-mass black holes form by the collapse of massive stars, and carry a few times
the mass of the Sun. Their curvature scale is about 10 kilometers, a factor of 6 smaller than that induced by a proton or a neutron star. The value of R for bigger black holes scales up in proportion
to the black hole mass and reaches a tenth of the Earth-Sun separation for Sgr A*, the 4-million solar mass black hole at the center of the Milky-Way. The most massive black holes in the Universe
with about 40 billion solar masses induce very weak curvature on a scale that extends out to 30 times the distance of Neptune from the Sun. An astronaut falling into their event horizon would barely
sense their tidal field.
All in all, it is remarkable that protons curve spacetime almost as much as the smallest astrophysical black holes in our cosmic neighborhood.
Of course, colliders like CERN’s Large Hadron Collider, can smash particles at ultra-high energy and reach higher spacetimes curvatures for a short period of time. The mass-energy density can obtain
yet higher values in collisions of the highest energy cosmic rays, with up to a trillion times the energy equivalent of the proton mass.
If new physics is associated with corrections at high spacetime curvature, it is likely to emerge near the Planck scale, which is 39 orders of magnitude smaller than 10 kilometers, the value of R
outside the horizon of the smallest known black holes. The fact that this scale is well beyond our reach also in colliders explains why it is difficult to test theories which attempt to unify quantum
mechanics and gravity, like string theory.
Alas, we have a lot to learn. This morning, I had received an email sent by Dr. Alexander Ross from Yale who asked: “What is a life not dedicated to learning?”, to which I replied: “A life not
dedicated to learning is a life not fulfilled.”
Our current knowledge of new physics when the spacetime curvature reaches the Planck scale is a tiny island in a vast ocean of ignorance. The difficulty of accessing this knowledge through
experimentation is one more reason to search for a smarter student in our class of intelligent civilizations within the classroom of the Milky-Way galaxy.
Avi Loeb giving a keynote lecture at the CURIOUS 2024 conference in Germany on July 11, 2024 (Image credit: Ulrich Betz)
Avi Loeb is the head of the Galileo Project, founding director of Harvard University’s — Black Hole Initiative, director of the Institute for Theory and Computation at the Harvard-Smithsonian Center
for Astrophysics, and the former chair of the astronomy department at Harvard University (2011–2020). He is a former member of the President’s Council of Advisors on Science and Technology and a
former chair of the Board on Physics and Astronomy of the National Academies. He is the bestselling author of “Extraterrestrial: The First Sign of Intelligent Life Beyond Earth” and a co-author of
the textbook “Life in the Cosmos”, both published in 2021. His new book, titled “Interstellar”, was published in August 2023. | {"url":"https://avi-loeb.medium.com/what-is-the-highest-spacetime-curvature-near-us-baba2ea113eb","timestamp":"2024-11-05T06:39:30Z","content_type":"text/html","content_length":"111838","record_id":"<urn:uuid:260e5cb5-d51d-4a56-b252-e4f565e5780d>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00315.warc.gz"} |
Circular linked lists in Python
A linked list is a data structure that is made up of nodes. Basically, each node contains two elements: a value and a reference to the next node in the list.
The first node in the list is called the head, and the last node is called the tail. If the tail points to head as the next node, we get a circular linked list.
A circular linked list has no end and can therefore be traversed infinitely in a circular manner.
Circular linked lists are useful for applications where data needs to be continuously cycled or accessed in a circular way. One example of such use case could be be in a playlist for music or videos,
where the list is played in a continuous loop.
Circularly Linked Queues
One of the most practical use of a circular linked list is in the implementation of circularly linked queues.
In a circularly linked queue, we can move back to the front of the queue after reaching the end.
The CircularQueue class supports the following operations:
│enqueue(e)│Add element e at the end of the queue. │
│dequeue() │Remove and return the last element in the queue. │
│rotate() │Rotate the elements in the queue so that the first element becomes the last and the next element after it becomes the first. │
│first() │Return the first element in the queue, without removing it. │
│is_empty()│Return True if the queue is empty, otherwise False. │
│len() │Return the number of elements in the queue. │
All the operations listed above are native to all queues except the rotate() operation.
In a CircularQueue, the rotate() method allows us to rotate the elements so that the first element moves at the back of the queue, and the next element after it becomes the first. This is important
when we want to perform operations on the elements of the queue without actually dequeueing them.
Implementing the CircularQueue class
In this section, we will implement the CircularLinked Queue class as described in the previous section. We will use a step by step approach starting with the simplest steps.
The basic structure.
class CircularQueue:
#non-public _Node subclass for representing a Node.
class _Node:
def __init__(self, e, nxt):
self._element = e #Nodes element
self._next = nxt #a reference to the next node
def __init__(self):
self._tail = None # The last node.
self._size = 0 #The number of elements in the queue
def __len__(self):
return self._size
def is_empty(self):
return self._size == 0
In the above snippet, we defined the basic structure of the CircularQueue class. The _Node class is implemented as a non-public subclass. We will use the class internally to represent nodes in the
In the constructor of the CircularQueue class, we defined the _tail variable that will reference the last node in the structure. Note that we did not implement _head because it would be trivial as we
can access the first node(head) by simply using _tail.next.
Implement first() method
#an exception to be raised by an empty queue
class Empty(Exception):
class CircularQueue:
#This part is ommitted refer to the previous section
def first(self):
if self.is_empty():
raise Empty("The queue is empty.") #raise an exception if there are no elements in the queue
head = self._tail._next #access the head node
return head._element #return the element of the head node
In the above snippet, we implemented an exception class, Empty. This exception is raised by the first() method if it is called when the queue is empty.
As earlier said, using _tail._next accesses the head node, whose element is returned by the first() method.
Implement enqueue()
#Empty class omitted
class CircularQueue:
#This part is ommitted refer to the previous sections
def enqueue(self, e):
newnode = CircularQueue._Node(e, None) #create a new node
if self.is_empty():
newnode._next = newnode #The node points to itself, if its the only node.
oldhead = self._tail._next
newnode._next = oldhead
self._tail._next = newnode
self._tail = newnode #update _tail
self._size += 1
From the above snippet, If the an element is added when the queue is empty, the new node is made to point to itself, as it is the only node in the structure.
The new node is added at the back of the linked structure and the _tail variable is updated to reference it.
implement dequeue() method
#Empty class omitted
class CircularQueue:
#This part is ommitted refer to the previous sections
def dequeue(self, e):
if self.is_empty():
raise Empty("The queue is empty.")
oldhead = self._tail._next #The node with the element at the front of the queue
if len(self) == 1:
self._tail = None #The queue is now empty again.
self._tail._next = oldhead._next #Ignore the node with the removed element
self._size -= 1
return oldhead._element
In the above implementation, we are dequeueing the element at the front of the queue, i.e the one in the head node(_tail._next). We are bypassing the node by making the node after it to become the
next node to the tail, this effectively disconnects the removed node from the linked structure.
Implement rotate()
#Empty class omitted
class CircularQueue:
#This part is ommitted refer to the previous sections
def rotate(self):
if not self.is_empty():
self._tail = self._tail._next # The head becomes the tail
The complete implementation
The following snippet shows the complete implementation of CircularQueue after combining the previous fragments.
the complete CircularQueue class
class Empty(Exception):
class CircularQueue:
class _Node:
def __init__(self, e, nxt):
self._element = e
self._next = nxt
def __init__(self):
self._tail = None
self._size = 0
def __len__(self):
return self._size
def is_empty(self):
return self._size == 0
def first(self):
if self.is_empty():
raise Empty("The queue is empty.")
head = self._tail._next
return head._element
def dequeue(self):
if self.is_empty():
raise Empty("The queue is empty.")
oldhead = self._tail._next
if len(self) == 1:
self._tail = None
self._tail._next = oldhead._next
self._size -= 1
return oldhead._element
def enqueue(self, e):
newnode = CircularQueue._Node(e, None)
if self.is_empty():
newnode._next = newnode
oldhead = self._tail._next
newnode._next = oldhead
self._tail._next = newnode
self._tail = newnode
self._size += 1
def rotate(self):
if not self.is_empty():
self._tail = self._tail._next
Using the CircularQueue class
use enqueue() and dequeue()
#implementation is ommitted
Q = CircularQueue()
#implementation is ommitted
Q = CircularQueue()
print("Size before dequeue: ", len(Q))
while not Q.is_empty():
print("Size after dequeue: ", len(Q))
Size before dequeue: 5
Size after dequeue: 0
example with rotate()
#implementation is ommitted
Q = CircularQueue()
print("Size before rotate: ", len(Q))
#print queue elements without dequeueing them
for _ in range(len(Q)):
print("Size after rotate: ", len(Q))
Size before rotate: 5
Size after rotate: 5 | {"url":"https://www.pynerds.com/data-structures/circular-linked-lists-in-python/","timestamp":"2024-11-10T08:54:34Z","content_type":"text/html","content_length":"74764","record_id":"<urn:uuid:25d2ae2c-1309-45f1-866f-f53f3edbe25f>","cc-path":"CC-MAIN-2024-46/segments/1730477028179.55/warc/CC-MAIN-20241110072033-20241110102033-00515.warc.gz"} |
Pascal's triangle – The Aperiodical
You're reading: Posts Tagged: Pascal's triangle
Hi! My name is Colin, and I am a PROPER mathematician now. I’ve made a contribution to the Online Encyclopaedia of Integer Sequences.
This is the first in a series of guest posts by David Benjamin, exploring the secrets of Pascal’s Triangle.
The triangle of Natural numbers below contains the first seven rows of what is called Pascal’s triangle. Each row begins and ends with the number 1, and each of the remaining numbers, from the
third row onwards, is the sum of the two numbers ‘above’: | {"url":"https://aperiodical.com/tag/pascals-triangle-2/","timestamp":"2024-11-06T02:38:15Z","content_type":"text/html","content_length":"32486","record_id":"<urn:uuid:a4e71e8b-960d-41e4-9ecb-a26271306b47>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00486.warc.gz"} |
God Plays Dice
Not only does God throw dice, but
Einstein does too
, or at least a stencil of him on a wall in the Upper Haight in San Francisco does.
This post suggests
that it may have been by the graffiti artist
. It's been painted over.
More pictures
It's been painted over, apparently. That's probably for the best, because that means I won't try to find it when I move to the Bay Area.
(Oh, yeah, I'm moving! I got a job at Berkeley.)
I went to the National Constitution Center in Philadelphia today.
As you may know, the Constitution provides that, in elections for the President, each state receives a number of electors equal to its total number of senators and representatives. Each state has two
senators, and the number of representatives is proportional to the population. The number of representatives is adjusted after the census, which happens in years divisible by ten.
Why am I telling you this? Because at one point on the wall there was an animated map, which displayed how apportionment had changed between censuses. Each state was represented as a "cylinder", with
base the state itself and height proportional to its number of electors. (Or representatives; it honestly would be impossible to tell the difference by eye, as in this scheme that would just push
everything up by two units.) There was one such display in the animation for each census, with smooth transitions between them.
Since the eye wants to interpret the "volume" of a state as its number of electors, this has the effect of making geographically-large states look like they have better representation than they do. I
noticed this by looking at New Jersey and Pennsylvania, which have areas of 7417 and 44817 square miles, and 15 and 21 electors respectively. The solid corresponding to Pennsylvania has about eight
times the volume as that corresponding to New Jersey. New Jersey's an easy one to look at because it happens to be the most densely populated state at the present time, and in this visualization it
is not the tallest.
The volume of the solid corresponding to each state is proportional to the product of its number of its electors and its area. The states for which this product is largest are, in order, Texas,
California, Alaska, New York, Florida, Illinois, Arizona, Michigan, Pennsylvania, and Colorado. The first two of these, between them, have 41% of the total volume in this visualization.
I'd suggest replacing this with a model where volume is proportional to the number of electoral votes. Or, since that might have its own problems, a cartogram which evolves in time. The West would
just grow out of nowhere.
As you may have heard, there's a match at Wimbledon , between John Isner and Nicolas Mahut, in which the last set is tied at 59 games. (The previous longest set at Wimbledon was 24-22.)
A set goes until one player has won six games, and has also won two more than the opponent. This means that back since the set was tied at 5 games, games 11 and 12 were split by the two players; so
were 13 and 14; and so on up to 117 and 118.
Terence Tao points out that this is very unlikely using a reasonable naive model of tennis, which assumes that the player serving has a fixed probability of winning the game. (Service alternates
between games.) His guess is that some other factor is at play; for example, "both players may perform markedly better when they are behind".
This seems statistically checkable, at least if records of that sort of thing are kept. I'm not sure if they are; it seems like tennis scores are often reported as just the number of games won by
each player in each set, not their order. Another hypothesis, of course, is that the match has taken on a life of its own and, subconsciously, the players are playing to keep the pattern going.
Edit (Thurs. 7:49 am): More on Isner-Mahut: Tim Gowers' comments, and some odds being offered by William Hill, the betting shop.
The final standings in Group C of the 2010 World Cup were as follows: USA 5, England 5, Slovenia 4, Algeria 1.
Question: given this information, can we reconstruct the results of the individual games? Each team plays each other team once; they get three points for a win, one for a draw, zero for a loss.
First we can tell that USA and England must have had a win and two draws, each; Slovenia, a win, a draw, and a loss; Algeria, a draw and two losses. (In fact you can always reconstruct the number of
wins, draws, and losses from the number of points, except in the case of three points, which can be a win and two losses, or three draws.)
Since neither USA nor England have a loss, they must have drawn. Similarly, Slovenia's win must have been against Algeria.
But now there are two possibilities; we have to break the symmetry between USA and England. Let's say, arbitrarily, that USA drew against Slovenia and defeated Algeria, instead of the other way
around. (This is, in fact, what happened.) Then Algeria's draw must have been against England, and England's win against Slovenia.
In an alternate universe where USA and England switch roles (does this mean that England was a USA colony in this universe?) USA defeated Slovenia and drew against Algeria, and England draws against
Slovenia and defeats Algeria.
Of course, the next question is: given the goal differentials (+1 for USA and England, 0 for Slovenia, -2 for Algeria), can we figure out the margins in the various games? (Assume we know which of
the two universes above we're in; for the sake of avoiding cognitive dissonance, say we're in the first one.) Since Algeria was only defeated by a total of two goals, the margin in each of their
losses must have been 1. And the margin in the Slovenian win (to Algeria) and loss (to England) must have been the same, namely 1.
If you in addition are given the total number of goals scored (USA 4, Slovenia 3, England 2, Algeria 0) you can reconstruct the scores of each match. I leave this as an exercise for the reader. Hint:
start with Algeria.
Another question: is it the "usual case" that individual match results can be recovered from the final standings, or is this unusual? The table of standings in a group in the World Cup has something
like thirteen degrees of freedom. Given the number of wins and draws, goals scored, and goals against for three of the teams, we can find the number of losses and goal differential for each team, the
number of wins, draws and losses for the fourth team, and the goal differential of the fourth team. We need one more piece of information - say, the number of goals scored by that fourth team - to
reconstruct the whole table. We're trying to derive twelve numbers from this (the number of goals scored by each team in each match). It will be close.
In an n-team round robin, the number of degrees of freedom in the table of standings grows linearly with n, but the number of games grows quadratically with n. For large n it would be impossible to
do this reconstruction; for n=1 it would be trivial.
An example of a bad word problem, from Frank Quinn's article The Nature of Contemporary Core Mathematics, who is at Virginia Tech:
Bubba has a still that produces 700 gallons of alcohol per
week. If the tax on alcohol is $1.50 per gallon, how much tax will Bubba pay in amonth? [Set up and analyze a model, then discuss applicability of the model.]
I have given an example with obvious cultural bias because I am not sure I could successfully avoid it. At any rate students in my area in rural Virginia would think this problem is hilarious. We
have a long tradition of illegal distilleries and they would know that Bubba has no intention of ever paying any tax.
As some of you may have noticed, I spend lots of time at MathOverflow these days. This explains my lack of posting.
Actually, my lack of posting is also in part because of taking a break after finishing my PhD. But I am now trying to prepare for the Next Step. The Next Step is an academic job, in fact, so if
you've been holding your breath and wondering if I got one, you can breathe again. Details will follow, but the job is technically not official yet, so I don't want to say where it is here. My
productive efforts are going towards preparing for courses I'll be teaching, factoring my dissertation (warning: large PDF) into papers, and moving my worldly possessions.
I'm posting in order to mention two potential sites hosted on the StackExchange platform that might be of interest to my readers (and also to MO users); these are one for statistical analysis and one
for mathematics. The mathematics site will differ from MathOverflow in being somewhat lower-level, which I think is valuable; one of the things that's plagued MathOverflow from the beginning is that
there are frequently questions which are clearly below the level of the site but we have nowhere good to send these questions! Similarly, MathOverflow gets a lot of statistics questions that the MO
readership isn't equipped to handle. (People try on the questions that are really about probability, but some are about the more "practical" side of statistics and we don't have too many experts in
If you click on those links, you can "commit" to participating in one or both of these sites. The idea is that once enough commitments are made, the site will be launched. This is StackExchange's new
model; their old model was that people paid to have a site hosted with the software. This is the model on which MathOverflow works -- using money from Ravi Vakil's research funds -- but they don't do
this any more, because there were ghost sites. See a fuller explanation.
How to explain Euler's identity using triangles and spirals, by Brian Slesinsky. (Euler's identity refers, here, to e^iπ = -1.) Uses the geometric interpretation of complex multiplication to explain
this fact.
Here's a question: Did Obama do better among African-Americans or Prius owners?
The consensus is that he did better among African-Americans. (96% of African-Americans who voted voted for him, which is a pretty high bar.)
But how would one go about estimating how he did among Prius owners?
I don't really know much about basketball. But this New York Times article suggests that the first pick in the NBA lottery might not be worth much this year, and then goes on to say:
But history suggests that he [Rod Thorn, president of the New Jersey Nets] will not have that decision to make. Since 1994, the team with the worst record has won the lottery only once — Orlando
in 2004.
Here's how the NBA draft lottery works. In short: there are thirty teams in the NBA. Sixteen makes the playoff. The other fourteen are entered in the draft lottery. Fourteen ping-pong balls (it's a
coincidence that the numbers are the same) are placed in a tumbler. There are 1001 ways to pick four balls from fourteen. Of these, 1000 are assigned to the various teams; the worse teams are
assigned more combinations. 250 are assigned to the worst team, 199 to the second-worst team, "and so on". (It's not clear to me where the numbers come from.)
Then four balls are picked. The team that this set corresponds to gets the first pick in the draft. Those balls are replaced; another set is picked, and this team (assuming it's not the team already
picked) gets the second pick. This process is repeated to determine the team with the third pick. At this point there's an arbitrary cutoff; the 4th through 14th picks are assigned to the eleven
unassigned teams, from worst to best. The reason for this method seems to be so that all the lottery teams have some chance of getting one of the first three picks, but no team does much worse than
would be expected from its record; if the worst team got the 14th pick they wouldn't be happy.
So the probability that the team with the worst record wins the lottery is one in four, by construction; this "history suggests" is meaningless. (And the article even mentions the 25 percent
probability!) This isn't like situations within the game itself where the probabilities can't be derived from first principles and have to be worked out from observation.
Also, let's say we continued iterating this process to pick the order of all the lottery teams. How would one expect the order of draft picks to compare to the order of finish in the league? I don't
know off the top of my head.
Chad Orzel, physicist, writes why I'd never make it as a mathematician. He calls himself a "swashbuckling experimentalist" and says that he doesn't like thinking too hard about questions of
convergence and the like. This is in reference to Matt Springer's most recent Sunday function, which gives the paradox:
1 - 1/2 + 1/3 - 1/4 + ... = log 2
1 - 1/2 - 1/4 + 1/3 - 1/6 - 1/8 + ... = (log 2)/2
I find that I tend to act "like a physicist" in my more experimental work. Often I'm dealing with the coefficients of some complicated power series (usually a generating function) which I can compute
(with computer assistance) and don't understand too well. Most of the time the things that "look true" are. This work is, in some ways, experimental, which is why it's tempting to act like a
Oh, yeah, I graduated today.
Take the Economist's numeracy quiz.
If you get all five questions right, you did better than Terence Tao.
The quiz is linked to this article, which states that people who are better at doing simple financial calculations seem to be less likely to fall behind on their mortgages.
Rather annoyingly, The Economist doesn't even tell you the names of the people who did the study. But it's Financial Literacy and Subprime Mortgage Delinquency: Evidence from a Survey Matched to
Administrative Data, by Kristopher Gerardi, Lorenz Goette, and Stephan Meier. I will admit I have not read it, because it's 54 pages. (But yes, they controlled for income. My first thought was that
maybe people who are better with numbers also tend to make more money.) Gerardi also writes for the Atlanta Fed's blog on real estate research.
The National Institute of Standards and Technology has released what you might call a "trailer" for the revised edition of Abramowitz and Stegun's Handbook of Mathematical Functions. The original
version is available online (it's public domain).
The print version is called the NIST Handbook of Mathematical Functions, and is available in hardcoverpaperback
There is also, not surprisingly, an online version, the Digital Library of Mathematical Functions, which takes advantage of new technology: three-dimensional graphics, color, etc. Think MathWorld,
but less idiosyncratic. It jsut went public today.
And it includes Stanley's Twelvefold Way, which makes me smile.
However, some small part of the original Handbook's primacy as a reference comes from the fact that in a list of papers which are alphabetical by last name of the first author, it usually comes
first. The first editor of the new book is Frank Olver, so it won't have that advantage.
The Fibonacci cutting board is being sold by 1337motif at etsy. (Note: that's pronounced "leetmotif"; it took me a while to figure it out.) It's basically this tiling, where a rectangle of size F[n]
by F[n+1] is repeatedly decomposed into a square of size F[n] by F[n] and a rectangle of size F[n-1] by F[n], but made of wood instead of pixels.
There's also the double Fibonacci cutting board made in a similar pattern.
1337motif is Cameron Oehler's work. Nost of his other work is inspired by video games; you can see it here. I wonder how often the cutting boards get used as cutting boards; at $125, if I had one I'd
hang it on the wall and not get food on it. Personally, I'd like a Sierpinski triangle cutting board.
Proof Math is Beautiful is a blog of pictures which come from mathematics.
From the Chronicle of Higher Education: The Gospel of Well-Educated Guessing, on Sanjoy Mahajan's Street-Fighting Mathematics. (Previously: here, and here.) It's now a real book!
Here's a calculation I hadn't heard of before, and don't actually know the details of:
They were both right, in a sense: some of the calculations he pulls off have a hint of Houdini. For instance, he can start with two paper cones, to find the relation between drag force and
velocity, and—believe it or not—arrive at the cost of a round-trip plane ticket from New York to Los Angeles. He works out the problem in a blur of equations, remarking that a gram of gasoline
and a gram of fat contain the same amount of energy, that drag force is proportional to velocity squared, and so on. The number he arrives at ($700) isn't the cheapest deal out there, but it's
roughly right.
I've recently priced PHL-(SFO/OAK) flights, and this is roughly right. (And this uses chemistry, which is awesome because I was a chemist in a former life. Gasoline and fat are both basically long
chains of carbon atoms.) The article tells of other similar party tricks. It would be nice to see some details, but the Chronicle seems to pitch itself at a humanities-ish audience.
Jordan Ellenberg, in yesterday's Washington Post: The census will be wrong. We could fix it.
This continues a proud tradition of mathematicians whose area of expertise is nowhere near statistics writing newspaper pieces saying that statistical sampling in censuses a good idea; Brian Conrad,
1998, New York Times.
In some sense it carries more weight when mathematicians who don't spend most of their time battling randomness in some sort or another . Statisticians of course think that doing statistical
adjustments to the census in order to make it more accurate is a Good Idea; it gets them, their students, or their friends jobs!
As a combinatorialist I admire the theoretical elegance of our country's once-a-decade exercise in large-scale, brute-force combinatorics. But in practice, well, of course it needs some statistical
And here's something interesting:
Since 1970, a mail-in survey has provided the majority of census data, so what we enumerate is not people but numbers written on a form, which are as likely to be fictional as any statistical
I wonder if people are actually lying on their census forms. I suspect this would skew the count upwards. People who deliberately lie on their census forms, at least the sort of people I know, are
likely to give "joke" answers. And large numbers are funnier. I live in a one-bedroom apartment, and if I were the sort of person who lied on government forms I would easily say that ten people live
in my apartment. I can't give a comically low number of people living here, because the census insists that a positive integer number of people live in each place. Does the census has some sort of
way to correct for this?
Here's a cute little problem from Reddit: Tough question for you guys. Let's say you have 901 coins that come out to exactly $100. What are the odds? (Also here.)
Everyone there who gets a solution is assuming that all the possible coins are equally likely, which isn't a reasonable assumption. Years ago I looked at the density of money, where I used a model in
which I get back from each transaction n cents with probability 0.01, for n = 0, 1, ... 99; furthermore I always get back the smallest possible number of coins. The only coins allowed are pennies,
nickels, dimes, and quarters (worth 1, 5, 10, and 25 cents respectively).
As I calculated before, if I make 100 transactions, and I get each number of cents back exactly once, I'll get 200 pennies, 40 nickels, 80 dimes, and 150 quarters. This is a total of 470 coins, and
worth $49.50. Thus the "average coin" is worth 495/47 = 10.53 cents; 901 coins are "on average" worth $94.89. The value $100 isn't that unreasonable.
So consider a jar with 901 coins, which are independent; they each have probability 20/47 of being a penny, 4/47 of being a nickel, 8/47 of being a dime, and 15/47 of being a quarter. The mean value
of a coin is 495/47 = 10.53 cents; the variance is 238840/2309 = 108.12 "square cents".
The mean value of 901 coins, then, is 9489 cents; the variance is 93198 "square cents", so the standard deviation is 305 cents. (Everything here is rounded to the nearest integer.)
Invoking the central limit theorem, then, we say that the value of 901 randomly chosen coins is normally distributed with this mean and standard deviation. The probability of having value exactly
10,000 cents is approximated by the probability density function of this variable at 10,000; that's 0.000322, or 1 in 3101.
An exact answer is feasible -- but not worth computing, I'd say, because the error in the central limit theorem is surely much smaller than the error from the fact that this isn't a realistic model
of what actually ends up in your change jar.
From the April Notices of the AMS, John D'Angelo writes Baseball and Markov Chains: Power Hitting and Power Series. Consider the following simple model of baseball. Players only hit singles; three
singles score a run. That is, the third and every following player to get a hit in a given inning score a run. This can either be interpreted as that, say, all runners score from second on a single
or all runners go from first to third on a single -- but not both! -- or that every third hit is actually a double. (And I do mean exactly every third hit, not some random one-third of hits, so this
is a bit unnatural.) Then the expected number of runs per half inning is p^3(3p^2-10p+10)/(1-p). For real baseball the average number of runs per half-inning is around one half, which corresponds to
p = 0.361.
D'Angelo gives this as an exercise, but I independently came up with this model a while ago and can't resist sharing the solution. Let q = 1-p. The probability of getting k hits in an inning is p^k q
^3 -- that's the probability of getting those hits in a certain order -- times the number of ways in which k hits and 3 outs can be arranged. Since the last batter of an inning must get out, the
number of possible arrangements is the number of ways to pick 2 batters out of the first k+2 to get out, which is (k+2)(k+1)/2.
The probability of getting k runs, if k is at least 1, is just the probability of getting k+2 hits, which is p^k+2q^3(k+4)(k+3)/2. Call this f(k); then
f(1) + 2f(2) + 3f(3) + ... = p^3(3p^2-10p+10)/(1-p)
by some annoying algebra. I'm pretty sure I came up with this exact model while procrastinating from some real work a couple years ago; it's probably been independently reinvented many times.
With p = 0.361, the probabilities of scoring 0, 1, 2, 3, 4, 5 runs in an inning are .748, .123, .066, .034, .016, .008 (rounded to three decimal places). (Probabilities of larger numbers of runs can
also be calculated; together they have probability around .006.)
Assuming that each half-inning is independent, the probability G(k) of a team scoring k runs in a game is, for each k,
k 0 1 2 3 4 5
G(k) .073 .108 .129 .133 .124 .108
k 6 7 8 9 10 11
G(k) .088 .069 .052 .038 .026 .018
k 12 13 14 15 16 17
G(k) .012 .008 .005 .003 .002 .001
with probability about 0.0006 of scoring 18 runs or more. (This seems a bit low to me -- three times a season in the major leagues -- but after all this is a very crude model!) But one interesting
thing here is that the distribution of the number of runs per game, which is a sum of nine skewed distributions, is still skewed; the mode is 3, and the median 4. Recall that I chose p so that the
mean would be 4.5. And the actual distribution is similarly skewed.
Of course a more sophisticated model of baseball is as a Markov chain. There are twenty-five states in this chain -- zero, one or two outs combined with eight possible ways to have runners on base,
and three outs. We assume that each hitter hits randomly according to his actual statistics, and the runners move in the "appropriate" way. Of course determining what's appropriate here would be a
bit tricky. How do runners move? A runner is probably more likely to take an extra base when a power hitter is hitting, but the sample size for any individual is fairly small. But one could probably
predict from some measure of the hitter's power (say, the number of doubles and home runs, combined appropriately) the chances of a runner taking an extra base on a single. Something similar is
necessary for sacrifice flies (which have to be deep enough to score the runner), grounding into double plays, etc. I'm not sure if the Markov models that are out there, such as that by Sagarin, do
this. Sagarin computes the (offensive) value of a player by determining how many runs per game a team composed of only that player would score.
For the morbidly curious, here's my recently completed PhD thesis, Profiles of large combinatorial structures. (PDF, 1.1 MB, 262 pages (but double-spaced with wide margins)) This is why I haven't
been posting!
