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Ten Examples
Go to molality problems #11-25
As is clear from its name, molality involves moles. Boy, does it!
The molality of a solution is calculated by taking the moles of solute and dividing by the kilograms of solvent.
moles of solute
Molality = –––––––––––––––––
kilograms of solvent
This is probably easiest to explain with examples.
Example #1: Suppose we had 1.00 mole of sucrose (it's about 342.3 grams) and proceeded to mix it into exactly 1.00 liter water. It would dissolve and make sugar water. We keep adding water,
dissolving and stirring until all the solid was gone. We then made sure everything was well-mixed.
What would be the molality of this solution? Notice that my one liter of water weighs 1000 grams (density of water = 1.00 g / mL and 1000 mL of water in a liter). 1000 g is 1.00 kg, so:
1.00 mol
Molality = ––––––––
1.00 kg
The answer is 1.00 mol/kg.
Notice that both the units of mol and kg remain. Neither cancels.
A symbol for mol/kg is often used. It is a lower-case m and is often in italics, m. Some textbooks also put in a dash, like this: 1.00-m. However, if you write 1.00 m for the answer, without the
italics, then that usually is correct because the context calls for a molality. Having said that, however, be aware that often m is used for mass, so be careful. (A lower-case m is also used for
meter, but the context should be clear that m means molality.) Maybe including the dash would be wise if there might be a potential misunderstanding
When you say it out loud, say this: "one point oh oh molal." You don't have to say the dash.
And never forget this: replace the m with mol/kg when you do calculations. The m is a symbol that stands for mol/kg. It is not the actual unit.
Example #2: Suppose you had 2.00 moles of solute dissolved into 1.00 L of solvent. What's the molality?
2.00 mol
Molality = ––––––––
1.00 kg
The answer is 2.00 m.
Notice that no mention of a specific substance is mentioned at all. The molarity would be the same no matter what the substance. It doesn't matter if it is sucrose, sodium chloride or any other
substance. One mole of anything contains 6.022 x 10^23 units.
Example #3: What is the molality when 0.750 mol is dissolved in 2.50 L of solvent?
0.750 mol
Molality = –––––––––
2.50 kg
The answer is 0.300 m.
Now, let's change from using moles to grams. This is much more common. After all, chemists use balances to weigh things and balances give grams, NOT moles.
Example #4: Suppose you had 58.44 grams of NaCl and you dissolved it in exactly 2.00 kg of pure water (the solvent). What would be the molality of the solution?
The solution to this problem involves two steps.
Step One: convert grams to moles.
Step Two: divide moles by kg of solvent to get molality.
In the above problem, 58.44 grams/mol is the molar mass of NaCl.
Step One: 58.44 g / 58.44 gr/mol = 1.00 mol.
Step Two: 1.00 mol / 2.00 kg = 0.500 mol/kg (or 0.500 m).
Sometimes, a book will write out the word "molal," as in 0.500-molal.
Example #5: Calculate the molality of 25.0 grams of KBr dissolved in 750.0 mL pure water.
25.0 g
––––––––– = 0.210 mol
119.0 g/mol
0.210 mol
––––––––– = 0.280 m
0.750 kg
Example #6: 80.0 grams of glucose (C[6]H[12]O[6], mol. wt = 180. g/mol) is dissolved in1.00 kg of water. Calculate the molality.
80.0 g
––––––––– = 0.444 mol
180. g/mol
0.444 mol
––––––––– = 0.444 m
1.00 kg
Example #7: Calcuate the molality when 75.0 grams of MgCl[2] is dissolved in 500.0 g of solvent.
Rather than a two-step approach, some teachers will want you to combined the steps into one diagram. This technique is called dimensional analysis.
1 mol 1
75.0 g x ––––––– x ––––––– = 1.58 m
95.2 g 0.500 kg
Example #8: 100.0 grams of sucrose (C[12]H[22]O[11], mol. wt. = 342.3 g/mol) is dissolved in 1.50 L of water. What is the molality?
1 mol 1
100.0 g x ––––––– x ––––––– = 0.195 m
342.3 g 1.500 kg
Example #9: 49.8 grams of KI is dissolved in 1.00 kg of solvent. What is the molality?
Two step:
49.8 g
––––––––– = 0.300 mol
166.0 g/mol
0.300 mol
––––––––– = 0.300 m
1.00 kg
Dimensional analysis:
1 mol 1
49.0 g x ––––––– x ––––––– = 0.300 m
166.0 g 1.00 kg
Example #10: You have 849.0 grams of water and you wish to make a 5.2 m (molality) solution of mercury(II) oxide, HgO. How many grams of the solute would you have to add to the water that you have?
5.2 mol 0.849 kg 216.589 g
––––––– x ––––––– x ––––––– = 956 g
1.00 kg 1 1 mol
HgO is rather insoluble in water, so the above solution could not be made. of course, the point of the question is not to make the solution, it is to learn how to do molality calculations.
In the molarity tutorial the phrase "of solution" kept showing up. The molarity definition is based on the volume of the solution. This makes molarity a temperature-dependent definition. However, the
molality definition does not have a volume in it and so is independent of any temperature changes. This will make molality a very useful concentration unit in the area of colligative properties.
Lastly, it is very common for students to confuse the two definitions of molarity and molality. The words differ by only one letter and sometimes that small difference is overlooked.
Bonus Example: The density of a solution of 3.69 g KCl in 21.70 g H[2]O is 1.11 g/mL. Calculate the molality of KCl in the solution.
1) Determine moles of KCl:
3.69 g / 74.551 g/mol = 0.049496 mol
2) Determine the molality:
0.049496 mol / 0.02170 kg = 2.28 m
3) Note that the density is not needed. It's a red herring.
Go to molality problems #11-25 | {"url":"https://web.chemteam.info/Solutions/Molality.html","timestamp":"2024-11-07T22:26:09Z","content_type":"text/html","content_length":"11909","record_id":"<urn:uuid:d9f783cb-5f06-4961-aee2-9dbcafec0a31>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00340.warc.gz"} |
Like terms worksheet
like terms worksheet
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Simplifying Expressions & Solving
Two Equations containing Two Variables Author Message
Solving Linear Equations not-manne Posted: Sunday 31st of Dec 09:40
Plane Curves Parametric Equation Hi! We just started discussing a new lesson in algebra regarding like terms worksheet and I did pretty well for most homeworks we got but
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SOLVING SYSTEMS OF EQUATIONS nxu Posted: Monday 01st of Jan 11:33
Exponential and Logarithmic Equations Being a professor , this is a comment I usually hear from children . like terms worksheet is not one of the most favorite topics amongst
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Literal Equations explains all the intermediary steps involved in reaching that final solution . That way apart from knowing the final answer, we also learn
Quadratic Equations how to go about solving questions right from the first step till the last, and it helps a lot in working on assignments.
Linear Equations in Linear Algebra
SOLVING LINEAR AND QUADRATIC EQUATIONS Registered:
Investigating Liner Equations Using 28.10.2001
Graphing Calculator From: Woudenberg,
represent slope in a linear equation Netherlands
Linear Equations as Models
Solving Quadratic Equations by
Factoring Double_J Posted: Wednesday 03rd of Jan 08:28
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Expressions piece of algebra software. I remember of going through difficulties with simplifying expressions, adding exponents and angle suplements. I
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Solve Quadratic Equations by
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Solving a Quadratic Equation From: Netherlands
nm3joj Posted: Friday 05th of Jan 08:09
Thank you very much for your response! Could you please tell me how to download this software ? I don’t have much time on hand since I
have to solve this in a few days.
Paubaume Posted: Friday 05th of Jan 11:51
Its really easy, just click on the following link and you are good to go – https://algebra-equation.com/solving-exponential-equations.html
. And remember they even give an unconditional money back guarantee with their software , but I’m sure you’ll love it and won’t ever ask
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From: In the stars...
where you left me,
and where I will wait
for you... always...
Home Linear Equations Literal Equations Simplifying Expressions & Solving Equations Two Equations containing Two Variables LinearEquations Solving Linear Equations Plane Curves Parametric Equation
Linear Equations and Matrices LinearEquations Test Description for EXPRESSIONS AND EQUATIONS Trigonometric Identities and Conditional Equations Solving Quadratic Equation Solving Systems of Linear
Equations by Graphing SOLVING SYSTEMS OF EQUATIONS Exponential and Logarithmic Equations Quadratic Equations Homework problems on homogeneous linear equations Solving Quadratic Equations
LinearEquations Functions, Equations, and Inequalities Solving Multiple-Step Equations Test Description for Quadratic Equations and Functions Solving Exponential Equations Linear Equations Linear
Equations and Inequalities Literal Equations Quadratic Equations Linear Equations in Linear Algebra SOLVING LINEAR AND QUADRATIC EQUATIONS Investigating Liner Equations Using Graphing Calculator
represent slope in a linear equation Equations Linear Equations as Models Solving Quadratic Equations by Factoring Solving Equations with Rational Expressions Solving Linear Equations Solve Quadratic
Equations by Completing the Square LinearEquations Solving a Quadratic Equation
Author Message
not-manne Posted: Sunday 31st of Dec 09:40
Hi! We just started discussing a new lesson in algebra regarding like terms worksheet and I did pretty well for most homeworks we got but the latest one my professor gave really
complex so I'd love if somebody will help me to understand it! It’s a problem solving homework my algebra professor gave out this day and it’s due next week and I tried solving
it but still can’t get it right. I just can’t finish it with ease unlike the other assignments. I had an easy time solving my past assignments but this particular assignment
with specific topic of factoring expressions just gives me difficulty just knowing how to start . I’m desperately in need of help. I’ll really be grateful if someone help me in
Registered: discussing the steps and how to solve it in a organized and clear way.
From: Marl, Germany
nxu Posted: Monday 01st of Jan 11:33
Being a professor , this is a comment I usually hear from children . like terms worksheet is not one of the most favorite topics amongst students . I never encourage my pupils
to get ready made answers from the web, however I do encourage them to use Algebrator. I have developed a liking for this tool over the years . It helps the students learn math
in a convenient way.
From: Siberia,
Russian Federation
Sdefom Koopmansshab Posted: Monday 01st of Jan 20:15
Algebrator truly is a masterpiece for us algebra students. As my dear friend said in the preceding post, it solves questions and it also explains all the intermediary steps
involved in reaching that final solution . That way apart from knowing the final answer, we also learn how to go about solving questions right from the first step till the last,
and it helps a lot in working on assignments.
From: Woudenberg,
Double_J Posted: Wednesday 03rd of Jan 08:28
Algebrator is the program that I have used through several math classes - Basic Math, Pre Algebra and Pre Algebra. It is a really a great piece of algebra software. I remember
of going through difficulties with simplifying expressions, adding exponents and angle suplements. I would simply type in a problem homework, click on Solve – and step by step
solution to my math homework. I highly recommend the program.
From: Netherlands
nm3joj Posted: Friday 05th of Jan 08:09
Thank you very much for your response! Could you please tell me how to download this software ? I don’t have much time on hand since I have to solve this in a few days.
Paubaume Posted: Friday 05th of Jan 11:51
Its really easy, just click on the following link and you are good to go – https://algebra-equation.com/solving-exponential-equations.html. And remember they even give an
unconditional money back guarantee with their software , but I’m sure you’ll love it and won’t ever ask for your money back.
From: In the stars...
where you left me,
and where I will wait
for you... always...
Posted: Sunday 31st of Dec 09:40
Hi! We just started discussing a new lesson in algebra regarding like terms worksheet and I did pretty well for most homeworks we got but the latest one my professor gave really complex so I'd love
if somebody will help me to understand it! It’s a problem solving homework my algebra professor gave out this day and it’s due next week and I tried solving it but still can’t get it right. I just
can’t finish it with ease unlike the other assignments. I had an easy time solving my past assignments but this particular assignment with specific topic of factoring expressions just gives me
difficulty just knowing how to start . I’m desperately in need of help. I’ll really be grateful if someone help me in discussing the steps and how to solve it in a organized and clear way.
Posted: Monday 01st of Jan 11:33
Being a professor , this is a comment I usually hear from children . like terms worksheet is not one of the most favorite topics amongst students . I never encourage my pupils to get ready made
answers from the web, however I do encourage them to use Algebrator. I have developed a liking for this tool over the years . It helps the students learn math in a convenient way.
Posted: Monday 01st of Jan 20:15
Algebrator truly is a masterpiece for us algebra students. As my dear friend said in the preceding post, it solves questions and it also explains all the intermediary steps involved in reaching that
final solution . That way apart from knowing the final answer, we also learn how to go about solving questions right from the first step till the last, and it helps a lot in working on assignments.
Posted: Wednesday 03rd of Jan 08:28
Algebrator is the program that I have used through several math classes - Basic Math, Pre Algebra and Pre Algebra. It is a really a great piece of algebra software. I remember of going through
difficulties with simplifying expressions, adding exponents and angle suplements. I would simply type in a problem homework, click on Solve – and step by step solution to my math homework. I highly
recommend the program.
Posted: Friday 05th of Jan 08:09
Thank you very much for your response! Could you please tell me how to download this software ? I don’t have much time on hand since I have to solve this in a few days.
Posted: Friday 05th of Jan 11:51
Its really easy, just click on the following link and you are good to go – https://algebra-equation.com/solving-exponential-equations.html. And remember they even give an unconditional money back
guarantee with their software , but I’m sure you’ll love it and won’t ever ask for your money back. | {"url":"https://algebra-equation.com/solving-algebra-equation/graphing-inequalities/like-terms-worksheet.html","timestamp":"2024-11-09T04:28:23Z","content_type":"text/html","content_length":"93176","record_id":"<urn:uuid:6f260518-dcb9-4bba-9536-5e7ef2d2d094>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00080.warc.gz"} |
Open Dialogues: How to Rethink Textbooks
In an effort to customize materials for their courses and save students money on rising textbook fees, instructors in the Math department at UBC have adopted open or freely accessible textbooks in
all first-year courses and most second-year courses. Since 1997, the widespread adoption of open or freely accessible resources in the department has impacted more than 23,000 students and saved them
between $1,315,000 and $2,429,000 on commercially available textbooks. While some math instructors began developing their own online textbooks in the late 90’s, most of the activity has taken place
in the last five years. This fall, instructors in five math courses replaced a required, traditional textbook with open resources. In the 2016 academic year, instructors in 16 math courses have
adopted open or free textbooks. 7,000 students are enrolled in these courses so far this year, and collectively they are saving between $608,000 and $1,024,000. (These estimates are based on a
formula suggested by
, a non-profit based at Rice University that publishes openly licensed textbooks, to take into account students who buy used textbooks, rent them or buy new textbooks.
, which manages the
B.C. Open Textbook Project
, also uses this
for calculating student savings.) In addition to the open or freely available textbooks, faculty members use a number of other freely accessible or open resources, such as
, a randomized online homework software that automatically grades student work; the
Math Exam/Education Resources wiki
with past exams and worked out solutions and videos; and course
wiki pages
which host additional resources. In some cases, students have helped with the production of the online textbooks written by faculty, and contributed to WeBWorK problems and the course wiki pages.
The motivation
Customizing the content to fit the courses and, in some cases, developing courses that are more suited to students in a particular field of study, have been main motivators for instructors who have
written their own online textbooks. Leah Keshet, a professor in the math department, along with her colleagues, Bill Casselman and David Austin, developed two calculus courses in the 90’s for
students interested in biology. The instructors began writing online
course notes
, which they used instead of a textbook. “The three of us felt that the calculus which was then taught was not really suitable for life science students, and we basically wanted to do it in a
slightly different way,” Keshet said. “During the first few years [the course notes were] essentially a few online notes supplemented by labs that we created ourselves, but as the technologies
improved, as YouTube has become current and as it’s been much easier to post things like PDF files, gradually the website that we had evolved to having course notes.” Eric Cytrynbaum, an associate
professor in the department, has coordinated one of the courses that Keshet helped develop, Math 102: Differential Calculus with Applications to Life Sciences, for the last five years. “I never
actually thought of what we’re doing as developing open resources. I just thought, these students are paying $200 for a textbook, and the textbooks don’t even really cover the stuff that I really am
trying to push in the course, which are the applications and modeling. So it’s more frustration with what’s available than an ideological standpoint.”
online textbook
that they could use. “That’s really what precipitated it was when the publisher…went from the 7th edition to the 8th edition with no obvious improvements and very minimal changes, and the price would
have been really quite ludicrous, for one, and for what the students were getting, say, over the third edition, which would have been $20 resale,” Rechnitzer said. “That’s what really pushed me in
that direction.” Feldman said that a number of textbooks could have been used for the course. “There are a lot of commercial calculus textbooks, and you have to be an expert to tell the difference
between them. Any one of them could be used as effectively as any other.” Now, in addition to their own textbook, the instructors point to three other open textbooks that students can use. Keshet,
herself, has written a traditional textbook, Mathematical Models in Biology, which was published in 1988. “That sort of taught me that dealing with publishers can be a little bit problematic, because
with my first book, the publisher essentially did nothing other than increase the price every year, which was disconcerting,” she said. “This way, I feel that I can reach the students more easily and
also change it on the go. So if one year I find that examples that were done previously were not particularly good, or I want to rearrange them slightly, I can do it.” Keshet’s published textbook is
written for third and fourth-year undergraduate or beginning graduate level courses. Regarding the online textbook she developed for her first-year courses, Keshet said, “I, myself, am a mathematical
biologist, so I’m very interested in applications of mathematics to biology, and so it’s very natural and appealing to me to think up examples that would work at a first-year level. And so, I felt
compelled to write them down. Coming up with the simplest possible examples of how math could be useful in the life sciences is a wonderful challenge, and I’m still finding great new examples and
being very excited by them, and that’s really my motivation.” Mark MacLean, a faculty member and undergraduate chair in the Math department and president of the UBC Faculty Association, said that the
efforts in the department have focused not just on cost, but also on the benefits to student learning. “We feel like, in certain courses, that we don’t need one of these expensive textbooks to
support student learning to the level we know we can,” he said. “And we’ve said, even Math 102/103 is a 20 year example. Students have come out of that course and done very well, and the materials
fit the course very well, and we’ve done well by students with that…We have to make sure that if we choose to build these resources to understand what they are and how they affect student experiences
in learning.”
Student response
Students who have taken Keshet’s course cite affordability and access as benefits of having a free, online textbook. Lydia Chen, now an assistant professor of chemistry at McMaster University, took
Keshet’s course in 2002. “We have to pay an exorbitant amount to get into university, and on top of that we have to pay $200 for a textbook. With open textbooks students will still be able to have
the learning experience. I think that’s what the student needs.”
#textbookbrokeBC campaign
last year. The AMS hopes to promote the use of open textbooks and other open resources on campus. Kevin Doering, associate vice president academic and university affairs at the AMS, said there are
three main benefits to using open textbooks and open educational resources. These include affordability, equal access to textbooks without barriers, and quality of content since open textbooks and
materials can be customized to a course. Referring to data from the 2016 AMS Academic Experience Survey, Doering said 75 percent of students surveyed have gone without a textbook or other course
resource due to cost, and 37 percent do this frequently or often. In the coming weeks, Doering and representatives from the Centre for Teaching, Learning and Technology and the UBC Library will be
talking to course coordinators and faculty members who teach high enrolment courses—courses with more than 900 students—to see what the barriers are to adopting more open textbooks and resources.
“Once we know what the barriers are to adoption of open educational resources, we’re hoping to provide solutions specific to UBC to make it easier for faculty,” Doering said.
What’s next
Faculty members in Math who have written their own online textbooks say that they plan to continue to refine them or add features to them to make them more interactive and appealing for students to
use. According to Keshet and Cytrynbaum, one of the issues that they’ve encountered with their students is that some have a hard time reading through the technical material in traditional textbooks
and in their course notes. Because of this, they would like to develop more links in their course notes to the videos and homework problems, as well as create more videos that explain concepts.
Keshet said that she’s noticed that the introduction of videos, along with clicker questions and two-stage quizzes, where students take a portion of a quiz on their own and then come together into
groups to finish the quiz, has been particularly effective with her students. “Teaching by the old chalk talk method is not very conducive to enthusiastic learning,” she said. “That’s why I’ve gone
into these. It does take a lot of time, but I think it’s worth it. And judging by the students’ reactions at the end of a course, and the fact that they come and say, ‘This was really unique. I
really appreciated it.’ It makes it worthwhile.” | {"url":"https://open.ubc.ca/open-dialogues-adopting-open-textbooks-in-the-math-department/","timestamp":"2024-11-01T23:49:49Z","content_type":"application/xhtml+xml","content_length":"71135","record_id":"<urn:uuid:e27a3353-a09f-4de4-a34b-61d129fa8f68>","cc-path":"CC-MAIN-2024-46/segments/1730477027599.25/warc/CC-MAIN-20241101215119-20241102005119-00256.warc.gz"} |
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• converting fractions to decimals calculator | {"url":"https://www.softmath.com/math-com-calculator/adding-matrices/algebra-balance-equations-5.html","timestamp":"2024-11-12T23:34:57Z","content_type":"text/html","content_length":"206596","record_id":"<urn:uuid:bbf5e07f-542e-40c0-8f4b-6007ef40cf47>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00416.warc.gz"} |
Confusion Matrix in cybersecurity
Confusion Matrix in Machine Learning-
The confusion matrix is a matrix used to determine the performance of the classification models for a given set of test data. It can only be determined if the true values for test data are known. The
matrix itself can be easily understood, but the related terminologies may be confusing. Since it shows the errors in the model performance in the form of a matrix, hence also known as an error matrix
• For the 2 prediction classes of classifiers, the matrix is of 2*2 table, for 3 classes, it is 3*3 table, and so on.
• The matrix is divided into two dimensions, that are predicted values and actual values along with the total number of predictions.
• Predicted values are those values, which are predicted by the model, and actual values are the true values for the given observations.
• It looks like the below table:
The above table has the following cases:
• True Negative: Model has given prediction No, and the real or actual value was also No.
• True Positive: The model has predicted yes, and the actual value was also true.
• False Negative: The model has predicted no, but the actual value was Yes, it is also called as Type-II error.
• False Positive: The model has predicted Yes, but the actual value was No. It is also called a Type-I error.
Need for Confusion Matrix in Machine learning
• It evaluates the performance of the classification models, when they make predictions on test data, and tells how good our classification model is.
• It not only tells the error made by the classifiers but also the type of errors such as it is either type-I or type-II error.
• With the help of the confusion matrix, we can calculate the different parameters for the model, such as accuracy, precision, etc.
Example: We can understand the confusion matrix using an example.
Suppose we are trying to create a model that can predict the result for the disease that is either a person has that disease or not. So, the confusion matrix for this is given as:
From the above example, we can conclude that:
• The table is given for the two-class classifier, which has two predictions “Yes” and “NO.” Here, Yes defines that patient has the disease, and No defines that patient does not has that disease.
• The classifier has made a total of 100 predictions. Out of 100 predictions, 89 are true predictions, and 11 are incorrect predictions.
• The model has given prediction “yes” for 32 times, and “No” for 68 times. Whereas the actual “Yes” was 27, and actual “No” was 73 times.
Cybercrime issue-
New technologies create new criminal opportunities but few new types of crime. What distinguishes cybercrime from traditional criminal activity? Obviously, one difference is the use of the digital
computer, but technology alone is insufficient for any distinction that might exist between different realms of criminal activity. Criminals do not need a computer to commit fraud, traffic in child
pornography and intellectual property, steal an identity, or violate someone’s privacy. All those activities existed before the “cyber” prefix became ubiquitous. Cybercrime, especially involving the
Internet, represents an extension of existing criminal behaviour alongside some novel illegal activities.
Hence it needs to be tackled, here confusion matrix plays its role.
Cybersecurity and Confusion Matrix
The rapid increase in connectivity and accessibility of computer system has resulted frequent chances for cyber attacks. Attack on the computer infrastructures are becoming an increasingly Serious
problem. Basically the cyber attack detection is a classification problem, in which we classify the normal pattern from the abnormal pattern (attack) of the system. Subset selection decision fusion
method plays a key role in cyber attack detection. It has been shown that redundant and/or irrelevant features may severely affect the accuracy of learning algorithms. The SDF is very powerful and
popular data mining algorithm for decision-making and classification problems. It has been using in many real life applications like medical diagnosis, radar signal classification, weather
prediction, credit approval, and fraud detection etc.
KDD CUP ‘’99 Data Set Description
To check performance of the proposed algorithm for distributed cyber attack detection and classification, we can evaluate it practically using KDD’99 intrusion detection datasets. In KDD99 dataset
these four attack classes (DoS, U2R,R2L, and probe) are divided into 22 different attack classes that tabulated in Table I. The 1999 KDD datasets are divided into two parts: the training dataset and
the testing dataset. The testing dataset contains not only known attacks from the training data but also unknown attacks. Since 1999, KDD’99 has been the most wildly used data set for the evaluation
of anomaly detection methods. This data set is prepared by Stolfo et al. and is built based on the data captured in DARPA’98 IDS evaluation program . DARPA’98 is about 4 gigabytes of compressed raw
(binary) tcpdump data of 7 weeks of network traffic, which can be processed into about 5 million connection records, each with about 100 bytes. For each TCP/IP connection, 41 various quantitative
(continuous data type) and qualitative (discrete data type) features were extracted among the 41 features, 34 features (numeric) and 7 features (symbolic). To analysis the different results, there
are standard metrics that have been developed for evaluating network intrusion detections. Detection Rate (DR) and false alarm rate are the two most famous metrics that have already been used. DR is
computed as the ratio between the number of correctly detected attacks and the total number of attacks, while false alarm (false positive) rate is computed as the ratio between the number of normal
connections that is incorrectly misclassified as attacks and the total number of normal connections.
In the KDD Cup 99, the criteria used for evaluation of the participant entries is the Cost Per Test (CPT) computed using the confusion matrix and a given cost matrix. A Confusion Matrix (CM) is a
square matrix in which each column corresponds to the predicted class, while rows correspond to the actual classes. An entry at row i and column j, CM (i, j), represents the number of misclassified
instances that originally belong to class i, although incorrectly identified as a member of class j. The entries of the primary diagonal, CM (i, i), stand for the number of properly detected
instances. Cost matrix is similarly defined, as well, and entry C (i, j) represents the cost penalty for misclassifying an instance belonging to class i into class j.
• True Positive (TP): The amount of attack detected when it is actually attack.
• True Negative (TN): The amount of normal detected when it is actually normal.
• False Positive (FP): The amount of attack detected when it is actually normal (False alarm).
• False Negative (FN): The amount of normal detected when it is actually attack.
In the confusion matrix above, rows correspond to predicted categories, while columns correspond to actual categories.
Confusion matrix contains information actual and predicted classifications done by a classifier. The performance of cyber attack detection system is commonly evaluated using the data in a matrix. | {"url":"https://vibhanshusharma13.medium.com/confusion-matrix-in-cybersecurity-dc93cb190fe5?source=user_profile_page---------9-------------4035829c71ee---------------","timestamp":"2024-11-03T15:49:15Z","content_type":"text/html","content_length":"132059","record_id":"<urn:uuid:16b8a22c-1a8f-4553-a126-1e0f3f727cf1>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00502.warc.gz"} |
Friday Riddle Answer
nudgie Member Posts: 1,478 Member
This week's winnder is PGLGreg. CONGRATS !!!!!!!!!!!!
RIDDLE: COUNTING TO 50
Henry and Gretchen are going to play a game. Henry explains, "You and I will take turns saying numbers. The first person will say a number between 1 and 10. Then the other person will say a number
that is at least 1 higher than that number, and at most 10 higher. We will keep going back and forth in this way until one of us says the number 50. That person wins. I'll start."
"Not so fast!" says Gretchen. "I want to win, so I will start."
What number should Gretchen say to start?
ANSWER: She wants to say "6". The series of numbers she will say is 6, 17, 28, 39, 50. Since she wants to say 50, she needs Henry to say a number between 29 & 38. So she wants to say 28. Following
this same logic recursively, she will want to say 17, and she will want to say 6 to start the game, and be assured victory.
Discussion Boards
• 6 CSN Information
• 121.8K Cancer specific | {"url":"https://csn.cancer.org/discussion/179416/friday-riddle-answer","timestamp":"2024-11-06T17:51:00Z","content_type":"text/html","content_length":"269072","record_id":"<urn:uuid:ce713a67-38d4-46ef-9a19-d33236dd3b6b>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00188.warc.gz"} |
merge_sort -- stably sort an array
template <class T>
T* merge_sort(T* b1,T* e1,T* b2);
template <class T>
T* merge_sort_r(int (*rel)(const T*,const T*),
T* b1,T* e1,T* b2);
(1) For the plain version, T::operator< defines a total ordering relation on T.
(2) For the relational version, rel defines a total ordering relation on T.
(3) The two arrays do not overlap.
(4) The second array has at least as many cells as the first array.
(5) T has operator=.
These functions stably sort an array using a merge sorting algorithm. The algorithm requires an auxiliary array of the same size, identified by the argument b2. They return a pointer to either (1)
the original array or (2) the auxiliary array, depending on which one contains the final result of the sort.
template <class T>
T* merge_sort(T* b1,T* e1,T* b2);
Uses T::operator< to define the ordering relation used by the sorting algorithm.
template <class T>
T* merge_sort_r(int (*rel)(const T*,const T*),
T* b1,T* e1,T* b2);
Uses rel to define the ordering relation used by the sorting algorithm.
If N is the size of the array, then complexity is O(NlgN). At most NlgN tests of the ordering relation and NlgN assignments are done.
Because a Block (see Block(3C++)) can always be used wherever an array is called for, Array Algorithms can also be used with Blocks. In fact, these two components were actually designed to be used
Array_alg(3C++), ins_sort(3C++), sort(3C++), Block(3C++) © 2004 The SCO Group, Inc. All rights reserved.
UnixWare 7 Release 7.1.4 - 25 April 2004 | {"url":"http://uw713doc.sco.com/en/man/html.3C%2B%2B/merge_sort.3C++.html","timestamp":"2024-11-05T13:02:50Z","content_type":"text/html","content_length":"5265","record_id":"<urn:uuid:532ce2ec-00f1-4a0e-b51d-6b0f466655d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027881.88/warc/CC-MAIN-20241105114407-20241105144407-00078.warc.gz"} |
Probably magic!
Why are we surprised when a magician correctly guesses cards picked from a deck, as in the trick shown in the video? Is it really that difficult to perform such a trick, or could it just be down to
The answer is most probably not, as the probabilities are vanishingly small. In this article we will do the basic calculations and we'll also show that whenever you shuffle a deck of cards you make
history: chances are you produce an order of cards that has never ever occurred before in the whole, long history of the Universe.
Picking cards
A deck of cards has 52 cards, divided into four suits, two red — hearts and diamonds — and two black – clubs and spades. Each suit has 13 cards: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K.
If you choose a card at random, there are 52 possibilities for what you might get. So, if a magician were to simply guess what card you have picked, there would be a chance of only 1/52 they guess it
correctly, or a little less than 2%. That means that if a fake magician repeats the same guessing trick around 100 times, they would be expected to guess correctly only twice. The audience would be
quite frustrated by the other attempts!
What's the probability of guessing two cards correctly? The chance of guessing the first card correctly is 1/52, as we have just seen. Then, after the first card has been picked, there are only 51
cards left for the second guess. The probability of correctly guessing the second card is therefore 1/51. Hence, the probability of correctly guessing two cards in a row is:
1/52 x 1/51 = 1/2652 ≈ 0.0004,
so guessing two cards correctly is a much more impressive trick.
This is the multiplication principle: the probability of two independent events (guessing two cards correctly) is the product of the two individual probabilities.
Similarly, the probability of correctly guessing three cards in a row, as in the video, is:
1/52 x 1/51 x 1/50 = 1/132600 ≈ 000000.8.
Counting combinations
The multiplication principle gives us the probabilities we are after because it counts how many possibilities there are: when picking three cards there are 52 possibilities for what the first one
might be, 51 for what the second one might be, and 50 for what the third one might be. This gives a total of 52 x 51 x 50 possible three-card combinations, which means that the probability of picking
one of them is
1/52 x 1/51 x 1/50.
Now imagine you have a whole deck of cards. Take the deck of cards and shuffle it several times (how many times can be considered "several times"?). Hold it in front of you and ask yourself:
Is it the first time ever in history that the cards are displayed in this exact order?
Unfortunately, there is no way that you could guarantee the answer. However, chances are that you are holding a never-ever-made sequence. Pretty amazing, huh? How can we say this?
First, consider the same technique we used to calculate how many three-card combinations there are. There are 52 cards to fill in a sequence of 52 spots for cards. There are 52 possibilities for the
first spot, 51 for the second (once the first spot is filled), and so on. Hence the total number is:
52 x 51 x 50 x 49 ... x 2 x 1 = 8.07 x 10 ^67.
If we were to write this number out in full, it would have 68 digits.
The chance of producing the order you have produced by your shuffling is therefore
1/(8.07 x 10 ^67).
This is an incredibly small probability. To show just how small it is, let's make our own metaphor. According to the Big Bang theory, the whole Universe was created about 13.7 billion years ago.
Also, as of right now, population of Earth is 7.5 billion people.
Now imagine a parallel world with 10 billion people who all live forever. Give one deck of cards to each person (it doesn't matter if the person is a newborn or elderly) and ask them to shuffle once
per second for the next 14 billion years. Also, consider that each year is a leap year (366 days). Further, assume that for every shuffle a completely new sequence of cards is created. How many new
combinations would be created in this process?