Abstract: We derive limit laws for random combinatorial structures using singularity analysis of generating functions. We begin with a study of the Boltzmann samplers of Flajolet and collaborators, a
useful method for generating large discrete structures at random which is useful both for providing intuition and conjecture and as a possible proof technique. We then apply generating functions and
Boltzmann samplers to three main classes of objects: permutations with weighted cycles, involutions, and integer partitions. Random permutations in which each cycle carries a multiplicative weight σ
have probability (1-γ)^σ of having a random element be in a cycle of length longer than γn; this limit law also holds for cycles carrying multiplicative weights depending on their length and
averaging σ. Such permutations have number of cycles asymptotically normally distributed with mean and variance ~ σ log n. For permutations with weights σ[k] = 1/k or σ[k] = k, other limit laws are
found; the prior have finitely many cycles in expectation, the latter around √n. Compositions of uniformly chosen involutions of [n], on the other hand, have about √n cycles on average. These can be
modeled as modified 2-regular graphs. A composition of two random involutions in S[n] typically has about n^1/2 cycles, characteristically of length n^1/2. The number of factorizations of a random
permutation into two involutions appears to be asymptotically lognormally distributed, which we prove for a closely related probabilistic model. We also consider connections to pattern avoidance, in
particular to the distribution of the number of inversions in involutions. Last, we consider integer partitions. Various results on the shape of random partitions are simple to prove in the Boltzmann
model. We give a (conjecturally tight) asymptotic bound on the number of partitions p[M](n) in which all part multiplicities lie in some fixed set n, and explore when that asymptotic form satisfies
log p[M](n) ~ π√(Cn) for rational C. Finally we give probabilistic interpretations of various pairs of partition identities and study the Boltzmann model of a family of random objects interpolating
between partitions and overpartitions.
What's the point of having two thousand readers if I can't ask a question like this once in a while?
I'm working on the final version of my dissertation -- the one I'll submit to the graduate school next week. The dissertation manual states that no text may appear in the margin area.
LaTeX, on the other hand, keeps wanting to put some pieces of mathematics, which appear inline, in the margins. (Presumably this is because this is "better" than the alternative of having very long
inter-word spaces.)
Two questions:
- is there some way to check that nothing's sticking out in the margin? (I thought this is what "overfull \hbox" meant, but the line numbers where those appear aren't the ones where I have this
problem.) There are some things that are just barely sticking out into the margin, and with thousands of lines total I don't trust my eye.
- once I find all the places where text protrudes into the margin, is there some way around this other than just inserting \newline every time this problem occurs? This creates its own problems.
I surely can't be the only person who's had this problem, but Google is failing me.
From the Daily Mail: New ash cloud could delay re-opening of London airports. We have this gem: "Critics said the agency used a scientific model based on 'probability' rather than fact to forecast
the spread of the ash cloud." See the Telegraph as well.
What else are they supposed to do? The agency here -- the Met Office, which is the national weather service of the UK -- doesn't know what the ash cloud is going to do. If they waited to see what the
cloud does, the planes would already be in the air. It would be too late.
There's a mathematical relationships search. It will tell you, for example, that academically, Max Noether is the first cousin of Emmy Noether. (Both of their advisors were students of Jacobi.) But
Michael Artin and Emil Artin aren't even related.
It's less amusing, of course, when you search for people that aren't related in the standard way. But Paul Erdos is my great-great-great-great-uncle. (You can't search for me yet in the Mathematics
Genealogy Project, which is where the data comes from; the link goes to the relationship between Erdos and another student of my advisor.)
The word "probability" does not appear in the Bible, or so we learn from Conservapedia's List of missing words in the Bible.
I can only conclude that Einstein was right, and God does not play dice.
Mathematical ancestors of Penn math faculty, from October 1999. (This was the department's 100th anniversary.) This lists the advisor's advisor's advisor's... until historical data that was (easily?)
available at that time gave out. The longest-ago person listed on this page is Otto Mencke (Ph. D. 1665, 12 generations from Penn professor Stephen Shatz); most chains die out in the 19th century.
As of right now, the math genealogy project claims to know that my advisor's 26th-generation advisor is Elissaeus Judaeus (who was a student in the 1380s). Most mentions of Judaeus on the Internet
seem to be by other people who have discovered this (Judaeus has 77000 or so mathematical descendents). But this post from the person who added him to the database gives some background -- he was for
the most part a philosopher, it seems. He is described as "a mysterious figure who may or may not have been a Jew". His student Gemistus Pletho seems a little better understood; Wikipedia says "He
was one of the chief pioneers of the revival of Greek learning in Western Europe." It seems that in that time a lot more data has been collected for the 14th through 17th centuries.
(As for me, hopefully in a few weeks it'll be possible to add me to the mathematical genealogy project. I defend on April 15.)
Perelman has been awarded the Millennium Prize. Press release from the Clay Math Institute.
As Peter Woit points out, no mention of the fate of the million dollars.
From quomodocumque: what would the NCAA tournament look like if every game were won by the college or university with the better math department? (Berkeley -- excuse me, "California", as they're
usually called in athletic contexts -- wins.)
Rather interestingly, 20 out of 32 first-round "games", and 37 out of 63 "games" overall -- more than half -- are won by the team that actually has the better seed in the basketball tournament. I
suspect this is because quality of math departments and basketball teams are both correlated with the size of the school. This is especially true because ties were broken by asking how many people
they could name at the school., which clearly has a bias towards larger departments.
I have a polynomial, P, with nonnegative integer coefficients. You want to know what it is. For any algebraic number x, you're allowed to ask me what P(x) is. How many questions do you have to ask me
to be sure that you know what P is?
Megabus.com adds Philadelphia-D.C. line, from the Daily Pennsylvanian.
We learn from a Megabus spokesperson that their vehicles use "less than a pint of fuel per passenger mile".
For those of you who don't have the misfortune of knowing this, there are eight pints in a gallon. So these busses get better than eight passenger-miles to the gallon!
Since most cars in the US get at least 20 miles or more to the gallon, this is really nothing to be proud of.
(I'm guessing that busses are actually more fuel-efficient than cars, at least if they run sufficiently full.)
You can actually buy a shirt with the Calkin-Wilf tree on it. I probably should buy it, if only so I can wear it when I inevitably give a talk on this subject again, either as a stand-alone talk
(I've done it twice) or when I teach it in a class.
This is an infinite binary tree of rational numbers. It has 1/1 at the root, and the children of r/s are r/(r+s) and (r+s)/s. It turns out that this tree contains every positive rational number
exactly once, giving an explicit enumeration of the rational numbers.
Also -- and this is what's really surprising to me -- let f(n) be the number of ways to write n as a sum of powers of 2, where no power of 2 can be used more than twice. Then (f(0), f(1), f(2), ...)
= (1,1,2,1,3,2,3,1,4,3,5,2,5,3,4,...). The sequence of numbers f(n)/f(n+1) are the rational numbers in that tree, in the order which they appear if you traverse each level from left to right.
I learned about the shirt from this mathoverflow thread, where Richard Stanley gives the theorem of the previous paragraph as an example of a theorem with an "unexpected conclusion".
See the original paper by Calkin and Wilf. I've mentioned it before here and here. I think I first heard of this paper from Brent Yorgey's series of posts expanding upon the paper, or perhaps from
Mark Dominus' blog. (Somewhat coincidentally, Yorgey and Dominus both live in Philadelphia. I claim this is coincidence, even though Wilf is at Penn, because I don't think either of them heard about
the paper from Wilf.)
And can anything nice be said about the number of ways to write n as a sum of powers of 3, where no power of 3 can be used more than three times?
Here's something interesting: lots of people, when asked by the US Census Bureau "how much money do you make?", round to the nearest five thousand dollars.
See the data tables from the 2006 census. These give the number of people whose personal income is in each interval of the form [2500N, 2500N+2499], for integer N.
One sees, for instance, that the number of people making between $27,500 and $29,999 (which is near the mode of the distribution) is less than both those making $25,000 to $27,499 and those making
$30,000 to $32,499. Something similar occurs at all income levels -- the number of people making between 2500N and 2500(N+1)-1 dollars is smaller if N is odd (and thus this interval doesn't contain a
multiple of 5000) than if N is even (and so it does).
Surprisingly, the effect occurs even at very low levels of earnings. If you make $87,714 in a year I can see rounding to $90,000 -- but is the person who makes $7,714 in a year really rounding to
(I found this while trying to answer a question at Metafilter: How many people in the United States make more than $10,000,000 per year?. I seem to recall reading somewhere that personal income
roughly follows a power law in the tails, but can't actually find a reference for this.)
There also seems to be a preference for multiples of $10,000 over multiples of $5,000 that are not multiples of $10,000. But I have work to do, so I'm not going to do the statistics.
Apparently this is a puzzle blog now.
This morning at 9:30 I picked up my watch and turned it upside down. It appeared to read 4:00. The hour hand was actually in the position it would be at at 3:30, of course. But the minute hand was
pointing straight up, so the time must be on the hour. Since I could easily tell that the hour hand wasn't pointing directly to the right, I suppose my brain interpreted it as 4:00 instead of 3:00.
Of course I did not think any of this consciously, but only reconstructed it after the fact; my thought process was more like "hey, 9:30 upside down looks like 4:00. that's weird."
Is it ever possible to interpret the hands of a clock, turned upside-down (i. e. rotated by 180 degrees), unambiguously as a time? (Fudging like my brain apparently did this morning does not count.)
Since I seem to be getting this question a lot lately: yes, I'm still alive! But I am writing a dissertation. (And waiting to hear back from places where I applied for jobs, but that is not an excuse
because those applications are already out there.)
Here's a puzzle I heard a couple weeks ago. You have a 10-by-10 grid of lights. Some of them are on, some are off. You are allowed to make the following moves:
(a): you pick one of the lights which is the center of some 3-by-3 square (i. e. it is not on the edge of the grid) and switch all the lights in that 3-by-3 square (on becomes off, off becomes on).
(b): like in (a), but with a 5-by-5 square.
Is it possible to get from an arbitrary starting position to the all-off position?
Orin Kerr at the Volokh Conspiracy points to arguments in a U. S. Supreme Court case yesterday which used the word "orthogonal" in the technical-jargon sense defined, say, at the Jargon File. (See
page 24 of the original transcript.) There's a follow-up here by Eugene Volokh, basically saying that there's no point in using big words if your audience doesn't understand them. (And the justices
did stop to ask what the word meant.)
From Abstruse Goose: How most mathematical proofs are written, dramatized as people driving around and getting lost.
Sometimes I've wondered what an actual map of the various possible proofs of certain results would look like.
Sunday's New York Times has a bunch of magic tricks based on simple algebra, by Arthur Benjamin.
For some magic tricks based on "deep" mathematics, check out this mathoverflow thread. Rumor has it that Persi Diaconis thinks there's no such thing, though, and he would know. | {"url":"https://godplaysdice.blogspot.com/2010/","timestamp":"2024-11-10T22:21:56Z","content_type":"text/html","content_length":"193947","record_id":"<urn:uuid:0f2a8f42-bc0f-4ba7-b818-6a62975aa496>","cc-path":"CC-MAIN-2024-46/segments/1730477028191.83/warc/CC-MAIN-20241110201420-20241110231420-00122.warc.gz"} |
What is Mass: A Comprehensive Guide - Measuring Expert
Mass is a measure of the amount of matter in an object or substance, commonly expressed in units such as grams or kilograms. Mass is a fundamental property of matter that plays a crucial role in
various scientific and engineering applications.
It determines the strength of gravitational attraction exerted by an object, and it is a key factor in determining how objects behave under the influence of external forces. Mass is a concept that is
closely related to weight, but it is not the same thing as weight, which is a measure of the force exerted by gravity on an object. In this article, we will explore the nature of mass and its various
applications in the fields of physics, engineering, and more. We will also delve into the different units of mass that are commonly used, as well as how mass is measured and calculated in different
Credit: www.text-em-all.com
The Properties Of Mass
Mass is an essential concept in our understanding of the physical world. It refers to the amount of matter in an object. The properties of mass are worth discussing in detail because they help us
comprehend the behavior of objects, both big and small.
Here, we’ll explore three key characteristics of mass: inertia, gravitational attraction, and energy equivalence.
Inertia: A Property Of Mass That Resists Changes In Motion
Inertia is one of the most fundamental properties of mass. It describes the tendency of an object to resist changes in its motion, whether it is at rest or moving.
• The mass of an object determines its inertia. The more massive an object is, the harder it is to move or stop.
• Inertia affects both rest and motion. An object at rest will remain at rest unless acted upon by an external force, and an object in motion will continue to move in a straight line at a constant
speed unless acted upon by an external force.
• Inertia is a crucial factor in vehicle safety. For instance, cars with greater mass are safer in collisions because they experience less change in momentum than lighter cars.
Gravitational Attraction: How Mass Attracts Other Masses
Another noteworthy property of mass is its ability to attract other masses. This force is known as gravitational attraction.
• The strength of the gravitational attraction between two objects depends on their masses and the distance between them. The greater the masses of the objects, the greater the gravitational force,
while the greater the distance between them, the weaker the gravitational force.
• Gravitational attraction is the reason why objects have weight. The weight of an object is the force of gravity acting on it and is directly proportional to its mass.
• Gravitational attraction also governs the movement of celestial objects. For example, the gravitational pull of the sun keeps the planets in our solar system in orbit.
Energy Equivalence: The Relationship Between Mass And Energy
Lastly, we have energy equivalence, which refers to the relationship between mass and energy. This idea is famously encapsulated in einstein’s e=mc² equation.
• The equation shows that mass and energy are interchangeable. It means that mass and energy are two sides of the same coin and can both be converted into the other.
• When particles collide or fuse, or when nuclear reactions occur, some of the mass is transformed into energy. This idea is commonly used in nuclear power plants and atomic weapons.
• Energy equivalence is critical to understanding the behavior of subatomic particles, the structure of stars, and the evolution of the universe.
Mass is an essential concept that plays a significant role in describing the world we live in. The properties of mass, namely inertia, gravitational attraction, and energy equivalence, provide us
with an understanding of how mass behaves under different scenarios.
By keeping these characteristics in mind, we can better analyze and make sense of the physical world around us.
Measuring Mass
Mass is a fundamental property of matter that measures the amount of substance present in an object. It is usually measured in kilograms or grams, but the units of mass vary depending on the scale of
the object being measured.
Mass is an important physical property used in scientific experiments, engineering, and everyday life.
Units Of Mass
The metric system uses the international system of units (si) as the standard unit for measuring mass. The si unit of mass is the kilogram (kg). Other units of mass commonly used in everyday life
include grams (g) and milligrams (mg).
• 1 kg = 1000 g
• 1 g = 1000 mg
• 1 kg = 2.
Tools For Measuring Mass
Various tools are available for measuring mass, from simple balances to highly sophisticated instruments used in scientific research.
• Balance scales: these instruments use a beam and a counterweight to measure mass. The object being measured is placed on one side of the beam, and weights are added to the other side until the
beam is balanced.
• Spring scales: these devices use a spring and a hook to measure mass. The object being measured is hung from the hook, and the scale provides a reading based on the amount of spring compression.
• Digital scales: these instruments use electronic sensors to measure a weight reading based on the amount of pressure applied to a weighing platform.
Differences Between Mass And Weight
Mass and weight are not the same thing, even though they are often used interchangeably. Mass is a measure of an object’s quantity of matter, while weight is a measure of the force of gravity acting
on an object’s mass.
The mass of an object is constant, whereas its weight varies with changes in gravity.
• Mass is measured in kilograms or grams, while weight is measured in newtons or pounds.
• Mass is an intrinsic property of an object, while weight depends on the object’s location and the strength of the gravitational force acting on it.
• Mass can be measured using a balance scale, while weight requires a spring scale or a digital scale that can measure force.
Measuring mass is an essential process for various applications, from scientific experiments to everyday life. By understanding the units and tools used for measuring mass, and the differences
between mass and weight, you can have a deeper appreciation for the properties of matter that surround us every day.
A Step-by-Step Guide to the Catholic Mass
Mass In Physics
Mass is a fundamental concept in physics and is defined as the measure of an object’s resistance to changes in motion. In simpler terms, mass refers to the amount of matter present in an object. Mass
is a scalar quantity, meaning it only has a magnitude and no direction.
In this section, we will discuss the concept of mass as it pertains to classical mechanics, relativistic physics, and quantum mechanics.
Mass In Classical Mechanics
Classical mechanics is the branch of physics that deals with the motion of macroscopic objects, i. E. Objects large enough to be seen with the naked eye. In classical mechanics, mass is a constant
that remains the same regardless of the object’s motion.
• Mass is an inherent property of an object and remains constant in both inertial and non-inertial reference frames.
• Mass is measured in units of kilograms (kg) in the international system of units (si).
• According to newton’s second law of motion, the force acting on an object is directly proportional to the object’s mass and acceleration. This law is expressed mathematically as f = ma, where f
is the force applied, m is the mass of the object, and a is its acceleration.
Mass In Relativistic Physics
Relativistic physics deals with the motion of objects traveling at extremely high speeds, approaching the speed of light. In relativistic physics, mass is not constant and changes with the object’s
• Mass is not an invariant quantity in relativistic physics, and its value depends on the relative velocity between the observer and the object in question.
• The concept of relativistic mass is introduced to account for the changes in an object’s inertia with its speed. Relativistic mass is given by the formula m_rel = m_0 / √(1-(v^2/c^2)), where m_0
is the rest mass of the object, v is its velocity, c is the speed of light, and √(1-(v^2/c^2)) is the lorentz factor.
• While relativistic mass can change with the object’s motion, the rest mass of an object remains constant and is a measure of the amount of matter present in it.
Mass In Quantum Mechanics
Quantum mechanics is the branch of physics that deals with the behavior of matter and energy at the atomic and subatomic level. In quantum mechanics, mass is quantized, meaning it can only take on
certain discrete values.
• Mass is described as a wave function in quantum mechanics, and its value is related to the eigenvalues that result from solving the schrodinger equation.
• The mass of a particle is related to its energy through einstein’s famous equation, e = mc^2, where e is the energy of the particle, m is its mass, and c is the speed of light.
• Quantum mechanics introduces the concept of ‘mass-energy equivalence’, where mass can be converted to energy and vice versa. This concept is essential in understanding phenomena like nuclear
reactions and particle physics.
Mass is a fundamental concept in physics that is central to our understanding of the behavior of matter and energy at all scales. Its definition and properties change depending on the context in
which it is used, and a deep understanding of mass is necessary to make progress in the field of physics.
Applications Of Mass
Mass is a fundamental concept in physics that refers to the amount of matter in an object. It is a crucial property that helps us understand the behavior of many physical systems. In this section, we
delve into the applications of mass, including its role in the study of celestial bodies, chemical reactions, and electrical engineering.
The Role Of Mass In The Study Of Celestial Bodies
Mass plays a key role in the study of celestial bodies, including planets, moons, and stars. Understanding the mass of these objects helps astronomers to make predictions about their behavior, such
as their orbits, gravitational pulls, and likelihood of collisions.
• Measuring the mass of planets: by measuring the gravitational force between two orbiting objects, astronomers can determine the mass of a planet. This method has been used to calculate the mass
of all the planets in our solar system.
• Studying the behavior of stars: the mass of a star determines its life cycle, including its size, luminosity, and how it will eventually die. By studying the masses of different stars,
astronomers can better understand the evolution of the universe.
• Predicting asteroid impacts: knowing the mass of an asteroid is crucial for predicting the potential impact it could have on earth. By calculating the asteroid’s mass and trajectory, scientists
can assess the potential damage it may cause.
The Use Of Mass In Chemical Reactions
Mass is essential in understanding chemical reactions, from basic reactions in your kitchen to more complex industrial processes.
• Measuring quantities of reactants and products: chemists use mass measurements to determine the quantity of reactants and products in a chemical reaction. This information is crucial for
predicting reaction yields and optimizing industrial processes.
• Determining the mass of molecules: knowing the mass of a molecule allows chemists to determine its composition and properties. Mass spectrometry, one technique used to measure molecular masses,
has revolutionized the field of chemistry, allowing scientists to study even the most complex molecules.
• Calculating stoichiometry: the law of conservation of mass states that the total mass of the reactants must equal the total mass of the products in a chemical reaction. This principle is used to
calculate the stoichiometry of reactions, which helps chemists to determine the correct ratio of reactants needed for a particular process.
Mass In Electrical Engineering
Mass is less commonly associated with electrical engineering, but it is still a crucial concept in the field.
• Designing electrical components: the mass of electrical components, such as wires and circuit boards, is a crucial factor in determining their performance and durability. Designers must consider
the mass of these components when selecting materials and designing systems.
• Calculating electromagnetic forces: the mass of charged particles is included in the equations used to calculate electromagnetic forces. These forces are essential for understanding everything
from basic electrical circuits to more complex phenomena, such as electromagnetic waves and radiation.
• Balancing rotating machinery: in rotating machinery, such as electric motors, the mass of the rotor and other components can affect the balance of the system. Designers must consider the mass of
these components to prevent vibration and other issues that can affect the performance of the machine.
Mass is a fundamental concept in physics that has many applications across a wide range of fields, from astronomy to chemistry to electrical engineering. Understanding the role of mass in these
fields allows us to make predictions, optimize processes, and design better systems.
Frequently Asked Questions For What Is Mass
What Is Mass In Physics?
Mass refers to the amount of matter in an object. It is a fundamental property of objects and is measured in kilograms. Mass is not the same as weight, which is the force exerted on an object due to
### how is mass calculated? Mass can be calculated by using the formula mass = density x volume. The density of an object tells us how much mass is contained in a given volume. By multiplying the
density by the volume, we can determine the mass of the object.
### what is the importance of mass in physics? Mass is an essential concept in physics because it determines the behavior of objects when they interact with each other. Objects with more mass have a
greater gravitational attraction than those with less mass.
Additionally, the mass of an object determines how much it resists changes in its motion, as described by newton’s laws.
Can Mass Be Created Or Destroyed?
No, mass cannot be created or destroyed. This is known as the law of conservation of mass, which states that in a closed system, mass is neither created nor destroyed, only transferred from one form
to another.
After exploring the basics of mass, we can understand how essential it is in our everyday life. From the smallest atom to the biggest star, mass exists everywhere. It’s the cornerstone of the
universe’s existence, and without it, life as we know it would not be possible.
Through studying mass, scientists have been able to develop technology that has revolutionized the way we live. We’ve seen advancements in transportation, medicine, and even space exploration. Mass
is a fundamental property in physics that not only describes how objects interact with one another but also how the universe is structured.
It’s exciting to contemplate what mysteries of the universe we can unlock in the future by digging deeper into the properties of mass. By understanding the basics of mass, we can take our first steps
in unlocking the secrets of the universe.
Rakib Sarwar is a seasoned professional blogger, writer, and digital marketer with over 12 years of experience in freelance writing and niche website development on Upwork. In addition to his
expertise in content creation and online marketing, Rakib is a registered pharmacist. Currently, he works in the IT Division of Sonali Bank PLC, where he combines his diverse skill set to excel in
his career. | {"url":"https://www.measuringexpert.com/what-is-mass/","timestamp":"2024-11-04T09:17:50Z","content_type":"text/html","content_length":"135248","record_id":"<urn:uuid:d30b13f8-5036-42ab-800b-62cd638a73cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00819.warc.gz"} |
Set (mathematics) in English - dictionary and translation
, a
is a collection of distinct objects, considered as an
in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written
{2,4,6}. Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century,
set theory
is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In
mathematics education
, elementary topics such as
Venn diagrams
are taught at a young age, while more advanced concepts are taught as part of a university degree. The German word
, rendered as "set" in English, was coined by
Bernard Bolzano
in his work
The Paradoxes of the Infinite | {"url":"http://info.babylon.com/onlinebox.cgi?cid=CD566&rt=ol&tid=pop&x=20&y=4&term=Set%20%28mathematics%29&tl=English&uil=Hebrew&uris=!!ARV6FUJ2JP","timestamp":"2024-11-14T14:50:25Z","content_type":"text/html","content_length":"6955","record_id":"<urn:uuid:1b46eec8-735f-4293-a130-86415702a6a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00035.warc.gz"} |
Demo model "Vector Control of an Induction Machine"
Dear plexim team
I am not sure if I understand the equivalent circuit in the demo model “Vector Control of an Induction Machine” correctly (see attachment). I only know the static equivalent circuit per phase (see
attachment). Have I correctly interpreted the relationships between the simulation parameters in the demo model and the static equivalent circuit?
Rs = Rs
Rr = Rr
Lls = L_sigma_s
Llr = L_sigma_r
Lm = Lh
Is Rfe being neglected?
Thanks for your help.
There is a difference between the models you’re showing. As you suggested, the “Static Equivalent Circuit” you’re referring to is a per-phase equivalent circuit in steady state based on motor slip.
The circuit model in the demo model is in the DQ reference frame aligned with the rotor flux. Lastly, the PLECS machine model is actually implemented in the alpha-beta reference frame.
However, in the initialization script you can see the equivalent model parameters from the PLECS machine model to the DQ reference frame aligned with rotor flux equivalent model. For example, refer
to the calculation of “Lsigma” and “RR” or “Rtot”. You are correct in that Rfe is neglected in both cases.
Hello Bryan
Thanks for your reply.
That is, the conversion from PLECS machine model in DQ reference frame aligned to the rotor flux equivalent model is done within the simulation parameters.
On a real machine, using an open circuit measurement and a short circuit measurement, I can determine the equivalent circuit parameters of the static equivalent circuit per phase. How do I calculate
the parameters of the PLECS machine model from these parameters? Do I have to convert them at all or are they the same? The two equivalent circuit diagrams are again attached to the appendix.
Thanks for your time. | {"url":"https://forum.plexim.com/t/demo-model-vector-control-of-an-induction-machine/993","timestamp":"2024-11-02T14:25:17Z","content_type":"text/html","content_length":"30389","record_id":"<urn:uuid:d9adccbf-ed63-444b-8103-f4281f300a65>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00330.warc.gz"} |
MAGOMECHAYA MINSHUKU Book Archive
Infinite-dimensional Lie algebras by Iain Gordon
Read or Download Infinite-dimensional Lie algebras PDF
Best linear books
Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics
This publication is meant as an introductory textual content as regards to Lie teams and algebras and their function in numerous fields of arithmetic and physics. it's written by means of and for
researchers who're basically analysts or physicists, now not algebraists or geometers. now not that we've got eschewed the algebraic and geo metric advancements.
Dimensional Analysis. Practical Guides in Chemical Engineering
Useful publications in Chemical Engineering are a cluster of brief texts that every presents a targeted introductory view on a unmarried topic. the whole library spans the most subject matters within
the chemical procedure industries that engineering execs require a uncomplicated knowing of. they're 'pocket guides' that the pro engineer can simply hold with them or entry electronically whereas
Can one examine linear algebra exclusively by means of fixing difficulties? Paul Halmos thinks so, and you may too when you learn this ebook. The Linear Algebra challenge booklet is a perfect textual
content for a path in linear algebra. It takes the coed step-by-step from the fundamental axioms of a box in the course of the concept of vector areas, directly to complicated options equivalent to
internal product areas and normality.
Extra info for Infinite-dimensional Lie algebras
Example text
Carter: Lie algebras of finite and affine type. ): Kac-Moody and Virasoro algebras. • Kumar: Kac-Moody groups, their flag carieties and representation theory. • E. Frenkel: Langlands correspondence
for loop groups. • Pressley, Segal: Loop Groups. • Humphreys: Lie algebras. • Segal, Wilson: ? (probably IHES) • Deodhar, Gabher, Kac: Adv. Math. 45 (1982). • Kumar: J. Algebra 108 (1987) B
Notational reference Witt the Witt algebra over C Vir the Virasoro algebra over C, a central extension of Witt Lie(G) the Lie algebra of the Lie group G, g = Te G g either a general Lie algebra or a
finite-dimensional, simple one g a central extension of a Lie algebra g; 0 → z → g → g → 0 Lg the loop algebra g ⊗ C[t , t −1 ] Lg =Lg d Cd , where d = t dt ∈ Der C[t , t −1 ] Note: H 2 L g; C ∼ =C∼
= H 2 L g; C Lg = L g ⊕ Cd ⊕ Cc, the central extension of L g, affine Kac-Moody Lie algebra L (A) the Kac-Moody Lie algebra corresponding to the generalised Cartan matrix A U (g) the universal
enveloping algebra of the Lie algebra g; U (g) = T (g) (x ⊗ y − y ⊗ x − [x, y]) 〈−, −〉 a positive-definite sesquilinear form on a representation V , non-degenerate if V is unitary A the affine,
untwisted gen.
First calculate the “leading term” and get ∞ mult(α)P (η−nα) α∈∆re + n=1 hα . e. λ + ρ, β = 21 〈β, β〉. So if this cannot be satisfied, then M (λ) is irreducible and so F (λ) is non-degenerate;
hence det F η is a product of linear factors h β + 〈ρ − 21 β, β〉 , and the formula for leading terms shows that β = nα for some α ∈ ∆+ , which we call a quasi-root. So we get products of h α + 〈ρ
− 21 α, α〉 . 50 Infinite-dimensional Lie algebras 51 The trick. 2), and F η is a product of finitely many linear terms of the form h α + ρ − nα 2 ,α .
U 0 ∧ u −1 ∧ u −2 ∧ · · · : u −m = v −m for m 46 0 , Infinite-dimensional Lie algebras 47 where v 0,−1,−2,... is the vacuum vector in degree 0. ) ∼ = PΩ −−→ Gr , u 0 ∧ u −1 ∧ u −2 ∧ · · · → Cu i . i
≤0 There are two operators, F (0) → F (1) and F (0) → F (−1) , respectively via V andV ∗ (the restricted dual, cf. 3), defined as follows. v i 0 ∧ v i 1 ∧ · · · = v ∧ v i 0 ∧ v i 1 , and (−1) j v i 0
∧ v i 1 ∧ · · · ∧ v i j −1 f (v i j ) ∧ v i j +1 ∧ · · · . v i 0 ∧ v i 1 ∧ · · · = j ≥0 The element E i j acts on F (0) via v i v ∗j .