Let's break the problem down into steps. After the first second, there would be 10 billion (10^10) new combinations: one for each of the 10 billion people who are shuffling. After the first minute,
there would be 10 billion times 60 new combinations, that's 10^10 x 60 = 6 x 10^11. We can continue like this:
After first... Number of combinations
Second: 10^10
Minute: 6 x 10^11
Hour: 6 x 10^11 x 60 = 3.6 x 10^13
Day: 3.6 x 10^13 x 24 = 8.64 x 10^14
Year: 8.64 x 10^14 x 366 ≈ 3.16 x 10^17
14 billion years: 3.16 x 10^17 x 14 x 10^9 = 4.424 x 10^27
But this number is less than a billionth of a billionth of a billionth of a billionth of 1 percent of the number of possible card sequences, 52 x 51 x 49 x ... x 2 x 1. Thus even after 14 billion
years of shuffling, our parallel world would only have produced a tiny fraction of all the possible combinations of cards!
So what about our card trick above? Well, it is a trick of course, and not just chance. To find out how it works, see the video:
About the authors
Ricardo Teixeira is a mathematician at the University of Houston-Victoria. He has been actively working in developing rigorous mathematical explanations for magic tricks and other recreational
activities. So far, he has published and worked in relating advanced topics such as probability, the theory of cyclic groups, linear algebra, coding theory, algorithms for data transmission, and
more, with magic tricks. To this date, over 2,000 students have witnessed him perform and explain his "mathemagics" tricks.
Gisele Teixeira is a second grader and gifted and talented student at Dudley Elementary. She has appeared in several YouTube videos performing mathematical magic tricks and explaining college algebra
material. She has also helped her father explain logarithms and matrices to college students.
"population of Earth is 7.5 million people" should be "billion" :)
Quite right! Now corrected.
"This gives a total of 52 x 51 x 50 possible three-card combinations, which means that the probability of picking one of them is
1/52 x 1/51 x 1/53."
I think that last 53 should be 50 (as it is slightly earlier)
Thank you for pointing out the error! We've corrected it.
In the section on "Counting Combinations" you have this:
1/52 x 1/51 x 1/53.
When it should be this:
1/52 x 1/51 x 1/50.
Likewise, thanks for spotting the typo!
I correctly guessed 3 cards in a row that my friend was holding. Nothing more than seeing the card in my mind. No trickery involved. I've practiced this alone by picking up random cards that were
face down.
It is impossible to practice this because it is down to probability. You got lucky once, that's it. I can almost guarantee you wont do that on your following attempt. | {"url":"https://plus.maths.org/content/comment/10407","timestamp":"2024-11-14T13:19:44Z","content_type":"text/html","content_length":"41335","record_id":"<urn:uuid:001c96f2-23e1-4d74-9469-216a48cffc5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028657.76/warc/CC-MAIN-20241114130448-20241114160448-00828.warc.gz"} |
Imad Aouali - Mixed-Effect Thompson Sampling
Mixed-Effect Thompson Sampling
AISTATS 2023
Looking for a practical algorithm, grounded in theory, that capitalizes on action similarities? Discover how to learn more efficiently in contextual bandits with large action spaces.
Mixed-Effect Thompson Sampling (meTS) leverages action correlations to explore the action space efficiently. It enjoys:
1. Efficient posterior updates.
2. Potential information sharing among actions.
3. Reduced Bayesian regret in comparison to standard methods.
4. Lower Bayes regret compared to standard methods.
5. Strong empirical performance without additional tuning.
6. Bridges the gap between offline learning of action correlations and online learning to act under uncertainty.
Think about movie recommendation, like when Netflix suggests movies to users. This situation can be seen as a contextual bandit problem. Here, we view user features as context, the movies they can
choose as actions, and the rating they give to a movie as the reward.
A common tactic is to learn how each user would rate each movie. So, we assess a movie's worth based on the ratings that other users gave to it. This is what standard methods like LinUCB and LinTS
do. But the issue is, when you have a lot of movies (a.k.a., large action space), this method isn't very efficient. Luckily, there's a silver lining: movies tend to be connected because they fit into
specific categories, like action, sci-fi, drama, and more.
Here's a key question: How can we leverage these categories to explore movies more efficiently? We'll explore some solutions, starting with the first one. Solution (A) involves learning how each user
rates each movie category based on the ratings of all movies in that category. Then, when predicting a user's rating for a movie, we simply use the rating they would have given to that movie's
category. This approach is more efficient because we're learning fewer parameters (ratings for a few categories instead of ratings for many individual movies) and we're using more data (ratings for
all movies within a category instead of just one movie). However, there's a catch: it tends to introduce a lot of bias because it assumes that all movies within the same category would have the same
rating from a user. In a nutshell, it might oversimplify things and we summrize this below.
To address the bias issue, we can take a similar approach as in (A), but with a twist. Here, we treat the rating that a user assigns to a movie as a random variable, with its average being the rating
that the user would typically give to the category that the movie falls under. This captures the connection between movies in the same category, as they all draw from the category's rating. However,
since a movie's rating is no longer exactly the same as its category rating, we need to learn both category ratings and individual movie ratings. The category ratings are learned by considering the
ratings of all movies within that category, while we fine-tune the movie ratings by focusing on the ratings for that specific movie. This method introduces some randomness to help reduce bias. This
is Solution (B) and it is summarized below.
There's a limitation with (B) – it assumes that a movie belongs to only one category. In reality, a movie can often fit into multiple categories. So, we take things a step further, much like (B), by
considering the rating a user gives to a movie as a random value. However, now, the average rating is like a mixture of the ratings that the user typically gives to all the categories the movie falls
under. The coefficients in this mixture show how strongly the movie is associated with each category. This adds a layer of complexity, creating even more connections between movies and making
exploration more statistically efficient. This final Solution (C) is what we call meTS, and it's a generalization of all the solutions we've discussed so far. It's like connecting all the dots.
Formally, we consider a mixed-effect model that writes
The corresponding graphical representation is given below.
A practical example is the Gaussian mixed-effect model where each action parameter is a Gaussian linear mixture of the effect parameters. That is, when
Even though the latent structure in this example is linear, the reward can be either linear or nonlinear.
Bridging the Gap Between Offline and Online Learning.
A question remains! How do we establish the mixed-effect model (with their structure) in the first place? Well, there are two possible routes. First, it can be inherently embedded in the problem
itself. For instance, in drug design, the actions represent different drugs, and the effects correspond to their components. The mixing weight (b) that links an action (drug) to an effect (component)
reflects the dosage of that component within the drug. In such cases, the relationship is apparent from the nature of the problem. However, in other situations, it might not be immediately clear how
these effects come about, and we need to learn this relationship. The good news is, this learning process can be carried out offline using off-the-shelf techniques like Gaussian Mixture Models (GMM).
These models can be trained on offline (noisy) estimates of the action parameters to create a proxy structure that's well-suited for meTS.
After formally presenting the model, we are in a position to present meTS, which is a hierarchical Thompson sampling algorithm that operates in two steps: it initially samples the effects, and then
it proceeds to sample the actions based on the sampled effects. Here's how it works
Theory is developed for meTS. We start with the derivations of the effect and action posteriors.
Then, the corresponding regret bounds capture the structure and they have the following form
The regret bound of meTS depends on both the number of actions (K) and the number of effects (L). But there are special cases where it only depends on one of these factors. For example, if the action
parameters are always the same for a given effect parameter, the agent only needs to learn the effect parameters, and the regret bound only depends on L. Similarly, if the effect parameters are
always the same, the agent only needs to learn the action parameters, and the regret bound only depends on K. In general, the regret depends on both K and L, but it's still better than the regret of
standard methods like LinTS, because meTS reduces the posterior variance.
To fully understand the benefits of meTS, we should pay close attention to the constants in the regret bound, as they can have a big impact. As we see above (refer to the paper for details), meTS can
significantly reduce regret, especially when the action space is large and when the effect parameters are more uncertain than the action parameters. Additionally, meTS has similar computational
efficiency to standard LinTS and it greatly improves the time and space complexities of TS that models the joint distribution of action parameters.
We start by simulating our mixed-effect model and observe that meTS and its factored variant (meTS-Fa) outperform all baseline methods that ignore the structure or incorporate it partially. The gain
is significant. Note that meTS-Fa is an approximation of meTS-Fa where effects are sampled independently. More details are in the paper.
But what if we can't simulate or access the mixed-effect model? This is the problem we address in MovieLens. Here, the rewards are not sampled from a mixed-effect model. However, we use a Gaussian
mixture model (GMM) trained on offline estimates of movie parameters to create a proxy mixed-effect model for meTS. Even though the true rewards are not generated from a mixed-effect model, meTS
still outperforms other methods. This shows that meTS is flexible and robust to model misspecification.
Main takeaways from experiments.
1. When the mixed-effect structure is available, it's better to design a TS algorithm that exploits it rather than marginalizing the effects and applying a standard TS algorithm. This is because
action information sharing reduces variance.
2. When the mixed-effect structure is not available, you can use offline estimates of the action parameters to create a proxy structure and apply meTS to it. In some cases, you may not have
offline estimates of the action parameters. In that case, you could use external datasets that are somewhat related to the problem. For example, you could use CLIP embeddings of movie images
and descriptions to get noisy estimates of the action parameters and train a GMM on them. However, the effectiveness of meTS with external data has not been proven empirically.
We have proposed a mixed-effect bandit framework that can be used to explore more efficiently. Our framework is represented by a two-level graphical model where actions can depend on multiple
effects. We have also designed an exploration algorithm, meTS, that leverages this structure. meTS has been shown to perform extremely well on both synthetic and real-world problems.
The core idea of our approach is that TS relies on prior distributions that can be chosen or learned in a clever fashion to improve computational and statistical efficiency. With the large
availability of offline data, we can use it to learn informative priors like the mixed-effect one. This can further improve the performance of TS and make it more efficient.
Our algorithmic ideas apply to the general mixed-effect model, which opens the door to richer models, such as general directed acyclic graphs (DAGs) or diffusion models. | {"url":"https://www.iaouali.com/blog/mixed-effect-ts","timestamp":"2024-11-05T02:52:35Z","content_type":"text/html","content_length":"115478","record_id":"<urn:uuid:5479fbd9-3e4e-45cd-aa66-55fd04a516c6>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00171.warc.gz"} |
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Math intuition, math without books
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Originally Posted by Kriston
If you don't get it, then you're forced to rely on rote memorization for the doing, and that's painful for a GT kid.
It's not just painful, rote memorization takes a whole lot more effort (at least for me!) and is less "sticky" than understanding something. I think it's because when you just memorize you may learn
a fact but it's just floating there in your brain, not connected to anything. It can be hard to retrieve that floating information. If I know how to reason something out it's like a trail of
breadcrumbs to follow back to the memory. Each time I repeat the reasoning process it becomes easier to find and follow the breadcrumbs. If you think about something enough, the trail becomes like a
superhighway leading straight to the answer.
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Math intuition, math without books Kriston 04/23/08 03:30 AM
Re: Math intuition, math without books OHGrandma 04/23/08 11:38 AM
Re: Math intuition, math without books squirt 04/23/08 01:49 PM
Re: Math intuition, math without books Kriston 04/23/08 01:53 PM
Re: Math intuition, math without books Kriston 04/23/08 01:59 PM
Re: Math intuition, math without books elh0706 04/23/08 02:30 PM
Re: Math intuition, math without books doodlebug 04/23/08 02:45 PM
Re: Math intuition, math without books Ania 04/23/08 03:33 PM
Re: Math intuition, math without books Kriston 04/23/08 03:58 PM
Re: Math intuition, math without books Kriston 04/23/08 04:16 PM
Re: Math intuition, math without books Ania 04/23/08 04:24 PM
Re: Math intuition, math without books LMom 04/23/08 04:29 PM
Re: Math intuition, math without books Kriston 04/23/08 04:36 PM
Re: Math intuition, math without books OHGrandma 04/23/08 04:40 PM
Re: Math intuition, math without books calizephyr 04/23/08 05:43 PM
Re: Math intuition, math without books pinkpanther 04/23/08 04:32 PM
Re: Math intuition, math without books LMom 04/23/08 04:27 PM
Re: Math intuition, math without books squirt 04/23/08 02:39 PM
Re: Math intuition, math without books cym 04/23/08 02:53 PM
Re: Math intuition, math without books Kriston 04/23/08 03:20 PM
Re: Math intuition, math without books doodlebug 04/23/08 03:38 PM
Re: Math intuition, math without books Dazed&Confuzed 04/23/08 03:44 PM
Re: Math intuition, math without books LMom 04/23/08 04:50 PM
Re: Math intuition, math without books Dazed&Confuzed 04/23/08 05:23 PM
Re: Math intuition, math without books Cathy A 04/23/08 07:14 PM
Re: Math intuition, math without books pinkpanther 04/23/08 07:19 PM
Re: Math intuition, math without books kimck 04/23/08 07:39 PM
Re: Math intuition, math without books st pauli girl 04/23/08 07:45 PM
Re: Math intuition, math without books Cathy A 04/23/08 07:51 PM
Re: Math intuition, math without books st pauli girl 04/23/08 08:28 PM
Re: Math intuition, math without books Ania 04/23/08 07:53 PM
Re: Math intuition, math without books Cathy A 04/23/08 08:02 PM
Re: Math intuition, math without books Dazed&Confuzed 04/23/08 07:25 PM
Re: Math intuition, math without books Cathy A 04/23/08 07:35 PM
Re: Math intuition, math without books Cathy A 04/23/08 07:38 PM
Re: Math intuition, math without books Kriston 04/23/08 08:05 PM
Re: Math intuition, math without books Cathy A 04/23/08 10:41 PM
Re: Math intuition, math without books Kriston 04/23/08 10:58 PM
Re: Math intuition, math without books Cathy A 04/23/08 11:10 PM
Re: Math intuition, math without books Kriston 04/23/08 11:15 PM
Re: Math intuition, math without books Cathy A 04/23/08 11:39 PM
Re: Math intuition, math without books EandCmom 04/23/08 11:45 PM
Re: Math intuition, math without books Kriston 04/24/08 12:02 AM
Re: Math intuition, math without books Cathy A 04/24/08 12:11 AM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 01:03 AM
Re: Math intuition, math without books squirt 04/24/08 01:13 AM
Re: Math intuition, math without books Kriston 04/24/08 01:55 AM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 02:11 AM
Re: Math intuition, math without books Cathy A 04/24/08 02:35 AM
Re: Math intuition, math without books Kriston 04/24/08 01:21 PM
Re: Math intuition, math without books AmyEJ 04/24/08 02:08 PM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 02:15 PM
Re: Math intuition, math without books Kriston 04/24/08 02:21 PM
Re: Math intuition, math without books Lori H. 04/24/08 03:13 PM
Re: Math intuition, math without books squirt 04/24/08 04:02 PM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 04:08 PM
Re: Math intuition, math without books Kriston 04/24/08 04:18 PM
Re: Math intuition, math without books Dazed&Confuzed 04/24/08 04:35 PM
Re: Math intuition, math without books questions 04/24/08 05:12 PM
Re: Math intuition, math without books Cathy A 04/24/08 06:41 PM
Re: Math intuition, math without books Cathy A 04/24/08 07:00 PM
Re: Math intuition, math without books Kriston 04/24/08 10:35 PM
Re: Math intuition, math without books incogneato 04/24/08 11:31 PM
Re: Math intuition, math without books Kriston 04/24/08 11:36 PM
Re: Math intuition, math without books incogneato 04/24/08 11:46 PM
Re: Math intuition, math without books Kriston 04/24/08 11:50 PM
Re: Math intuition, math without books incogneato 04/24/08 11:56 PM
Re: Math intuition, math without books Kriston 04/25/08 12:05 AM
Re: Math intuition, math without books doodlebug 04/27/08 12:16 PM
Re: Math intuition, math without books Kriston 04/27/08 01:12 PM
Re: Math intuition, math without books Kriston 04/27/08 07:46 PM
Re: Math intuition, math without books Kriston 04/27/08 11:10 PM
Re: Math intuition, math without books Dazed&Confuzed 04/27/08 11:22 PM
Re: Math intuition, math without books incogneato 04/28/08 02:33 AM
Re: Math intuition, math without books Kriston 04/28/08 04:28 PM
Re: Math intuition, math without books Kriston 04/29/08 01:16 PM
Re: Math intuition, math without books Kriston 04/29/08 02:19 PM
Re: Math intuition, math without books incogneato 04/29/08 04:13 PM
Re: Math intuition, math without books snowgirl 04/24/08 02:40 AM
Re: Math intuition, math without books Cathy A 04/24/08 02:48 AM
Re: Math intuition, math without books Cathy A 04/24/08 02:52 AM
Re: Math intuition, math without books OHGrandma 04/24/08 11:25 AM
Re: Math intuition, math without books Cathy A 04/24/08 02:45 AM
Re: Math intuition, math without books OHGrandma 04/23/08 11:10 PM
Re: Math intuition, math without books Kriston 04/23/08 08:20 PM
Re: Math intuition, math without books Kriston 04/23/08 08:27 PM
Re: Math intuition, math without books Kriston 04/23/08 09:40 PM
Re: Math intuition, math without books Kriston 04/23/08 08:10 PM
Re: Math intuition, math without books Kriston 04/23/08 08:32 PM
Re: Math intuition, math without books LMom 04/23/08 11:11 PM
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Kurt Gödel: Life, Work, and Legacy
Courtesy of the Kurt Gödel Papers, Institute for Advanced Study, on deposit at Princeton University
Kurt Gödel (1906–78) in the Mathematics–Natural Sciences Library
Looking back over that century in the year 2000, TIME magazine included Kurt Gödel (1906–78), the foremost mathematical logician of the twentieth century among its top 100 most influential thinkers.
Gödel was associated with the Institute for Advanced Study from his first visit in the academic year 1933–34, until his death in 1978. He was Professor in the School of Mathematics from 1953 until
1976, when he became Professor Emeritus.
The Early Years
Kurt Friedrich Gödel was born on April 28, 1906, in what is now Brno in the Czech Republic. His father, Rudolf Gödel, was originally from Vienna; his mother, Marianne Handschuh, came from the German
Rhineland. Rudolf Gödel managed and was part owner of one of Brno’s major textile companies and the family lived in middle-class comfort with servants and a governess for Kurt and his older brother,
Rudolf, born in 1902.
The young Gödel was known affectionately by the nickname, der Herr Warum (Mr. Why). In Logical Dilemmas: The Life and Work of Kurt Gödel, biographer John W. Dawson, Jr., describes him as “an
earnestly serious, bright, and inquisitive child who was sensitive, often withdrawn and pre-occupied, and who, already at an early age, exhibited certain signs of emotional instability.” At age
eight, after reading a medical book, Gödel became convinced that he had a weak heart, a possible complication of the rheumatic fever that he had recovered from at age six. Hypochondriac concerns for
his health would become a lifelong preoccupation.
Academic Life
In 1923, Gödel enrolled in the University of Vienna with the intention of studying physics. After attending lectures on number theory by the charismatic professor Philipp Furtwängler, brother of
famed German conductor Wilhelm Furtwängler, he switched to mathematics. Furtwängler was paralyzed from the neck down and lectured from his wheel chair with an assistant writing his formulae on the
board. He made an impression much like that of Stephen Hawking today.
As a student, Gödel attended meetings of what would later become Der Wiener Kreis (Vienna Circle), a group of mainly philosophers who met to discuss foundational problems and who were inspired by
Ludwig Wittgenstein’s Tractatus Logico-Philosophicus. The group focused on questions of language and meaning and logical relations such as entailment, and originated Logical Positivism. Led by Moritz
Schlick, a professor at the University who was later murdered by a deranged former student as he climbed the steps to the main lecture hall (1936), its members included Rudolf Carnap, Otto Neurath,
Carl Hempel, Hans Reichenbach, and others.
Kurt Gödel Papers, Institute for Advanced Study, on deposit at Princeton University
Kurt Gödel and Adele Porkert were married in Vienna in 1938.
In 1927, at age 21, Gödel met dancer Adele Nimbursky (née Porkert), in the Viennese night club, Der Nachtfalter (The Moth). Because Adele had been married and was six years older than Kurt, his
parents disapproved of the match (his mother Marianne was 14 years younger than his father, Rudolf). This was the second time they had disapproved of Kurt’s liaison with an older woman and it was not
until the autumn of 1938 that Kurt and Adele were married.
Gödel pursued studies in mathematics and logic with Hans Hahn and Karl Menger. His doctoral thesis was completed in 1929, the year in which his father Rudolf died, leaving the family in comfortable
circumstances. Gödel’s mother bought an apartment in Vienna where she lived with both of her sons and enjoyed the cultural life of the city, especially musical theater. Gödel developed a lifelong
love of operetta.
After receiving his doctorate, Gödel became a privatdozent (unpaid lecturer) at the University of Vienna. Like many of the young scholars who later found their way to the Institute for Advanced Study
from Europe in the 1930s, Kurt Gödel was brilliant. Unlike many, he was not Jewish, although he moved in circles of Jewish intellectuals and was sometimes thought to be Jewish. He had once been
attacked as such by a gang of youths while walking with Adele on a street in Vienna. During the 1930s it was not unusual for university students who were Jewish or had socialist leanings to be
forcibly removed from classes. Many of Gödel’s contemporaries were fleeing Europe.
Incompleteness Theorems
In 1931, Gödel published results in formal logic that are considered landmarks of 20th-century mathematics. Gödel demonstrated, in effect, that hopes of reducing mathematics to an axiomatic system,
as envisioned by mathematicians and philosophers at the turn of the 20th century, were in vain. His findings put an end to logicist efforts such as those of Bertrand Russell and Alfred North
Whitehead and demonstrated the severe limitations of David Hilbert’s formalist program for arithmetic.
In his introduction to his 1931 paper, Gödel stated: “It is well known that the development of mathematics in the direction of greater precision has led to the formalization of extensive mathematical
domains, in the sense that proofs can be carried out according to a few mechanical rules.... It is reasonable therefore to make the conjecture that these axioms and rules of inference are also
sufficient to decide all mathematical questions, which can be formally expressed in the given systems. In what follows it will be shown that this is not the case.”
By the age of 25 Kurt Gödel had produced his famous “Incompleteness Theorems.” His fundamental results showed that in any consistent axiomatic mathematical system there are propositions that cannot
be proved or disproved within the system and that the consistency of the axioms themselves cannot be proved. In addition to his proof of the incompleteness of formal number theory, Gödel published
proofs of the relative consistency of the axiom of choice and the generalized continuum hypothesis (1938, 1940). His findings strongly influenced the (later) discovery that a computer can never be
programmed to answer all mathematical questions.
In 1938, Gödel’s application for a paid position at the University of Vienna was turned down. Fearing conscription into the German army, he applied for a visa to the United States. In late 1939, Kurt
and Adele fled Nazi Germany, traveling via the trans-Siberian railway and ship to San Francisco, where they arrived on March 4, 1940. They settled in Princeton where Gödel’s position at the Institute
was renewed annually until 1946, when he became a permanent Member until appointed to the Faculty.
The Later Years
After suffering from severe bleeding from a duodenal ulcer, Gödel maintained an extremely strict diet that led to severe weight loss. By several accounts, Adele Gödel was a loving support to her
husband, whom she addressed as strammer bursche (strapping lad). Mathematical logician Georg Kreisel, a Member in the School of Mathematics (1955–57), records their relationship in Biographical
Memoirs of Fellows of the Royal Society [1980, Volume 26]: “I visited them quite often in the fifties and sixties. It was a revelation to see him relax in her company. She had little formal
education, but a real flair for the mot juste, which her somewhat critical mother-in-law eventually noticed too, and a knack for amusing and apparently quite spontaneous twists on a familiar ploy: to
invent—at least, at the time—far-fetched grounds for jealousy. On one occasion, she painted the I.A.S., which she usually called Altersversorgungsheim (home for old-age pensioners), as teeming with
pretty girl students who queued up at the office doors of permanent professors. Gödel was very much at ease with her style.”
When Gödel became convinced that he was being poisoned, Adele became his food taster. His digestive ailments and, particularly, his refusal to eat led ultimately to his death on January 14, 1978. He
died in Princeton at age 71 and is buried in the Princeton Cemetery.
Gödel's Awards
One of the most significant acknowledgements of Gödel’s accomplishments came in 1974, when he received the National Medal of Science in the discipline of mathematics and computer science from
President Ford in a ceremony at the White House.
The award citation read: “For laying the foundation for today’s flourishing study of mathematical logic.”
Prior to the National Medal of Science, Gödel received the Institute’s Einstein Award in 1951, which consisted of a gold medal (pictured) and the sum of $15,000. The gift of Institute Trustee Lewis
L. Strauss, it was presented to Gödel by Einstein at a ceremony in Princeton. Following this award, several other accolades came to Gödel, including honorary doctorates from Yale, Harvard, and
Rockefeller universities and from Amherst College. Gödel was a Member of the National Academy of Sciences of the United States, a Foreign Member of the Royal Society of London, a Corresponding Member
of the Institut de France, a Corresponding Fellow of the British Academy, and an Honorary Member of the London Mathematical Society. | {"url":"https://www.ias.edu/kurt-g%C3%B6del-life-work-and-legacy","timestamp":"2024-11-10T00:15:19Z","content_type":"text/html","content_length":"65761","record_id":"<urn:uuid:092be883-6434-4052-908e-e1aa7b967481>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00577.warc.gz"} |
Using Multivariate Analysis Techniques in PhD Research: A Critical Analysis of Their Capabilities and Limitations
First, let us understand what multivariate analysis is. In multivariate analysis (MVA), numerous variables (more than two) are analysed to find any potential relationships between them. By examining
every potential independent variable and how it relates to other independent variables, the multivariate analysis provides a more thorough interpretation of the data. In this blog, we are going to
describe multivariate analysis in depth along with using and analysing it in a PhD research.
Defining Multivariate analysis
In a broad sense, multivariate analysis refers to a group of statistical techniques used to examine many datasets at once. The goal of multivariate analysis is to use various statistical techniques
to apply and interpret the inherent structure and meaning that are revealed within large collections of variables.
When using a multivariate technique, two important considerations must be made: the data matrix's multidimensionality and the goal of maintaining its intricate structure. This is predicated on the
idea that because the variables are connected, only a set of the same test may give a more comprehensive picture of the topic being researched while gathering data that univariate and bivariate
statistical approaches cannot. Let me give you an example to simplify multivariate analysis.
An investigation is being conducted to identify potential causes of a medical illness, such as heart disease. Data on age, height, weight, serum cholesterol, phospholipids, blood glucose, nutrition,
and many other potential determinants are gathered from a preliminary survey of healthy guys. These boys' medical histories are monitored in order to ascertain whether and when they might be given a
heart disease diagnosis.
Figure 1: Multivariate analysis
Using multivariate analysis in PhD research
The multivariate technique is being used in 8 factors in a PhD research such as cluster analysis, discriminant analysis, factor analysis, CHAID, regression analysis, correspondence analysis,
structural modelling of equations and statis. The types of analysis are described below:
Cluster analysis -
It is a technique used for categorising distinct groupings of cases that are connected. Another use is to depict groups of related cases in a data compilation. Dissection is the name of this cluster
analysis method.
Discriminant analysis -
It is a statistical technique used to group exhaustively and mutually exclusive individuals or things based on a set of independent factors.
Factor analysis -
It is a Statistical approach used to explain the variability between studied variables in terms of a possibly smaller set of unknown variables that are referred to as factors."
CHAID -
It is a form of decision tree strategy that is based on the Bonferroni testing approach to adjusted significance testing.
Regression analysis -
It refers to any techniques for modelling and examining multiple variables where the emphasis is on the link between a dependent variable and one or more independent variables.
Correspondence analysis -
The process of creating visual representations of interactions between the modalities (or "categories") of two categorical variables.
Structural modelling of equation -
Using a combination of statistical information and qualitative causal hypotheses, structural equation modelling (SEM) is a statistical technique for testing and estimating causal links.
Statis -
When at least one three-way table dimension is shared by all tables, Statis is used. The method's initial step involves doing a PCA on each table and creating a table that compares the units in each
Capabilities and limitations of multivariate analysis in PhD research
Figure 2: Regression of Multivariate analysis
Researchers can use multivariate techniques to quantify the link between variables and examine relationships between variables in a broad sense. Cross tabulation, partial correlation, and multiple
regressions can be used to control the association between variables, and they can also introduce new variables to establish the relationships between the independent and dependent variables or to
define the circumstances under which the association occurs. Multivariate analysis has the benefit of providing a more accurate image than single variable analysis. Comparatively to univariate
procedures, multivariate techniques offer a strong test of significance.
A statistical programme is needed to examine the data using multivariate approaches because they are intricate and involve advanced mathematics. It can be expensive for an individual to purchase
these statistical applications. The fact that the results of statistical modelling are not always simple for students to understand is one of the main drawbacks of multivariate analysis. Large
samples of data are required for multivariate approaches to produce conclusions that are relevant; otherwise, results with high standard errors are useless. The level of confidence in the results is
determined by the standard errors, and the results from a large sample can be trusted more than those from a small sample. Although running statistical algorithms is pretty simple, understanding the
data requires statistical training.
Finally, we can conclude that multivariate analysis is helpful to examine datasets and also to make relationships between them. The different uses of multivariate analysis have been described along
with a critically analysing of its limitations and capacities of it. We, at Regent Statistics, provide you with high-quality PhD dissertations at an affordable price that covers all your needs. | {"url":"https://www.regentstatistics.co.uk/blog/post/using-multivariate-analysis-techniques-in-phd-research:-a-critical-analysis-of-their-capabilities-and-limitations","timestamp":"2024-11-11T17:17:20Z","content_type":"text/html","content_length":"50526","record_id":"<urn:uuid:3729766e-4f92-410f-bc2e-4c09b5c5438b>","cc-path":"CC-MAIN-2024-46/segments/1730477028235.99/warc/CC-MAIN-20241111155008-20241111185008-00719.warc.gz"} |
Time-independent PDE solution and derived quantities
A StationaryResults object contains the solution of a PDE and its gradients in a form convenient for plotting and postprocessing.
• A StationaryResults object contains the solution and its gradient calculated at the nodes of the triangular or tetrahedral mesh, generated by generateMesh.
• Solution values at the nodes appear in the NodalSolution property.
• The three components of the gradient of the solution values at the nodes appear in the XGradients, YGradients, and ZGradients properties.
• The array dimensions of NodalSolution, XGradients, YGradients, and ZGradients enable you to extract solution and gradient values for specified equation indices in a PDE system.
To interpolate the solution or its gradient to a custom grid (for example, specified by meshgrid), use interpolateSolution or evaluateGradient.
There are several ways to create a StationaryResults object:
• Solve a time-independent problem using the solvepde function. This function returns a PDE solution as a StationaryResults object. This is the recommended approach.
• Solve a time-independent problem using the assempde or pdenonlin function. Then use the createPDEResults function to obtain a StationaryResults object from a PDE solution returned by assempde or
pdenonlin. Note that assempde and pdenonlin are legacy functions. They are not recommended for solving PDE problems.
NodalSolution — Solution values at the nodes
vector | array
This property is read-only.
Solution values at the nodes, returned as a vector or array. For details about the dimensions of NodalSolution, see Dimensions of Solutions, Gradients, and Fluxes.
Data Types: double
XGradients — x-component of gradient at the nodes
vector | array
This property is read-only.
x-component of the gradient at the nodes, returned as a vector or array. For details about the dimensions of XGradients, see Dimensions of Solutions, Gradients, and Fluxes.
Data Types: double
YGradients — y-component of gradient at the nodes
vector | array
This property is read-only.
y-component of the gradient at the nodes, returned as a vector or array. For details about the dimensions of YGradients, see Dimensions of Solutions, Gradients, and Fluxes.
Data Types: double
ZGradients — z-component of gradient at the nodes
vector | array
This property is read-only.
z-component of the gradient at the nodes, returned as a vector or array. For details about the dimensions of ZGradients, see Dimensions of Solutions, Gradients, and Fluxes.
Data Types: double
Object Functions
evaluateCGradient Evaluate flux of PDE solution
evaluateGradient Evaluate gradients of PDE solutions at arbitrary points
interpolateSolution Interpolate PDE solution to arbitrary points
Obtain a StationaryResults Object from solvepde
Create a PDE model for a system of three equations. Import the geometry of a bracket and plot the face labels.
model = createpde(3);
title("Bracket with Face Labels")
title("Bracket with Face Labels, Rear View")
Set boundary conditions such that face 4 is immobile, and face 8 has a force in the negative z direction.
Set coefficients that represent the equations of linear elasticity. See Linear Elasticity Equations.
E = 200e9;
nu = 0.3;
Create a mesh.
Solve the PDE.
results = solvepde(model)
results =
StationaryResults with properties:
NodalSolution: [14093x3 double]
XGradients: [14093x3 double]
YGradients: [14093x3 double]
ZGradients: [14093x3 double]
Mesh: [1x1 FEMesh]
Access the solution at the nodal locations.
u = results.NodalSolution;
Plot the solution for the z-component, which is component 3.
Results from createPDEResults
Obtain a StationaryResults object from a legacy solver together with createPDEResults.
Create a PDE model for a system of three equations. Import the geometry of a bracket and plot the face labels.
model = createpde(3);
title("Bracket with Face Labels")
title("Bracket with Face Labels, Rear View")
Set boundary conditions such that F4 is immobile, and F8 has a force in the negative z direction.
Set coefficients for a legacy solver that represent the equations of linear elasticity. See Linear Elasticity Equations.
E = 200e9;
nu = 0.3;
c = elasticityC3D(E,nu);
a = 0;
f = [0;0;0];
Create a mesh.
Solve the problem using a legacy solver.
u = assempde(model,c,a,f);
Create a StationaryResults object from the solution.
results = createPDEResults(model,u)
results =
StationaryResults with properties:
NodalSolution: [14093x3 double]
XGradients: [14093x3 double]
YGradients: [14093x3 double]
ZGradients: [14093x3 double]
Mesh: [1x1 FEMesh]
Access the solution at the nodal locations.
u = results.NodalSolution;
Plot the solution for the z-component, which is component 3.