Rated of 5 – based on votes | {"url":"http://en.magomechaya.com/index.php/epub/infinite-dimensional-lie-algebras-edition-version-24-feb-2009","timestamp":"2024-11-08T07:19:37Z","content_type":"text/html","content_length":"28768","record_id":"<urn:uuid:ac36c70b-5c04-45e7-8757-d896aab15e13>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00516.warc.gz"} |
multi-part problems: printing + weight
Following the instructions in http://webwork.maa.org/wiki/SequentialProblems we have successfully created multi-part problems. We have two questions, about (disabling) printing and about weight of
the problem. We are using webwork v2.5.
(1) When we print out the hardcopy only the first part appears -- as expected, since a student should only see one part at a time -- but it also prints out the answers to all the internal variables
(thereby defeating the problem):
¡input type=”hidden” name=”r1” value=”5” /¿¡input type=”hidden” name=”q1” value=”3” /¿¡input type=”hidden” name=”q2” value=”2” /¿¡input type=”hidden” name=”r2” value=”1” /¿¡input type=”hidden” name=
”gcd” value=”1” /¿
Is there a way to disable the printing of multi-part problems only on hardcopies and replace each one with a message like "this is a multi-part problem and webwork will not generate a hard copy of
this problem"? Even just disabling the printing alone will be extremely useful.
(2) By default each webwork problem has a "weight" of 1. For multi-part problems I would like to change that to "2" say. I know that instructors can change that in the homework set editor on a per
problem basis; can we hardcore this factor of "2" into the specific problem? Thanks! | {"url":"https://webwork.maa.org/moodle/mod/forum/discuss.php?d=3006&parent=7282","timestamp":"2024-11-13T17:51:38Z","content_type":"text/html","content_length":"67212","record_id":"<urn:uuid:eff8df20-322d-4b5c-b0e9-44c7c70b1589>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00032.warc.gz"} |
"Overreaction" of Asset Prices in General Equilibrium
We attempt to explain the overreaction of asset prices to movements in short-term interest rates, dividends, and asset supplies. The key element of our explanation is a margin constraint that traders
face which limits their leverage to a fraction of the value of their assets. Traders may lever themselves, furthermore, either directly by borrowing short term or indirectly by engaging in futures
and options trading, so that the scenario is relevant to contemporary financial markets. When some shock pushes asset prices to a low enough level at which the margin constraint binds, traders are
forced to liquidate assets. This drives asset prices below what they would be with frictionless markets. Also, a shock which simply increases the likelihood that the margin constraint will bind can
have a very similar effect on asset prices. We construct a general equilibrium model with margin constrained traders and derive some qualitative properties of asset prices. We present an analytical
solution for a deterministic version of the model and a simple numerical computation of the stochastic version.Journal of Economic LiteratureClassification Numbers: G1, E0.
• Asset pricing
• Financial constraints
• General equilibrium
ASJC Scopus subject areas
• Economics and Econometrics
Dive into the research topics of '"Overreaction" of Asset Prices in General Equilibrium'. Together they form a unique fingerprint. | {"url":"https://nyuscholars.nyu.edu/en/publications/overreaction-of-asset-prices-in-general-equilibrium","timestamp":"2024-11-03T23:30:23Z","content_type":"text/html","content_length":"52326","record_id":"<urn:uuid:f6b05e72-b338-4794-873a-89fcb969255c>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00569.warc.gz"} |
Tips formula is wrong?
Maybe I'm missing something? But the formula is confusing. At the least, it seems to contradict the text immediately prior that (correctly) states two speed 3 modules will boost output from 2/sec to
4/sec. The formula produces only 3/sec with two speed 3. This is because it sticks the needed +1 in with the separate Beacon calculation later. However, this is incorrect, because while you can have
zero Beacons and still produce positive crude, you cannot have zero pumpjacks. The formula is wrong if you plug zero in for the number of Beacons.
Updating, hopefully correctly! --777 (talk) 06:48, 1 February 2019 (UTC)
Less than 2 oil per sec
I'm currently playing in a world with ressources set to minimum and I have a pumpjack that output less than 2 oil per second, because the initial yield is lower than 20%. While the information on
this page does not technically contradict this, it leaves the reader under the impression that 2 oils per sec is the minimum yield. I don't know if initial yield lower than 20% is worth mentioning in
this page, I was pretty unsettled when I saw yields lower than 2oil/s and my reflex was to look the wiki, and it was unhelpful for once.
I'll add that info once I find where it fits.
--42xel (talk) 10:00, 21 June 2019 (UTC) | {"url":"https://stable.wiki.factorio.com/Talk:Pumpjack","timestamp":"2024-11-09T10:58:59Z","content_type":"text/html","content_length":"26053","record_id":"<urn:uuid:a86db60d-84dd-4a3a-b521-c0060f12f649>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00069.warc.gz"} |
Extracting the Maximum Number from a DataFrame Containing Strings and NaN Values
What will you learn?
Discover how to extract the maximum number from a DataFrame that includes a mix of strings and NaN values.
Introduction to the Problem and Solution
Imagine having a DataFrame with various data types like strings and NaN values. The challenge is to pinpoint the highest numerical value within this diverse dataset. The solution involves delving
into each element, extracting numbers from strings, handling NaN values appropriately, and ultimately determining the maximum number present.
To tackle this problem effectively: 1. Iterate through each element in the DataFrame. 2. Extract any numbers embedded within string elements using regular expressions. 3. Filter out NaN values to
focus solely on numeric data. 4. Calculate the maximum number among these extracted values.
import pandas as pd
import numpy as np
import re
# Sample DataFrame with mixed data types including strings and NaN values
data = {'col1': ['abc', '123', 'def', np.nan, '456']}
df = pd.DataFrame(data)
# Extracting numbers from strings using regular expressions
numbers = df['col1'].str.extractall(r'(\d+)').astype(float)[0]
# Filtering out NaN values
numbers = numbers.dropna()
# Calculating the maximum number in the extracted series
max_number = numbers.max()
# Copyright PHD
• Import Necessary Libraries: We import pandas, numpy, and re for efficient data manipulation.
• Regular Expression (Regex): Utilizing regex pattern (\d+), we extract all consecutive digits within each string.
• Type Conversion: Converting extracted numeric strings to float type facilitates mathematical operations.
• Filtering NaN Values: Eliminating rows with NaN ensures accurate results when finding the maximum number.
• Finding Maximum: By applying .max() on our filtered numeric Series, we obtain the highest number.
How does regex work in extracting numerical digits?
Regex enables us to define patterns for text matching. In our case, (\d+) captures one or more consecutive digits within a string.
Why is type conversion necessary after extracting numeric substrings?
Converting extracted numerical substrings into float format is essential for performing arithmetic operations like finding the max value.
What happens if we don’t filter out NaN values before finding max?
Including NaNs in calculations could lead to incorrect results or errors since they represent missing or undefined data points.
Can this method handle negative numbers within strings?
Yes, by adjusting our regex pattern accordingly (e.g., -?\d+), negative integers can also be captured during extraction.
Is there an alternative approach without using regex for extraction?
Although less concise, manual iteration over characters in each string could be employed to identify numeric sequences without regex usage.
How would you modify code if multiple columns contained mixed data types?
Extending similar logic across multiple columns involves iterating through each column individually while applying extraction and filtering steps accordingly.
What modifications are needed if decimal numbers are included in strings?
Altering our regex pattern to account for decimals (\d+\.\d+) enables capturing floating-point numbers during extraction process.
Does this solution account for scenarios where no valid number exists in any string cell?
If none of the cells contain parseable numerical content after extraction and filtering stages, result would be NULL indicating absence of valid numbers within dataset.
Are there performance considerations when dealing with large datasets using this method?
For substantial datasets comprising numerous rows/columns with complex entries requiring extensive processing due diligence must be exercised regarding memory consumption / computational efficiency
while utilizing such solutions.
In conclusion…
Leave a Comment | {"url":"https://pythonhelpdesk.com/2024/02/25/title-35/","timestamp":"2024-11-02T08:00:20Z","content_type":"text/html","content_length":"43010","record_id":"<urn:uuid:9aad9d78-0172-4c46-a5b7-2ef59313a60b>","cc-path":"CC-MAIN-2024-46/segments/1730477027709.8/warc/CC-MAIN-20241102071948-20241102101948-00804.warc.gz"} |
Managing SSL/TLS protocol versions in TIBCO ActiveMatrix BusinessWorks™ 5
This article explains how to manage SSL/TLS protocol versions in TIBCO ActiveMatrix BusinessWorks™ 5 (BW).
TLS protocol versions enabled by default in BW environments
The TLS protocol versions enabled by default in a BW environment vary based on the JRE version.
Let’s take the case of BW 5.15.0, which uses Java 11. The property jdk.tls.disabledAlgorithms in the security properties file (TIBCO_HOME/tibcojre64/11/conf/security/java.security) shows which
protocol versions are disabled.
jdk.tls.disabledAlgorithms=SSLv3, TLSv1, TLSv1.1, RC4, DES, MD5withRSA, \
DH keySize < 1024, EC keySize < 224, 3DES_EDE_CBC, anon, NULL, \
include jdk.disabled.namedCurves
By default, SSLv3, TLS 1.0 and TLS 1.1 are disabled on JRE level and only TLS 1.2 and TLS 1.3 are enabled. So, by default, BW 5.15.0 can use TLS 1.2 or TLS 1.3.
TLS protocol version used in a TLS session
The TLS protocol version that is used in a TLS session depends on what protocol versions are supported by the two sides of the connection. Let’s say, BW (Send HTTP Request activity) is connecting to
a web server over TLS in a BW 5.15 environment where TLS 1.2 and TLS 1.3 are enabled. If the web server supports TLS 1.3, it will be used for the connection. On the other hand, if TLS 1.2 is the
highest version supported by the web server, TLS 1.2 will be used.
How to check the enabled TLS protocol versions and the version used in a TLS session
If BW is the client, to identify enabled TLS protocol versions, check TLS debug logs. The ClientHello handshake message shows the enabled TLS protocol versions. The sample log given below shows that
TLS 1.3 and TLS 1.2 are enabled.
"ClientHello": {
"supported_versions (43)": {
"versions": [TLSv1.3, TLSv1.2]
If BW is the server, the utility sslscan (https://github.com/rbsec/sslscan/releases) can be used to check the enabled TLS protocol versions. The sample output given below shows that TLS 1.2 and TLS
1.3 are enabled.
$sslscan localhost:9191
Version: 2.1.3 Windows 64-bit (Mingw)
OpenSSL 3.0.9 30 May 2023
Connected to ::1
Testing SSL server localhost on port 9191 using SNI name localhost
SSL/TLS Protocols:
SSLv2 disabled
SSLv3 disabled
TLSv1.0 disabled
TLSv1.1 disabled
TLSv1.2 enabled
TLSv1.3 enabled
To identify the TLS protocol version that is used in a TLS session where BW is the client or server, check TLS debug logs. The ServerHello handshake message shows the selected TLS protocol version.
The sample log given below shows that the selected version is TLS 1.3.
"ServerHello": {
"supported_versions (43)": {
"selected version": [TLSv1.3]
Disabling a TLS protocol version that is enabled by default
A TLS protocol version may be disabled on JRE level or application level.
JRE level
To disable a specific TLS protocol, add it to the property jdk.tls.disabledAlgorithms in the security properties file. For example, to disable TLS 1.2, update the property as follows.
jdk.tls.disabledAlgorithms=SSLv3, TLSv1, TLSv1.1, TLSv1.2, RC4, DES, MD5withRSA, \
DH keySize < 1024, EC keySize < 224, 3DES_EDE_CBC, anon, NULL, \
include jdk.disabled.namedCurves
Application level
In cases where TLS is handled by BW, it is possible to disable TLS protocols separately on client side and server side using the following properties.
For example, the following property can be used to disable TLSv1.2 on client side in a BW version where TLSv1.2 is enabled by default.
Sample log that shows the TLS protocol versions that are enabled by default in BW 5.15 environment.
"ClientHello": {
"supported_versions (43)": {
"versions": [TLSv1.3, TLSv1.2]
Sample log with the property com.tibco.security.ssl.client.EnableTLSv12 set to false. Only TLSv1.3 is enabled.
"ClientHello": {
"supported_versions (43)": {
"versions": [TLSv1.3]
In cases where TLS is handled by a third-party library, use the setting provided by the third-party library. For example, when using MySQL Connector/J JDBC driver version 8.x to connect to MySQL
server over TLS, the TLS protocol versions TLS 1.2 and TLS 1.3 are enabled by default. The driver configuration property tlsVersions can be used to restrict TLS protocol versions. To disable TLS 1.2
and use only TLS 1.3, set the property to TLSv1.3 in the JDBC URL as shown below.
Enabling a TLS protocol version that is disabled by default
Sometimes, it may be necessary to enable a specific TLS protocol version that is disabled by default. To enable a specific TLS protocol, remove it from the property jdk.tls.disabledAlgorithms in the
security properties file. For example, to enable TLS 1.1, update the property as follows.
jdk.tls.disabledAlgorithms=SSLv3, TLSv1, RC4, DES, MD5withRSA, \
DH keySize < 1024, EC keySize < 224, 3DES_EDE_CBC, anon, NULL, \
include jdk.disabled.namedCurves
Sample log with the property updated to enable TLSv1.1. TLS 1.3, TLS 1.2 and TLS 1.1 are enabled.
"ClientHello": {
"supported_versions (43)": {
"versions": [TLSv1.3, TLSv1.2, TLSv1.1]
Note that any changes made to the default security properties file affect all the BW applications running under the TIBCO_HOME. If the requirement to enable a TLS protocol version is specific to an
application, a better option would be to make a copy of the security properties file, make the change in the new file and configure the application to use the new file. More information on specifying
an alternate security properties file can be found in the comments section of the security properties file.
Recommended Comments
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Millihenries to Megahenries Conversion (mH to MH)
Millihenries to Megahenries Converter
Enter the electrical inductance in millihenries below to convert it to megahenries.
Do you want to convert megahenries to millihenries?
How to Convert Millihenries to Megahenries
To convert a measurement in millihenries to a measurement in megahenries, divide the electrical inductance by the following conversion ratio: 1,000,000,000 millihenries/megahenry.
Since one megahenry is equal to 1,000,000,000 millihenries, you can use this simple formula to convert:
megahenries = millihenries ÷ 1,000,000,000
The electrical inductance in megahenries is equal to the electrical inductance in millihenries divided by 1,000,000,000.
For example,
here's how to convert 5,000,000,000 millihenries to megahenries using the formula above.
megahenries = (5,000,000,000 mH ÷ 1,000,000,000) = 5 MH
Millihenries and megahenries are both units used to measure electrical inductance. Keep reading to learn more about each unit of measure.
What Is a Millihenry?
One millihenry is equal to 1/1,000 of a henry, which is the inductance of a conductor with one volt of electromotive force when the current is increased by one ampere per second.
The millihenry is a multiple of the henry, which is the SI derived unit for electrical inductance. In the metric system, "milli" is the prefix for thousandths, or 10^-3. Millihenries can be
abbreviated as mH; for example, 1 millihenry can be written as 1 mH.
Learn more about millihenries.
What Is a Megahenry?
One megahenry is equal to 1,000,000 henries, which are the inductance of a conductor with one volt of electromotive force when the current is increased by one ampere per second.
The megahenry is a multiple of the henry, which is the SI derived unit for electrical inductance. In the metric system, "mega" is the prefix for millions, or 10^6. Megahenries can be abbreviated as
MH; for example, 1 megahenry can be written as 1 MH.
Learn more about megahenries.
Millihenry to Megahenry Conversion Table
Table showing various millihenry
measurements converted to
Millihenries Megahenries
1 mH 0.000000001 MH
2 mH 0.000000002 MH
3 mH 0.000000003 MH
4 mH 0.000000004 MH
5 mH 0.000000005 MH
6 mH 0.000000006 MH
7 mH 0.000000007 MH
8 mH 0.000000008 MH
9 mH 0.000000009 MH
10 mH 0.00000001 MH
100 mH 0.0000001 MH
1,000 mH 0.000001 MH
10,000 mH 0.00001 MH
100,000 mH 0.0001 MH
1,000,000 mH 0.001 MH
10,000,000 mH 0.01 MH
100,000,000 mH 0.1 MH
1,000,000,000 mH 1 MH
More Millihenry & Megahenry Conversions | {"url":"https://www.inchcalculator.com/convert/millihenry-to-megahenry/","timestamp":"2024-11-07T12:40:10Z","content_type":"text/html","content_length":"65667","record_id":"<urn:uuid:704f570c-40ae-4162-bd5c-a38ab0f0d19a>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00442.warc.gz"} |
Paper Abstract: 2002g
Title: Linearization of affine connection control systems (30 pages)
Author(s): David Tyner
Detail: MSc Thesis, Queen's University
Original manuscript: 2002/09/09
A simple mechanical system is a triple (Q,g,V) where Q is a configuration space, g is a Riemannian metric on Q, and V is the potential energy. The Lagrangian associated with a simple mechanical
system is defined by the kinetic energy minus the potential energy. The equations of motion given by the Euler-Lagrange equations for a simple mechanical system without potential energy can be
formulated as an affine connection control system. If these systems are underactuated then they do not provide a controllable linearization about their equilibrium points. Without a controllable
linearization it is not entirely clear how one should deriving a set of controls for such systems.
There are recent results that define the notion of kinematic controllability and its required set of conditions for underactuated systems. If the underactuated system in question satisfies these
conditions, then a set of open-loop controls can be obtained for specific trajectories. These open-loop controls are susceptible to unmodeled environmental and dynamic effects. Without a controllable
linearization a feedback control is not readily available to compensate for these effects.
This report considers linearizing affine connection control systems with zero potential energy along a reference trajectory. This linearization yields a linear second-order differential equation from
the properties of its integral curves. The solution of this differential equation measures the variations of the system from the desired reference trajectory. This second-order differential equation
is then written as a control system. If it is controllable then it provides a method for adding a feedback law. An example is provided where a feedback control is implemented.
542K pdf
Last Updated: Fri Mar 15 08:08:35 2024
Andrew D. Lewis (andrew at mast.queensu.ca) | {"url":"https://mast.queensu.ca/~andrew/papers/abstracts/2002g.html","timestamp":"2024-11-10T10:55:54Z","content_type":"text/html","content_length":"2886","record_id":"<urn:uuid:d4b8e894-99fa-44d1-a1f8-07dc8d9897a0>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00874.warc.gz"} |
On the Entropy of a Noisy Function
Let 0 < ϵ < 1/2 be a noise parameter, and let T[ϵ] be the noise operator acting on functions on the Boolean cube {0,1}^n. Let f be a nonnegative function on {0,1}^n. We upper bound the entropy of T
[ϵ] f by the average entropy of conditional expectations of f, given sets of roughly (1-2 ϵ )^2. n variables. In information-theoretic terms, we prove the following strengthening of Mrs. Gerber's
Lemma: let X be a random binary vector of length n, and let Z be a noise vector, corresponding to a binary symmetric channel with crossover probability ϵ . Then, setting v = (1-2 ϵ)^2. n, we have (up
to lower order terms): H (X ⊕ Z) ≥ n H[2] ( ϵ + (1-2 ϵ) H[2]^-1 (E[|B|] = v H (X[i]{i ∈ B)/v). Assuming ϵ ≥ 1/2-δ, for some absolute constant δ > 0, this inequality, combined with a strong version of
a theorem of Friedgut et al., due to Jendrej et al., shows that if a Boolean function f is close to a characteristic function g of a subcube of dimension n-1, then the entropy of T[ϵ] f is at most
that of T[ϵ] g. Taken together with a recent result of Ordentlich et al., this shows that the most informative Boolean function conjecture of Courtade and Kumar holds for high noise ϵ ≥ 1/2-δ.
Namely, if X is uniformly distributed in {0,1^n} and Y is obtained by flipping each coordinate of X independently with probability ϵ, then, provided ϵ ≥ 1/2-δ, for any Boolean function f holds I (f
(X);Y ) ≤ 1-H(ϵ).
Bibliographical note
Publisher Copyright:
© 2016 IEEE.
• Boolean functions
• extremal inequality
• mutual information
Dive into the research topics of 'On the Entropy of a Noisy Function'. Together they form a unique fingerprint. | {"url":"https://cris.huji.ac.il/en/publications/on-the-entropy-of-a-noisy-function","timestamp":"2024-11-03T10:28:35Z","content_type":"text/html","content_length":"49962","record_id":"<urn:uuid:cde1b36c-2900-4874-b578-7b500cc16787>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00534.warc.gz"} |
What is the significance of the Burrows-Wheeler Transform in data compression using data structures? | Hire Someone To Do Programming Assignment
What is the significance of the Burrows-Wheeler Transform in data compression using data structures?
What is the significance of the Burrows-Wheeler Transform in data compression using data structures? A couple of months ago I wrote an answer to this question by Christopher Burrows-Wheeler that
presents some interesting results about the transformation of a their website structure (which might initially look like a data structure model) into a data structure model. To mention some of the
benefits of using data structures in this question is to suggest data retrieval techniques (as @Frydling2015b notes in this post) that greatly simplify and improve the writing process, however
significantly improve the time to produce data structures that (mostly) correspond to what we call the “transformers”. Moreover, as @Brunton2019a suggested, it’s possible to develop systems that
explicitly convert data structures over a data space into a data structure, there are plenty of interesting ways to use the transforming, but one very interesting data set contains a significant
amount of a variety of non-representative data structures only. The paper is entitled “Decoding & Decoding Data Structures by Data Structures.” It starts by reviewing how data structures can be
transformed into a data structure in order to construct a “transform*”, while carefully explaining how to split data representations into multiple data structures. In Section \[reduction\], we
describe how data structures are transformed: a data structure can encode data information into a data structure. We then present a Bonuses method called uniting the data representation and the data
representation and use it to transform see this data representation to another data structure. Then, in Section \[data\_sub\], we describe an alternate approach to data structures that uses data
structure fusion to convert data representations into a data representation of the transform. Then, in Section \[formula\], we discuss examples of data structures that transform into data
representations. Sections \[split\_data\] and \[transform\_data\] present several transforms and apply this new method to the rest of the paper. These approachesWhat is the significance of the
Burrows-Wheeler Transform in data compression using data structures? Seth Krivtsev has written the first proof of the Burrows-Wheeler Transform in the compressed data structures that are built into
KITV1-based compression compression apparatus. It is a second proof that compresses a stream from an initial compressed state. The you can look here proof was first published in KITV1 (or a similar
publication called the KITV1-POPX-X) for example but it has been implemented in several other distributions and is in the intermediate form. You could of course also use the first post to verify that
the data compression is really done inside data structures. The data compression is done with data structures like you have heard about (data structures in other places and many other places) or even
a very common data structure of data compression. Imagine you have a full audio file written into a large Minkowski table that you use to speed things up. There will be some noise that will be added
or removed and that is not the fault of the compressed stream. This is also kind of the non-compression or even (in reality) lossy compression method we used in later presentations of KITV1. However,
having a few copies of the Jitter file and the compression effect on the audio output that you had written for the audio file it still stands the possibility that your audio programming assignment
taking service will simply end up having something that was added into the end only to arrive the new file. The timing and timing of this destruction will not go on the audio output and sound
reproduced in the compressed stream but before it comes back and will take a smaller “buffer” length which is why we would normally say “encrypt” to stop.
Why Do Students Get Bored On Online Classes?
So when you put your audio file back it will take some time to rebuild the audio file on decompression which in turn is what you want the resulting compressed stream to be. If the audio file is
decoded or decompressed into three blocks that the player can then process (What is the significance of Home Burrows-Wheeler Transform in data compression using data structures? The basic concept
that can be accomplished at the 3D compute front using this image example is as follows: Compute a sample by creating a structure on/from the web page to load into a linear image and compressing that
structure into a compressed image. This image example described above is in fact valid enough to work in a transform as well as not to introduce too many errors in the operations except in those that
are not necessary to compress. I’ve used the WPT3.1 image and presented it in the last couple of tutorials about it’s construction and use on the Web page for rendering. The image is comprised
primarily of pixels available from the web page, but displays a well known pattern of continuous lines and stripes arranged in a high resolution pattern. This is a typical image definition and can go
all imp source way to 400 in a single image. The procedure that it takes to obtain the WPT3 image, starting from the start (width = 2 * height) and scaling down, is as follows: 1. First transform an
image into a 5 lines surface by one of the transformation transformors, converting it into a 20 and a half image, where the center of the 20, the centre of the image, the points left and right of the
center of the 20, the points up and down, the point left and right bottom, the point bottom left, and the point bottom right. 2. Compress and transform it into a regular image, such as 20 × 10 in a
rectangular box inside of which lines may go as shown in a little image, that is, a 2 × 2 + 1 rectangle. I wrote this in C, and is able to create my own image with each transformation possible in
only one place. After full processing, I restored my original image. 3. Convert to a 7 × 7 rectangle (without line) 4. Compress the image to the desired resolution and create 8 lines and | {"url":"https://progassignments.com/what-is-the-significance-of-the-burrows-wheeler-transform-in-data-compression-using-data-structures","timestamp":"2024-11-03T09:32:35Z","content_type":"text/html","content_length":"112303","record_id":"<urn:uuid:adde6477-a75a-4e3f-a64f-13bc8414b38c>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00519.warc.gz"} |
Use a Vector to Specify Variable Bounds for SubModel Optimization
When using a SubModel for optimization, it would be nice to be able to store the optimization lower and upper bounds and initial value in a vector Data element rather than three different scalar
elements. This is especially the case if you have a lot of optimization parameters. If, for example, you have four optimization parameters, it would make sense to have 4 3-item vector data elements
rather than 12 separate scalar data elements. However, as shown below, you have to use scalar data elements.
I have an optimization SubModel and I am passing the bounds for an optimization parameter, C1, through the SubModel interface. I pass the bounds and initial value both as a 3-item vector called
C1_bounds and also as individual scalar data elements C1_Lower, C1_Initial and C1_Upper.
In the optimization settings for C1, I can reference the scalar values as shown below. It allows me to close the dialog and I can run an optimization.
On the other hand, when I instead reference subitems of the vector C1_bounds (as shown below), I cannot run the optimization. In fact, I can't even close out of this dialog without an error message.
It appears (as seen above) that everything is valid. But as soon as you try to close out of the optimization variable setup, it gives the message below.
It would be nice to be able to consolidate lower, initial and upper bound values for a given optimization parameter into a vector data element rather than having to use three separate scalar data
1 comment
By the way, it would be nice to be able to attach model files. We can insert images and we can include hyperlinks, but we can't attach anything. I suppose people will have to upload models to a file
sharing system and then provide a link.
Comment actions | {"url":"https://support.goldsim.com/hc/en-us/community/posts/210434087-Use-a-Vector-to-Specify-Variable-Bounds-for-SubModel-Optimization","timestamp":"2024-11-12T20:23:35Z","content_type":"text/html","content_length":"24888","record_id":"<urn:uuid:00bc80e8-ee96-4ae5-a0b1-b04be0447992>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00191.warc.gz"} |
How to determine even or oddHow to determine even or odd 🚩 Even or odd function 🚩 Math.
Record function in the form of the dependence y=y(x). For example, y=x+5.
Substitute the argument x the argument (-x) and see what happened in the end. Compare with the original function y(x). If y(-x)=y(x) are even function. If y(-x)=-y(x) are odd function. If y(-x) is
not equal to y(x) and not equal to -y(x), have the function of the General form.
Write the output to this step of the research function. Possible output:y(x) is an even function,y(x) is an odd function,y(x) is a function of the General form.
Proceed to the next step of the study of a function using the standard algorithm. | {"url":"https://eng.kakprosto.ru/how-47799-how-to-determine-even-or-odd","timestamp":"2024-11-02T01:38:46Z","content_type":"text/html","content_length":"31666","record_id":"<urn:uuid:2089f212-f85f-4e6e-b899-c5f39d2d0165>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00587.warc.gz"} |
How do you estimate 4975 times 78 by rounding? | HIX Tutor
How do you estimate #4975 times 78# by rounding?
Answer 1
$5 , 000 \cdot 80 = 400 , 000$
Now, mentally multiply 5,000 by 80 to get a product of 400,000. Round 4,975 to 5,000 and 78 to 80.
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Answer 2
To estimate ( 4975 \times 78 ) by rounding, you can round each number to the nearest multiple of 10 or 100 and then perform the multiplication. For example, you can round ( 4975 ) to ( 5000 ) and (
78 ) to ( 80 ). Then, multiply ( 5000 \times 80 ) to get an estimate.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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• Readily available 24/7 | {"url":"https://tutor.hix.ai/question/how-do-you-estimate-4975-times-78-by-rounding-8f9afa4d11","timestamp":"2024-11-07T16:47:48Z","content_type":"text/html","content_length":"573769","record_id":"<urn:uuid:b9d28cfa-49ef-4859-b592-0de46aeba20c>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00118.warc.gz"} |
Additional Concepts for Posets
Bob sees a pattern linking the first two posets shown in Figure 6.8 and asserts that any poset of one of these two types is isomorphic to a poset of the other type. Alice admits that Bob is right—but
even more is true. The four constructions given in Example 6.7 are universal in the sense that every poset is isomorphic to a poset of each of the four types. Do you see why? If you get stuck
answering this, we will revisit the question at the end of the chapter, and we will give you a hint. | {"url":"https://rellek.net/book-2016.1/s_posets_additional-concepts.html","timestamp":"2024-11-07T03:06:07Z","content_type":"text/html","content_length":"36609","record_id":"<urn:uuid:a36c288d-3d9a-4f5e-bc5d-a77c962047e5>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00363.warc.gz"} |
Count the Cost, Part 1: Increasing Resolution by Increasing Column Efficiency
John W. Dolan, LC Troubleshooting Editor
When considering column efficiency, more is not always better. We look at some ways to quickly estimate the effects of changes in column length and particle diameter rather than trying the
experiments in the laboratory.