Version History
Introduced in R2016a
R2016b: Added evaluateCGradient function
You can now evaluate flux of PDE solution as a tensor product of c-coefficient and gradient of PDE solution. | {"url":"https://au.mathworks.com/help/pde/ug/pde.stationaryresults.html","timestamp":"2024-11-13T19:49:36Z","content_type":"text/html","content_length":"98186","record_id":"<urn:uuid:824d1c91-a02c-4c77-bac6-c309001e3932>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00301.warc.gz"} |
6.2 Adjusting Nominal Values to Real Values - Principles of Macroeconomics 2e | OpenStax
By the end of this section, you will be able to:
• Contrast nominal GDP and real GDP
• Explain GDP deflator
• Calculate real GDP based on nominal GDP values
When examining economic statistics, there is a crucial distinction worth emphasizing. The distinction is between nominal and real measurements, which refer to whether or not inflation has distorted a
given statistic. Looking at economic statistics without considering inflation is like looking through a pair of binoculars and trying to guess how close something is: unless you know how strong the
lenses are, you cannot guess the distance very accurately. Similarly, if you do not know the inflation rate, it is difficult to figure out if a rise in GDP is due mainly to a rise in the overall
level of prices or to a rise in quantities of goods produced. The nominal value of any economic statistic means that we measure the statistic in terms of actual prices that exist at the time. The
real value refers to the same statistic after it has been adjusted for inflation. Generally, it is the real value that is more important.
Converting Nominal to Real GDP
Table 6.5 shows U.S. GDP at five-year intervals since 1960 in nominal dollars; that is, GDP measured using the actual market prices prevailing in each stated year. Figure 6.7 also reflects this data
in a graph.
Year Nominal GDP (billions of dollars) GDP Deflator (2005 = 100)
1960 543.3 19.0
1965 743.7 20.3
1970 1,075.9 24.8
1975 1,688.9 34.1
1980 2,862.5 48.3
1985 4,346.7 62.3
1990 5,979.6 72.7
1995 7,664.0 81.7
2000 10,289.7 89.0
2005 13,095.4 100.0
2010 14,958.3 110.0
If an unwary analyst compared nominal GDP in 1960 to nominal GDP in 2010, it might appear that national output had risen by a factor of more than twenty-seven over this time (that is, GDP of $14,958
billion in 2010 divided by GDP of $543 billion in 1960 = 27.5). This conclusion would be highly misleading. Recall that we define nominal GDP as the quantity of every final good or service produced
multiplied by the price at which it was sold, summed up for all goods and services. In order to see how much production has actually increased, we need to extract the effects of higher prices on
nominal GDP. We can easily accomplish this using the GDP deflator.
The GDP deflator is a price index measuring the average prices of all final goods and services included in the economy. We explore price indices in detail and how we compute them in Inflation, but
this definition will do in the context of this chapter. Table 6.5 provides the GDP deflator data and Figure 6.8 shows it graphically.
Figure 6.8 shows that the price level has risen dramatically since 1960. The price level in 2010 was almost six times higher than in 1960 (the deflator for 2010 was 110 versus a level of 19 in 1960).
Clearly, much of the growth in nominal GDP was due to inflation, not an actual change in the quantity of goods and services produced, in other words, not in real GDP. Recall that nominal GDP can rise
for two reasons: an increase in output, and/or an increase in prices. What is needed is to extract the increase in prices from nominal GDP so as to measure only changes in output. After all, the
dollars used to measure nominal GDP in 1960 are worth more than the inflated dollars of 1990—and the price index tells exactly how much more. This adjustment is easy to do if you understand that
nominal measurements are in value terms, where
$Value = Price Ă— Quantity orNominal GDP = GDP Deflator Ă— Real GDPValue = Price Ă— Quantity orNominal GDP = GDP Deflator Ă— Real GDP$
Let’s look at an example at the micro level. Suppose the t-shirt company, Coolshirts, sells 10 t-shirts at a price of $9 each.
$Coolshirt"s nominal revenue from sales = Price Ă— Quantity = 9 Ă— 10 = 90Coolshirt"s nominal revenue from sales = Price Ă— Quantity = 9 Ă— 10 = 90$
$Coolshirt"s real income = Nominal revenuePrice = 909 = 10Coolshirt"s real income = Nominal revenuePrice = 909 = 10$
In other words, when we compute “real” measurements we are trying to obtain actual quantities, in this case, 10 t-shirts.
With GDP, it is just a tiny bit more complicated. We start with the same formula as above:
$Real GDP = Nominal GDPPrice IndexReal GDP = Nominal GDPPrice Index$
For reasons that we will explain in more detail below, mathematically, a price index is a two-digit decimal number like 1.00 or 0.85 or 1.25. Because some people have trouble working with decimals,
when the price index is published, it has traditionally been multiplied by 100 to get integer numbers like 100, 85, or 125. What this means is that when we “deflate” nominal figures to get real
figures (by dividing the nominal by the price index). We also need to remember to divide the published price index by 100 to make the math work. Thus, the formula becomes:
$Real GDP = Nominal GDPPrice Index / 100Real GDP = Nominal GDPPrice Index / 100$
Now read the following Work It Out feature for more practice calculating real GDP.
Computing GDP
It is possible to use the data in Table 6.5 to compute real GDP.
Step 1. Look at Table 6.5, to see that, in 1960, nominal GDP was $543.3 billion and the price index (GDP deflator) was 19.0.
Step 2. To calculate the real GDP in 1960, use the formula:
$Real GDP = Nominal GDPPrice Index / 100 = 543.3 billion19 / 100 = 2,859.5 billionReal GDP = Nominal GDPPrice Index / 100 = 543.3 billion19 / 100 = 2,859.5 billion$
We’ll do this in two parts to make it clear. First adjust the price index: 19 divided by 100 = 0.19. Then divide into nominal GDP: $543.3 billion / 0.19 = $2,859.5 billion.
Step 3. Use the same formula to calculate the real GDP in 1965.
$Real GDP = Nominal GDPPrice Index / 100 = 743.7 billion20.3 / 100 = 3,663.5 billionReal GDP = Nominal GDPPrice Index / 100 = 743.7 billion20.3 / 100 = 3,663.5 billion$
Step 4. Continue using this formula to calculate all of the real GDP values from 1960 through 2010. The calculations and the results are in Table 6.6.
Year Nominal GDP (billions of dollars) GDP Deflator (2005 = 100) Calculations Real GDP (billions of 2005 dollars)
1960 543.3 19.0 543.3 / (19.0/100) 2859.5
1965 743.7 20.3 743.7 / (20.3/100) 3663.5
1970 1075.9 24.8 1,075.9 / (24.8/100) 4338.3
1975 1688.9 34.1 1,688.9 / (34.1/100) 4952.8
1980 2862.5 48.3 2,862.5 / (48.3/100) 5926.5
1985 4346.7 62.3 4,346.7 / (62.3/100) 6977.0
1990 5979.6 72.7 5,979.6 / (72.7/100) 8225.0
1995 7664.0 82.0 7,664 / (82.0/100) 9346.3
2000 10289.7 89.0 10,289.7 / (89.0/100) 11561.5
2005 13095.4 100.0 13,095.4 / (100.0/100) 13095.4
2010 14958.3 110.0 14,958.3 / (110.0/100) 13598.5
There are a couple things to notice here. Whenever you compute a real statistic, one year (or period) plays a special role. It is called the base year (or base period). The base year is the year
whose prices we use to compute the real statistic. When we calculate real GDP, for example, we take the quantities of goods and services produced in each year (for example, 1960 or 1973) and multiply
them by their prices in the base year (in this case, 2005), so we get a measure of GDP that uses prices that do not change from year to year. That is why real GDP is labeled “Constant Dollars”
or, in this example, “2005 Dollars,” which means that real GDP is constructed using prices that existed in 2005. While the example here uses 2005 as the base year, more generally, you can use any
year as the base year. The formula is:
$GDP deflator = Nominal GDPReal GDP Ă— 100GDP deflator = Nominal GDPReal GDP Ă— 100$
Rearranging the formula and using the data from 2005:
$Real GDP = Nominal GDPPrice Index / 100 = 13,095.4 billion100 / 100 = 13,095.4 billionReal GDP = Nominal GDPPrice Index / 100 = 13,095.4 billion100 / 100 = 13,095.4 billion$
Comparing real GDP and nominal GDP for 2005, you see they are the same. This is no accident. It is because we have chosen 2005 as the “base year” in this example. Since the price index in the
base year always has a value of 100 (by definition), nominal and real GDP are always the same in the base year.
Look at the data for 2010.
$Real GDP = Nominal GDPPrice Index / 100 = 14,958.3 billion110 / 100 = 13,598.5 billionReal GDP = Nominal GDPPrice Index / 100 = 14,958.3 billion110 / 100 = 13,598.5 billion$
Use this data to make another observation: As long as inflation is positive, meaning prices increase on average from year to year, real GDP should be less than nominal GDP in any year after the base
year. The reason for this should be clear: The value of nominal GDP is “inflated” by inflation. Similarly, as long as inflation is positive, real GDP should be greater than nominal GDP in any
year before the base year.
Figure 6.9 shows the U.S. nominal and real GDP since 1960. Because 2005 is the base year, the nominal and real values are exactly the same in that year. However, over time, the rise in nominal GDP
looks much larger than the rise in real GDP (that is, the nominal GDP line rises more steeply than the real GDP line), because the presence of inflation, especially in the 1970s exaggerates the rise
in nominal GDP.
Let’s return to the question that we posed originally: How much did GDP increase in real terms? What was the real GDP growth rate from 1960 to 2010? To find the real growth rate, we apply the
formula for percentage change:
$2010 real GDP – 1960 real GDP1960 real GDP × 100 = % change13,598.5 – 2,859.52,859.5 × 100 = 376%2010 real GDP – 1960 real GDP1960 real GDP × 100 = % change13,598.5 – 2,859.52,859.5 ×
100 = 376%$
In other words, the U.S. economy has increased real production of goods and services by nearly a factor of four since 1960. Of course, that understates the material improvement since it fails to
capture improvements in the quality of products and the invention of new products.
There is a quicker way to answer this question approximately, using another math trick. Because:
$Nominal = Price × Quantity % change in Nominal = % change in Price + % change in Quantity OR % change in Quantity = % change in Nominal – % change in PriceNominal = Price × Quantity % change in
Nominal = % change in Price + % change in Quantity OR % change in Quantity = % change in Nominal – % change in Price$
Therefore, real GDP growth rate (% change in quantity) equals the growth rate in nominal GDP (% change in value) minus the inflation rate (% change in price).
Note that using this equation provides an approximation for small changes in the levels. For more accurate measures, one should use the first formula. | {"url":"https://openstax.org/books/principles-macroeconomics-2e/pages/6-2-adjusting-nominal-values-to-real-values","timestamp":"2024-11-14T12:23:36Z","content_type":"text/html","content_length":"445344","record_id":"<urn:uuid:80c9fdb3-9d82-43c9-9ca7-0f1c20c8f94d>","cc-path":"CC-MAIN-2024-46/segments/1730477028558.0/warc/CC-MAIN-20241114094851-20241114124851-00163.warc.gz"} |
Understanding (a+b) x (c+d)
The expression (a+b) x (c+d) represents the multiplication of two binomial expressions. This type of multiplication is a fundamental concept in algebra and is frequently encountered in various
mathematical contexts. Let's break down the process and explore its significance:
The Distributive Property
The core principle underlying the multiplication of binomials is the distributive property. This property states that multiplying a sum by a number is the same as multiplying each term of the sum by
that number individually and then adding the results.
In the case of (a+b) x (c+d), we can apply the distributive property twice:
1. First distribution: We multiply the first term of the first binomial (a) by each term of the second binomial (c and d):
2. Second distribution: We then multiply the second term of the first binomial (b) by each term of the second binomial:
Combining the Results
Finally, we add all the individual products obtained from the distribution steps:
(a+b) x (c+d) = ac + ad + bc + bd
Let's illustrate the process with an example:
Suppose a = 2, b = 3, c = 4, and d = 5.
(2+3) x (4+5) = (2 x 4) + (2 x 5) + (3 x 4) + (3 x 5)
= 8 + 10 + 12 + 15
= 45
The multiplication of binomials is essential in various mathematical areas, including:
• Algebraic manipulations: Simplifying and solving equations involving polynomials.
• Calculus: Finding derivatives and integrals of functions.
• Geometry: Calculating areas and volumes of geometric figures.
• Physics: Modeling physical phenomena and solving equations in physics.
Understanding the concept of multiplying binomials through the distributive property is crucial for proficiency in algebra and other related fields. The expanded form of (a+b) x (c+d) as ac + ad + bc
+ bd provides a structured approach to handling these expressions and facilitates further mathematical operations. | {"url":"https://jasonbradley.me/page/(a%252Bb)x(c%252Bd)","timestamp":"2024-11-02T23:56:53Z","content_type":"text/html","content_length":"59771","record_id":"<urn:uuid:556c545a-3b6b-49e6-b13d-be716ed8a661>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00763.warc.gz"} |
Unique 2D Barcodes with Orientation Detection
A friend of mine is studying to be a teacher and was recently explaining how she uses a service called
to administer multiple choice tests in classes. Every child gets their own square matrix style barcode on a card approximately 100mm x 100mm (that can vary though). Each side of the code is labelled
with the letters A, B, C, or D and when a question is asked, the kids hold up their card with the letter that corresponds to their answer at the top. The teacher then uses an app to scan the class,
recording each child's answer. It works because the set of cards provided are unique no matter what orientation they are viewed in. It's quite a clever idea, not only can the software say "that card
belongs to student X", it also knows what side is pointing up. The cards can be downloaded from the website in sets of up to 63 cards and look like this.
When I heard 63 I immediately thought that somehow the number of cards was related to 2^6 - 1, but that was just a hunch. Let's actually work out how many cards are possible.
Each code is made of a 5x5 grid of black and white squares. By looking through them, you soon notice that the centre square is always white and the 8 surrounding it are black. This is most likely
similar to the finder pattern in QR codes. The 4 corner squares are always black as well. This means that a bounding box for each code can be easily found. Both of these features make it easier for
the the image recognition algorithms to locate and identify each code. The 12 squares remaining are used to encode data. These are shown below as blue squares.
In the image above you can see that each side has three data squares, allowing each set of three to be coloured black or white 8 different ways. From this point on in the analysis instead of refering
to individual squares I'll just refer to the decimal encoding of bits on each side. For instance, code 47 above can be described as (2,4,5,5) starting at the top, going clockwise and assuming white
squares and 0 and black are 1. As there are 4 different orientations for the card you could also describe it as (4,5,5,2) , (5,5,2,4), or (5, 2, 4, 5).
For this system to work, you need to be able to determine what side of a card is facing up. If a card looks the same after rotating it either 90, 180, or 270 degrees, it's useless. What conditions
need to be met to ensure this? Lets take a card with the encoding (a, b,c d) and rotate it 90 degrees to obtain the card (d, a, b, c). If these two orientations are to be confused (a, b, c, d) is to
equal (d, a, c, d). In the image below you can see that only way that this can happen is if a=b=c=d. A symmetrical argument can be made for rotation of 270 degrees.
What if the card (a, b, c, d) is rotated 180 degrees to get the card (c, d, a, b)? This time things are a little different. The only way the two orientations can be confused is if a=c and b=d. This
can be seen in the image below.
So to make sure that the orientation of a card can be determined, all we need to do is make sure that the cards aren't rotationally symmetric by 180 degrees. This covers the cases of rotation by 90,
180, and 270 degrees. This means the the first two positions can contain any data we want. This gives rise to an n squared term. To make sure that the cards aren't rotationally symetrtic, the last
two positions can also contain anything we want, except for them being equal to the first two positions. This gives rise to an n squared minus one term. This gives rise to an equation of the number
of orientations possible.
Dividing this by 4 gives an equation for the number of unique cards possible. This is because each card covers four different orientations.
In this case n is equal to 8 (2^3) as there are 8 different ways to arrange the squares on each side. This means there are 1008 different possible cards. So why do they only use 63 of them? Your
guess is as good as mine. Maybe they have another condition to make sure that cards have a certain number of differing squares to minimise the chances of cards being misidentified. Maybe their
software only uses one byte to store each scanned card, 6 bits for the card and 2 for the orientation. Who knows.
I'll just point out that by expanding that equation above, we can easily see that the result will be an integer for any integer input. If n is even, the n squared term is divisible by 4, if n is odd,
(n-1) (n+1) is divisible by 4.
You may also ask why they didn't use QR codes? Uniqueness of codes is easy and orientation can be determined simply. A rule of thumb for scanning QR codes with a phone is that you can only scan codes
that have a width equal to one tenth of your distance from it. So lets say a teacher is 5 meters from a student, the code would have to be 50cm wide. That's too big. This is because QR codes have a
lot of detail. The Plickers codes are small enough to fit inside the large finder pattern on a QR code.
It'd be interesting to see if it's possible to allow more than 4 options for multiple choice questions. Maybe holding the card at 45 degrees like a diamond could allow this. Just a thought.
I've add a small python script that calculates all the possible barcodes just to check the maths. Everything checks out.
No comments: | {"url":"https://www.grant-trebbin.com/2016/05/unique-2d-barcodes-with-orientation.html","timestamp":"2024-11-03T23:33:29Z","content_type":"application/xhtml+xml","content_length":"66059","record_id":"<urn:uuid:d41b6c2f-0e36-4f72-b5a4-08611c0fd7ad>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00895.warc.gz"} |
VPython Glowscript
Go Straight to The Teaching Resources
Follow the link to go straight to my UbD unit "AP Physics: Modeling with Computer Simulations." It is published on Trinity University's Digital Commons. The unit has a general outline as well as
supplementary materials and everything you need to know to start getting set up with Glowscript. Lots of sample programs are there as well. It is published under Creative Commons license, so feel
free to use it and modify!
What is VPython Glowscript?
Vpython is a version of the Python programming language that makes it extremely easy to render and animate 3-D objects. Even if you have no coding experience, the syntax is intuitive and it's easy to
get the hang of it by looking through the sample programs and playing around a bit. If your goal is to teach physics through coding (rather than teaching coding through physics), this is a perfect
Glowscript the free site on which you can store, share, and run VPython programs. Think of it like google drive, but for storing and running your own code. Once you create a login, you can edit and
run your code right in your web browser without having to download anything to your computer. This is really handy because it is easy to help students get setup. Another HUGE benefit is the program
sharing. Once you've written some code, you can share it just by copying a link.
Take a look at my VPython Glowscript video tour below:
What Are The Benefits of Computer Modeling?
(1) Computer modeling emphasizes the creation and evaluation of physical models over the application of those models to arbitrary scenarios.
A ball is launched on level ground with a speed of 25 m/s at a 30 degree angle above the horizontal. Neglecting air drag, (a) how far does it go, (b) how much time does it spend in the air, and (c)
what maximum height does it achieve?
This type of projectile-motion problem shows up in almost every introductory physics class. To solve it, students must apply vector mathematics, constant velocity kinematics, and constant
acceleration kinematics. However, the problem itself is completely arbitrary. The numbers were randomly chosen and even if the question was given in terms of variables instead of numbers, there is
very little emphasis placed on actually understanding the physical models that must be applied to solve it.
For many students, problems like this become about choosing the right equations and doing algebra until they get a right answer. They're not about really understanding why a particular combination of
mathematical equations accurately models the system’s behavior. Giving problems like this can encourage students to seek out tricks/shortcuts that allow correct answers to be obtained, but completely
bypass the skills and understandings we hope for them to develop.
If the goal is to create a working computer simulation, emphasis transfers to understanding a physical models rather than getting a right answer. After all, a computer program is really just one way
of writing a mathematical model. Rather than students thinking “how do I solve this problem?” they should be thinking, “How do I model this type of system?” There's nothing wrong with solving
problems, but once a simulation for a system is developed, solving a specific scenario, becomes as simple as changing a couple values in their code or adding a few lines.
(2) Through time-iterative programming, even algebra-based physics allows students to model systems with non-constant force interactions.
Without the application of calculus, students cannot build complete mathematical models of systems where objects undergo non-constant forces. This is extremely problematic from a pedagogical point of
view because most intro physics classes include analysis of some non-constant force systems (like mass-spring oscillators and pendula). Introducing equations that model these systems is typically
problematic for teachers because calculus is required to derive them from more basic equations.
Through computer simulations, we can take a system like this and “cheat” by assuming the force on objects IS constant, but over very small time intervals. Then we update the force value appropriately
over each small interval. Essentially, time-iterative programming allows us to turn a non-constant force problem into many neighboring constant force problems. The same idea can be applied to other
systems with non-constant force, like projectiles with air drag.
(3) Physics modeling through computer programming provides students with an opportunity to develop basic coding skills.
I tried to keep this blog post short because I have so much related material published in my unit on the Trinity University Digital Commons site. Make sure to check it out if you're interested. | {"url":"https://www.sphericalcowblog.com/post/vpython-glowscript","timestamp":"2024-11-13T21:55:36Z","content_type":"text/html","content_length":"932640","record_id":"<urn:uuid:a9b816cb-30e3-4c6b-b8d3-cc09832b225c>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00753.warc.gz"} |
Page 2 - What are the real chances of an asteroid hitting Earth?
It just dawned upon me, there may be actuarial tables/estimates for the very question that started this thread. Just something to ponder.
In the meantime, here is a fun read:
That's interesting.
is an update (2020) for Fig. 4.
I'm puzzled, however, with the dip in the blue line for Fig. 1. Why would a larger 4km object have a lower percentage of being found than a 2km asteroid? What am I missing?
Sorry Helio, that takes me to discovery statistics.
Where is the Fig.?
The odds of asteroid collision have been worked out by serious astronomers long ago.
The odds of the Earth being struck by an asteroid larger than the one that did in the dinosaurs is somewhat distressing. The odds are 1.000 plus or minus o.oo1. That a 100% chance.
But that isn't for any one year, it's over the lifetime of the planet.
It's happened at least five times over the past Billion years, or maybe just the past 3/4 of a billion years. On average it's once every hundred million years. It's been 65 Million years since the
last one, so we are about due. Sometime in the next 35 million years that is.
For larger objects the odds go down dramatically. For smaller objects they go up quite dramatically as well.
Down at the bottom of the scale, the Earth gets hit several thousand times a day up to several million times a day by dust sized bits of space debris.
Rocks large enough to destroy a city strike somewhere between once a decade and three times a decade. The last one was over a city in Russia. Before that, several years before, there was one in
Africa. There was also one reported over the open ocean.
Most strikes will be over the ocean because there is three times as much surface that is ocean as there is that is land.
So strikes by space rocks is a real thing and it happens several times a decade and always has.
The Tunguska Siberia strike in the early 1900's is one famous example. A Tunguska sized event we currently believe happens once every hundred years or so. Like the Chelyabinsk event more recently, it
was an air burst, meaning that the actual body exploded high in the air and there was no crater to find on the ground. Still, it exploded like a hydrogen bomb blast.
So the real answer to 'Will it happen?' is Yes. It will happen. But when is very different question. But before 35 million years from now, there will be a planet wide disaster caused by a falling
space rock. Before that, there will be hundreds of city sized disasters and a few continent sized disasters all caused by falling space debris.
Its all really just a consequence of us living in a somewhat littered solar system.
"In 2006 Toutatis came closer than 2 AU to Jupiter; its orbit lies inside of Jupiter's.
In the 2100s,
it will approach Jupiter many times at a similar distance
" Wiki
And they still say Toutatis' orbit can be calculated hundreds of years into the future? And with the Earth Jupiter resonance as well?
Remember Shoemaker–Levy 9?
Do you think the dark matter component of Shoemaker -Levy is still orbiting or has slowed down and dissipated since its normal matter collided with Jupiter?
"Do you think the dark matter component of Shoemaker -Levy is still orbiting or has slowed down and dissipated since its normal matter collided with Jupiter? "
Please define the dark matter component of Shoemaker-Levy and I will reply to your question.
Y A Bob: "The Tunguska Siberia strike in the early 1900's is one famous example. A Tunguska sized event we currently believe happens once every hundred years or so. Like the Chelyabinsk event more
recently, it was an air burst, meaning that the actual body exploded high in the air and there was no crater to find on the ground. Still, it exploded like a hydrogen bomb blast. " My emphasis.
Bob, are you passing off as inconsequential the "H bomb blast", which flattened a vast area of trees, as having very little effect on New York skyscrapers?
I am just pointing out one relatively recent relatively common event, and the effect it would have over a city as opposed to almost totally uninhabited deserted Siberia (shame about the trees).
Sorry Helio, that takes me to discovery statistics.
Where is the Fig.?
I had tried to find something quickly but couldn't, but I had planned to look further and forgot.
There should be some nice formula out there that gives a better idea of the frequency per size, but I haven't seen any yet.
There is this
Wiki page on impact frequency
but it doesn't do that great a job when extrapolating it to the smaller impacts.
The inverse square of size to yield frequency seems to be the rough rule of thumb.
I found that there are about 500 meteorites impacting per year. These need to be > 1 meter in diameter to do so. I assume these are mostly the stony ones, but they are only about 1/6th of all
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Shoemaker -Levy had a gravity well which I assume had a higher concentration of dark matter than the surrounding space. This matter has mass so it also has momentum. How long would it take to
disperse after the baryonic matter stopped orbiting when it collided with Jupiter?
I was not asking for your assumptions.
May I please politely request some factual data if you would like my reasoned response?
I Am sure a lot of people would be happy if someone could better define dark matter.
Then I am torn between 'nothing about something' and ''something about nothing'.
There are all sorts of calculations saying nothing (asteroid, bolide, comet, what you will) is going to hit us in the next 20 minutes (Ooops sorry, I meant years) but how reliable are they?
How are we to see objects with an albedo less than soot coming 'out of the Sun'?
NEAs, especially those sharing our plane, and especially coming around in short order are affected on each return by our gravity. Large error margins start creeping into those calculations.
Are we doing enough to protect our planet? Is the Late Heavy Bombardment repeatable?
What are your views?
What we need to remember is that our Sun is in a 250 million year or so galactic orbit, of which some parts of the milky way are more crowded than others. It's coming no doubt about it. So when it
happens we will need to find a way off for at least until the oceans fall in their holes again otherwise we vanish.
The solar system’s orbital angle to the galactic plane is somewhere around 30 degrees +5 -10. The solar system crosses the galactic plane approximately every 30 million years (as measured on Earth).
The physics for this orbit uses the real properties of space-time. The property of inertia does change producing the galactic velocity curve which are not anomalous if you use the correct
gravitational equations.
This effect occurs because the reduced mass density of space affects the energy exchange of the inertial field. Your speed increases in empty space. The velocity of our solar system moving up and out
of the dense galactic plane increases so that the half orbit which should have taken 120 million years (within the galactic plane) only takes about 30 million years in the empty space above that
Since the major extinctions appear to coincide with these galactic plane crossings the main threat appears to be the higher velocity interstellar impactors.
While the impact of a 2 kilometer diameter local asteroid would produce a really bad day. The ecological and other weather effects might last as little as a decade (to a few hundred years). The
Deccan and Siberian flood basalts are the markers for interstellar impacts.
The best dates obtainable for the Siberian impact indicates that all of the extinctions at the Permian- Triassic boundary occurred within about 30,000 years of the impact. Not bad for an extinction
mechanism which does not exist within the scientific literature (Interstellar Impacts).
The solar system’s orbital angle to the galactic plane is somewhere around 30 degrees +5 -10. The solar system crosses the galactic plane approximately every 30 million years (as measured on
I didn't look hard and I'm not sure
this link's info
is correct, but if it is, our orbital plane with the galaxy is only inclined by less than 1 degree.
There is an angular momentum vector used that is near 60 degrees, which can make things confusing.
As for debris from hitting spiral arms or whatever, since the free-fall time for something out of Oort Cloud ranges from 178,000 years (inner region) to 2.8 million years (outer region) -- these are
my calculations so they're likely off a tad -- we should have plenty of time to get out there where we can watch them and guide them to a happier ending.
I found this interesting article on this topic:
Widespread\u00a0extinctions of land-dwelling animals – which include amphibians, reptiles, mammals and birds – follow a cycle of about 27 million years.
I found this interesting article on this topic:
Widespread\u00a0extinctions of land-dwelling animals – which include amphibians, reptiles, mammals and birds – follow a cycle of about 27 million years.
You are quite correct in pointing this out, but it is by no means a new idea.
I think I first came across it a few years ago. But you can't repeat an important observation too often!
Thanks for that link.
According to Google. Didymos weighs 52.7 billion kg and is travelling at 48,000 km / h
I don't know if anyone has calculated the effect of hitting it with that object?
The fly-by in Sept. 2022 brings no risk of impact. Looks like it won’t get closer than ~ 26x the Earth-Moon distance.
More here.
NASA Horizons JPL program
, the closest approach is on Oct. 4th at 0.07124 AU (matching my estimate above).
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Thanks for that link.
According to Google. Didymos weighs 52.7 billion kg and is travelling at 48,000 km / h
I don't know if anyone has calculated the effect of hitting it with that object?
E= MV*2 convert km/h to M/Sec, multiply kg by 1000 to convert to grams, there you are.
57,200,000,000 kg * 48000 km/sec * 1000 m/km /3600 s/hr gives kg-m/sec^2
That's roughly 8749.5 Megatons, roughly the same as half the US arsenal or a quarter of Russia's total bombs. It will create quite an impression.
Bob, are we talking momentum or force here?
kg-m/sec^2 = mass x acceleration ? = force ?
Am I missing something?
E= MV*2 convert km/h to M/Sec, multiply kg by 1000 to convert to grams, there you are.
57,200,000,000 kg * 48000 km/sec * 1000 m/km /3600 s/hr gives kg-m/sec^2
That's roughly 8749.5 Megatons, roughly the same as half the US arsenal or a quarter of Russia's total bombs. It will create quite an impression.
I get 1215 megatons. KE = 1/2 mv^2. Still huge, of course.
Bob, are we talking momentum or force here?
kg-m/sec^2 = mass x acceleration ? = force ?
Am I missing something?
Impacts use KE.
Bob, are we talking momentum or force here?
kg-m/sec^2 = mass x acceleration ? = force ?
Am I missing something?
From Wikipedia, " linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. "
Force is "By substituting the definition of
, the algebraic version of
Newton's second law
is derived:
F → = m a → . {\displaystyle {\vec {F}}=m{\vec {a}}.}
Newton never explicitly stated the formula in the reduced form above." Again from Wikipedia.
But here we are discussing Energy, which is Force through a distance, or MV^2. The limit possible is of course when the velocity is the speed of light, or E = MC^2
So no, I was not really discussing force, but rather energy. The velocity is squared to give energy as output. Momentum would be if the velocity were not squared.
Physics is still the same as it was when we were in School, at least for things like this. I have totally ignored relativistic effects naturally. the speeds involved are minor for that.
I used an on-line unit converter to convert from Newton-Meters to Megatons. I haven't checked that, and don't offhand know the conversion numbers. But it's still quite a lot of energy. I haven't
fact checked the numbers, just threw it into a spreadsheet then ran the result through the converter to get something to compare to bomb numbers.
Oh, and I also assumed the mass of Didymus reported was correct. I don't know if that also incuded Didymoon, which while much smaller would still make quite an impression.
BTW, Catastrophe, I do continue to enjoy your occasional comments on these forums. It's specifically appropriate on this particular question for your Nominal Nomus. Please continue posting. You
are one of the rare few who both enlighten and entertain.
Bob, Thank you for your kind comments. They are much appreciated.
With regard to the Didymus stats, as I recall they came directly from Googling "Didymus asteroid mass / speed" as two separate searches. I just checked the mass and it is close to the figure stated.
I always try to use best sources when quoting stats.
Apart from my chemical orientated B.Sc. or (BS), my scientific interest covers the "history" of the Universe down to local level viz. from (before) BB through cosmology/astronomy/planetary science/
geology. My Physics was O/A/S levels GCEs but very rusty. I was wondering whether the figure was available for the lateral component of the impacting body and how this affected the deviation from
current trajectory. I must admit that I ignored Didymoon. It seemed to me (in my ignorance) that it might be difficult to affect Didymus, considering its momentum.
Wiki gives
Inertia is the resistance of any physical object to any change in its velocity. This includes changes to the object's speed, or direction of motion.
Having only just found that quote, I shall go back and read further. | {"url":"https://forums.space.com/threads/what-are-the-real-chances-of-an-asteroid-hitting-earth.33040/page-2","timestamp":"2024-11-12T20:17:50Z","content_type":"text/html","content_length":"360563","record_id":"<urn:uuid:13a7730f-0040-4a70-a662-14924d6e53fe>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00660.warc.gz"} |
Need help: what are the ways of reusing curtain panel or generic model instances?
Hey there,
On daily-basis work, we insert several curtain walls, each containing the generic-model-based curtain panels, each being counted as one on a final table sheet. As you can see through the arrows, this
are the assembly direction of the walls, therefore the last two panels (circled) are not on their full lenght.
The green circled one, actually, are at half of their lenght, leaving half of the panel to be used on other walls, such as the orange-circled one.
What would be the best ways to make Revit recognize that, in real life, one panel can be cutted and used on other curtain walls?
There would be some rules to it, also, but I’m stuck on the first step of it, yet.
@v.monteiro ,
what do you want exactly in this corner ?
a certan angle? or both grid patterns are the same?
This is essentially the pipe/spool optimization problem. If you have a given number of known fractional lengths, what’s the minimum number of full lengths needed to produce your required cuts.