Those of you who have been reading “LC Troubleshooting” for several years will have realized that I am strongly in favour of avoiding work. Call me lazy . . . , or perhaps just efficient? That is, I
don’t believe it is a very good use of my time to perform physical experiments when a mental experiment will yield the same, or approximately the same, result with less time, effort, and cost. As
such, I often begin my discussion of solving a liquid chromatography (LC) problem with a review of the mental processes that I go through. This is not a new concept, by any means. The Bible includes
a 2000-year-old quote from Jesus, “suppose one of you wants to build a tower. Won’t you first sit down and estimate the cost to see if you have enough money to complete it?” (1). Although Jesus was
talking about spiritual matters, it makes good sense (pun intended) to evaluate the cost and likelihood of success before undertaking chromatography experiments, too.
For the next few instalments of “LC Troubleshooting”, I will be considering some of the choices we make when developing or modifying an LC method. Which are the good choices, and which aren’t so
good? What will a particular choice cost in terms of time, finances, or likelihood of solving the separation problem? I will try to help answer these, and other questions, without doing actual
laboratory experiments by using information available to us all.
The Resolution Equation-A Practical Guide
The separation of two peaks in a chromatogram, which we refer to as resolution, is one of the most important factors when developing a separation. Often resolution, Rs, is expressed in what is
commonly referred to as the fundamental resolution equation:
Rs = ¼N0.5 (α-1) [k/(1+k)] [1]
(i) (ii) (iii)
where N is the column plate number (or column efficiency), k is the retention factor, and α is the selectivity:
α = k2/k1 [2]
where k1 and k2 are the k-values for two adjacent peaks.
Equation 1 is an extremely powerful tool to help guide method development. It is used as one of the organizing features of the classic, Practical HPLC Method Development book by Snyder, Kirkland, and
Glajch (2). In our contract laboratory we found that its use as a guide for quickly developing LC methods saved us time and money. It has formed the foundation of LC Resources’ popular Advanced LC
Method Development class for more than 30 years. So it is not surprising that this equation is one of my go-to tools when considering challenges associated with LC separation methods.
You’ll notice that I’ve divided equation 1 into three pieces. Factor i has to do with what I often refer to as the column conditions: length, particle size, and flow rate. Factor ii relates primarily
to the chemistry of the system: column chemistry, mobile phase chemistry, and temperature. The final factor (iii) has to do with retention, and is influenced primarily by the mobile phase strength
and the column temperature. Over the next few instalments of “LC Troubleshooting”, we’ll look at each of these factors in detail and see how some mental experiments can help us to count the cost of
investing time and resources in a particular variable with a goal of increasing resolution.
The (Lack of) Power of N
For the rest of this month’s discussion, let’s focus on term i of equation 1, and for simplicity, we’ll focus on isocratic separations. If we hold all other variables constant except the plate
number, resolution is a function of the square root of the column plate number:
Rs = f(N0.5) [3]
With a large emphasis today on ultrahigh-pressure LC (UHPLC) and superficially porous particles (SPP), it is easy to get the impression that we can solve all our separation problems by going to a
sub-2 µm particle UHPLC column or an SPP column with similar performance. Unfortunately, this is not true, either in theory or practice. This is information easily obtained from equation 1 or 3.
Consider the partial chromatograms of Figure 1. During method development, we would not be discouraged if we were able to obtain the separation of Figure 1(a) during the process. The separation is
not satisfactory for many applications, but it does show promise. Usually we would like to obtain a separation like the one in Figure 1(c), where the peaks are separated to baseline with a small bit
of baseline between the two peaks. Figure 1(c) has Rs = 2.0, which is the target for the minimum resolution for most methods that fall under regulatory oversight. Getting from Figure 1(a) (Rs = 1.0)
to Figure 1(c) shouldn’t be too hard-just increase the plate number, right? Wrong! According to equation 3, to double the resolution, we would have to increase the plate number by fourfold (40.5 =
2). How are we going to do that? Perhaps we might be tempted by the Siren of UHPLC-smaller particles.
The relationship between N and the particle size, dp is:
N = c (L/dp) [4]
where c is a constant, assuming only column length, L, and dp are changed. In other words, to increase N fourfold, we would have to reduce dp fourfold. If we were using the most popular column
configuration, a 150 mm × 4.6 mm, 5-µm dp column, it would mean switching to a 150 mm × 4.6 mm, 1.25-µm column. This presents two additional problems. First, no one that I know of makes columns
packed with 1.25-µm particles. And if they did, the pressure would surely be quite high. We can estimate the column pressure, because we know that the change in column pressure is related to the
square of the change in particle diameter:
â pressure = c (dp2/dp1)2 [5]
where dp1 and dp2 are the diameters of the original and new particles, respectively. If I were running a conventional LC system with the 150 mm × 4.6 mm, 5-µm column, I might be seeing
150–200 bar of column back pressure. A fourfold reduction in particle size would mean a 16-fold increase in column pressure, to pressures of more than twice the pressure limits of any of the
currently available UHPLC systems. Clearly, a change in particle size alone will not allow us to double resolution for the example of Figure 1(a).
An alternative might be to increase the column length. According to equation 4, the plate number increases in proportion with an increase of column length. To increase N fourfold, we would have to
increase L fourfold. That is, connect four columns in series. This could be done, but it is not very practical. The first challenge might be to get your supervisor to sign off on a purchase order to
buy four columns for this purpose! If you did connect four columns in series, the pressure would increase fourfold, as well, likely exceeding the pressure limits of the LC system you are using. It
would also increase the run-time fourfold, something that is not very compatible with the goal that most of us have to have a fast separation. To reduce the pressure to some acceptable pressure
increase-for example, double-we would have to reduce the flow rate by an equal amount, further doubling the run time to eightfold. So attempting to double resolution by changing column length is not
a fruitful approach, either.
At this point, you might be wondering if there is ever any advantage to changing the column length or packing particle size to improve resolution. Yes, either of these approaches can be practical
ways to marginally improve separations. Although changing N by a factor of four is not very practical, doubling N is quite doable. For example, if you had a separation like the one of Figure 1(b) (Rs
= 1.5), you could switch from a 150 mm × 4.6 mm, 5-µm dp column to a 250 × 4.6 mm, 5-µm column. This change would increase N by 250/150 = 1.7â fold, giving a separation with Rs ≈ 2.5. The price
you’d pay would be an increase in both pressure and run time by the same 1.7-fold. Alternatively, you could change from the 150 mm × 4.6 mm, 5-µm column to a 150 mm × 4.6 mm, 3-µm column. This would
give the same increase in N (5/3 = 1.7â fold) for Rs ≈ 2.5 with no penalty in run time, but a 1.72 = 2.8â fold increase in pressure, which may or may not be acceptable. If you were working with a
100 mm × 4.6 mm, 3-µm column and changed to UHPLC conditions with a 100 mm × 4.6 mm, 1.7-µm column, you should observe a 3/1.7 = 1.8-fold increase in resolution to Rs ≈ 2.6 and an increase in
pressure by (3/1.7)2 = 3.1-fold, which should not be a problem with a UHPLC instrument. (I must repeat my standard caution here: If you try to replicate my calculations, please be aware that I have
often rounded the numbers for ease of presentation.)
And what about flow rate? For particles 5 µm in diameter and real samples, there is not a significant change in N or Rs with a twofold change in flow rate. As the particle size is reduced below 5 µm,
the effect of flow diminishes. This means that for most separations, the main effect of a change in flow rate is a change in retention times and column pressure, but not a significant change in the
All this is to say that a few simple calculations show us that it is not reasonable to expect that we can double resolution by changing the primary factors that influence the column plate number:
column length and packing particle diameter. However, it may be possible to improve a marginal resolution by 1.7-fold or so by changing to the next longer column or next smaller particle size
available. In this case, the practical penalty is a longer run time or higher pressure, or both.
Where to Start?
Another bit of useful information can be obtained with estimates from the above equations-this information is related to our initial choice of columns during method development. For reasons I won’t
go into here (see references 3 and 4 for more details), a column with N ≈ 9000–10,000 should separate a sample of 10–15 components without too much trouble. Of course, every sample differs, but if we
use this plate number as a reasonable starting place, we can evaluate our options. Remember that whatever the plate number is for the column you choose, you can easily calculate the effect of
increasing or decreasing N using the process discussed above.
To help visualize the tradeoffs between resolution and column plate number, I’ve plotted data generated from equation 3 in Figure 2, where the column plate number is shown on the x-axis and relative
resolution (normalized to 100) on the y-axis. Although well-packed columns operated under ideal conditions with well-behaved samples should produce reduced plate heights of two particle diameters, it
is more conservative to use a value of three particle diameters for real samples under real operating conditions. If this is assumed, we can then estimate the column plate number from the column
length (in millimetres) and the particle diameter (in micrometres) as follows:
N ≈ 300 L/dp [6]
Based on values obtained from equation 6, I’ve shown where several popular columns fall on the curve of Figure 2 (black dots). The column diameter does not influence the plate number (assuming
appropriate injection techniques, minimal extracolumn band broadening, and adjustment of flow rate to keep the linear velocity of the mobile phase constant), so only L and dp are shown (as L/dp) in
Figure 2. I have also arbitrarily stopped plotting data at N = 15,000, which corresponds to the 250 mm and 5 µm, 150 mm and 3 µm, or 100 mm and 1.7–2 µm columns. Combinations of smaller particles and
longer columns than these are not very practical for routine use on today’s instrumentation when pressure and run time are considered.
Considering the earlier statement that a column with N ≈ 9000–10,000 is a good place to start with many samples, we can understand some of the reasons that the 150 mm × 4.6 mm, 5-µm dp column, and
more recently, its 100 mm × 4.6 mm, 3-µm dp counterpart, is so popular. These columns generate the desired plate number and can be operated with conventional LC equipment at flow rates of 1.5–2.0 for
reasonably short run times and acceptable pressures. These columns also give about 80% of the maximum resolution relative to the 15,000-plate columns, so there is not much to gain from going to a
higher-plate number column considering the run time and pressure tradeoffs.
Figure 2 also helps us understand why the 50 mm × 2.1 mm, 3-µm dp column is so popular for LC–mass spectrometry or LC–tandem mass spectrometry (LC–MS or LC–MS/MS). Although the plate number
(N ≈ 5000) is half of the 150 mm and 5 µm or 100 mm and 3 µm column, the resolution is reduced by only 30% or so. The added selectivity of the MS or MS/MS detector more than makes up for this
potential loss in resolution, and the shorter run times make for more cost-effective use of these expensive detectors.
As we’ve “counted the cost” of the column plate number, we’ve seen that practical changes in N (for example, twofold) tend not to have large influences on the resolution of two peaks. This is because
resolution is influenced only by the square root of the plate number. We also saw that a column generating approximately 9000–10,000 plates is a reasonable starting place for samples of 10–15
components. These columns have a high chance of success, give relatively fast separations, and generate reasonable pressures. This confirms practical experience by helping to explain why the
150 mm × 4.6 mm, 5-µm dp column is so popular. So a good starting point for conventional LC systems is a 150 mm × 4.6 mm, 5-µm column or a 100 mm × 4.6 mm, 3-µm one. For UHPLC, 50–100 mm × 2.1 mm
columns packed with 1.7–2.0 µm particles make the most sense in many cases. We did not consider superficially porous particle columns, but 2.5–2.7 µm SPP particle columns have been demonstrated to
generate plate numbers similar to 2-µm columns packed with totally porous particles (TPP). However, 2.5–2.7 µm SPP columns have back pressures more typical of 3-µm dp columns, so can be operated on
conventional or UHPLC equipment for fast and efficient separations.
As we’ll see in future instalments, it is best to pick a column with sufficient separation power (N) to separate the complexity of sample you have, then concentrate on optimizing retention and
selectivity before making further adjustments in column length or particle size, because the effect of these latter changes can be accurately calculated without additional experimentation.
1. The Holy Bible, New International Version (Zondervan, Grand Rapids, Michigan, USA, 1995), Luke 14:28.
2. L.R. Snyder, J.J. Kirkland, and J.L. Glajch, Practical HPLC Method Development, 2nd edition, (J. Wiley & Sons, Hoboken, New Jersey, USA, 1997).
3. L.R. Snyder, J.J. Kirkland, and J.W. Dolan, Introduction to Modern Liquid Chromatography, 3rd edition, (J. Wiley & Sons, Hoboken, New Jersey, USA, 2010), pp. 76–77.
4. J.W. Dolan, L.R. Snyder, N.M. Djordjevic, D.W. Hill, and T.J. Waeghe, J. Chromatogr. A857, 1–20 (1999).
“LC Troubleshooting” Editor John Dolan has been writing “LC Troubleshooting” for LCGC for more than 30 years. One of the industry’s most respected professionals, John is currently a principal
instructor for LC Resources in McMinnville, Oregon, USA. He is also a member of LCGC Europe’s editorial advisory board. Direct correspondence about this column via e-mail to LCGCedit@ubm.com | {"url":"https://www.chromatographyonline.com/view/count-cost-part-1-increasing-resolution-increasing-column-efficiency","timestamp":"2024-11-04T03:54:15Z","content_type":"text/html","content_length":"446263","record_id":"<urn:uuid:91f11c8d-46e9-40af-aeea-e1048b00d6ab>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00153.warc.gz"} |
Complex Systems Certificate
Program: Complex Systems Certificate
Program Description
The field of complex systems is relatively young and evolving, encompassing a wide range of disciplines in the sciences, engineering, computer science, and mathematics. In this context, a “complex
system” is defined as an engineered or natural system that is characterized by dynamical properties and unobservable interactions that may not be apparent from observations of its emergent behaviors.
Moreover, those dynamics can be intelligent and potentially adversarial. With a strong emphasis on the application of mathematical theory, computational techniques, and modeling in the program,
students and faculty will engage in research on complex systems.
The goals of this certificate are to:
• Train and prepare the next-generation scientists and engineers to formulate and solve complex systems problems.
• Advance CSUN’s mission to engage students in creative activities, particularly those from underrepresented in STEM.
• Provide the needed background training for prospective Ph.D. students in complex systems.
• Generate opportunities for industry employees to continue their interdisciplinary education and advance their careers.
This program will prepare students to apply mathematical theory, data-enabled modeling, and computational techniques to the study of complex systems in the natural and engineered world. Coursework
will focus on the theoretical foundations of complex systems analysis with an emphasis on applications.
Program Requirements
A. Admission Requirements
A master’s degree in STEM fields is required for admission to the certificate program. Students enrolled in the certificate program are eligible to apply to the Ph.D. program. The coursework will be
transferred, if admitted. The students in the Ph.D. program will automatically be enrolled in the certificate program and, therefore, will be granted this certificate along with their Ph.D. degree in
complex systems.
B. Course Requirements
Students must complete 19 units in three semesters as follows.
Semester 1 (6 units)
Select three courses from the following:
CPLX 701 Mathematical Foundations for Complex Systems (2)
CPLX 702 Physics Foundations for Complex Systems (2)
CPLX 703 Chemistry Foundations for Complex Systems (2)
CPLX 704 Biology Foundations for Complex Systems (2)
CPLX 705 Computer Science Foundations for Complex Systems (2)
CPLX 706 Engineer Foundations for Complex Systems (2)
Semester 2 (8 units)
Complete two additional courses from the above list and the following:
CPLX 710A Complex Systems I (4)
Semester 3 (5 units)
Complete the following two courses:
CPLX 710B Complex Systems II (4)
CPLX 791 Research Seminar (1)
Total Units for the Certificate: 19
Department of Mathematics
Chair: Katherine Stevenson
Live Oak Hall (LO) 1300
(818) 677-2721
Program Learning Outcomes
Students receiving a Complex Systems Certificate will be able to:
1. Understand the nature of complex systems.
2. Model or mathematically describe complex systems or systems of systems.
3. Analyze processes and phenomena with respect to agents, complex systems of agents, or systems of systems using applied mathematical and computational frameworks.
4. Demonstrate competency in quantitative and scientific reasoning.
5. Demonstrate a depth of understanding of the essential content and principal modes of inquiry in engineering, computer science, and/or the sciences.
6. Evaluate research findings and the appropriate literature. | {"url":"https://catalog.csun.edu/academics/math/programs/certificate-complex-systems/","timestamp":"2024-11-03T09:54:50Z","content_type":"text/html","content_length":"39668","record_id":"<urn:uuid:c9476729-5f0a-4176-9b56-e91559627b67>","cc-path":"CC-MAIN-2024-46/segments/1730477027774.6/warc/CC-MAIN-20241103083929-20241103113929-00179.warc.gz"} |
Comparing Numbers Worksheets , Free Simple Printable - BYJU'S
Frequently Asked Questions
The prerequisites and difficulty level for the topic are taken into consideration when creating worksheets. Each worksheet has the word ‘easy’, ‘medium’, or ‘hard’, depending on how difficult the
problems are in that worksheet. It is suggested that a student starts with the easy level worksheets before moving on to the medium and hard levels.
No, there isn’t. All BYJU’S Math interactive worksheets are absolutely free.
The comparing numbers worksheets is a building block for any student aiming to master advanced topics in math. It is useful in a number of everyday situations like comparing a certain quantity to
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In math, the greater than symbol is represented by‘ > ’.
In math, the greater than symbol is represented by‘ < ’.
After the successful submission of worksheets, the student can instantly see how much they scored on a certain worksheet along with the respective answer key.
BYJU’S Math offers worksheets of various math concepts that are available in PDF format. A student or parent can easily open and print them using any free PDF tool.
Both offline and online worksheets are available. Interactive worksheets are offered for the student who chooses the online mode. Students can download the PDF for offline worksheets, print it, and
start answering the questions. | {"url":"https://byjus.com/us/math/comparing-numbers-worksheets/","timestamp":"2024-11-03T12:21:30Z","content_type":"text/html","content_length":"164121","record_id":"<urn:uuid:996987d1-2209-403e-b988-910f8548074c>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00075.warc.gz"} |
Other than "Jupyter Notebook", is there any other way to use Sage?
Other than "Jupyter Notebook", is there any other way to use Sage?
I am searching for a "compiler" way to use Sage.
Something like TeXShop, I write everything in a single file and press compile to get the results.
Is there an easy way to accomplish this?
Currently I am using the "Notebook" form of Sage which is neat but kind of gets in the way of more complicated programming.
(To be more clear, I am looking for something like an IDE "Integrated development environment", something like PyCharm for Python.)
Thanks a lot.
2 Answers
Sort by » oldest newest most voted
Ways to use SageMath
On my computer
• the sage REPL or command-line-interface
□ launch Sage in a terminal, wait for the sage: prompt
□ type a command then hit the ENTER or RETURN key, get output
• the Jupyter Notebook with the Sage kernel
□ ways to launch it
☆ in a terminal, type sage -n jupyter
☆ or use a launcher such as
○ the macOS "Sage app"
○ the Windows launcher
○ a desktop launcher for Linux
• JupyterLab
□ install using sage --pip install jupyterlab
□ then in the terminal run sage -n jupyterlab
• the legacy SageNB notebook
□ ways to launch it
☆ in a terminal, type sage -n sagenb
☆ or in the Sage REPL, type notebook()
☆ or use a launcher
• run a command with sage -c
□ example: in a terminal, run sage -c "print(2 + 2)"
• run a file with extension .py or .sage
□ put the commands in a file such as myfile.sage or myfile.py
□ in the terminal, run sage myfile.sage or sage myfile.py
□ difference between .sage and .py files
☆ for .sage files, the Sage preparser will be used
☆ for .py files, the Sage preparser will not be used
• use external files
□ commands
☆ load, runfile, attach, %load, %runfile, %attach
☆ examples
○ suppose the file myfile.sage contains def sq(a): return a*a
○ in the terminal, run sage -c "load('myfile.sage'); print(sq(2))"
• from another program
□ SageTeX, a LaTeX package to use Sage within TeX/LaTeX documents
□ Cantor
□ TeXmacs
• use OpenMath and SCSCP to exchange mathematical objects with other mathematics software; work in progress, see
• SageCell
□ one-off computations on SageCell page
□ use it to include compute cells in a webpage
□ use it with PreTeXt
• CoCalc
□ see dedicated item below
• JupyterHub
□ there are some deployments of JupyterHub which offer the Sage kernel
• mybinder
□ there are some mybinder instances that include Sage
• Sage Notebook servers
□ there some deployments of the SageNB notebook; the one formerly at sagenb.org is no longer active
• in a CoCalc terminal, run sage for the Sage REPL
• use sage_select to select which version of Sage to use by default
• CoCalc Sage worksheets (.sagews)
• Jupyter Notebook worksheets (.ipynb)
□ using CoCalc's version of the Jupyter Notebook
□ using the Classic Jupyter Notebook
☆ go to Project preferences and launch Classic Jupyter Notebook; this will open a separate browser tab which will connect to your project using the classic Jupyter Notebook protocol;
☆ allows to use Jupyter Notebook extensions that are not yet implemented in "CoCalc Jupyter", such as widgets, RISE, ...
Sage: file formats, data formats
• common file extensions
□ .py, .sage, .pyx, .spyx, ...
□ .sobj
□ SageNB notebook worksheets: .sws
□ CoCalc Sage worksheets: .sagews
□ .rst, .txt, .html
• converters
□ rst2ipynb, ...
• viewing, saving, copying, transferring worksheets
Run shell commands from within sage
• any command starting with ! is executed as a shell command
• many basic shell commands are available in IPython without !: ls, cd, pwd...
• see also the Python modules os and sys
Read from, write to, append to file
• this uses standard Python functionality
• 'r' for read, 'w' for write (overwriting file), 'a' for append
• read from file, all at once or line by line
with open('/path/to/file.txt', 'r') as f:
s = f.read()
with open('/path/to/file.txt', 'r') as f:
for line in f:
<do something with line>
• write to file, bit by bit or line by line
with open('/path/to/file.txt', 'a') as f:
f.writelines(['haha', 'hehe', 'hihi'])
• see also the csv module for "comma-separated-values" data files
edit flag offensive delete link more
This is ridiculously comprehensive and should be on the main Sage website in a prominent place ...
kcrisman ( 2018-07-12 19:11:57 +0100 )edit
If one wants to use Sage in Jupyterlab on macOS Catalina 10.15.1 (19B88), this worked for me:
1. download Sage binaries
2. unpack download
3. move SAGE_ROOT to the desired location
4. set SAGE_ROOT and any other environment variables (e.g. SAGE_KEEP_BUILT_SPKGS) in ~/.zshrc
(a) export SAGE_KEEP_BUILT_SPKGS="yes"
(b) export SAGE_NUM_THREADS=3
(c) export SAGE_ROOT="$HOME/SageMath"
5. Update $PATH by editing /etc/paths Note: You’ll need to use SUDO.
6. cd $SAGE_ROOT && ./sage -i openssl && ./sage -f python3 && make ssl && sage –pip3 install jupyterlab
when all this is done, sage -n jupyterlab, should work.
Update: Avoid updating Python packages using pip. I broke the sage kernal in jupyterlab doing this.
edit flag offensive delete link more
This has been discussed a few times on sage-devel. Given the ongoing switch to Python 3, which shall enable the upgrade of a lot of Python libraries in Sage (namely ipython, jupyterund so weider...),
the addition of an "official" Jupyterlab interface is probably a question of time...
Emmanuel Charpentier ( 2020-01-15 18:55:09 +0100 )edit | {"url":"https://ask.sagemath.org/question/42876/other-than-jupyter-notebook-is-there-any-other-way-to-use-sage/?answer=42877","timestamp":"2024-11-11T03:05:59Z","content_type":"application/xhtml+xml","content_length":"70177","record_id":"<urn:uuid:f4086ba6-16b5-4b01-bada-8c41cdd378a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00845.warc.gz"} |
DIY Presolve
In certain models, we can choose some level of "presolve" at the model level.
are part of LP/MIP solvers. Their main task is to inspect the model and look for opportunities to make it smaller. Some of this is done by simple operations such as: change a singleton constraint
into a bound, remove variables that are not needed, etc [2,3].
Note that the meaning of "
" in this context is not totally obvious: some substitutions will decrease the number of decision variables and constraints, while increasing the number of non-zero coefficients in the LP matrix.
Usually larger and sparser is better than smaller and denser (most LP and MIP solvers exploit sparsity), so I tend to focus on nonzero counts.
The question I am exploring here: how many reductions do we apply at
the modeling level
opposed to leave it to solver? If a solver is able to reduce the size by a large amount (loosely defined), I always feel I did not do a good job as a modeler. I just did not pay attention.
The model below demonstrates how we can apply different reduction levels to the model. The model becomes smaller, but at the expense of more complex modeling. What is the right level to choose? Of
course there is no objective answer to this. Your "optimal level" may be different from mine.
Problem description
The problem is from [1]. Consider the board:
We need to fill the board with integers: \(x_{i,j} \in Z\). The following rules apply:
1. Green cells must contain strictly positive values, \(x_{i,j}\ge 1\).
2. Red cells must contain strictly negative values, \(x_{i,j}\le -1\).
3. White and blue cells have a value of zero, \(x_{i,j}= 0\).
4. The red and green cells form a symmetric pattern: if \(x_{i,j}\) is a green cell, \(x_{j,i}\) is a red cell, and the other way around.
5. Skew-symmetry or Anti-symmetry: we have the restriction \(x_{i,j} = -x_{j,i}\). Putting it differently: \(X^T = -X\).
6. Row and column sums are equal to zero: \[ &\sum_i x_{i,j} =0 & \forall j\\& \sum_j x_{i,j} = 0 & \forall i\]
There are multiple solutions. We may choose the solutions with the smallest sum of green values:\[\min \sum_{\mathit{Green}(i,j)} x_{i,j}\]
In the board above we have the following statistics:
Cell type Count
Green cells 57
Red cells 57
Blue cells 20
White cells 266
Total 400
Presolve level 0
A direct formulation for all \(x_{i,j}\) is: \[\min\> & z=\sum_{\mathit{Green}(i,j)} x_{i,j}\\ & x_{i,j}\ge 1 & \mathit{Green}(i,j)\\ & x_{i,j} \le -1 & \mathit{Red}(i,j)\\ & x_{i,j} =0 & \mathit
{WhiteBlue}(i,j)\\ & x_{i,j} = -x_{j,i} & \forall i,j\\ &\sum_i x_{i,j} =0 & \forall j\\&\sum_j x_{i,j} =0 & \forall i \\ & x_{i,j} \in Z\]
This model, with all equations stated as explicit constraints, has the following sizes:
Model Size Count
rows 840
columns 400
nonzero elements 1980
The counts here exclude the objective function. Although the solver will automatically convert singleton equations into bounds, I never write these as explicit constraints. I prefer to specify
singleton equations as bounds.
Presolve level 1
The first three constraints can be implemented as bounds: \[\ell_{i,j} = \begin{cases} 1 & \mathit{Green}(i,j)\\ -\infty & \mathit{Red}(i,j)\\ 0 & \text{otherwise}\end{cases}\] and \[u_{i,j} = \begin
{cases} \infty & \mathit{Green}(i,j)\\ -1& \mathit{Red}(i,j)\\ 0 & \text{otherwise}\end{cases}\] Now the model can read: \[\min \> & z=\sum_{\mathit{Green}(i,j)} x_{i,j}\\ & x_{i,j} = -x_{j,i} & \
forall i\lt j\\ &\sum_i x_{i,j} =0 & \forall j\\&\sum_j x_{i,j} =0 & \forall i\\ & x_{i,j} \in [\ell_{i,j}, u_{i,j}] \\& x_{i,j} \in Z\] I also reduced the number of skew-symmetry constraints \(x_
{i,j}=-x_{j,i}\): we only need these for \(i\lt j\). This reduces the model size to:
Model Size Count
rows 230
columns 400
nonzero elements 1180
All singleton equations have been formulates as bounds. This model has a large number of variables fixed to zero (all variables corresponding to blue and white cells). The solver will presolve those
variables away, but I prefer to do this myself.
Presolve level 2
The next level is to remove all \(x_{i,j}\) that are known to be zero from the model. \[\min \> & z= \sum_{\mathit{Green}(i,j)} x_{i,j}\\ & x_{i,j} = -x_{j,i} & \forall \mathit{Green}(i,j)\\ &\sum_{i
|\mathit{GreenRed}(i,j)} x_{i,j} =0 & \forall j\\&\sum_{j|\mathit{GreenRed}(i,j)} x_{i,j} =0 & \forall i\\ & x_{i,j} \in [\ell_{i,j}, u_{i,j}] & \forall \mathit{GreenRed}(i,j)\\& x_{i,j} \in Z\] The
only cells we model here are the red and green ones. Our counts are:
Model Size Count
rows 97
columns 114
nonzero elements 342
This was my first actual implementation. However, we can go further, and use some more reductions. Here the model starts to become less intuitive.
Presolve level 3
We can implicitly deal with the red cells: a red cell \(x_{i,j}\) has a corresponding green cell \(-x_{j,i}\). \[\min \> & z=\sum_{\mathit{Green}(i,j)} x_{i,j}\\ & \sum_{i|\mathit{Green}(i,j)} x_
{i,j} - \sum_{i|\mathit{Green}(j,i)}x_{j,i} =0 & \forall j\\&\sum_{j|\mathit{Green}(i,j)} x_{i,j} - \sum_{j|\mathit{Green}(j,i)} x_{j,i} =0 & \forall i\\ & x_{i,j} \ge 1 & \forall \mathit{Green}(i,j)
\\& x_{i,j} \in Z\] We only solve for the green cells here. The value of the red cells can be recovered afterwards.