Keep in mind, as an optimization problem, there are many different ways to go about finding a solution depending on how complex/accurate you want to be. Here’s an example of a similar topic:
The simplest solution (and still a pretty decent one) would be to optimize from longest to shortest. Start with the longest piece you have, and then look for the next largest piece that still fits
within the remainder of a “full length” section. Continue this for a single section until there are no pieces left that fit within the constraints. That becomes one section of cuts and the process
begins again with a new section. | {"url":"https://forum.dynamobim.com/t/need-help-what-are-the-ways-of-reusing-curtain-panel-or-generic-model-instances/93296","timestamp":"2024-11-02T21:16:17Z","content_type":"text/html","content_length":"33149","record_id":"<urn:uuid:b2396c3c-21a0-46af-9c77-8b90fc801506>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00826.warc.gz"} |
Young’s Modulus of Alumina - ceramic atijolart
Young’s Modulus of Alumina
Young’s modulus is an essential mechanical property for engineers in creating materials that resist stress and damage, showing how much force is necessary to break a material while also serving as an
indicator of its strength.
This research investigates the effect of porosity on Young’s modulus for sintered alumina. Utilizing an innovative synthesis technique, this study doubled Young’s modulus.
It is a measure of the material’s strength
Young’s modulus measures the elastic force that materials exert when subjected to specific strain. It is an isotropic material’s property and applies only under uniaxial stress (tensile or
compressive). We can use Young’s modulus to predict how much a sample will extend under tension or compress under compression, and also determine its elasticity structure; this information can be
invaluable in predicting deflection for statically determinate beams, as well as creating structures which withstand large forces without cracking under compression or tension.
Mechanical properties of g-alumina were evaluated through Hertzian indentation tests and finite element modelling (FEM). Stress-strain curves were plotted, with Young’s modulus E being determined by
slope of linear portion of curve. Yield stress (Y) and strain-hardening parameter n were estimated by correlating FEM results with experimental curves.
Alumina’s primary advantage lies in its plasticity, which enables it to deform without breaking, making it ideal for applications requiring high dimensional stability. Unfortunately, as temperature
rises the Young’s modulus decreases with increasing temperatures – an effect which may be partially explained by thermal expansion and particle dislocations.
Poisson’s ratio of alumina can also help measure its toughness; this property measures the material’s ability to absorb energy under cyclic loading and is an integral part of material design and
manufacture for applications involving large tensile or compressive loads, as well as fatigue testing of materials.
Poisson’s ratio can be calculated from its Young’s modulus and other materials’ Poisson’s ratios; however, its morphology is complex, and requires extensive knowledge of porosity and grain size for
accurate calculations. Therefore, we employed a novel synthetic technique to synthesize g-alumina granules with defined pore sizes and shapes and studied their morphologies by scanning electron
microscopy and field-emission scanning electron microscopy; their relationships between Young’s modulus modulus Poisson’s ratio, and pore size were analysed using Finite Element Method (FEM); overall
this doubled Young’s modulus thus increasing overall strength overall.
It is a measure of its stiffness
Young’s modulus of alumina is an indicator of stiffness that measures material resistance to deformation. It is measured by applying force (load) to a small sample of material and its displacement
(displacement/force ratio). Young’s modulus can differ depending on temperature, alloy composition, crystal structure and manufacturing processes as well as its geometry; its behaviour also being
affected by different orientation of granules within it and by porosity within the material itself.
Contrary to most metals and ceramics, alumina is anisotropic; meaning its mechanical properties vary depending on the direction in which a force is applied. You can make it more isotropic by treating
it with certain impurities or mechanically working it; however, doing so will result in decreased strength and toughness.
Nanoindentation can be used to accurately determine Young’s modulus of alumina as well as its stiffness, hardness and elastic resilience. The force versus deflection curve of samples makes the method
more accurate than standard tensile tests while smaller samples reduce risk and produce regular distribution curves.
Young’s Modulus of alumina depends on its production process and has been observed to vary non-linearly with load. This phenomenon may be explained by changes in interatomic bonding caused by
temperature variations; fitting and theory have enabled prediction of this phenomena.
Alumina’s high Young’s Modulus makes it an extremely stiff material that resists deformation, yet its brittle nature precludes its use for applications that require plasticity such as structural
components and cutting tools. Furthermore, without a yield point it will break under compressive or tensile loads almost instantly rather than gradually over time.
Young’s modulus of alumina changes asymmetrically under load, with its shape determined by its granule geometry. This occurs because individual alumina granules may be at various angles to each other
and to the indenter; this results in non-monotonic dependencies between its modulus of elasticity and hardness of material.
It is a measure of its hardness
Mechanical properties of alumina are integral to many applications, from abrasion resistance and wear resistance to antimicrobial protection. Accurate measurement of these properties is critical,
especially when exposed to elevated temperatures that compromise structural integrity of materials such as Young’s modulus testing methods – nondestructive testing techniques provide an effective
means of accomplishing this objective.
Impulse excitation indentation tests are an ideal method for estimating Young’s modulus of alumina. These tests create a stress-strain curve which is used to calculate elastic modulus of material;
its value corresponds with yield strength. Furthermore, slope of curve also helps determine strain hardening characteristics; results from indentation tests may be compared with predictions made
using finite element modeling (FEM).
In this study, impulse excitation and FEA were employed to measure the Young’s modulus of g-alumina with various diameters using Impulse Excitation Method and to perform Young’s Modulus Measurements
of different diameters of Alumina material. Alumina’s Young’s modulus can vary widely; thus it is essential that one fully understands its specific properties prior to selecting or using any material
in an application.
Alumina is one of the most widely-used technical oxide ceramics, boasting superior levels of strength, hardness, corrosion and wear resistance as well as low thermal expansion rate, good
conductivity, chemical inertness and low thermal expansion rate for use in harsh environments. Furthermore, due to its high Young’s modulus and low density characteristics it makes an excellent
material choice for electrical applications due to its excellent Young’s modulus performance and low density density properties.
This research applies a novel synthesis method to create granular g-alumina with enhanced mechanical properties. Produced via a new sintering process, the resultant material features higher Young’s
modulus than other samples with similar diameter. Furthermore, hardness measurements show increased durability.
Monitors have studied the dynamic Young’s modulus of partially sintered alumina at temperatures ranging from 1200-1600 degC. At room temperature, its Young’s modulus decreases linearly as temperature
rises; however, at 1600 degC its Young’s modulus begins to sharply increase as densification takes place – in accordance with its master curve for alumina.
It is a measure of its density
Alumina boasts the highest Young’s modulus rating among ceramic materials, making it ideal for applications that demand strength. Engineers rely on this value to assess how much stress a material can
withstand before deforming permanently or breaking. The higher its Young’s modulus rating is, the stiffer its characteristics are.
Alumina’s microstructure and chemistry play an integral part in its properties. Alumina’s atomic configuration is defined by pair radial distribution functions and bond angle distributions, while
density estimates can be made using simplex statistics.
Simulations of amorphous alumina reveal its tetrahedral structure with an interstitial space, linking together its atoms through oxygen bonds and creating perfect tetrahedra (PTEs) with an average
density of 2.84 g cm-3. Furthermore, these PTEs may link with one another through common oxygen to form large poly-PTEs with a density of 3.81 g cm-3.
Young’s modulus for alumina depends on its purity; higher purity means greater Young’s modulus due to reduced impurities and lower self-diffusion coefficient. Unfortunately, producing pure alumina
with the desired purity can be challenging due to low melting and boiling points; one solution could be adding carbon to increase purity further and significantly raise Young’s modulus.
Young’s modulus of alumina also depends on its temperature. As it heats, its elastic properties disintegrate; upon reaching its original firing temperature it begins to sinter (densify), increasing
Young’s modulus drastically.
Three and four point bending tests provide engineers with a means of characterizing the elastic properties of alumina, enabling them to measure its bending capacity under compressive and tensile
stresses. As shown by these results, 12.6 GPa was determined as its intrinsic elastic modulus value – consistent with theoretical predictions and useful for predicting its flexural properties in
various applications – an important step toward creating nonferrous alloys containing this element. | {"url":"https://ceramicatijolart.com/lt/youngs-modulus-of-alumina/","timestamp":"2024-11-12T19:50:07Z","content_type":"text/html","content_length":"133246","record_id":"<urn:uuid:e9475aa7-758b-43a3-8b3a-1c208e00a0d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028279.73/warc/CC-MAIN-20241112180608-20241112210608-00519.warc.gz"} |
Cite as
Fedor V. Fomin, Petr A. Golovach, Ignasi Sau, Giannos Stamoulis, and Dimitrios M. Thilikos. Compound Logics for Modification Problems. In 50th International Colloquium on Automata, Languages, and
Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 61:1-61:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
Copy BibTex To Clipboard
author = {Fomin, Fedor V. and Golovach, Petr A. and Sau, Ignasi and Stamoulis, Giannos and Thilikos, Dimitrios M.},
title = {{Compound Logics for Modification Problems}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {61:1--61:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-278-5},
ISSN = {1868-8969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.61},
URN = {urn:nbn:de:0030-drops-181137},
doi = {10.4230/LIPIcs.ICALP.2023.61},
annote = {Keywords: Algorithmic meta-theorems, Graph modification problems, Model-checking, Graph minors, First-order logic, Monadic second-order logic, Flat Wall theorem, Irrelevant vertex technique} | {"url":"https://drops.dagstuhl.de/search?term=Stamoulis%2C%20Giannos","timestamp":"2024-11-05T06:50:37Z","content_type":"text/html","content_length":"157017","record_id":"<urn:uuid:03480feb-ab46-4c40-8e80-88e1e0256c5b>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00681.warc.gz"} |
How to teach moles at post-16
Use these classroom ideas to help your students make sense of moles and avoid misconceptions
Whether making plastics, medicines or preparing materials for a chemical reaction, we have to measure the amounts we need. Atoms react in fixed proportions represented by the stoichiometry – the
ratio of moles in a balanced equation for the reactants and products – of the reaction. Before conducting a reaction, we need to measure the number of these reactants. But, counting atoms is
impossible, so we use the concept of the mole, which allows us to count atoms by measuring their mass.
One atom of helium has a relative atomic mass number of four (two protons and two neutrons), so it is four times as heavy as one atom of hydrogen (one proton). It follows that four grams of helium
and one gram of hydrogen have the same number of atoms. Similarly, there is the same number of atoms in 12 grams of carbon. In fact, the relative atomic mass of all elements will have the same number
of atoms. Experiments show that the number is 6.02214076 × 10^23, known as Avogadro’s constant. We call one lot of this number one mole.
What students need to know
The mole is the SI unit for the amount of substance. One mole contains exactly 6.02214076 x 10^23 elementary entities. For students to fully understand the mole we need to expand beyond this core
definition using class discussion to understand how to use a few specific definitions. Definitions to discuss and explore with students include:
1. A mole is a number consisting of exactly 6.02 x 10^23 elementary entities.
2. The mole (symbol: mol) is the unit for the amount of substance.
3. The relationship between the number of moles, molar mass and mass is represented by this equation: number of moles (mol) = (mass (g))/(molar mass (g mol^-1))
4. Historically, a mole was described as the number of atoms in exactly 12 g of carbon-12. However, on 20 May 2019 the General Conference on Weights and Measures set a mole as exactly 6.02214076 ×
10^23, as recommended by IUPAC, replacing the classic definition.
If you are new to teaching this topic at post-16, or need a refresher, the RSC on-demand CPD course on quantitative chemistry (rsc.li/3yyUvz8) has sections devoted to the mole.
Common misconceptions
The mole is a difficult idea for students to grasp as a number because Avogadro’s constant is so large that they can’t imagine it meaningfully. Common student misconceptions include:
• mole refers only to molecules.
• mole is a certain mass (rather than a number).
• different substances of the same mass have the same number of particles.
So, at all stages, reaffirm that the mole is a number. Questioning is the best way to probe students’ understanding and discover misconceptions early before they’re embedded. BEST resources from STEM
learning have a great selection of student worksheets on the mole that use diagnostic questions and tasks to identify and resolve misconceptions for pre-16 students. You can use these to provide a
solid base to build on and unpick any embedded misconceptions.
It’s a tricky topic, but wonderful to see it click with students
Use examples and comparisons to ensure students understand, develop, question and test the concept of a mole. For example, directly address the question of the difference between one mole of hydrogen
atoms and molecules. You can explore misconceptions in more depth and a possible teaching route by looking at this CPD article on moles and titrations.
Use examples and comparisons to ensure students understand, develop, question and test the concept of a mole. For example, directly address the question of the difference between one mole of hydrogen
atoms and molecules. You can explore misconceptions in more depth and a possible teaching route by looking at this CPD article on moles and titrations: rsc.li/3ZHETVQ.
Ideas for your classroom
Visualising moles can help students link ideas. Providing physical examples helps them link the idea that all have the same number of elementary entities; this can be counted by measuring mass, and
the coefficients of a reaction link to the proportions of moles. To help my students, I have a box of moles that contains the following in sealed containers:
• 12 g of carbon
• Two lots of 32.1 g of sulfur
• 55.8 g of iron
• 63.5 g of copper
• 24.3 g of magnesium
• 18 g of water
• 100.1 g of calcium carbonate
• 65.4 g of zinc
• 58.5 g of sodium chloride
• 87.9 g of iron sulfide
I can use my box of moles to physically represent one mole, as well as how mass and number are connected. I try to keep these in familiar forms where possible – iron as iron nails and magnesium as
magnesium ribbon.
Analogies can be helpful when students are struggling with the concept of the mole. A common analogy is coins at a bank. A bank clerk doesn’t count pennies to calculate their total value. Instead,
they measure the mass of the coins to count the number present. In the same way as with moles, the clerk needs to know the mass of each coin (molar mass, M) and the total mass of coins (mass, m)
before using the same relationship we use to count atoms. Different denominations (1p, 2p, 5p, etc) have a different mass for each coin, so the relationship between mass and number varies depending
on the mass of each coin (molar mass, M).
You could also use the analogy of bags of rice or cups. Or consider a resource that summarises the mole and Avogadro with the idea that one mole of moles has a mass only a little less than the moon.
You could also use the analogy of bags of rice or cups. Or consider a resource that summarises the mole and Avogadro with the idea that one mole of moles has a mass only a little less than the moon (
Download this
Moles infographic and resources, for age range 16–18
Get your students to apply their mathematical skills to three different problems involving moles.
Download the poster, fact sheet and worksheet from the Education in Chemistry website: rsc.li/3GvyR32
Checking for understanding
Questioning is important to identify misconceptions before they become embedded. Use a combination of flashcards or a fact sheet to embed the meaning of the mole alongside other definitions. Then,
follow up with practice questions and calculations.
Once students grasp the mole, regular retrieval and interleaving is essential to ensure they hold on to this understanding. For example, develop their understanding by asking them to determine the
relative atomic mass of magnesium experimentally, and you can follow up with questioning to check their understanding.
Take-home points
• It is impossible to count individual atoms, so we count them by measuring their mass.
• We use Avogadro’s constant to represent one mole.
• The molar mass has the same numerical value as the relative atomic mass, but in g mol^-1.
• Students need to develop a conceptual understanding of the mole and how it relates to number, mass and proportions. It’s a tricky topic, but wonderful to see it click with students.
William Barron de Burgh is a chemistry teacher at Dame Elizabeth Cadbury School | {"url":"https://edu.rsc.org/cpd/how-to-teach-moles-at-post-16/4017128.article","timestamp":"2024-11-06T08:19:24Z","content_type":"text/html","content_length":"239775","record_id":"<urn:uuid:da6f6830-2f32-4643-977a-abd0d7027358>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00281.warc.gz"} |
Derivative Rules - MathCracker.com
Instructions: Use this derivative calculator to find the derivative of any function you provide, using the most common derivative rules, showing all the steps. Please type the function you want to
apply derivative rules in the form box below.
About Derivative Rules
This calculator will allow you compute the derivative of a function you provide by applying the required basic differentiation rules, showing all the steps of the process, and noting where each rule
is applied.
You just need to provide a valid function that is differentiable (which means that it has a derivative). For example, a valid function could be f(x) = 1/3*x*sin(x), just to mention one example.
Then, when you have already typed your function, you just click on "Calculate" to get all the steps of the differentiation shown.
The simplicity of the rules of derivatives makes the process of differentiation one that is recognized as 'easy', a judgment that is perhaps an overstatement.
Basic derivative rules
There are four basic derivative rules for you to learn
• Linearity Rule: For functions \(f(x)\) and \(g(x)\), and a constant \(a\), then the derivative is a linear operation: \((af(x)+g(x))' = af'(x)+g'(x)\)
• Product Rule For functions \(f(x)\) and \(g(x)\), the derivative of the product is \((f(x)g(x))' = f'(x)g(x) + f(x)g'(x)\)
• Quotient Rule: For functions \(f(x)\) and \(g(x)\), the derivative of the quotient is is \(\left(\frac{f(x)}{g(x)}\right)' = \left(\frac{f'(x)g(x) - f(x)g'(x)}{g^2(x)}\right)\)
• Chain Rule For functions \(f(x)\) and \(g(x)\), the derivative of the composite function is \((f(g(x)))' = f'(g(x))g'(x)\)
This rules will work like a charm and will help you find the derivative of any basic function.