Model Size Count
rows 40
columns 57
nonzero elements 228
The row and column sums now become more difficult to recognize. In addition we need to add some code to recalculate the value of the red cells after the solve.
Presolve level 4
Finally, we can also remove one of the summations because of symmetry. We end up with: \[\min \> & z=\sum_{\mathit{Green}(i,j)} x_{i,j}\\ & \sum_{i|\mathit{Green}(i,j)} x_{i,j} - \sum_{i|\mathit
{Green}(j,i)}x_{j,i} =0 & \forall j\\ & x_{i,j} \ge 1 & \forall \mathit{Green}(i,j)\\& x_{i,j} \in Z\]
Model Size Count
rows 20
columns 57
nonzero elements 114
Again, the value of the red cells need to be calculated after the solve. This model is now very compact, but we moved away from the original problem statement. When reading this model, we would not
immediately see the correspondence with the problem.
Does it make a difference?
No. The first model shows:
Presolved: 13 rows, 48 columns, 96 nonzeros
Variable types: 0 continuous, 48 integer (0 binary)
The last model gives:
Presolved: 13 rows, 48 columns, 96 nonzeros
Variable types: 0 continuous, 48 integer (0 binary)
So should we even worry? I still like to generate models that are somewhat small. For me this is not even a performance issue, but rather a question of paying attention. I see sometimes sloppy
modeling causing an excessive number of variables, equations and nonzero elements. How far I take this DIY presolve effort is determined by readability and understandability: the limit is when the
formulation becomes less obvious and when readability starts to suffer.
In any case, if the solver can presolve large parts of the model away, I want to be able to explain this. May be the model is largely triangular, or there are many singleton equations. Things may be
more complex, and such an explanation is not always obvious to find.
The solutions looks like:
The minimum sum of the green cells is 87.
Update: fixed reference: use proper names of authors.
1. Antisymmetric Table Puzzle where the rows/columns sum to zero. https://math.stackexchange.com/questions/2794165/antisymmetric-table-puzzle-where-the-rows-columns-sum-to-zero
2. A.L. Brearley, G. Mitra, H.P. Williams, Analysis of mathematical programming problems prior to applying the simplex algorithm, Mathematical Programming, 8 (1975), pp. 54-83.
3. E.D. Andersen, K.D. Andersen, Presolving in Linear Programming, Mathematical Programming, 71 (1995), pp. 221-245.
No comments: | {"url":"https://yetanothermathprogrammingconsultant.blogspot.com/2018/05/diy-presolve.html","timestamp":"2024-11-15T03:16:29Z","content_type":"text/html","content_length":"131147","record_id":"<urn:uuid:217b9e7d-3d37-4318-b346-13bc5aa532f9>","cc-path":"CC-MAIN-2024-46/segments/1730477400050.97/warc/CC-MAIN-20241115021900-20241115051900-00001.warc.gz"} |
A056510 - OEIS
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See
for pdf file of Chap. 2]
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. Permuting the symbols will not change the structure. | {"url":"https://oeis.org/A056510","timestamp":"2024-11-02T20:31:47Z","content_type":"text/html","content_length":"15052","record_id":"<urn:uuid:8f45ad0a-5bd8-4ce8-a7bf-af644a2bde84>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00299.warc.gz"} |
Basics of Hydrostatic Level Measurement - Inst Tools
Basics of Hydrostatic Level Measurement
A vertical column of fluid generates a pressure at the bottom of the column owing to the action of gravity on that fluid. The greater the vertical height of the fluid, the greater the pressure, all
other factors being equal. This principle allows us to infer the level (height) of liquid in a vessel by pressure measurement.
Pressure of a fluid column
A vertical column of fluid exerts a pressure due to the column’s weight. The relationship between column height and fluid pressure at the bottom of the column is constant for any particular fluid
(density) regardless of vessel width or shape. This principle makes it possible to infer the height of liquid in a vessel by measuring the pressure generated at the bottom:
The mathematical relationship between liquid column height and pressure is as follows:
Where, P = Hydrostatic pressure ρ = Mass density of fluid in kilograms per cubic meter (metric) or slugs per cubic foot (British) g = Acceleration of gravity γ =Weight density of fluid in newtons per
cubic meter (metric) or pounds per cubic foot (British) h = Height of vertical fluid column above point of pressure measurement
For example, the pressure generated by a column of oil 12 feet high (h) having a weight density of 40 pounds per cubic foot (γ) is:
Note the cancellation of units, resulting in a pressure value of 480 pounds per square foot (PSF). To convert into the more common pressure unit of pounds per square inch, we may multiply by the
proportion of square feet to square inches, eliminating the unit of square feet by cancellation and leaving square inches in the denominator:
Thus, a pressure gauge attached to the bottom of the vessel holding a 12 foot column of this oil would register 3.33 PSI. It is possible to customize the scale on the gauge to read directly in feet
of oil (height) instead of PSI, for convenience of the operator who must periodically read the gauge. Since the mathematical relationship between oil height and pressure is both linear and direct,
the gauge’s indication will always be proportional to height.
An alternative method for calculating pressure generated by a liquid column is to relate it to the pressure generated by an equivalent column of water, resulting in a pressure expressed in units of
water column (e.g. inches W.C.) which may then be converted into PSI or any other unit desired.
For our hypothetical 12-foot column of oil, we would begin this way by calculating the specific gravity (i.e. how dense the oil is compared to water). With a stated weight density of 40 pounds per
cubic foot, the specific gravity calculation looks like this:
The hydrostatic pressure generated by a column of water 12 feet high, of course, would be 144 inches of water column (144 ”W.C.). Since we are dealing with an oil having a specific gravity of 0.641
instead of water, the pressure generated by the 12 foot column of oil will be only 0.641 times (64.1%) that of a 12 foot column of water, or:
We may convert this pressure value into units of PSI simply by dividing by 27.68, since we know 27.68 inches of water column is equivalent to 1 PSI:
As you can see, we arrive at the same result as when we applied the P = γh formula. Any difference in value between the two methods is due to imprecision of the conversion factors used (e.g. 27.68
”W.C., 62.4 lb/ft3 density for water).
Any type of pressure-sensing instrument may be used as a liquid level transmitter by means of this principle. In the following photograph, you see a Rosemount model 1151 pressure transmitter being
used to measure the height of colored water inside a clear plastic tube:
In most level-measurement applications, we are concerned with knowing the volume of the liquid contained within a vessel, and we infer this volume by using instruments to sense the height of the
fluid column. So long as the vessel’s cross-sectional area is constant throughout its height, liquid height will be directly proportional to stored liquid volume. Pressure measured at the bottom of a
vessel can give us a proportional indication of liquid height if and only if the density of that liquid is known and constant. This means liquid density is a critically important factor for
volumetric measurement when using hydrostatic pressure-sensing instruments. If liquid density is subject to random change, the accuracy of any hydrostatic pressure-based level or volume instrument
will be correspondingly unreliable.
It should be noted, though, that changes in liquid density will have absolutely no effect on hydrostatic measurement of liquid mass, so long as the vessel has a constant cross-sectional area
throughout its entire height. A simple thought experiment proves this: imagine a vessel partially full of liquid, with a pressure transmitter attached to the bottom to measure hydrostatic pressure.
Now imagine the temperature of that liquid increasing, such that its volume expands and has a lower density than before.
Assuming no addition or loss of liquid to or from the vessel, any increase in liquid level will be strictly due to volume expansion (density decrease). Liquid level inside this vessel will rise, but
the transmitter will sense the exact same hydrostatic pressure as before, since the rise in level is precisely countered by the decrease in density (if h increases by the same factor that γ
decreases, then P = γh must remain the same!). In other words, hydrostatic pressure is seen to be directly proportional to the amount of liquid mass contained within the vessel, regardless of changes
in liquid density. This is useful to know in applications where true mass measurement of a liquid (rather than volume measurement) is either preferable or necessary
Differential pressure transmitters are the most common pressure-sensing device used in this capacity to infer liquid level within a vessel. In the hypothetical case of the oil vessel just considered,
the transmitter would connect to the vessel in this manner (with the high side toward the process and the low side vented to atmosphere):
Connected as such, the differential pressure transmitter functions as a gauge pressure transmitter, responding to hydrostatic pressure exceeding ambient (atmospheric) pressure. As liquid level
increases, the hydrostatic pressure applied to the “high” side of the differential pressure transmitter also increases, driving the transmitter’s output signal higher.
Some pressure-sensing instruments are built specifically for hydrostatic measurement of liquid level in vessels, eliminating with impulse tubing altogether in favor of a special kind of sealing
diaphragm extending slightly into the vessel through a flanged pipe entry (commonly called a nozzle). A Rosemount hydrostatic level transmitter with an extended diaphragm is shown here:
The calibration table for a transmitter close-coupled to the bottom of an oil storage tank would be as follows, assuming a zero to twelve foot measurement range for oil height, an oil density of 40
pounds per cubic foot, and a 4-20 mA transmitter output signal range:
Credits : by Tony R. Kuphaldt – Creative Commons Attribution 4.0 License
1 thought on “Basics of Hydrostatic Level Measurement”
1. When is about measurement, a professional site (… as this site claims to be) must use SI (“metric”) units in every example, every presentation and every time. Using a “British”, “American”,
“Canadian”, “Indian”, “Eskimo”, etc…. units system is confusing and non-professional …………
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13. In a survey of 100 students, the number of students studyin... | Filo
Question asked by Filo student
13. In a survey of 100 students, the number of students studying the various languages is found as: English only 18; English but not Hindi 23; English and Sanskrit 8; Sanskrit and Hindi 8; English
26; Sanskrit 48 and no language 24. Find (i) how many students are studying Hindi, (ii) how many students are studying English and Hindi both. 14. In a town of 10,000 families, it was found that of
the families buy newspaper buy newspaper buy newspaper C, 5\% buy and buy and and buy and . If buy all the three newspapers, find the number of families which buy (i) A only, (ii) Bonly, (iii) none
of and .
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13. In a survey of 100 students, the number of students studying the various languages is found as: English only 18; English but not Hindi 23; English and Sanskrit 8; Sanskrit and Hindi 8;
Question English 26; Sanskrit 48 and no language 24. Find (i) how many students are studying Hindi, (ii) how many students are studying English and Hindi both. 14. In a town of 10,000 families, it
Text was found that of the families buy newspaper buy newspaper buy newspaper C, 5\% buy and buy and and buy and . If buy all the three newspapers, find the number of families which buy (i) A
only, (ii) Bonly, (iii) none of and .
Updated Nov 3, 2022
Topic Algebra
Subject Mathematics
Class Class 11
Answer Video solution: 1
Upvotes 72
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358,200 research outputs found
We present our latest results on the glueball spectrum of SU(N) gauge theories in 2+1 dimensions for spins ranging from 0 to 6 inclusive, as well as preliminary results for SU(3) in 3+1 dimensions.
Simple glueball models and the relation of the even-spin spectrum to the 'Pomeron' are discussed.Comment: LAT03 proceedings (spectrum), 3 pages, 3 figures, talk by H.Meye
We develop a new method to describe the accretion flow in the corona above a thin disk around a black hole in vertical and radial extent. The model is based on the same physics as the earlier
one-zone model, but now modified including inflow and outflow of mass, energy and angular momentum from and towards neighboring zones. We determine the radially extended coronal flow for different
mass flow rates in the cool disk resulting in the truncation of the thin disk at different distance from the black hole. Our computations show how the accretion flow gradually changes to a pure
vertically extended coronal or advection-dominated accretion flow (ADAF). Different regimes of solutions are discussed. For some cases wind loss causes an essential reduction of the mass
flow.Comment: 8 pages, 4 figures, accepted for publication in A&
We define the filtrated K-theory of a C*-algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of
filtrated K-theory. For finite spaces with totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same
consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe a space with four points and two C*-algebras over this space in the
bootstrap class that have isomorphic filtrated K-theory but are not KK(X)-equivalent. For this particular space, we enrich filtrated K-theory by another K-theory functor, so that there is again a
Universal Coefficient Theorem. Thus the enriched filtrated K-theory is a complete invariant for purely infinite, stable C*-algebras with this particular spectrum and belonging to the appropriate
bootstrap class.Comment: Changes to theorem and equation numbering
Group cohomology of polynomial growth is defined for any finitely generated discrete group, using cochains that have polynomial growth with respect to the word length function. We give a geometric
condition that guarantees that it agrees with the usual group cohomology and verify this condition for a class of combable groups. Our condition involves a chain complex that is closely related to
exotic cohomology theories studied by Allcock and Gersten and by Mineyev.Comment: 19 pages, typo corrected in version
Observations of the black hole X-ray binaries GX 339-4 and V404 Cygni have brought evidence of a strong correlation between radio and X-ray emission during the hard spectral state; however, now more
and more sources, the so-called `outliers', are found with a radio emission noticeably below the established `standard' relation. Several explanations have already been considered, but the existence
of dual tracks is not yet fully understood. We suggest that in the hard spectral state re-condensation of gas from the corona into a cool, weak inner disk can provide additional soft photons for
Comptonization, leading to a higher X-ray luminosity in combination with rather unchanged radio emission, which presumably traces the mass accretion rate. As an example, we determined how much
additional luminosity due to photons from an underlying disk would be needed to explain the data from the representative outlier source H1743-322. From the comparison with calculations of Compton
spectra with and without the photons from an underlying disk, we find that the required additional X-ray luminosity lies well in the range obtained from theoretical models of the accretion flow. The
radio/X-ray luminosity relation resulting from Comptonization of additional photons from a weak, cool inner disk during the hard spectral state can explain the observations of the outlier sources,
especially the data for H1743-322, the source with the most detailed observations. The existence or non-existence of weak inner disks on the two tracks might point to a difference in the magnetic
fields of the companion stars. These could affect the effective viscosity and the thermal conductivity, hence also the re-condensation process.Comment: 7 pages, 2 figures. Accepted for publication in
A &
The white dwarf in AM Her systems is strongly magnetic and keeps in synchronous rotation with the orbit by magnetic coupling to the secondary star. As the latter evolves through mass loss to a cool,
degenerate brown dwarf it can no longer sustain its own magnetic field and coupling is lost. Angular momentum accreted then spins up the white dwarf and the system no longer appears as an AM Her
system. Possible consequences are run-away mass transfer and mass ejection from the system. Some of the unusual cataclysmic variable systems at low orbital periods may be the outcome of this
evolution.Comment: 6 pages, 1 figure, Proceedings of "Cataclysmic Variables", Symposium in Honour of Brian Warner, Oxford 1999, eds. P.Charles, A.King, O'Donoghue, to appea | {"url":"https://core.ac.uk/search/?q=author%3A(Meyer)","timestamp":"2024-11-03T01:37:52Z","content_type":"text/html","content_length":"125381","record_id":"<urn:uuid:688d1855-e92d-4312-8797-74baad922377>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00555.warc.gz"} |
Carnival of Mathematics 162 - Chalkdust
Carnival of Mathematics 162
This month’s round up of mathematical blog posts from all over the internet
Welcome to the 162nd Carnival of Mathematics, the monthly round up of maths blogs organised by The Aperiodical. Next month’s Carnival will be hosted by Elias at The Math Section. You can submit items
for next month here.
But before we begin, it is customary to share some interesting facts about the number 162.
• 162 is a Harshad number: it is a mutiple of the sum of its digits.
• 162 cannot be written in the form $ab+a+b$, where $a$ and $b$ are strictly positive integers.
• 162 is an abundant number: the sum of its factors (including 1 but not including 162 itself) is greater than 162.
• Issue 162 of Chalkdust will be released in Autumn 2095. If this is too long for you to wait, you can get your hands on issue 08 much sooner.
While waiting for the release of issue 08, you can read the following highlights from the last month of the internet.
When measuring the size of sets in the conventional way, the sets $\{1,2,3,4,…\}$ and $\{2,3,4,5,…\}$ have the same size. If this makes you unhappy—as of course these is one less thing in the second
set—then you should read James Propp’s post about an alternative way to measure the size of infinite sets.
John Baez wrote about the 5/8 theorem: in a group, if the probability that two randomly chosen elements ($x$ and $y$) commute (ie $xy=yx$) is greater than 5/8, then all pairs of elements in the group
must commute. If you want to know why this is true, then read John’s post.
Tai-Danae Bradley wrote en explanation of what applied category theory is. To understand how category theory can be applied, you’ll need to know something about what it is: but don’t panic, Tai-Danae
has written another post telling you everything you need to know.
Jordan Ellenberg interviewed mathematician and retired American football player John Urchel. It’s a really interesting interview and a highly recommended read.
Mark Dominus wrote about how to find lines and curves that approximate data. Speaking of approximation, some guy I’ve never heard of called Matthew Scroggs posted last month about @RungeBot, a
Twitter bot that he made.
Over on Twitter, Justin Lanier posted a thread about a recently discovered result involving points moving around inside circles or spheres.
John D Cook wrote about primes in the digits of $\pi$. John noticed that the number 314159 is prime while reading a post by Evelyn Lamb. This got him wondering how many different primes could be
formed by the first digits of $\pi$, and whether there are more or less than you would expect.
Chalkdust issue 08 is released, October is also Black History Month in the UK and Black Mathematician Month. Over the course of the next month, we will publish articles written by black
mathematicians about their work, as well as pieces that explore current initiatives tackling the lack of diversity.
With this in mind, we offer you a challenge to complete during October: pick a black mathematician, write about them or their work, then submit what you’ve written to next month’s Carnival of
Mathematics. (If you don’t have your own blog to post on, why not submit a guest post to us or The Aperiodical?) If you need some inspiration before getting started, Faith Uwadiae is tweeting about
one black scientist every day this month.
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Study on the stability of a shear-thinning suspension used in oil well drilling
© F.M. Fagundes et al., published by IFP Energies nouvelles, 2018
List of notation
a,b,c,e: Estimated parameters [M^0L^0T^0]
d: Estimated parameter [M^−1L^−1T^−2]
β: Calibration constant [M^0L^0T^0]
ϵ[s]: Volumetric solids concentration [M^0L^0T^0]
ϵ[s0]: Initial volumetric solids concentration [M^0L^0T^0]
ϵ[sm]: Maximum volumetric solids concentration [M^0L^0T^0]
I: Intensity of gamma-ray beams [M^0L^0T^0]
m: Consistency index [M^1L^−1T^−1]
P[s]: Pressure on solids [M^1L^−1T^−2]
ρ[f]: Specific mass of fluid [M^1L^−3T^0]
ρ[s]: Specific mass of solids [M^1L^−3T^0]
R: Corrected counting of radiation pulses [M^0L^0T^−1]
R[0]: Corrected counting of radiation pulses crossing the test tube without solids [M^0L^0T^−1]
t[m]: Resolution time of the system [M^0L^0T^−1]
z: Monitoring position [M^0L^1T^0]
1 Introduction
In the drilling oil wells, different formulations of fluids have been used to present specific properties desirable in each stage of the process. The substances usually added are emulsifying,
thickening and gelling agents that are pumped to the surface; lift the cuttings to the surface; cool, lubricate and reduce friction between machinery and the wellbore; apply a hydrostatic pressure on
well walls to prevent the invasion of fluid in the reservoir rock [1–7].
Corrective or preventive operational stops interrupt the fluid flow used in drilling, and the solids concentration profile changes due to sedimentation of particles in the fluid and the cuttings
generated in the operation. In this context, drilling fluids that can form gel structures in the absence of shear stress have been developed. Gelling prevents particle sedimentation and,
consequently, damages caused by the accumulation of solids on the drill bit [7–9].
Although some studies have been carried out [10–17] on the behavior of settling particles in shear-thinning fluids, there is lack of information on this phenomenon in drilling fluids, especially in
the compression zone.
Thus, the general aim of this study was to investigate solid-liquid separation in a fluid developed by an oil company, which has been already used in the field during an operational stop. We also
intended to present a constitutive equation for pressure on solids. With these results and the equation of motion for solids, we aim to reproduce in the future a real situation of inactive oil wells
filled with the fluid studied.
2 Experimental procedure
2.1 Fluid characterization
The estimated parameters were: specific mass of solid (ρ[s]=2709kg/m^3) by helium pycnometer technique in the Micromeritics Gas Pycnometer, AccuPyc model 1330; specific mass of fluid (ρ[f]=
1145.9kg/m^3) by simple pycnometry; and determination of initial solids concentration in suspension (ϵ[s0]≈14%) by retort analysis using the FANN Model 210463 Kit with a 50mL capacity.
2.2 Solid characterization
The solid content of suspension is formed by solids additives such as barite and cut solids from rock drilling.
The characterization of solids size was: D[0.1]=3.008µm, D[0.5]=40.803µm and D[0.9]=232.247µm (laser granulometer, Malvern Mastersizer MicroPlus MAF 5001®).
2.3 Rheology
To characterize the rheological properties of fluid, the hysteresis and flow curve were determined. We performed tests in triplicate and with new samples, at 25°C in the Brookfield model R/S Plus
rheometer, geometrically in rotary cone and fixed plate, with a Brookfield thermostatic bath, programmable controller model TC-6021.
Samples were submitted to intense pre-shear (rate of 1050s^−1) for 1min. To assess viscosity dependence over time, we conducted tests varying the strain rates from 1s^−1 to 1050s^−1. Then, we
analyzed the area of hysteresis between the ascending curves followed by a decrease in the strain rate. To construct the flow curve, we submitted the samples to constant shear rates of 200, 400, 600,
800, and 1000s^−1 until they reached the steady state, when the shear stress values were obtained and adjusted to the Power-law model (Eq. 1). $τ=mγn,$(1) where m and n are indices of consistency
and behavior, respectively, γ is the shear stress, and γ is the shear rate.
2.4 Gamma-ray attenuation technique
Gamma-ray attenuation measurements is a technique used to obtain indirect volumetric concentration of solids in batch sedimentation tests without interfering in the configuration and stability of the
medium. This technique was applied in studies of gravitational sedimentation in Newtonian fluids [18–20] and laboratory produced shear-thinning fluids [16].
The radioisotope application unit, represented by the flow chart of Figure 1, consisted of a radiation targeting/detection system coupled to a metal structure. This metal structure had a mobile
platform that allowed the horizontal beam of radiation to be positioned at different heights, allowing monitoring of concentration from the bottom of the test vessel (z=0cm) to the top of the
suspension column (z=21cm). We placed the homogenized fluid in a glass test tube with the following dimensions: 350-mm high, 55-mm internal diameter, and 3-mm thick. We positioned the glass tube
in the center of the radioisotope application unit.
In order to guarantee that the experimental unit operated in optimal condition, we used pre-scaled results [19]: the high voltage source was 900V, and the radioisotope energy range was 500–800mV
To obtain local solids concentration by means of gamma-ray attenuation technique, we previously corrected the intensity of gamma-ray beam (I) with the system's resolution time (t[m]=240±50µs)
(Eq. 2). Then, we used Lambert's equation [21], (Eq. 3): $R=I1−tmI,$(2) $ln(R0R)=βϵs,$(3) where R and R[0] represent the corrected counting of pulses arriving at the detection system after passing
through a solution with and without suspended solids, respectively; ϵ[s] is the volumetric concentration of solids; and β is a calibration constant.
To determine β, we used the pulse counting when the suspension was homogenized, which corresponded to the initial volumetric solids concentration (ϵ[s0]), and the pulse counting in the region of zero
solids concentration. The value of the calibration constant was 0.205.
2.5 Constant concentration curves
We organized the volumetric solids concentration values obtained through gamma-ray attenuation technique with the time of appearance in each position monitored, which allowed the construction of
constant concentration curves.
2.6 Pressure on solids
Determining the constitutive law of a fluid is a characterization very important to maintain the process security [6]. The determination of a constitutive equation for pressure on solids (P[s])
started with an adjustment of the values of volumetric solids concentration in the sediment as a function of the monitoring position, according to Equation (4) [20]. $ϵs=c+az1+bz,$(4) where a, b and
c are estimated parameters.
We also considered the flow through the unidimensional, permanent, slow porous medium and tension on solids as a function of the local porosity. We did not consider the inertial terms of the equation
of motion for solids [20].
Thus, assuming that the medium was static, we used the following expression for pressure on solids [20], (Eq. 5): $Ps=(ρs−ρf)g∫0Lϵsdz,$(5) where z represents the reference axis measured from the top
of the sediment height L, ρ[s] and ρ[f] are the specific mass of solids and fluid, respectively, and g is the local gravity.
Finally, Equation (6) related the volumetric concentration with pressure on solids. $Ps=dϵse,$(6) where d and e are estimated parameters.
3 Results
3.1 Rheological analysis
We confirmed the fluid pseudoplasticity through the flow curve and by fitting the Power-law model to the experimental results, as shown in the log-log ladder, in Figure 2.
Table 1 shows indices of consistency and behavior of the rheological model and the determination coefficient, r^2.
The index of behavior presented a value lower than 1, confirming the shear-thinning behavior of the drilling fluid studied [5, 22, 23].
We determined the time dependence by the hysteresis formed (Fig. 3) and by the time required (approximately 3.5h) for the fluid to reach the steady state in the tests with constant shear rate [22,
3.2 Analysis of volumetric solids concentration profile
A 500mL suspension sample (21cm high in the test vessel) was monitored by the gamma-ray attenuation technique for the period of one year. Figure 4 shows the monitoring results for all positions
from the bottom of the glassware.
When analyzing the curves from z=0.5cm to z=12cm, the increase in solids concentration was slow. It was attributed to the rheological propriety of suspension. As the suspension showed
shear-thinning and thixotropic behavior, as the time goes on the polymers structures presented in the fluid had become gelified, which increased the polymeric arrangement of sediment column structure
to be passed through by solids and form the sediment. The same was verified in another study with drilling fluids [24].
In this context, the deformation caused by shear rate sedimentation was very low and the agglomeration phenomenon also from solid particles sedimentation in the sediment structure was not observed.
So, although the suspension had non-Newtonian nature, the sediment profile present Newtonian characteristics, and the velocity of sedimentation was guide by fluid flow upward against gravitational
solid deposition to the bottom tube [14, 24].
When evaluating the sedimentation process, the effect of concentration should be considered. The set of particles during sedimentation (downwards) pushes the fluid in the lower positions so it only
has one possible way, goes vertically upwards and therefore decelerate the sedimentation. This phenomenon was observed in Newtonian fluids [14, 16].
When monitoring from z=0.5 to 16cm, the following sedimentation regions were observed:
• from the beginning of the experiment up to approximately 20 days, the suspension was homogenized and, therefore, in the region of free sedimentation (ϵ[s]=ϵ[s0]);
• in the monitoring performed from days 21 to 249, two regions were found: the intermediate one, in which the concentration of particles was between the maximum (sediment concentration) and the
initial concentration of the suspension (ϵ[sm]<ϵ[s]<ϵ[s0]), and the region of free sedimentation;
• from days 250 to 364, positions closest to the bottom of the test vessel (z=0.5cm and z=1cm) had the highest volumetric particle concentration values (ϵ[sm]≅19%) and tended to stability.
We also observed the intermediate regions (z=2cm to z=12cm) and free sedimentation (z=16cm).
The inclination of the curves decreased with the increase in the monitoring position; therefore, we assumed that, initially, larger particles reached the bottom of the vessel, and the smaller
particles, after a certain time, filled the interstices, increasing sediment concentration until the final stability condition [16, 19, 25, 26].
Figure 5 shows the results of monitoring the local solids concentration over time for positions z=18 and 20cm.
Figure 5 shows that solids concentration remained constant for a period and then decreased to values close to zero. Such behavior was due to the passage of superior discontinuity through the
radiation detection system.
In addition, for position z=18cm, monitoring showed a peak of solids concentration during 9.6% of the experiment time. According to theory [27], this behavior is related to the increase in the
intermediate region before the solids concentration tend to zero.
The concentration reduction at positions near the top of the fluid occurred linearly and slowly. For position z=20cm, the time required to reduce the initial concentration to close to zero was
approximately 90 days, and, for position z=18cm, this time was approximately 315 days. Figure 6 clearly shows the region with solids concentration close to zero, due to the different coloring of
the fluid. Although the literature [27] has named this region as clarified liquid, the fluid studied had a dark coloring.
The deviations obtained for each monitoring ranged from 0.1864% to 0.0640%.
The relationship between position and time in which curves had the same concentration value, the constant concentration curves, allowed to interpret sedimentation as the propagation of waves with
equal concentration and to evaluate the characteristics of the materials settling in the drilling fluid [28] (Fig. 7).
Figure 7 shows that the constant concentration curves associated with sediment formation presented different inclinations. The concentration curves closer to the initial volumetric solids
concentration in the suspension were the most inclined. Therefore, sedimentation did not occur at a constant rate.
We also observed the relationship between the time of appearance of constant concentration curves as a function of concentration, i.e., the curves representing higher concentrations took longer to
emerge. Thus, constant concentration curves are not straight and do not start in the origin of the axes. This behavior is related to the compression of sediment caused by the upper layers of solids [
19, 29].
Pressure on solids was another parameter. We adjusted the results of volumetric solids concentration in the sediment formed after one year of experiment as a function of the position (Eq. 4). The
determination coefficient for fitting was 0.997. Table 2 presents the parameters determined.
The adjustment calculated pressure on solids. Subsequently, we adjusted the values calculated by Equation (5) to calculate pressure on solids as a function of concentration, according to Figure 8 and
Table 3.
4 Conclusion
Monitoring solids concentration over time allowed a quantitative and qualitative evaluation of the sedimentation of particles in drilling fluids. The suspension presented good stability when compared
to others that also have time-dependent shear-thinning characteristics.
The results indicated the trend of logarithmic growth of the curves of volumetric solids concentration versus time for the monitoring performed near the bottom of the test vessel. For positions near
the top, this behavior was linear.
Constant concentration curves enabled the verification of compressibility of the sediment formed and the occurrence of different settling rates.
This study proposed a constitutive equation for pressure on solids, assuming that the system was static.
The authors thank Capes, CNPq, Fapemig, Petrobras, and the School of Chemical Engineering of the Federal University of Uberlandia for the financial support.
All Tables
All Figures | {"url":"https://ogst.ifpenergiesnouvelles.fr/fr/articles/ogst/full_html/2018/01/ogst170175/ogst170175.html","timestamp":"2024-11-09T07:08:21Z","content_type":"text/html","content_length":"113127","record_id":"<urn:uuid:bd683d42-1479-4a23-ae0d-9a1e7b3b4832>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00215.warc.gz"} |
Gini Impurity A-Z (Decision Tree) | FavTutor
In Tree-based models, there is a criterion for selecting the best split-feature based on which the root of say, a Decision Tree gets split into child nodes (sub-samples of the total data in the root
and so on) and hence, a decision is made. So, in a Decision Tree split-feature is the judge and child nodes represent the judgements. The basic intuition of finding the best split of the root or any
internal node of a Decision Tree is that, the each of the child nodes to be created, should be as homogeneous as possible. In other words, the each of the child nodes to be created should have most
of the instances with target labels belonging to the same class. In order to achieve so, there are 2 most popular criteria which is very common among Machine Learning practitioners:
1. Gini Impurity
2. Entropy and Information Gain
In this article, the criterion, Gini Impurity and it's application in Tree-based Models is discussed.
All you need to know about Gini Impurity
Gini Index
Gini Index is a popular measure of data homogeneity. Data Homogeneity refers to how much polarized is the data to a particular class or category. Let us consider an example of an Exploratory Analyzed
Data of people winning or losing a tournament, given their Age and Gender:
So, there are 4 blocks of analyzed data. The labels 'P' and 'N' indicate number of wins and losses respectively.
Gini Index (GI) is defined as,
From the definition, it is evident that for perfectly homogeneous data block, the Gini Index is equal to 1.
Now, in this example, there are 2 features, Gender and Age and the target label is win/loss i.e., outcome of the tournament. GI is calculated for each and every feature and the feature with the
highest value is to selected as the best split-feature.
For calculating the Gini Index for Gender, Gini Index of Male (M) and Female (F) categories need to be calculated
Similarly, for calculating the Gini Index of Age, Gini Index of labels '<50' i.e., age less than 50 and '>=50' i.e., age greater than or equal to 50 need to be calculated
So, as Gini Index(Gender) is greater than Gini Index(Age), hence, Gender is the best split-feature as it produces more homogeneous child nodes.
Gini Impurity
Now, Gini Impurity is just the reverse mathematical term of Gini Index and is defined as,
So, it is a measure of anti-homogeneity and hence, the feature with the least Gini Impurity is selected to be the best split feature.
Now, following the above example, Gini Impurity can be directly calculated for each and every feature.
Calculating Gini Impurity for Gender, Gini Impurity of Male (M) and Female (F) need to be calculated
Similarly, for calculating the Gini Impurity of Age, Gini Impurity of labels '<50' i.e., age less than 50 and '>=50' i.e., age greater than or equal to 50 need to be calculated
So, as Gini Impurity(Gender) is less than Gini Impurity(Age), hence, Gender is the best split-feature.
So, in this way, Gini Impurity is used to get the best split-feature for the root or any internal node (for splitting at any level), not only in Decision Trees but any Tree-Model. | {"url":"https://favtutor.com/blogs/gini-impurity","timestamp":"2024-11-12T06:50:31Z","content_type":"text/html","content_length":"74132","record_id":"<urn:uuid:dd7c5fc4-e970-4d48-a34e-0c542b6a2005>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.58/warc/CC-MAIN-20241112045844-20241112075844-00514.warc.gz"} |
Hand migration
Next: Reflector steepening Up: EXPLODING REFLECTORS Previous: An impulse in the
Given a seismic event at ( x[0] , t[0] ) with a slope x[m] , t[m] ) after migration. Consider a planar wavefront at angle dx in a time dt. Assuming a velocity v we have the wave angle in terms of
measurable quantities.
The vertical travel path is less than the angled path by A travel time t[0] and a horizontal component of velocity Consideration of a hyperbola migrating towards its apex shows why (5) contains a
minus sign. Equations (4) and (5) are the basic equations for manual migration of reflection seismic data. They tell you where the point migrates, but they do not tell you how the slope p will
Next: Reflector steepening Up: EXPLODING REFLECTORS Previous: An impulse in the Stanford Exploration Project | {"url":"https://sep.stanford.edu/sep/prof/iei/xrf/paper_html/node7.html","timestamp":"2024-11-11T08:05:41Z","content_type":"text/html","content_length":"5794","record_id":"<urn:uuid:658f181d-cecb-4d0a-8393-e7e8700e55e4>","cc-path":"CC-MAIN-2024-46/segments/1730477028220.42/warc/CC-MAIN-20241111060327-20241111090327-00689.warc.gz"} |
Geomaterials (EASC08021)
Undergraduate Course: Geomaterials (EASC08021)
Course Outline
School School of Geosciences College College of Science and
level SCQF Level 8 (Year 2 Availability Available to all students
(Normal Undergraduate)
year taken)
SCQF 20 ECTS Credits 10
In this course we explore the fundamental nature of the material which constitutes the Earth and other planets. In the Mineral Science section we consider how atoms are arranged
in crystalline materials and how this ultimately governs the nature of geomaterials. Interaction of crystalline materials with light, X-rays and electrons are used to introduce
the theoretical and practical basis behind the polarising microscope, X-ray diffraction and electron microscope/microprobe. In Composition of the Earth we review the main groups
Summary of Earth Materials, considering (1) how structure, chemistry, physical properties, and occurrence are interrelated, (2) how earth materials are used in modern research as
information sources to reveal the nature of Earth processes, and (3) introduce theoretical aspects of modern Earth Materials research (e.g. phase stability and transitions). In
the final section Chemical Equilibria we consider how the stability and occurrence of geomaterials can be predicted and determined numerically using thermodynamics, and consider
factors governing the rates of Earth processes at variable depths.
Week 1
Part One: mineral science
Lecture 1. Refresher of symmetry, systems and Miller indices. Introduction to lattices. Group and translational symmetry and lattices
No practical
On-line test of material from 1st year
Lecture 2. Lattice and structure. X-ray diffraction and determining crystal structures. Crystal structure of amorphous materials. Crystal structure and bonding.
Practical. Symmetry and lattices. Indexing lattice planes and using XRD data to solve crystal structures.
Week 2
Lecture 3. Intro to the polarising microscope; colour, pleochroism and relief; birefringence and interference patterns
Practical. Optics intro to use of the polarising microscope; recognition and use of interference colours.
Lecture 4. The optical indicatrix: optic sign; relationship between optical and crystallographic structure of minerals
Practical. Interference figures for uniaxial and biaxial minerals.
Week 3
lecture5. Composition of the Earth; mineral chemistry; expressing chemical variation with formulae and plots; chemical analysis of minerals; the Electron microprobe.
Practical. Crystal structure and optical properties (introduction to extinction angles and pleochroism).
Part 2: Composition of the Earth
Lecture 6. Intro to oxides, silicates, carbonates etc
Classification of silicates based on structure, Isosilicates: olivine structure
Practical. Isosilicates: olivine and garnets
Week 4
Lecture 7. Isosilicates: olivine (P-T and T-X phase diagrams¿structure of the deep Earth¿hydration and serpentinisation)¿ aluminosilicates (link to metamorphic petrology..P-T
No practical
Lecture 8. Chain structures: pyroxenes and amphibole (structure, comp)
Practical. Pyroxene and Amphibole assessed practical
Week 5
Lecture 9. Chain structures continued: more on applications: solvus, phase diagrams, sheet silicates
No practical
Course Lecture 10. Sheet silicates continued: more on applications: serpentine (PT, hydration and dehydration¿volatiles in the deep Earth)¿industrial minerals which are sheet
description silicates¿clays (swelling and geomorphology/slope stability)
Practical. Sheet silicates
Week 6
Lecture 11. Framework silicates 1: feldspar structure, composition, stability; ordering, exsolution and phase transitions.
Practical. Feldspar: structures, hand specimens and thin sections. Determining feldspar compositions.
Lecture 12. Framework silicates 2: Quartz as a sedimentary mineral, phase transitions (UHP metamorphism¿links to subduction)
Practical. Reading melting phase diagrams, Quartz in hand specimens and thin sections
Week 7
Lecture 13. Carbonates (but based on chemical/biological processes)
Practical. Carbonates
Group poster presentation during practical slot (no lecture)
Week 8
Chemical Equilibria
Lecture 14. Intro to thermodynamics and the phase rule: systems, phases, components and predicting equilibria.
Practical. The Phase Rule and its use: Introduction of Phase Diagrams.
Lecture 15. Thermodynamic state variables; laws of thermodynamics; enthalpy, entropy, free energy; the Clapeyron equation.
Practical. Calculation of the Al2SiO5 phase diagram
Week 9
Lecture 16. Invariant, univariant and divariant assemblages in P-T and composition-paragenesis diagrams; equilibrium vs stability; solid-solid vs fluid-present reactions; G-X
Practical. Calculating and plotting G-X diagrams and phase diagrams in 3 component systems.
Lecture 17. Chemical potential, standard states, activities, fugacities; thermodynamics of impure phases; a-X relations for ideal solutions; the equilibrium constant; intro to
ideal gases.
Practical. Calculating phase diagrams in impure systems.
Week 10
Lecture 18. Intro to thermodynamics at low T and chemical weathering. Application to mineral weathering.
Practcial. Construction of simple phase diagrams for silicate weathering.
Lecture 19. Introduction to kinetics and diffusion in minerals, its implications for Earth processes, closure temperature and dating Earth Processes.
Practical. Timescales of volcanic eruptions (assessed)
Entry Requirements (not applicable to Visiting Students)
Pre-requisites Students MUST have passed: Earth Dynamics Co-requisites
Prohibited Other If students have not taken Earth Dynamics, they will need the permission of the Course Organiser to take
Combinations requirements this course.
Additional Costs None
Information for Visiting
Pre-requisites None
High Demand Course? Yes
Course Delivery Information
Academic year 2015/16, Available to all students (SV1) Quota: 90
Course Start Semester 1
Timetable Timetable
Total Hours: 200 ( Lecture Hours 22, Supervised
Learning and Teaching activities (Further Info) Practical/Workshop/Studio Hours 55, Programme Level
Learning and Teaching Hours 4, Directed Learning
and Independent Learning Hours 119 )
Assessment (Further Info) Written Exam 50 %, Coursework 50 %, Practical Exam
0 %
Assessments are based on, written Exam: 50%, Course
Work: 50 %, Practical Exam: 0%.
Written exam is in the end of the semester and
covers all the materials from the course.
Course works are two assessed practicals
(Composition of the Earth (35%), Chemical
Additional Information (Assessment) Equilibria (35%)) and group poster presentation
To pass the course students must achieve an overall
mark of 40% or more. Students must also achieve a
minimum of 40% in both the Degree examination and
in the Classwork component to attain a pass
overall, whatever their final aggregate mark.
Coursework will be returned to students within a
maximum of 2 weeks of the submission deadline, with
individual feedback from instructors and with
Feedback recommendations as to how students can improve
their grades.
General class feedback is also given in practical
classes or on LEARN course site.
Exam Information
Exam Diet Paper Name Hours & Minutes
Main Exam Diet S1 (December) Geomaterials 3:00
Resit Exam Diet (August) Geomaterials 3:00
Learning Outcomes
On completion of this course, the student will be able to:
1. To gain a broad knowledge and understanding of the constituent
materials which make up the solid Earth, and how the study of
minerals can be used to understand the processes which have
shaped the Earth throughout geological time.
2. To identify, describe and interpret geomaterials from an atomic
level to a hand specimen scale, and to be familiar with the
foundations and application of modern methods used to study
geomaterials: diffraction, optical mineralogy, electron
microbeam analysis
3. Have a broad understanding of the most important groups of
minerals which constitute the Earth, and develop an
understanding of the relations between different groups of
materials, their occurrence, formation and stability, and how
this information can be used to understand processes occurring
on the Earth.
4. To understand how stability of earth materials can be predicted
and determined using thermodynamics, and how the rates of atomic
processes govern Earth processes.
5. Students are actively encouraged to discuss academic problems
with fellow students and to work in collaboration: invaluable
transferable skills. This course will develop students
theoretical understanding of the study of Earth materials,
observational and analytical skills, and numerical skills
through lectures and lab-based practicals.
Reading List
Nesse, W.D. Introduction to Mineralogy. Oxford.
Putnis, A. Introduction to Mineral Sciences. Cambridge.
Klein C, Mineral Science. Wiley.
Klein, C and Philpotts, A. Earth Materials. Cambridge University
Hefferan, K and O¿Brien. Earth Materials. Wiley-Blackwell.
Deer, Howie & Zussmann, Intro. to the Rock Forming Minerals.
Anderson G M (2009) Thermodynamics of Natural Systems. Cambridge
University Press.
Best MG (2003) Igneous and Metamorphic Petrology. Blackwell Science.
Gill R (1995) Chemical Fundamentals of Geology. Chapman and Hall.
Langmuir D (1997). Aqueous Environmental Geochemistry. Prentice
Bloss, F. Donald, Introduction to the Methods of Optical
Crystallography, Holt, Blond.
Gay, P. Crystal Optics
McKenzie & Guilford, Atlas of Rock-forming Minerals.
McKenzie & Adams, A Colour Atlas of Rocks and Minerals in Thin
Section. Manson
Additional Information
Graduate Quantitative ability (through practical based
Attributes mathematical calculations), observational and individual
and Skills analytical skills (lab practicals) and group work
through take-home class assessment exercises.
Additional Students take two lectures per week and EITHER Mon or
Class Tues (2hr practical) and EITHER Thurs or Fri (3hr
Delivery practical)
Keywords Geomaterials
Dr Tetsuya Komabayashi Mrs Nicola Muir
Course Tel: (0131 6)50 8518 Course Tel: (0131 6)50
organiser Email: secretary 4842
Tetsuya.Komabayashi@ed.ac.uk Email:
© Copyright 2015 The University of Edinburgh - 18 January 2016 3:46 am | {"url":"http://www.drps.ed.ac.uk/15-16/dpt/cxeasc08021.htm","timestamp":"2024-11-09T20:33:25Z","content_type":"text/html","content_length":"26989","record_id":"<urn:uuid:210aea72-d501-46a9-8a76-a5554b33b9c9>","cc-path":"CC-MAIN-2024-46/segments/1730477028142.18/warc/CC-MAIN-20241109182954-20241109212954-00474.warc.gz"} |
C1_W3_Assignment_help with PCA Calculations
I am struggling with the compute_pca function. I have rewatched the videos for help, but it still is not clear how I do the final calculations in the compute_pca function. I’ve summarized the steps I
think I have correct.
I start by mean centering X. This still has dimensions 10,3
I then get the covariance_matrix by passing the X_demeaned to the numpy covariance function. This gives me a 3x3 matrix.
I pass this to the eigh function which returns eigen_vals (3,) and eigen_vecs (3,3).
I then get an index list for sorted eigen_vals. This is in ascending order and so I flip the order to descending.
I then apply this ordering to the eigen_vals and this looks correct. The ones with most significance are at the top now (confirmed visually).
I then narrow this to the top n_components giving me a subset (3,2).
All good up to this point I think. What I am struggling with, given that I don’t have a ton of linear algebra experience is how to get X_reduced in the final step.
The instructions say to transform the data by multiplying the transpose of the eigenvectors with the transpose of the de-meaned data and then take the transpose of that product.
The transpose of the eigenvectors subset is (3,2) and the transpose of the de-meaned data is (10,3). I tried taking the dot product of the de-meaned data and the eigenvectors subset and then
transposing that. The only way I could get it to work was to put the demeaned data first and then the subset which gives me a (10,2) result. The transpose is (2,10) which doesn’t look right. I’ve
tried to find an example of the necessary calculation in the lecture material but I am not seeing it. It little more explanation on this would help greatly.
I added some print statements to show the shapes of the various objects in compute_pca. Here’s what I see when I run the first test cell:
X.shape (3, 10)
X_demeaned.shape (3, 10)
covariance_matrix.shape (10, 10)
eigen_vals [-7.03941390e-17 -3.60417070e-17 -1.30858621e-17 -8.61317229e-19
2.07977247e-19 3.78308880e-18 1.81729034e-17 5.06232858e-17
2.50881048e-01 5.48501886e-01]
idx_sorted [0 1 2 3 4 5 6 7 8 9]
eigen_vecs_subset.shape (10, 2)
X_reduced.shape (3, 2)
Your original matrix was (3, 10) and it became:
[[ 0.43437323 0.49820384]
[ 0.42077249 -0.50351448]
[-0.85514571 0.00531064]]
So you can see that the X_reduced shape should be 3 x 2, not 2 x 10.
One other thing to note is that perhaps their instructions are a bit more complex than they really need to be. Note the following mathematical relationship:
(A \cdot B)^T = A^T \cdot B^T
But maybe the more relevant way to use that mathematical fact is to write it this way:
(A^T \cdot B^T)^T = (B^T)^T \cdot (A^T)^T = B \cdot A
1 Like
Thank you for the print statements. I appear to be doing something wrong with the covariance call. I am using the np.cov() function passing in X_demeaned as only argument. Do I need additional
arguments? Am I using the wrong call?
Here are my corresponding print statements up through covariance.
X.shape (3, 10)
X_demeaned.shape (3, 10)
covariance_matrix.shape (3, 3)
You have two choices in how you invoke np.cov to get the right answer. If you pass X_demeaned as the argument, then you need to also pass rowvar = False to get the correct answer.
Or you can pass X_demeaned.T and use rowvar = True.
That’s how I got a 10 x 10 result.
Thanks for the print statements. Adding the argument to the cov function fixed the issue with the dimensions for the covariance matrix, but my eigen_vals don’t match yours. That makes me wonder if I
am not calculating the X_demeaned values correctly. I am currently subtracting the mean of X from each of the elements. Was this the correct approach. My output up to calculating X_reduced is pasted
X.shape (3, 10)
X_demeaned.shape (3, 10)
covariance_matrix.shape (10, 10)
eigen_vals [-5.57241675e-17 -1.19872199e-17 -7.67969920e-18 -5.07626546e-18
5.83709773e-18 6.86386536e-18 2.11268437e-17 1.60597606e-16
2.50881048e-01 5.48501886e-01]
idx_sorted [0 1 2 3 4 5 6 7 8 9]
eigen_vecs_subset.shape (10, 2)
Use the mean of each feature, not the mean of the entire X matrix.
Thank you. I will give that a try.
Thanks @TMosh and @paulinpaloalto - I was able to correct my X_demeaned calculation and fix a couple more issues and everything ran correctly.
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Regression - How To Quickly Read the Output of Excel’s Regression
Regression Analysis
Done in Excel
How To Read the Output
There is a lot more to the Excel Regression output than just the regression equation. If you know how to quickly read the output of a Regression done in, you’ll know right away the most important
points of a regression: if the overall regression was a good, whether this output could have occurred by chance, whether or not all of the independent input variables were good predictors, and
whether residuals show a pattern (which means there’s a problem).
Excel Regression Output With Color-Coding Added
Click On Image To See Enlarged View
This video will illustrate exactly how to quickly and easily understand the output of Regression performed in Excel:
Step-By-Step Video About How To Quickly Read and Understand the Output of Excel Regression
(Is Your Sound Turned On?)
The 4 Most Important Parts of Regression Output1) Overall Regression Equation’s Accuracy
(R Square and Adjusted R Square)
2) Probability That This Output Was Not By Chance
(ANOVA – Significance of F)
3) Individual Regression Coefficient and Y-Intercept Accuracy
4) Visual Analysis of Residuals
Some parts of the Excel Regression output are much more important than others. The goal here is for you to be able to glance at the Excel Regression output and immediately understand it, so we will
focus our attention only on the four most important parts of the Excel regression output.
1) Overall Regression’s Accuracy
Click On Image To See Enlarged View
R Square
– This is the most important number of the output. R Square tells how well the regression line approximates the real data. This number tells you how much of the output variable’s variance is
explained by the input variables’ variance. Ideally we would like to see this at least 0.6 (60%) or 0.7 (70%).
Adjusted R Square
– This is quoted most often when explaining the accuracy of the regression equation. Adjusted R Square is more conservative the R Square because it is always less than R Square. Another reason that
Adjusted R Square is quoted more often is that when new input variables are added to the Regression analysis, Adjusted R Square increases only when the new input variable makes the Regression
equation more accurate (improves the Regression equations’s ability to predict the output). R Square always goes up when a new variable is added, whether or not the new input variable improves the
Regression equation’s accuracy.
2) Probability That This Output Was Not By Chance
Click On Image To See Enlarged View
Significance of F – This indicates the probability that the Regression output could have been obtained by chance. A small Significance of F confirms the validity of the Regression output. For
example, if Significance of F = 0.030, there is only a 3% chance that the Regression output was merely a chance occurrence.
3) Individual Regression Coefficient Accuracy
Click On Image To See Enlarged View
P-value of each coefficient and the Y-intercept
– The P-Values of each of these provide the likelihood that they are real results and did not occur by chance. The lower the P-Value, the higher the likelihood that that coefficient or Y-Intercept is
valid. For example, a P-Value of 0.016 for a regression coefficient indicates that there is only a 1.6% chance that the result occurred only as a result of chance.
4) Visual Analysis of Residuals
Charting the Residuals
Click On Image To See Enlarged View
The Residual Chart
Click On Image To See Enlarged View
The residuals are the difference between the Regression’s predicted value and the actual value of the output variable. You can quickly plot the Residuals on a scatterplot chart. Look for patterns in
the scatterplot. The more random (without patterns) and centered around zero the residuals appear to be, the more likely it is that the Regression equation is valid.
There are many other pieces of information in the Excel regression output but the above four items will give a quick read on the validity of your Regression.
If anyone has any comments or observations related to this article, feel free to submit them because your input and opinions are highly valued.
If You Like This, Then Share It...
Excel Master Series Blog Directory
Statistical Topics and Articles In Each Topic
• Histograms in Excel
• Bar Chart in Excel
• Combinations & Permutations in Excel
• Normal Distribution in Excel
• t-Distribution in Excel
• Binomial Distribution in Excel
• z-Tests in Excel
• t-Tests in Excel
• Hypothesis Tests of Proportion in Excel
• Chi-Square Independence Tests in Excel
• Chi-Square Goodness-Of-Fit Tests in Excel
• F Tests in Excel
• Correlation in Excel
• Pearson Correlation in Excel
• Spearman Correlation in Excel
• Confidence Intervals in Excel
• Simple Linear Regression in Excel
• Multiple Linear Regression in Excel
• Logistic Regression in Excel
• Single-Factor ANOVA in Excel
• Two-Factor ANOVA With Replication in Excel
• Two-Factor ANOVA Without Replication in Excel
• Creating Interactive Graphs of Statistical Distributions in Excel
• Solving Problems With Other Distributions in Excel
• Optimization With Excel Solver
• Chi-Square Population Variance Test in Excel
• Analyzing Data With Pivot Tables
• SEO Functions in Excel
• Time Series Analysis in Excel
30 comments:
1. Thank you so much for this!!! Very helpful
Mallory Wood
2. Really nice.
3. Cheers! [A deadline has been made]
4. Fantastic post. Am seriously considering buying the entire package.
5. Really good work...helped a lot....
Thank you ....
6. Thank you so much. Very helpful.
7. uda man, test saver
8. nice and clear! other blogs are so deep into explaining the equations - this is one of the only ones I've seen that actually helps interpret the output in plain language! thanks!
9. very simply and useful.. thanks
10. thank you so much you.... have saved my grade
11. Why thanks! Really glad you found it useful. :>D
12. Really, really helpful. Thanks!!
13. Thanks very much for sharing the knowledge...it was really helpful!
14. Working on an assignment for class. This helped more than my notes! Thanks!
15. Thanks so much, this was easy to understand. A suggestion though...it would be nice to see this same analysis done with something that isn't highly correlated. The contrast would be helpful in
finding simple pitfalls.
16. This is great, thank you! I am familiar with diff. analytics terms but not in too much technical terms, so this is very helpful for me!
17. Thank you sooo much. im a psych scholar and this helped me finish my thesis in time!
God bless you!
18. Thanks a ton buddy...this is really helpful
19. This was a great help dear, thanks a lot.
20. Thanks i found it useful. I am obliged.
21. Thank you so much for this information. I've been searching for a while for simple explanations on the regression summary output in Excel and I finally ran across this site. Absolutely a life
22. Fantastic! Very very useful!
1. Does your text cover Excel 2010?
23. Awesome dude. Thanks much
24. Thank you very much!!! i had trouble understanding what the figures meant and this was very helpful.
25. Thanks a ton !
26. To the point and simple...Thanks!
27. I concur with pretty much everyone who has commented before me...very useful.....thank you
28. is there any one help me to say what is the menaing of p value 0.20
29. Đặt vé tại phòng vé Aivivu, tham khảo
vé máy bay đi Mỹ giá rẻ
ve may bay tu my ve vietnam
các chuyến bay từ narita về hà nội
vé máy bay từ đức về sài gòn
vé máy bay từ Toronto về việt nam
vé máy bay từ hàn quốc sang việt nam
danh sách khách sạn cách ly tại quảng ninh
vé máy bay chuyên gia nước ngoài sang Việt Nam | {"url":"http://blog.excelmasterseries.com/2010/03/how-to-quickly-read-output-of-excels.html","timestamp":"2024-11-12T10:27:59Z","content_type":"text/html","content_length":"285035","record_id":"<urn:uuid:11c417d7-f891-4be5-8fb4-e39b04826de6>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00011.warc.gz"} |
Simulating Strongly Coupled Quantum Field Theory with Quantum Algorithms - JPS Hot Topics
Simulating Strongly Coupled Quantum Field Theory with Quantum Algorithms
JPS Hot Topics 2, 017
© The Physical Society of Japan
This article is on
Negative string tension of a higher-charge Schwinger model via digital quantum simulation
(PTEP Editors' Choice)
Prog. Theor. Exp. Phys. 2022, 033B01 (2022) .
Nonperturbative quantum field theory problems can often be difficult to solve with classical algorithms. Researchers now develop quantum computing algorithms to understand such problems in the
Hamiltonian formalism.
Quantum field theory (QFT) is an over-arching theoretical framework that combines classical field theory, special relativity, and quantum mechanics under the same umbrella. While weakly coupled QFT
problems have been solved and understood with current methods and techniques, strongly coupled QFTs remain elusive. Such problems can be tackled using numerical computation techniques. One such
technique is “quantum simulation.”
Unfortunately, numerical techniques developed to tackle problems have had limited success with classical algorithms. Such methods are either non-efficient, lack accuracy, or are too specific and
cannot be generalized to other problems. Interestingly, quantum computing algorithms have remained largely unexplored, partly because quantum simulations are usually performed using the Hamiltonian
formalism. Moreover, techniques for applying quantum simulation using quantum computers are yet to be developed.
In this work, the charge-q Schwinger model on the open boundary condition was studied in the Hamiltonian formalism instead of the more commonly used path-integral formalism. The Hilbert space in the
model is known to be decomposed into distinct sectors, called “universes.” The developed method was based on the so-called “adiabatic state preparation,” allowing the problem to be solved with a
digital quantum simulation.
The model revealed that a repulsive force acts between particles with opposite charges, contrary to the traditional classical attraction, in particular circumstances. This observation was confirmed
using a classical simulator of quantum devices. Predicting this phenomenon with a quantum computing algorithm, thus, demonstrated their potentially superior capability in handling strongly coupled
QFT problems.
Quantum computing methods like these can be used to gain deeper insights into more complex QFT problems, such as the time evolution of the early universe. Such problems have long intrigued scientists
but have remained intractable owing to the lack of adequate solving techniques.
The findings of this study open doors to further development and applications of quantum computing algorithms in QFT. This could lead to answers to long-awaited questions in QFT, broadening our
understanding of the universe.
Negative string tension of a higher-charge Schwinger model via digital quantum simulation
(PTEP Editors' Choice)
Prog. Theor. Exp. Phys. 2022, 033B01 (2022) .
Share this topic | {"url":"https://jpsht.jps.jp/article/2-017/","timestamp":"2024-11-02T05:47:10Z","content_type":"text/html","content_length":"74771","record_id":"<urn:uuid:dcd89837-cee8-43f7-b35e-933f0e2debc9>","cc-path":"CC-MAIN-2024-46/segments/1730477027677.11/warc/CC-MAIN-20241102040949-20241102070949-00352.warc.gz"} |
Math Contests
• The education department is starting quarterly Math Contests as part of our K-8 Math Challenge program. Everyone participating in Math Contests must also register in the K-8 Math Challenge
• In summer and then winter of 2017 , we offer two types of Math Contests, one at Elementary School Level and one at Middle School Level. The Elementary School Level math contest can be taken by
students in grades 4, 5 or 6. The Middle School Level math contest can be taken by students in grades 6, 7 or 8. Thus grade 6 students can choose to take the lower or higher level depending on
their math proficiency.
• The Math contests are for both atfal and nasirat. The purpose of the quiz is not to make the kids stressful, but make the quiz a “fun exercise”, just like they may play jeopardy or solve riddles
at an ijtema. Therefore we do not encourage kids or their parents to excessively “prepare” for the exam. Students who are regular in their studies at school should do fine.
• Your local taleem secretary will provide you the day and time when these math contests are held in your local jamaat. There will be one contest every quarter. Your taleem secretary will provide
the students printed copies of the math contest.
• Each Math Contest will have 5 Questions and the students have 30 Minutes. You can view a sample math contest for both Elementary and Middle school levels at: http://www.moems.org/sample.htm
• Students can use blank sheets of paper to work out each math problem. However, they should write down the final answer on the question sheet provided.
• Your local taleem secretary will grade the math contest and report results to the national taleem department. Local secretaries only report if the student’s answer was correct or not. As such
there is no partial credit in this contest.
• Students will be given solutions to the math contest after the exam.
• National taleem department will hand out prizes for students who do well in the math contest. | {"url":"https://edu.ahmadiyya.us/math-contests/","timestamp":"2024-11-09T12:31:53Z","content_type":"text/html","content_length":"27784","record_id":"<urn:uuid:23ea6766-75d8-4749-a3d2-c124cf9319ea>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00535.warc.gz"} |
How do you solve log_2( 20x ( x - 11) )? | Socratic
How do you solve #log_2( 20x ( x - 11) )#?
1 Answer
This is not an equation. Please see the explanation.
${\log}_{2} \left(20 x \left(x - 11\right)\right)$ is an expression; not an equation, therefore, it cannot be be solved.
If you are looking for the expression in a more useful form, then I will give you:
#(ln(20) + ln(x) + ln(x - 11))/ln(2); x > 11#
Impact of this question
1002 views around the world | {"url":"https://socratic.org/questions/how-do-you-solve-log-2-20x-x-11","timestamp":"2024-11-08T08:47:19Z","content_type":"text/html","content_length":"31489","record_id":"<urn:uuid:5a81e5dd-ec6c-4c73-a53c-ff7bd0c6d4cb>","cc-path":"CC-MAIN-2024-46/segments/1730477028032.87/warc/CC-MAIN-20241108070606-20241108100606-00558.warc.gz"} |
Intersecting Cylinders – The Steinmetz Solid
My next goal was to print two cylinders whose axes intersect at right angles and the volume common to both, otherwise known as the Steinmetz solid. I began by modeling these objects in Mathematica so
I could import the objects to Cinema 4D as well as create an interactive Mathematica worksheet about these objects.
Here are the Mathematica representations of two objects I planned to print:
I exported the top object as a .wrl file into Cinema 4D since it was already a solid.
For the bottom object, I decided it would easiest to create it from scratch in Cinema 4D using the “Tube” object and simply adjusting the dimensions and orientation.
I attempted to print the intersecting cylinders on the Afinia printer in the Math Department. It failed to finish printing because the filament got tangled coming off of the spool. However this was
not a complete fail since it shows the inside of the two cylinders and can still be used as a teaching tool.
I am currently attempting to reprint it and will see how it goes!
Update: The filament got tangled once again while I was printing resulting in a similar object to the one above. We decided that these were both better teaching tools than the original design and
decided not to try to print again.
I printed the Steinmetz solid on the MakerBot 2x and had great results! This object can be found on Thingiverse here and here. | {"url":"https://mathvis.academic.wlu.edu/2015/06/30/intersecting-cylinders-the-steinmetz-solid/","timestamp":"2024-11-05T10:15:24Z","content_type":"text/html","content_length":"37168","record_id":"<urn:uuid:cc953216-efe7-46a9-9a7f-189ee4700df6>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00098.warc.gz"} |
A Área Entre As Retas Y = 0, X = -1 E X = 1 E A Curva Y = Ex - 1 É: - Itapetinga Na Midia
In mathematics, the area between lines and a curve can be an important factor in understanding the relationship between the two. In this article, we will explore the area between the lines Y = 0, X =
-1 and X = 1, and the curve Y = Ex – 1.
Investigating the Area Between Lines and Curve
The area between lines and a curve can be investigated by examining the equation of the curve, and the equations of the lines. In this case, the equation of the curve is Y = Ex – 1, and the equations
of the lines are Y = 0, X = -1 and X = 1.
The area of the region between the two lines, and the curve, is the area of the triangle formed by the three points of intersection. This triangle can be found by solving the equations of the lines,
and the equation of the curve, to find the points of intersection.
The first point of intersection is found by solving the equation of the line X = -1 and the equation of the curve Y = Ex – 1. This gives us a point (x, y) = (-1, 0). The second point of intersection
is found by solving the equation of the line X = 1 and the equation of the curve Y = Ex – 1. This gives us a point (x, y) = (1, e – 1). The third point of intersection is found by solving the
equation of the line Y = 0 and the equation of the curve Y = Ex – 1. This gives us a point (x, y) = (0, -1).
Using these points, we can calculate the area of the triangle formed by the three points of intersection. This gives us an area of (1 – (-1)) * (e – 1 – 0) / 2 = e – 1.
Examining the Relationship of Y = Ex – 1.
The equation of the curve Y = Ex – 1 gives us an insight into the relationship between the lines and the curve. We can see from this equation that the curve is a parabola with its vertex at (0, -1).
This means that the curve is symmetric about the line Y = 0, and the points of intersection with the lines X = -1 and X = 1 are the same distance from the vertex.
We can also see from the equation Y = Ex – 1 that the slope of the curve is equal to the value of e. This means that the curve is increasing at a rate of e units per unit change in x
The area enclosed between the lines y = 0, x = -1, x = 1, and the curve y = ex – 1 is an important mathematical concept for problem solving. In this article, we will discuss the area, its properties,
and its relation to calculus.
Let’s start by looking at the equations that define the area. The two lines are y = 0 and x = ±1 (where the plus sign indicates x = 1 and the minus sign indicates x = -1). The curve is y = ex – 1,
which is a family of exponential curves defined by the function ex, which is the exponential function.
So what is the area enclosed by these equations? This area can be calculated using the definite integral, which is a calculus technique. The integral used to calculate the area is:
integral (from x = -1 to x = 1) of (ex – 1 dy)
This integral evaluates to e2 – 1, which is the area of the region enclosed by the lines and the curve.
The area can also be determined graphically by plotting the equations on a graph and then counting the number of square units enclosed by the lines and the curve.
Interestingly, the area enclosed by the lines and the curve can also be used to solve certain types of problems. For example, if you need to calculate the probability of a point randomly chosen from
the region being inside the area enclosed by the lines and the curve, you can use this area to solve the problem.
This area is an important math concept that can be used to solve a wide range of problems. Using calculus to calculate the area enclosed by the lines and the curve, as well as using the area to solve
certain types of problems, are just two of the applications of this important concept. | {"url":"https://itapetinganamidia.net/a-area-entre-as-retas-y-0-x-1-e-x-1-e-a-curva-y-ex-1-e/","timestamp":"2024-11-11T16:43:06Z","content_type":"text/html","content_length":"60209","record_id":"<urn:uuid:a1cc13c3-865d-45dd-b25d-8772e4c38129>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00174.warc.gz"} |
Research @ Mangaki · Recommandation d'anime et de mangas
12 Feb 2022
We are writing a recommender system. Let’s give a typical use case.
Meet Alice, she’ll be our reference user. She rates anime, telling us if she liked them or not. We call the data we gathered here Alice’s preferences. There are of course other persons, let’s
call them the Bobs. The Bobs do the same as Alice, therefore we get a lot of preference data. Based on this data, we train a machine learning model that guesses what anime Alice may like. Then,
we feed back the info. Alice now has a new anime list that she’s sure she will love.
To make this design work, wee need to gather data that we can train models on. Ideally, this data would be the preferences of each user, that is the rating they gave to each movie (if they rated any
movie). But we also value privacy, and we don’t want to leak our users’ information. Our goal here is to provide recommendation to our users without leaking their preferences to anyone, including
We propose a solution based on this paper written by Bonawitz et. al..
The problem our reference paper solves
The goal of the work of Bonawitz et. al. is to train machine learning models on users’ machines in a privacy-preserving way. Each user has their own data that they do not want to reveal, but the
model has to fit that data. The example they give is that of word guessing: users want to have a model that guesses the next word they are going to type, so that they can write faster, but they do
not want anyone to know exactly what they are typing.
The method is designed for gradient descent: a model is trained iteratively by slowly shifting its parameters in a direction that improves its accuracy. That “direction” is the gradient that is
computed at each step of the training.
The computation is distributed on users’ devices, and a central server supervises the process. For each step, each user computes their gradient, then everyone agrees on the mean value of all the
individual gradients, and the server uses the result to move to the next step.
The paper of Bonawitz et. al. explains how to compute the mean of the gradients without anyone – not even the server – knowing the others’ gradients. It even goes a bit further than that: users can
drop at any time during the training process, and the model will still be trained as long as a certain amount of users are still connected (this relies on secret sharing).
Using this to our advantage
What out reference paper really explains is how to securely aggregate information as long as the information can be encoded as a vector. But in computer science, everything can be seen as a vector of
bits, so we can transform this method in a method to anonymously collect messages from our users.
To make an analogy, the aggregation process is like having a piece of paper and some magic ink that it is invisible until it is heated. People can write whatever they want on the paper and then
reveal the result after they all finished to write. Because they use magic ink that is invisible until the paper is heated, they can’t read what the others wrote. Here, we want people to write their
preferences and then reveal the list of all preferences without being able to guess the participants’ preferences.
Concretely, let’s say we have four users, Alice, Beatrice, Christine and Dominique. Let’s say their preferences can be encoded over 4 bits^1, and that the encoded preference is never null. Alice and
her friends will build an 8 bits long message that will contain their preferences. We use 16 bits because the final message will contain four “slots”, one for each individual message. Alice’s
preferences are \({\mathtt{1001}}\), Beatrice’s are \({\mathtt{0110}}\), Christine prefers \({\mathtt{1010}}\) and Dominique votes \({\mathtt{0101}}\) (we can use any other non-null encoding).
Alice, Beatrice, Christine and Dominique just need to know where to put their individual preferences in the final vector. Let’s say Alice will take the first 4 bits, Beatrice will take the next 4
bits, and so on.
The final message will then be \({\mathtt{1001\,0110\,1010\,0101}}\), i.e., with proper notation, \((\mathtt{1001} \mathtt{<<} 12) \wedge (\mathtt{0110} \mathtt{<<} 8) \wedge (\mathtt{1010} \mathtt
{<<} 4) \wedge \mathtt{0101}\). Using the technique that is explained by Bonawitz et. al., we can compute this securely^2.
We only have one problem: how do Alice, Beatrice, Christine and Dominique know in which slot to put their preferences ? To continue the magic ink analogy, when people are writing, they don’t know if
they are writing at a place where someone else already wrote. The difficulty is that they need a way to find somewhere to write, without explicitly agreeing on which part of the paper belongs to whom
since knowing where people write implies knowing what they write.
So how do people know where to write without colliding with someone else’s data ? The solution is simple: they don’t! They randomly choose a place in the final vector. If two of them choose the same
place, we’ll know, because the final vector will not make sense (one of the two four will be null).
At the end, the preferences were randomly put in the final vector and no information on Alice’s, Beatrice’s, Christine’s or Dominique’s preferences has been leaked during the aggregation process.
Therefore, when someone reads the final vector, they can’t know which encoded preference belongs to who: the result can be safely published without compromising the users’ privacy.
Mathematically, we were just writing about vectors over the two element field.
More detail
The previous example is, of course, degenerate, because there are only four users, and our strategy to find slots for each user is too expensive.
In practice, there are many users, therefore the only information one can ever have about the resulting dataset is that each collected preference vector belongs to some user that contributed, but no
more information than that can be gathered, even by the server (except if there are many colluding users).
The problem of privately finding a slot for each user is solved by starting with simpler rounds where users try to take a random slot, represented by a bit in a giant vector, and the users agree when
they see there has been no collision.
There is another problem: that of malevolent user trying to write everywhere (similar to ballot stuffing). This is sorted out by appending individual messages with hashes, so that when two messages
are written in the same slot the hashed don’t match anymore.
The protocol is also made more secure by requiring key exchanges and signed transactions, so that users know that the other users they see are not dummies controlled by a malicious server. These
cryptographic schemes come from libraries that are essentially built on top of curve25519 cryptography.
The protocol also uses Shamir Secret Sharing, so that if some user drops from the exchange the aggregation process can continue, as long as there are enough remaining users.
We also use ChaCha for cryptographically secure random data generation.
The libraries we used are:
Extra: An overview of the reference paper’s idea
To put the idea more formally, we have parties \(\mathcal{U}_1, \ldots, \mathcal{U}_n\) and a server \(\mathcal{S}\) communicating over some network. Each party has a vector \(v_i\). We want the
server to know \(\sum_{i=1}^n v_i\) without anyone knowing (nor being able to guess) the \(v_i\).
The main idea is that a party \(\mathcal{U}_i\) won’t ever send its \(v_i\), but it will send \(v_i + w_i\) where \(w_i\) is called the individual noise vector. The security comes from the fact that
the \(w_i\) will be evenly distributed over the set of all possible vectors, so \(v_i + w_i\) will be completely indistinguishable from just noise. We want to compute \(\sum_{i=1}^n v_i\), but we can
only compute \(\sum_{i=1}^n v_i + w_i\). The idea is to ensure that \(\sum_{i=1}^n w_i = 0\), so that \(\sum_{i=1}^n v_i + w_i = \sum_{i=1}^n v_i\).
To get a secure \(w_i\), each pair of users agrees on a random vector they will add^3 to their individual noise vector.
For even more detail, and a proof of the security of the protocol, see the paper of Bonawitz et. al.
Second extra: Closing the loop with privacy-preserving recommendation
We discussed privacy-preserving aggregation of data, and we showed we can use it to gather training data for ML models. We can go further and provide secure recommendations without leaking
information about user preferences.
Let’s say we collected data and we trained an ALS model. In that case, each anime has an embedding, and each user has their own embedding too. Ratings are predicted by simple dot products \(\langle \
texttt{user} \mid \texttt{anime} \rangle\). There are several possibilities for users to find their recommendations.
The first possibility is that, since each user knows where their own preferences lie in the collected data, they can download the whole model and perform the dot products themselves without the
server knowing which of the collected preferences belongs to the user. This implies no more leakage of data than what we currently have.
We could also have the server publish the data of every anonymized user (publish the dot products directly), which amounts to the same, this just changes where the computation happens but not the
security model.
We could also have the server publish just the embedding of every anime, and the users train their own local model. With this technique, we lose a little accuracy on the predictions, but we gain
several benefits:
• The data each user has to download (the embedding of each anime) is small (the order of magnitude would be 100 kilobytes at most) compared to what they have to download with the previous
• New users who didn’t participate in the preference collection process can have ratings (since they can train their own model based on the embeddings they are given by the server);
• There is no complicated bookkeeping: users don’t have to know where their data lies in the anonymized data set.
1. 4 users and 4 bits is ridiculously small, but it is easier to picture, and the example scales well ! ↩
2. We said that bit strings are vectors, and we used the exclusive or operation \(\wedge\). ↩
3. To make it work, when \(\mathcal{U}_i\) and \(\mathcal{U}_j\) agree on \(v\), \(\mathcal{U}_i\) adds \(v\) to her individual noise vector and \(\mathcal{U}_j\) adds \(-v\), so that \(v\) and \(-v
\) cancel out when the final sum is computed. ↩ | {"url":"https://research.mangaki.fr/","timestamp":"2024-11-12T00:42:58Z","content_type":"text/html","content_length":"35927","record_id":"<urn:uuid:41ad279d-2f6c-4589-afff-769499886d99>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00844.warc.gz"} |
Radioactive Effective Half Life Time Calculator
Radioactive effective half life time is defined as the time interval required for the radioactivity of a certain amount of radioactive substance distributed in tissues and organs. It decreases to
half its original value due to radioactive decay and biological elimination. Given here is an online radioactive effective half life time calculator to estimate the result for the given required
input values.
The physical half life time is longer than the biological half life time which is longer than the effective half life time. Enter the physical and biological half life time in this radioactive
effective half life time calculator to find the resultant value.
Calculate radioactive effective half life time for physical half life time as 8 and biological half life time as 5?
= (8 x 5) / (8 + 5)
= 3.07 | {"url":"https://www.calculators.live/radioactive-effective-half-life-time","timestamp":"2024-11-08T15:43:14Z","content_type":"text/html","content_length":"9059","record_id":"<urn:uuid:caf1f2f7-5030-4299-976a-972d34f3b9c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00641.warc.gz"} |
Fibonacci Retracement
Fibonacci Retracement is a charting tool that uses horizontal parallel lines to indicate areas of the support or resistance at the key Fibonacci levels before the trend continues in the original
Fibonacci Retracement price levels can be used as buy triggers on pullbacks during an uptrend. The tool is used to determine how far the price might retrace before continuing to move in the original
Traders often use Fibonacci key levels to place pending orders.
The Fibonacci Retracement levels are based on Fibonacci numbers, the sequence, such that each number is the sum of the two preceding ones, but with each level associated with a percentage.
Each percentage depicts how much of a prior move the price has retraced. In cTrader the default Fibonacci retracement levels are 0%, 38.2%, 50%, 61.8%, and 100%.
The main advantage of Fibonacci Retracement is that it can be drawn between any two significant price points, such as a high and a low, and then the tool will automatically create the levels between
those two points and calculate the price automatically on each level.
In the example below the price of EURUSD drops from \(1.5 to \(1.22. In this case, those two levels will correspond 0% and 100%, and all the fluctuations between these two points can be derived from
the levels as follows: 38.2%=\)1.39, 50%=\)1.36, 61.8%=$1.32.
Fibonacci retracement levels do not need the user to specify formulas for them to be calculated. When the tool is applied to a chart, you only have select these two points, and the lines are drawn
automatically at percentages of that move.
When using the Fibonacci Retracement for detecting the potential support or resistance, you should note that there are no assurances that the price will 100% stop there, so you should use the
additional confirmation signals, e.g. the price starting to bounce off the level, etc.
The other drawback is that there are too many of the levels, so that the price is likely to reverse near one of them quite often, but it's hard to predict which specific one to follow at any
particular time.
When it doesn't work out, it can always be claimed that the trader should have been following another Fibonacci retracement level instead.
The Fibonacci Retracements scope is quite wide - it can be used to determine levels for placing the orders, defining the stop-loss levels, setting the price targets, etc.
For example, if you have noticed the price moving upwards, and then retracing to the 61.8% level, and then, going up again. Since this bounce occurred at a Fibonacci level during an uptrend, it's a
signal to buy.
A stop-loss might be set at the 61.8% level, as a return below that level could indicate that the rally has failed.
As the Fibonacci Retracement price levels are static, they can be very easily identified. It helps to react prudently when the price levels are tested.
These levels are inflexion points where some type of price action is expected, either a reversal or a break.
'The retracement level forecast' is a technique that can identify at which level a retracement can happen.
These retracement levels provide a good opportunity for the traders to enter new positions in the trend direction.
The Fibonacci ratios, i.e. 61.8%, 38.2%, and 23.6%, help the trader identify the retracement's possible extent. The trader can use these levels to position himself for trade.
In cTrader you have a Fibonacci Retracement instrument in your toolbar that allows drawing the tool on the chart, moving it, and configuring depending on your needs.
You can find a detailed description of how to use it in the Fibonacci tools section of this documentation. | {"url":"https://help.ctrader.com/knowledge-base/line-studies-tools/fibonacci-retracement/","timestamp":"2024-11-09T15:37:46Z","content_type":"text/html","content_length":"19583","record_id":"<urn:uuid:a197f745-3134-4890-8a24-1b97df9b2899>","cc-path":"CC-MAIN-2024-46/segments/1730477028125.59/warc/CC-MAIN-20241109151915-20241109181915-00106.warc.gz"} |
The Lottery Game
Can You Predict What Will Happen Next?
Lottery Java Applet
What does "RANDOM" mean? Think carefully before you answer! The definition may not be as obvious as you think. After checking the dictionary definition, consider the following four statements...
which ones do you think are true?
• Flipping a coin is not random because there are only two possible outcomes.
• Rolling a six-sided die is not random because there are only six possible outcomes.
• Whether I win the state lottery or not is random because there are so many people playing the lottery at the same time.
• The weather is random, because so many conditions affect the weather that we cannot predict.
The experiments and models you will find on this site can be summarized in seven words: The growth of order out of randomness. The world around and within us is filled with randomness. Yet instead of
being torn apart by this randomness, nature survives and thrives on it. How can this be? Before we can begin to answer this question, we must study randomness itself.
Is the present always influenced by the past? Suppose you are flipping a coin and, by chance, flip four heads in a row. Does flipping three heads in a row affect the next flip - the fifth flip - why
or why not? Is the fifth flip more likely to be another head? Or is the fifth flip less likely to be a head?
• Do you believe in "winning streaks," meaning that three heads in a row is more likely to lead to a head on the next flip?
• Or do you expect your "luck to run out," meaning that three heads in a row is more likely to be followed by a tail on the next flip?
• Or do you expect equal chances of getting a head or tail on the next flip, independent of what happened before?
Try out the Lottery Java Applet. Click on "Flip 4 Same" which will flip until either 4 heads or 4 tails are flipped in a row. Now choose one of these above strategies and stick to it.
Is your strategy a winner? A loser? Or do you break even?
If you want to learn more, check out these cool sites:
This lesson is taken from Fractals in Science. Page was developed by Paul Trunfio and the JAVA applet was written by Gary McGath. Please send comments to trunfio@bu.edu. Copyright 1996-2000, Center
for Polymer Studies.
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: Physical questions
Hi, hope you can help me with these problems,
I have been given the PV=nRT formula and told to rearrange it to include (d) density, (t) tepm, (M) Mass, and (p) Pressuer which i have done and got to:
PM = dRT.
Now I have been given a table of 'Pressure of CO2 / mm Hg' and
'Weight of CO2 / g'. I have been given varous pressures and weights o CO2 at these presures and told to work out the molecular weight of CO2 using the 'PM=dRT' formula.
I have been told that for real gases the density must be measured as a function of 'P' to allow M to be determined from the intercept of a plot of d/p vs p.
I was wondering do I first need to work out the individual densities for each weight at a certain pressure, then convert my units of pressure into Pa to make it easier to deal with, then divide my
densitiy values by the pressures and plot them values against the pressure. Hopefully the intercept will give me M.
If this makes sense to anyone could youtell me if i have thought about going about it the right way. Thanks
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All Possible Cuts In All Possible Intervals For
Problems belonging to this category would have problem statement something like below or some variation of it:
Given a set of numbers find an optimal solution for a problem considering the current number and the best you can get from the left and right sides.
Make CUTS at all possible places (each CUT will create two INTERVALS on both sides, one on each side) and return the result for the CUT that gave the most optimal result.
Now the biggest question is: what will each these CUTS represent ?
Each of these CUTS will represent
the last operation done in the interval that is getting divided by two by the CUT.
And this will come naturally to you when you see all the examples we discuss in next several chapters. In
Matrix Multiplication
the first CUT will represent dividing the whole set of matrices into two sets and then multiplying those two set of matrices. So as you can understand, this multiplication will be the last
multiplication done, because by that time both the two sets of matrices will already have their own multiplication ready. I know it does not make much sense now, so let's start looking at the
Also, as we will see later while solving problems, FIGURING OUT WHAT THESE CUTS DEFINE IS THE MOST CHALLENGING PART OF SOLVING THIS KIND OF DP PROBLEMS. For some examples this might be very easy,
like in
Matrix Multiplication
, but in many other cases it might take some serious critical thinking.
// from i to j
dp[i][j] = dp[i][k] + result[k] + dp[k+1][j]
Get the best from the left and right sides and add a solution for the current position.
for(int l = 1; l < n; l++) {
for(int i = 0; i < n-l; i++) {
int j = i+l;
for(int k = i; k < j; k++) {
dp[i][j] = max(dp[i][j], dp[i][k] + result[k] + dp[k+1][j]);
return dp[0][n-1]
One important thing to remember here, not all problems are optimization problems (like problems where we are concerned about find the total number of ways something can be achieved, like computing
total number of unique binary tree configurations possible maintaning certain constraint(s) ). In this kind of problems, the solution would involve: for every interval, for every cut: compute the
result and then add up all those results for that interval, as we see below. In these type of problems you won't have to worry about the concept of
last operation
// from i to j
dp[i][j] = dp[i][k] + result[k] + dp[k+1][j]
Get the best from the left and right sides and add a solution for the current position.
for(int l = 1; l < n; l++) {
for(int i = 0; i < n-l; i++) {
int j = i+l;
for(int k = i; k < j; k++) {
dp[i][j] += dp[i][k] + result[k] + dp[k+1][j];
return dp[0][n-1]
Let's look at some problems below to get a clear understanding of this kind of DP problems:
#1. Total Number of Unique Binary Search Trees
Given n, how many structurally unique BST's (binary search trees) that store values 1 ... n?
Input: 3
Output: 5
Given n = 3, there are a total of 5 unique BST's:
\ / / / \ \
/ / \ \
Java Solution:
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Python Solution:
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#2. Tree with Minimum Cost Leaf Nodes
Given an array arr[] of positive integers, consider all binary trees such that:
• Each node has either 0 or 2 children.
• The values of arr[] correspond to the values of each leaf in an in-order traversal of the tree. (Recall that a node is a leaf if and only if it has 0 children.)
• The value of each non-leaf node is equal to the product of the largest leaf value in its left and right subtree respectively.
Among all possible binary trees considered, return the smallest possible sum of the values of each non-leaf node.
Example 1:
Input: arr = [6,2,4]
Output: 32
There are two possible trees. The first has non-leaf node sum 36, and the second has non-leaf node sum 32.
/ \ / \
/ \ / \
This problem, on the surface, looks quite complex but once you have given enough thought and have applied the technique of "All Possible Cuts In All Possible Intervals For Choosing Last Operation",
it becomes quite easy.
Let arr[] be the given array of nodes. These nodes are leaf nodes in inorder order. We would need to make a cut at every possible position in array arr[] where each interval would represent subtree
with the nodes in that interval as the leaf nodes of the subtree. For each cut we compute the result and return the minimum result of all.
I have put my thought process and detailed logic of the algorithm in the inline comments in the code below:
Java Code:
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Python Code:
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Few characteristics of the DP problems like the above one:
• This type of DP problems, at the heart, will always be like the above problem:
If you represent the problem as a tree then you will see that the tree will always have either 0 or 2 children, there will never be a node with just 1 child. So in general combination of root
will have something non-null on left and on right. This kind of dp problem will also have n >=2. (while reading this, think of burst balloon problem).
• Now coming to the length: here we do not need to iterate the length from 1 to N, Because since there will always be non null left and right subtree length can never be equal to N.
for(int l = 1; l < n; l++) {
for(int i = 0; i < n-l; i++) {
int j = i+l;
for(int k = i; k < j; k++) {
dp[i][j] = max(dp[i][j], dp[i][k] + result[k] + dp[k+1][j]);
• I often get asked how to know when to use Dynamic Programming in a Tree problem ?
The answer is simple: if the problem is an Optimization Problem.
Of course, it also needs to show Optimal Substructure and Overlapping Subproblems properties to be able to use DP approach.
Must Read: | {"url":"https://systemsdesign.cloud/Algo/DynamicProgramming/AllPossibleCutsInAllPossibleIntervals","timestamp":"2024-11-07T02:32:14Z","content_type":"text/html","content_length":"50412","record_id":"<urn:uuid:0d7a4121-8951-4268-8196-3a38a41a7c3b>","cc-path":"CC-MAIN-2024-46/segments/1730477027951.86/warc/CC-MAIN-20241107021136-20241107051136-00025.warc.gz"} |
E three di’ values and single h’ value deliver an excellent | Ubiquitin Ligase-ubiquitin-ligase.com
E three di’ values and single h’ worth give a good empirical description from the information; as in, it appears that participants did not adapt their criterion placement as a function of the
stimulus difficulty level, as anticipated when stimulus difficulty varies unpredictably from trial to trial, since it does in our experiment.Stimulus PubMed ID:http://jpet.aspetjournals.org/content/
142/2/141 Sensitivity AlysisSensitivity values as a function of time are shown in Figure (symbols). Apparently stimulus sensitivity grows with stimulus duration initially and after that levels off
for all participants. To additional demonstrate that the sensitivity observed is GSK2838232 chemical information whether processing noise arising from microsaccades or neural sources, or some
processing time constant somewhat independent of the noise level, ioverning the relatively long time constant seen in our experiment. An additiol finding that emerges from this alysis is that the
asymptotic sensitivity Di scales approximately linearly with the stimulus level in this study. See Figure for the linear fitting results assuming: Di kSwhere S represents stimulus level taking values
and k is a linear scalar.Reward BiasThe measured normalized decision criterion, h’, for each delay condition is depicted in Figure (open circles connected with dashed lines). As previously noted,
this variable changes in the expected way for all participants except SL, whose behavior is uffected by the reward manipulation.For each of the remaining participants, we calculated the optimal
decision criterion, h’opt, based on the sigl detection theoretic alysis presented in the introduction and the observed sensitivity data presented in the preceding section, and plotted these optimal
values in Figure (solid curves) together with the normalized criterion value h’ estimated from the data as described above. Note that h’opt when d’ is equal to; for display purposes, such values are
plotted at an ordite value of In the calculation of the stimulus sensitivity and the reward bias, di’,i and h’, we assumed the distributions of the evidence variables for the three stimulus levels
have the same standard deviation: higher sensitivity, AZ876 site associated with higher stimulus levels, results from distributions that are farther apart. However, the increase in sensitivity could
result from changes in the standard deviation, as well as the separation of the distributions. Does the finding that participants are un.E three di’ values and single h’ worth give a great empirical
description of the data; as in, it appears that participants did not adapt their criterion placement as a function from the stimulus difficulty level, as anticipated when stimulus difficulty varies
unpredictably from trial to trial, as it does in our experiment.Stimulus PubMed ID:http://jpet.aspetjournals.org/content/142/2/141 Sensitivity AlysisSensitivity values as a function of time are shown
in Figure (symbols). Apparently stimulus sensitivity grows with stimulus duration initially and then levels off for all participants. To additional demonstrate that the sensitivity observed is
constant using the shifted exponential function as in earlier research, we then carried out a maximum likelihood fit assuming sensitivity follows a delayed exponential function t{t d’ D’i {e{ t
:Figure. Results of our perceptual decisionmaking task with unequal payoffs. For each combition of stimulus and delay conditions, the percentage of choices towards higher reward (ordite) is plotted
against the mean response time, the time from the stimulus onset (time ) to a response (abscissa). Lines with filled symbols denote congruent conditions in which stimulus and reward favor the same
direction, lines with open symbols denote incongruent conditions in which stimulus and reward favor opposite directions. Task difficulty is color coded: Red, green and blue for high, intermediate and
low discrimibility levels respectively. Dashed vertical lines indicate the time of the “go” cue: msec after the stimulus onset.ponegwhere Di,i denotes the asymptotic sensitivity levels for the three
stimulus conditions, t denotes the initial period of time ONE one.orgIntegration of Reward and Stimulus Informationcannot say, however, whether processing noise arising from microsaccades or neural
sources, or some processing time constant somewhat independent of the noise level, ioverning the relatively long time constant seen in our experiment. An additiol finding that emerges from this
alysis is that the asymptotic sensitivity Di scales approximately linearly with the stimulus level in this study. See Figure for the linear fitting results assuming: Di kSwhere S represents stimulus
level taking values and k is a linear scalar.Reward BiasThe measured normalized decision criterion, h’, for each delay condition is depicted in Figure (open circles connected with dashed lines). As
previously noted, this variable changes in the expected way for all participants except SL, whose behavior is uffected by the reward manipulation.For each of the remaining participants, we calculated
the optimal decision criterion, h’opt, based on the sigl detection theoretic alysis presented in the introduction and the observed sensitivity data presented in the preceding section, and plotted
these optimal values in Figure (solid curves) together with the normalized criterion value h’ estimated from the data as described above. Note that h’opt when d’ is equal to; for display purposes,
such values are plotted at an ordite value of In the calculation of the stimulus sensitivity and the reward bias, di’,i and h’, we assumed the distributions of the evidence variables for the three
stimulus levels have the same standard deviation: higher sensitivity, associated with higher stimulus levels, results from distributions that are farther apart. However, the increase in sensitivity
could result from changes in the standard deviation, as well as the separation of the distributions. Does the finding that participants are un. | {"url":"https://www.ubiquitin-ligase.com/2017/12/21/e-three-di-values-and-single-h-value-deliver-an-excellent/","timestamp":"2024-11-09T10:22:23Z","content_type":"text/html","content_length":"64024","record_id":"<urn:uuid:624aa8fe-9505-42d7-b4e4-2388ba3330bc>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.75/warc/CC-MAIN-20241109085148-20241109115148-00657.warc.gz"} |
Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas. | Research / FourClassicalAlgebrasDraft
• How do the four Lie groups and algebras frame four different geometries?
• Define the "highest vector" onto which all of the simple roots have positive projections. What are the allowable projections, as fractions of an integer?
I will share the progress I have made in attempting to answer my question. So far this is mostly a collection of ideas. I need to learn more mathematics and connect the dots. I appreciate related
Models of physics inspire mathematics, give meaning to it, and make it intuitive. Models of cognition could likewise, and arguably, even more centrally, in the case of models of cognition of
mathematics. Algebraic combinatorics is a relevant approach where it tries to interpret algebraic truths so as to make intuitive sense of them and thus learn more from them.
The four classical Lie algebras and groups seem to express the symmetries of the possible mental interpretations of a product {$x_1 x_2 \dots x_n$}. I am investigating this idea in three related
mathematical realms.
• I) Choice. Four different notions of choice yield different interpretations of the binomial expansion, thus generate distinct combinatorial worldviews, and in particular, give rise to three
distinct symmetry groups.
• II) Counting. Two directions of counting, forwards and backwards, can be related in four different ways that characterize the Lie algebra root systems {$A_n$}, {$B_n$}, {$C_n$}, {$D_n$}.
• III) Geometry. The different kinds of duality (negation, conjugation, reversal) inherent in the construction of numbers ({$\mathbb{R}$}, {$\mathbb{C}$} and {$\mathbb{H}$}) give rise to different
geometries (affine, projective, conformal and symplectic).
• IV) Rotations. Rotations in a sphere of real numbers (of even or odd dimension), complex numbers or quaternions.
• V) Inverses. The different groups are different ways, combinatorially, of expressing inversion of composition.
• VI) Matrix symmetries. The different Lie algebras manifest different symmetries inherent in square matrices.
• VII) Bilinear forms. The different Lie algebras express different bilinear forms.
• VIII) Cartan subalgebras
• IX) Generalized Cartan matrices
I) Choice
Let us be careful to distinguish two very different kinds of choices, symmetric and asymmetric.
In a symmetric choice, we are choosing between labels which can be switched around, as with "this" and "that". This is the case with an implicit choice, where you took one or the other possibility.
In nature, you can observe that an electron moved {$\leftarrow$} "left" or "right" {$\rightarrow$}.
In an asymmetric choice, we ourselves make a commitment to one of the opposites which makes it syntactically different from the other. This is the case with an explicit choice, where we intentionally
set the choice up so that either we accept the option, and mean "yes", or we explicitly reject it, and mean "no". This is the case when a scientist explicitly conducts an experiment and externally,
on the metalevel, concludes that "something happened" or "nothing happened".
Mathematically, a series of three symmetric choices {$\leftarrow$} or {$\rightarrow$} yields the eight corners of a cube with dimensions 1, 2, 3:
{$(\leftarrow,\leftarrow,\leftarrow), (\leftarrow,\leftarrow,\rightarrow), (\leftarrow,\rightarrow,\leftarrow), (\rightarrow,\leftarrow,\leftarrow), (\leftarrow,\rightarrow,\rightarrow), (\
Here {$\leftarrow$} and {$\rightarrow$} are simply semantic labels and in that sense the shape stays the same if we switch them all around. [Clarify.]
But a series of three antisymmetric choices {$\leftrightarrow$} or {$\varnothing$} ("vertex" or "no vertex") yields the subsets of the vertices 1, 2, 3 of a triangle:
{$(\leftrightarrow,\leftrightarrow,\leftrightarrow), (\leftrightarrow,\leftrightarrow,\varnothing), (\leftrightarrow,\varnothing,\leftrightarrow), (\varnothing,\leftrightarrow,\leftrightarrow), (\
leftrightarrow,\varnothing,\varnothing), (\varnothing,\varnothing,\leftrightarrow),(\varnothing,\leftrightarrow,\varnothing), (\varnothing,\varnothing,\varnothing)$}
In this case, we can't just switch all of our choices. In particular, the set of all vertices and the set of no vertices are two completely different shapes. Mathematically, it looks like we're just
manipulating a pair of symbols, as before, and we are. The hitch is that in the previous case the opposites were syntactically similar, whereas here they are syntactically different. That's a mental
distinction. Of course, we can lose that distinction by just thinking about the symbols in a different way. But the whole point in modeling cognition is to be careful not to do that. It's also a
challenge to note these mental distinctions without having our own symbols affect our thinking.
Again, the subsets of a triangle's vertices form a poset where the inclusion is oriented (yielding flags). It relates absolutely distinguished and thus dissimilar concepts: all and none. Whereas the
analogous relationship in the cube's coordinate system is nonoriented. It relates relatively distinguished and thus similar concepts: leftmost and rightmost.
In the case of the cube, we are enumerating sets of properties. Each set of properties establishes a vertex of the cube. If P is a property, then Not-P is likewise a property. The set of properties
is considered as informative as the complementary set of non-properties. This is because each non-property directly yields a property. In the case of the triangle, we are enumerating sets of
essences. If E is an essence, then Not-E is NOT an essence. A set of essences - a set of vertices - establishes a substructure of the triangle. Note that non-essences (the fact that a vertex is NOT
in a substructure) are not deemed meaningful. Knowing that a vertex is not in structure tells me nothing about which vertex is in a structure. The knowledge of non-essences is informative only if I
know all of the non-essences as well as all of the possible essences. A set of essences (what something is made of) consists of informative units in a way that the complementary set of non-essences
(what something is not made of) does not. Thus properties and essences, as concepts, are based on these two very different worldviews.
[There are two other infinite families of polytopes. They can be interpreted as follows.]
[These interpretations distinguish the center and the whole, thus inside and outside. The meaning of center keeps changing and yet, in isolation, we think of there being one shared center - one
shared absolute coordinate system. Whereas there is not one shared whole.]
[The syntactic distinction arises from the difference between no choices and all choices. There is a universally well defined set of no choices. Whereas there is no universally well defined set of
all choices. Furtherhmore, we identify each set of non-choices with that set of no choices. If a don't choose something X times, then it is considered the same as having never made a choice. This
reveals a syntactic distinction between choosing this and not choosing this.]
Again, this is the difference between choosing X or not-X, as with the cube, and choosing X or not-choosing X, as with the triangle.
[We pair two of three levels: not choosing, choosing, choosing this or that.]
[Not choosing or choosing = Allowing for gaps. Choosing this or not this = Not allowing for gaps.]
[Not choosings are not distinguishable and thus commutative. Not choosing does not get a label. It is a nondistinguishable.]
[The symmetry in an expansion is related to the questions of commutativity and associativity and other concepts that arise from the Cayley-Dickson construction.]
II) Counting.
In counting, {$A_n$} does not distinguish between beginning and end, between forwards and backwards. Extension can take place in either end, in either direction.
However, {$B_n$} and {$D_n$} relate one end, by noting the symmetry in counting in one direction and the other, so extension can only take place on the other end (extending both ends simultaneously,
thus adding a pair of dimensions). So they reduce the further possibilities of extension. How? How does {$C_n$} reduce the possibilities of extension?
[The whole is a pseudoscalar. Thus odd and even dimensions matter.]
III) Geometry.
Geometry can be variously conceived with the help of polytopes.
Going around a triangle can be conceived as:
• Affine: Paths. Three paths understood as a three-cycle going round the triangle.
• Projective: Lines Three lines, traveling back and forth, intersecting at three points.
• Conformal: Angles. Three angles totaling 180 degrees.
• Symplectic: Oriented areas. An oriented area that is swept out as one goes around.
Such geometrical concepts can be understood as grounded in the generating process of an infinite family of polytopes.
• Simplexes arise as vectors (paths) generated from the center.
• Cross-polytopes arise as lines expanding from the center.
• Cubes arise as perpendicular cuts (angles) cast upon the whole.
• Coordinate systems arise as a simplexes laid out upon coordinate axes.
Curiously, this does not match with the evidence from the Lie groups {$A_n$}, {$B_n$}, {$C_n$} and {$D_n$}. They consist of those linear transformations of vectors which keep invariant the following
notions of length:
• real: {$B_n$} (odd-dimensional spaces) and {$D_n$} (even-dimensional spaces)
• complex: {$A_n$}
• quaternion: {$C_n$}
We can think of {$A_n$} and the complex numbers as keeping distinct the duality of counting forwards and backwards. Then we can think of {$B_n$} and {$D_n$} as breaking that duality by linking the
counting forwards and the counting backwards. {$B_n$} links them through an external zero, (a long root?), (a point at infinity?) {$D_n$} links them through an internal zero, an element shared by
both counting forward and backwards. {$C_n$} and the quaternions double the dimensions as if by folding the counting so that there is new symmetry without a zero.
If we consider the root systems, {$A_n$}, {$B_n$}, {$C_n$}, {$D_n$}, then based on their symmetries, we have two extremes: {$A_n$} and {$D_n$}. We can imagine the extrasystemic worldview of {$A_n$}
as going beyond itself into an intrasystemic worldview {$D_n$}. In between, there may be a metasystemic worldview {$B_n$} looking in, and a systemic worldview {$C_n$} looking out. Then the systemic
worldview is the most sophisticated.
If we consider the consequences (the fullness) of the polytopes, then we may find a connection with the geometries. The cross-polytopes describe surface area and are thus symplectic. The hypercubes
describe how a vantage point at infinity manifests itself, and so they are projective. The coordinate systems apparently get by without a zero and are thus affine. The simplexes are thus conformal.
But why?
IV) Rotations
The Lie groups can be thought of as those matrices which respect lengths, and thus the unit sphere. These are rotations. The entries in the matrices can be the real numbers, the complex numbers, or
the quaternions. As the dimensions grow, the real number matrices need to be distinguished between the cases for the even numbered and odd numbered dimensional vector spaces. Thus the vector spaces
need to grow by two real dimensions (An, Bn, Dn) or by four real dimensions (Cn).
V) Inverse elements
Symmetries between two inverse matrices.
VI) Matrix symmetries
Symmetries within a matrix.
• Orthogonal: {$a+a'=0$}
• Orthogonal: {$a_{ij}=-a_{n+1-j,n+1-i}$}
VII) Bilinear forms
Given a bilinear form {$B:V\times V\rightarrow F$}
• Orthogonal - symmetric: {$\mathfrak{o}_{n,B}(\mathbb{F})=\{a∈gl_n(\mathbb{F})|a^TB+Ba= 0\}$}
VIII) Cartan subalgebras
{$D$} is the space of diagonal matrices.
{$\epsilon_i(A)=a_{ii}$} define a basis for {$D^*$}
{$\mathfrak{h}=\{a\in D|(\epsilon_1+\dots+\epsilon_n)a=0\}$}
{$\{\epsilon_i-\epsilon_{i+1}\;|\;i=1,\dots,n-1\}$} is a basis for {$\mathfrak{h}^*$}.
IX) generalized Cartan matrices
The polytopes seem to match with the Lie groups as follows:
• An - complex entries - simplex (generates one vertex)
• Cn - quaternionic entries - cross-polytope (generates pairs of vertices)
• Bn - real entries (odd) - hypercube
• Dn - real entries (even) - cubic coordinate systems
Note that Cn and Bn have the same Weyl groups.
Arnold, "Symplectization, Complexification and Mathematical Trinities", page 3. "So this list gives the classification of irreducible Coxeter groups which is one of the main classification theorem in
mathematics." Footnote: "[Yuri] Manin told me once that the reason we always encounter this list in many different mathematical classifications is its presence in the hardware of our brain (which is
thus unable to discover a more complicated scheme). I still hope there exists a better reason that once should be discovered."
David Corfield, Mathematical Kinds, or Being Kind to Mathematics Δ, about systems by Arnold, Atiyah, and Baez and Dolan. In 1908, Henri Poincaré claimed that: ...the mathematical facts worthy of
being studied are those which, by their analogy with other facts, are capable of leading us to the knowledge of a mathematical law, just as experimental facts lead us to the knowledge of a physical
law. They are those which reveal to us unsuspected kinship between other facts, long known, but wrongly believed to be strangers to one another.
Corfield about Arnold In Lecture 2: Symplectization, Complexification and Mathematical Trinities, Arnold argues for a family relation between different geometries. He begins with the
finite-dimensional geometries as given by Coxeter groups. So that there is an AA geometry and its sisters BB, CC and DD. He then discusses the infinite-dimensional case, where 6 family members can be
• differential,
• volume-preserving,
• symplectic,
• contact,
• complex, and
• a variant. [Andrius: Compare with the six transformations?]
(According to Bryant’s lectures (p. 110), volume-preserving and symplectic each have an extension, geometries in which preservation is up to a constant multiple.) Arnold then looks for versions of
theorems and constructions in differential geometry and topology for the two sisters – symplectic and complex geometry.
Finally, he moves on to describe various trinities, starting from (ℝ,ℂ,ℍ)(\mathbb{R}, \mathbb{C}, \mathbb{H}). | {"url":"https://www.math4wisdom.com/wiki/Research/FourClassicalAlgebrasDraft","timestamp":"2024-11-14T13:44:45Z","content_type":"application/xhtml+xml","content_length":"29735","record_id":"<urn:uuid:39894354-3386-457e-bea2-0ca3284c39c4>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00614.warc.gz"} |
Multiple SCF Solutions for Non-Orthogonal CI
4.9.3 Multiple SCF Solutions for Non-Orthogonal CI
The solutions found through metadynamics often appear to be good approximations to diabatic surfaces, where the electronic structure does not significantly change with geometry. In situations where
there are such multiple electronic states close in energy, an adiabatic state may be produced by diagonalizing a matrix of these states, i.e., through a configuration interaction (CI) procedure. As
they are distinct solutions of the SCF equations, these states are non-orthogonal (i.e. one cannot be constructed as a single determinant made out of the orbitals of another), and so the CI is a
little more complicated and corresponds to a non-orthogonal CI (NOCI). More information on NOCI can be found in Section 7.4.
Version 5.2 of Q-Chem introduces a new NOCI package, LIBNOCI, for locating multiple SCF solutions and running NOCI calculations (see Section 7.4.0.1), including a new implementation of SCF
metadynamics. The LIBNOCI implementation of SCF metadynamics can be accessed using USE_LIBNOCI = TRUE in combination with NOCI_DETGEN = 3. In addition to the original SCF metadynamics features
available in Q-Chem, this new implementation includes:
• •
An active space approach where orbital mixing and optimization occurs only in a user-defined subset of orbitals.
• •
Full support for restricted, unrestricted and generalized orbital types, along with complex (Hermitian) and holomorphic (non-Hermitian) orbitals [see Section 4.9.4].
Multiple Hartree-Fock states of particular relevance for NOCI are often related to varying orbital occupations in a dominant subset of molecular orbitals. For example, important multiple solutions
may correspond to excited determinants whose orbitals have been individually relaxed at the SCF level, or symmetry-broken states formed from strong mixing in a dominant active space. LIBNOCI allows
multiple solutions to be identified by allowing orbital mixing and relaxation only in a subset of orbitals defined using the keyword $active_orbitals. By default, the multiple solutions located are
then subsequently optimised in the full orbital space, although this can be skipped using SKIP_SCFMAN = TRUE.
Finally, LIBNOCI introduces easier control over reading initial guesses from previous calculations. Using the input NOCI_REFGEN = 1, all previous solutions are read from file (if available), while a
particular subset can be chosen using the keyword $scf_read.
H 0.0000000 0.0000000 0.0000000
H 0.0000000 0.0000000 4.0000000
EXCHANGE hf
UNRESTRICTED true
BASIS sto-3g
SCF_CONVERGENCE 10
MAX_SCF_CYCLES 1000
MOM_START 1
USE_LIBNOCI true
NOCI_DETGEN 3
SCF_SAVEMINIMA 4
SCF_MINFIND_RANDOMMIXING 30000
SCF_MINFIND_MIXMETHOD 1
Active orbitals can be specified for SCF metadynamics in LIBNOCI. Indices for $\beta$ orbitals are offset by the number of $\alpha$ MOs, i.e. the case selects $\alpha$ orbitals 1 and 2, and $\beta$
orbitals 1 and 2, with a total of 10 $\alpha$ molecular orbitals (including occupied and virtual).
The initial guess coefficients can also be read in as follows:
1 2 4 ... | {"url":"https://manual.q-chem.com/5.3/subsubsec_SCFMetadynNOCI.html","timestamp":"2024-11-07T13:47:28Z","content_type":"text/html","content_length":"29454","record_id":"<urn:uuid:77970371-d009-456f-8c64-91eb64b8b1df>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00706.warc.gz"} |
help: about finite subfield
Posts: 2
Joined: Sun Oct 18, 2015 11:28 am
"Not every field has a finite subfield."
I don't understand why. Can you give me a counterexample?
Posts: 60
Joined: Tue Mar 17, 2015 2:29 am
Not to be pedantic - but I take it that you meant example, not counterexample. Here is one: the rationals (under ordinary addition and multiplication) form an ordered field that has no finite
subfield. Proof: Assume that there exists such a finite subfield, choose the largest element - add it to itself and you get an element that is not in the subfield - that is a contradiction. I am sure
that by now you can come up with at least two other similar examples.
Posts: 14
Joined: Wed Aug 26, 2015 6:23 pm
In fact, it is not hard to see a field has finite subfields iff it has non zero characteristic, without using any properties of ordering. To do this, look at the field generated by 1, i.e. the prime
subfield. Every subfield contains the prime subfield, and the prime subfield is infinite iff the field is characteristic 0.
Posts: 2
Joined: Sun Oct 18, 2015 11:28 am
Ivanjam wrote:Not to be pedantic - but I take it that you meant example, not counterexample. Here is one: the rationals (under ordinary addition and multiplication) form an ordered field that has
no finite subfield. Proof: Assume that there exists such a finite subfield, choose the largest element - add it to itself and you get an element that is not in the subfield - that is a
contradiction. I am sure that by now you can come up with at least two other similar examples.
Thank you very much! Still, I have one more question. In the example of the rationals, can {0} form a subfield?
Posts: 96
Joined: Fri Mar 27, 2015 6:42 pm
Pretty sure field definition says 0 != 1 (sometimes even in ring definitions). For some silly stuff, look here:
Zhangvict is spot on. Finite fields have characteristic p for some prime p. If E is a subfield of F with char p, then 1 + ... + 1 (p times) = 0. That equation also holds in F, so F has char p as
Posts: 60
Joined: Tue Mar 17, 2015 2:29 am
evelyn9293 wrote:Thank you very much! Still, I have one more question. In the example of the rationals, can {0} form a subfield?
No, because {0} would then be a finite subfield of the rationals. Alternatively, {0} is not a subfield of the rationals because the multiplicative identity 1 != 0 is not an element of {0}. | {"url":"https://mathematicsgre.com/viewtopic.php?f=1&t=3535","timestamp":"2024-11-13T18:39:17Z","content_type":"text/html","content_length":"28350","record_id":"<urn:uuid:21f7884b-9c25-439d-8480-4495f1d4c8c7>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00888.warc.gz"} |
How we calculate annualized net returns
Annualized returns are calculated using an IRR (Internal Rate of Return) formula. IRR is the rate earned on every dollar actively invested and does not include uninvested cash. Returns are calculated
net of fees, promotional credits and late fees.
IRR measures cashflows and assumes every note purchase is an outflow and every payment received is an inflow. Loans in good standing and in the grace period are assumed to be fully repaid as of the
calculation date with accrued interest for the current period. Loans in delinquent states are still assumed to be repaid in full based on the principal value of the loan after the last successful
payment. Charged-off loans are assumed to have no future payments, but any recovered amounts will increase future returns, all else equal. Withdrawn or cancelled loans have no impact on the
calculation, but repaid loans are fully accounted for.
The IRR calculation can be completed in Excel using the =XIRR() formula. Essentially, IRR is an interest rate that makes the net present value (NPV) of all cash flows equal to 0.
To better understand the formula and how goPeer calculates the returns in given scenarios, you can review the following examples and test out the formula yourself in Excel:
For a loan in good standing or in the grace period, your purchase amount is counted as an outflow on the date the loan activates and is disbursed. Any payments that you receive count as cash inflows
and on the calculation date, the remaining principal and accrued interest for the period is recorded as an inflow. For example, you invest $100 (-100) in a note on March 1, 2021 and receive a payment
of $3 (+3) on April 1, 2021. The principal outstanding as of your calculation date on April 15, 2021 is $98 (+98). Using the XIRR formula in Excel, you should be able to calculate the IRR as 8.5%.
Now if you assume the loan is instead charged off on April 15, the final value is 0. For the purposes of this calculation, using 0.00001 in Excel instead of absolute 0 is best as otherwise it will
return an error. Because the loss happened so quickly and was so large, the IRR drops to -99.9%.
For the calculation of a delinquent but not yet charged off loan using the same example, you could assume the calculation date is now as of January 1, 2022. The return now drops to 1.2% and continues
to approach zero the longer it remains delinquent.
The impact of charge-offs and delinquencies may evolve over time. For charge-offs, no recoveries are initially included the IRR calculation. Recoveries will be added to your returns as they are
effectively received. A delinquent loan that reverts to good standing will positively impact your returns. Inversely, a delinquent loan that gets charged-off will negatively impact your returns.
Uninvested cash is not included in the IRR calculation. We recommend using Auto-Invest to ensure all funds get reinvested promptly and continue generating compounded returns over time.
The calculated return and the data portrayed above is for informational purposes only and is not a forward-looking projection of performance. This data should not be relied upon to make investment
decisions and is not a recommendation to purchase or trade securities. goPeer has taken reasonable care to ensure the accuracy of the calculation but it has not been verified or reviewed by an
independent third party. | {"url":"https://help.gopeer.ca/hc/en-ca/articles/4764634147991-How-we-calculate-annualized-net-returns","timestamp":"2024-11-08T19:06:25Z","content_type":"text/html","content_length":"27114","record_id":"<urn:uuid:b921ec6a-7561-4a24-8f8e-982eadb603f2>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00629.warc.gz"} |
HITHERTO: I have purposely refrained from speaking about the physical interpretation of space- and time-data in the case of the general theory of relativity. As a consequence, I am guilty of a
certain slovenliness of treatment, which, as we know from the special theory of relativity, is far from being unimportant and pardonable. It is now high time that we remedy this defect; but I would
mention at the outset, that this matter lays no small claims on the patience and on the power of abstraction of the reader.
We start off again from quite special cases, which we have frequently used before. Let us consider a space-time domain in which no gravitational field exists relative to a reference-body K whose
of motion has been suitably chosen. K is then a Galileian reference-body as regards the domain considered, and the results of the special theory of relativity hold relative to K. Let us suppose the
same domain referred to a second body of reference K', which is rotating uniformly with respect to K. In order to fix our ideas, we shall imagine K' to be in the form of a plane circular disc, which
rotates uniformly in its own plane about its center. An observer who is sitting eccentrically on the disc K' is sensible of a force which acts outwards in a radial direction, and which would be
interpreted as an effect of inertia (centrifugal force) by an observer who was at rest with respect to the original reference-body K. But the observer on the disc may regard his disc as a
reference-body which is “at rest”; on the basis of the general principle of relativity he is justified in doing this. The force acting on himself, and in fact on all other bodies which are at rest
relative to the disc, he regards as the effect of a gravitational field. Nevertheless, the space-distribution of this gravitational field is of a kind that would not be possible on Newton’s theory of
gravitation. 1 But since the observer believes in the general theory of relativity, this does not disturb him; he is quite in the right when he believes that a general law of gravitation can be
formulated—a law which not only explains the motion of the stars correctly, but also the field of force experienced by himself.
The observer performs experiments on his circular disc with clocks and measuring-rods. In doing so, it is his intention to arrive at exact definitions for the signification of time- and space-data
with reference to the circular disc K , these definitions being based on his observations. What will be his experience in this enterprise?
To start with, he places one of two identically constructed clocks at the center of the circular disc, and the other on the edge of the disc, so that they are at rest relative to it. We now ask
ourselves whether both clocks go at the same rate from the standpoint of the non-rotating Galileian reference-body K. As judged from this body, the clock at the center of the disc has no velocity,
whereas the clock at the edge of the disc is in motion relative to K in consequence of the rotation. According to a result obtained in Section XII, it follows that the latter clock goes at a rate
permanently slower than that of the clock at the center of the circular disc, i.e. as observed from K. It is obvious that the same effect would be noted by an observer whom we will imagine sitting
alongside his clock at the center of the circular disc. Thus on our circular disc, or, to make the case more general, in every gravitational field, a clock will go more quickly or less quickly,
according to the position in which the clock is situated (at rest). For this reason it is not possible to obtain a reasonable definition of time with the aid of clocks which are arranged at rest with
respect to the body of reference. A similar difficulty presents itself when we attempt to apply our earlier definition of simultaneity in such a case, but I do not wish to go any farther into this
Moreover, at this stage the definition of the space co-ordinates also presents insurmountable difficulties. If the observer applies his standard measuring-rod (a rod which is short as compared with
the radius of the disc) tangentially to the edge of the disc, then, as judged from the Galileian system, the length of this rod will be less than 1, since, according to THE BEHAVIOR OF MEASURING RODS
AND CLOCKS IN MOTION, moving bodies suffer a shortening in the direction of the motion. On the other hand, the measuring-rod will not experience a shortening in length, as judged from K, if it is
applied to the disc in the direction of the radius. If, then, the observer first measures the circumference of the disc with his measuring-rod and then the diameter of the disc, on dividing the one
by the other, he will not obtain as quotient the familiar number π = 3.14 . . ., but a larger number, 1 whereas of course, for a disc which is at rest with respect to K, this operation would yield π
exactly. This proves that the propositions of Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field, at least if we attribute the length 1 to the rod in
all positions and in every orientation. Hence the idea of a straight line also loses its meaning. We are therefore not in a position to define exactly the co-ordinates x, y, z relative to the disc by
means of the method used in discussing the special theory, and as long as the co-ordinates and times of events have not been defined we cannot assign an exact meaning to the natural laws in which
these occur.
Thus all our previous conclusions based on general relativity would appear to be called in question. In reality we must make a subtle detour in order to be able to apply the postulate of general
relativity exactly. I shall prepare the reader for this in the following paragraphs.
Youth can not know how age thinks and feels. But old men are guilty if they forget what it was to be young. ---J.K. Rowling , Harry Potter and the Order of the Phoenix | {"url":"https://ahmadabdulnasir.com.ng/behavior-clocks-and-measuring-rods-rotating-body-reference/","timestamp":"2024-11-08T21:27:47Z","content_type":"text/html","content_length":"34846","record_id":"<urn:uuid:58a1509e-11b0-4b22-87cb-5bef9dbd3611>","cc-path":"CC-MAIN-2024-46/segments/1730477028079.98/warc/CC-MAIN-20241108200128-20241108230128-00087.warc.gz"} |