How to use the derivative rules?
• Step 1: Identify the function f(x) you want to differentiate, simplify if needed
• Step 2: Try to break the function into smaller derivative chunks, using linearity
• Step 3: Depending on the structure of the function f(x), use any of the available rules (product, quotient and chain rule), and be aware that you may need to apply many of the rules
Usually you will end up a combination of several differentiation rules, until you reach of point where you find an elementary function, of which you already know how to differentiate.
Can I solve all derivatives
Saying that using differentiation rules can lead you to solving ALL derivatives can be an overstatement. You will be able to solve MOST derivatives, and certainly all basic ones, but there are
functions that have a less intuitive behavior that could be defined, though they are no typically dealt with in basic Calculus courses.
For what basic functions are concerned, most of them will be differentiated without a problem.
A product rule derivative, quotient rule derivative or chain rule derivative are unlikely to be in isolation, and will likely come in a sequence of several rules that need to be used together.
Example: Derivative Rules
Using basic derivative rules, compute the following derivative: \(\frac{d}{dx}\left( x^2 \cos(x^2) \right)\)
Solution:Let us consider the following given function for which the derivative needs to be computed \(\displaystyle f(x)=x^2\cos\left(x^2\right)\)
The function does not need simplification, so we can go straight into computing its derivative:
\( \displaystyle \frac{d}{dx}\left(x^2\cos\left(x^2\right)\right)\)
Now, using the Product Rule: \(\frac{d}{dx}\left( x^2\cos\left(x^2\right) \right) = \frac{d}{dx}\left(x^2\right) \cdot \cos\left(x^2\right)+x^2 \cdot \frac{d}{dx}\left(\cos\left(x^2\right)\right)\)
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x^2\right) \cdot \cos\left(x^2\right)+x^2 \cdot \frac{d}{dx}\left(\cos\left(x^2\right)\right)\)
We have to use the Chain Rule: \(\frac{d}{dx}\left( \cos\left(x^2\right) \right) = \frac{d}{dx}\left(x^2\right)\cdot \left(-\sin\left(x^2\right)\right)\) and using the Power Rule for polynomial
terms: \(\frac{d}{dx}\left( x^2 \right) = 2x\)
\( \displaystyle = \,\,\)
\(\displaystyle \left(2x\right) \cos\left(x^2\right)+x^2 \cdot 2x\cdot \left(-\sin\left(x^2\right)\right)\)
\( \displaystyle = \,\,\)
\(\displaystyle x^2\cdot 2x\cdot \left(-\sin\left(x^2\right)\right)+2x\cos\left(x^2\right)\)
By reordering some of the numerical values, and then grouping the terms with \(x\) in the term \(x^2\cdot 2x\)
\( \displaystyle = \,\,\)
\(\displaystyle 2x^3\cdot \left(-\sin\left(x^2\right)\right)+2x\cos\left(x^2\right)\)
But we get \((2x^3) \cdot (-\sin\left(x^2\right)) = -2x^3\sin\left(x^2\right) = -2x^3\sin\left(x^2\right)\), due to the fact that we can use the distributive property on each term of the expression
on the left, with respect to the terms on the right
\( \displaystyle = \,\,\)
\(\displaystyle -2x^3\sin\left(x^2\right)+2x\cos\left(x^2\right)\)
And finally, grouping the terms together
\( \displaystyle = \,\,\)
\(\displaystyle -2\left(x^2\sin\left(x^2\right)-\cos\left(x^2\right)\right)x\)
The corresponding graph of the function and its derivative is shown below:
Example: More derivative rules
Compute the following derivative: \(\frac{d}{dx}\left( x \cos(x^2+1) \right)\) using basic derivative rules.
Solution:Now, the task at hand is to differentiate the function \(\displaystyle f(x)=x\cos\left(x^2+1\right)\)
\( \displaystyle \frac{d}{dx}\left(x\cos\left(x^2+1\right)\right)\)
In this case, we have to use the Product Rule: \(\frac{d}{dx}\left( x\cos\left(x^2+1\right) \right) = \frac{d}{dx}\left(x\right) \cdot \cos\left(x^2+1\right)+x \cdot \frac{d}{dx}\left(\cos\left(x^2+1
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x\right) \cdot \cos\left(x^2+1\right)+x \cdot \frac{d}{dx}\left(\cos\left(x^2+1\right)\right)\)
The Chain Rule for this composition: \(\frac{d}{dx}\left( \cos\left(x^2+1\right) \right) = \frac{d}{dx}\left(x^2+1\right)\cdot \left(-\sin\left(x^2+1\right)\right)\)
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x\right) \cdot \cos\left(x^2+1\right)+x \cdot \frac{d}{dx}\left(x^2+1\right)\cdot \left(-\sin\left(x^2+1\right)\right)\)
By linearity, we know \(\frac{d}{dx}\left( x^2+1 \right) = \frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\), so plugging that in:
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x\right) \cdot \cos\left(x^2+1\right)+x \left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)\cdot \left(-\sin\left(x^2+1\right)\right)\)
The derivative of a constant is 0, so then:
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x\right) \cdot \cos\left(x^2+1\right)+x \left(\frac{d}{dx}\left(x^2\right)\right)\cdot \left(-\sin\left(x^2+1\right)\right)\)
Using the Power Rule for polynomial terms: \(\frac{d}{dx}\left( x^2 \right) = 2x\)
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x\right) \cdot \cos\left(x^2+1\right)+x \left(2x\right)\cdot \left(-\sin\left(x^2+1\right)\right)\)
\( \displaystyle = \,\,\)
\(\displaystyle x\cdot 2x\cdot \left(-\sin\left(x^2+1\right)\right)+\cos\left(x^2+1\right)\)
Putting together the numerical values and grouping the terms with \(x\) in the term \(x\cdot 2x\)
\( \displaystyle = \,\,\)
\(\displaystyle 2x^2\cdot \left(-\sin\left(x^2+1\right)\right)+\cos\left(x^2+1\right)\)
Observe that \((2x^2) \cdot (-\sin\left(x^2+1\right)) = -2x^2\sin\left(x^2+1\right) = -2x^2\sin\left(x^2+1\right)\), due to the fact that we can use the distributive property on each term of the
expression on the left, with respect to the terms on the right
\( \displaystyle = \,\,\)
\(\displaystyle -2x^2\sin\left(x^2+1\right)+\cos\left(x^2+1\right)\)
\( \displaystyle = \,\,\)
\(\displaystyle -2x^2\cos\left(x^2\right)\sin\left(1\right)-2x^2\cos\left(1\right)\sin\left(x^2\right)+\cos\left(1\right)\cdot \cos\left(x^2\right)-\sin\left(1\right)\cdot \sin\left(x^2\right)\)
Example of derivative rules
For the function \( f(x) = (x-1)(x^2+1) \), use derivative rules to find its derivative.
Solution: For this final example, we need to differentiate: \(\displaystyle f(x)=\left(x-1\right)\left(x^2+1\right)\).
Initial Step: In this case, we first need to expand the given function \(\displaystyle f(x)=\left(x-1\right)\left(x^2+1\right) \), and in order to do so, we conduct the following simplification
\( \displaystyle f(x)=\left(x-1\right)\left(x^2+1\right)\)
Note that \((x-1) \cdot (x^2+1) = x\cdot x^2+x-x^2-1^2 = x^3-x^2+x-1\), due to the fact that we can use the distributive property on each term of the expression on the left, with respect to the terms
on the right
\( \displaystyle = \,\,\)
\(\displaystyle x^3-x^2+x-1\)
After expanding the function, we can proceed to the calculation of the derivative:
\( \displaystyle \frac{d}{dx}\left(x^3-x^2+x-1\right)\)
By linearity, we know \(\frac{d}{dx}\left( x^3-x^2+x-1 \right) = \frac{d}{dx}\left(x^3\right)-\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x\right)-\frac{d}{dx}\left(1\right)\), so plugging that
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x^3\right)-\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x\right)-\frac{d}{dx}\left(1\right)\)
The derivative of a constant is 0, so then:
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x^3\right)-\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(x\right)\)
We know that \(\frac{d}{dx}\left(x\right) = 1\)
\( \displaystyle = \,\,\)
\(\displaystyle \frac{d}{dx}\left(x^3\right)-\frac{d}{dx}\left(x^2\right)+1\)
Using the Power Rule for polynomial terms: \(\frac{d}{dx}\left( x^2 \right) = 2x\) and \(\frac{d}{dx}\left( x^3 \right) = 3x^2\)
\( \displaystyle = \,\,\)
\(\displaystyle 3x^2-2x+1\)
Graphically, this is how the function and its derivative look:
More derivative calculators
One of the magic about differentiation is that you can find the derivative of any function using some basic and simple rules, including the product rule, quotient rule and naturally, the chain rule.
This small arsenal is usually sufficient to compute any derivative you need
Differentiation and integration are the main lanes in Calculus, without any dispute, as they are the center of so many applications, in all aspects of science. From related rates to implicit
differentiation, with partial derivatives in Physics and Economics | {"url":"https://mathcracker.com/derivative-rules","timestamp":"2024-11-12T23:44:49Z","content_type":"text/html","content_length":"134457","record_id":"<urn:uuid:2d8d5a04-ec74-4ddd-a020-f496f3855ccc>","cc-path":"CC-MAIN-2024-46/segments/1730477028290.49/warc/CC-MAIN-20241112212600-20241113002600-00110.warc.gz"} |
A238281 - OEIS
Conjecture: (i) a(n) > 0 for all n > 1. Moreover, if n > 1 is not equal to 8, then there is a positive integer k < n with 2*k + 1 prime such that the two intervals ((k-1)*n, k*n) and (k*n, (k+1)*n)
contain the same number of primes.
(ii) For any integer n > 4, there is a positive integer k < prime(n) such that all the three intervals (k*n, (k+1)*n), ((k+1)*n, (k+2)*n), ((k+2)*n, (k+3)*n) contain the same number of primes, i.e.,
pi(k*n), pi((k+1)*n), pi((k+2)*n), pi((k+3)*n) form a 4-term arithmetic progression.
a(8) = 1 since each of the two intervals (7*8, 8*8) and (8*8, 9*8) contains exactly two primes.
d[k_, n_]:=PrimePi[(k+1)*n]-PrimePi[k*n]
a[n_]:=Sum[If[d[k, n]==d[k+1, n], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 80}] | {"url":"https://oeis.org/A238281","timestamp":"2024-11-06T05:16:29Z","content_type":"text/html","content_length":"14738","record_id":"<urn:uuid:826f67a4-a796-4d09-9330-c07384ed4659>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00576.warc.gz"} |
Math classroom decoration ideas
I've written a lot about the benefits I've seen come from hanging student work and math word walls. But I didn't always have my classroom bulletin boards covered. It wasn't until a few years after my
first year teaching that I started decorating my classroom.
This post is a collection of math classroom decoration ideas, some from my own classroom, some printables I've made for teachers, and photos that teachers have sent to me of their own math classroom
There was a positive change in my classroom once I started hanging student work and math word wall references to our classroom walls.
Even my seniors liked seeing their work displayed. Here is a photo of our classroom Fridge where students hung their graded papers:
Most of the time even my juniors and seniors choose to hang their papers on our classroom Fridge instead of bringing them home. I also like having their papers displayed. On tough days, days that my
students feel they "can't", I can point to their wonderful work on the wall and say, "Yes you can!".
At the beginning of the school year as students enter your classroom for the first time, this Welcome, Math Person! poster poster is a reminder that we are all math people.
This Practice Makes Better poster reminds students that perfect is overrated. Great things happen when we work hard, get better and stop worrying about being perfect. Making mistakes is important and
the way we grow, especially in math class. I hope this poster helps students remember this.
To me, math confidence is everything. When students are confident, they can push through that tough math topic or at least be more willing to try. That confidence can be tough to build and comes from
more than just classroom décor. But I do believe that the decorations we hang in our classrooms can help.
There are letters included to make GEMDAS, GEMS and a few other acronyms. The mobile has since been updated with an "Order of Operations" card for the top of the mobile. It also now includes letters
to make GEMS.
Teachers have sent me so many great photos of their classrooms and seeing them all has made me so happy. This one was sent over by Ms. Koehler of her geometry word wall.
Mr. Urzua shared a photo of his math word wall on Instagram. I love the black background.
Ms. Woodworth brought her math classroom décor outside and hung hundreds of matholution math pennants in her school's courtyard.
I've made over 100 math pennants, with a bunch being for the holidays. They're fun activities that cover math topics and then double as student-created math classroom décor.
How many times have you heard, "I'm just not a math person" from students when they get stuck? No way. We are all math people. This How to be a Math Person poster reminds all students that math is
for all of us.
How cute is Ms. Lyons's How to be a Math Person mural? Her friend painted it on her classroom wall and totally nailed it!
Here is an "I Know I Can" motivational classroom poster with words from Nas. I heard from a bunch of teachers that their students thought the words were from Lil Nas X. Nothing makes a person feel
older, lol. There are directions to enlarge PDFs in this post.
This back to school math pennant with a Golden Spiral theme helps students get to know each other at the beginning of the year. It comes with a short lesson on the Golden Ratio and optional glyph
directions for students to color based on get-to-know-you questions.
My friend Karrie from Mrs. E Teaches Math sent this photo of her students' back to school math pennants. How about that wood paneling? There are more math pennants to see here.
If you're into Halloween and teach algebra or algebra 2, this is a free set of 12 posters with skeletons posing as algebraic functions. It's a fun set to display in the fall.
Ms. Page sent this photo with little Santa hats on her skeleton functions to extend their life through the winter holidays!
I made this billions place value reference for a 4th grade teacher who needed it for her classroom's math word wall. You can grab it from my dropbox here.
How do you like to decorate your walls? Do you believe less is better, go all out or are you somewhere in the middle? If you're part of the Visual Math Facebook group, I'd love to see your photos!
18 comments:
1. Thank you so it's a wonderful idea i will use it this year ....i wish i can find someone here to share with me ideas about math ....i'm from Algeria.......thanks in advance
1. Algeria! Wow! Thank you for your comment! We have a math group on Facebook called Visual Math where teachers share ideas about teaching math to visual learners. I hope you will join!
2. Nice indeed!
I would love to share the walls from my classroom too. We mostly have math formula as students want to remember for solving problems.
1. Thank you Keshab! I hope you have a wonderful school year!
3. I love these a lot! I just finished my first year and only wish I had found this post before. I will be incorporating some of these ideas next year! When you give tests, do you cover over the
word walls or leave them up?
1. Thank you for asking, Allison. I never covered the walls for tests. Anything that I gave in class wasy fair game to use on any assessments I gave. That being said, I had the luxury of
creating the curriculum we used in class because there was no self-contained Algebra 2 or Consumer Math curriculum. If the state test was given in my room, the walls would definitely have to
be covered though.
4. Thank you for share your ideas!! I really like it! I will do it in my classroom (im a math teacher from the Patagonia Argentina).
Thanks again!
1. Hello from Massachusetts, US! Thank you for your comment! I hope you have a wonderful school year!
5. Do you have the parent function people somewhere that I can purchase them or access them elsewhere?
1. Thanks for asking, Elisabeth. They are a free download on the OK Math website here: http://okmathteachers.com/algebra-aerobics/
6. Thank you so much. I am from Nigeria,hope to learn more from you .
1. Hello from Massachusetts, US!
7. I thought I saw on your blog a wall decoration for a daily or weekly calculation/word problem posting. Do you still have it?
1. I don't think it was here, but I do know that my friend Alex over at Middle School Math Man has a Challenge of the Week: https://www.middleschoolmathman.com/middleschoolmathmanblog/
8. AnonymousJanuary 17, 2023
what font do you use?
1. I use a lot of fonts from Kimberly Geswein. Her fonts are free for personal use on TPT.
9. AnonymousJune 03, 2023
Thank you for the ideas. I am a new Mathematics teacher here in the Philippines. You are a very great help.
1. Congratulations on your new math teaching job! | {"url":"https://www.scaffoldedmath.com/2018/06/math-classroom-decoration-ideas.html","timestamp":"2024-11-06T08:16:10Z","content_type":"application/xhtml+xml","content_length":"150605","record_id":"<urn:uuid:22634e64-ed4d-41bf-9f03-2ecb02a98ff6>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00774.warc.gz"} |
Money Puzzles
There were two families living next door to one another at Tooting
Bec--the Jupps and the Simkins. The united ages of the four Jupps
amounted to one hundred years, and the united ages of the Simkins also
amounted to the same. It was found in the case of each family that the
sum obtained by adding the squares of each of the children's ages to the
square of the mother's age equalled the square of the father's age. In
the case of the Jupps, however, Julia was one year older than her
brother Joe, whereas Sophy Simkin was two years older than her brother
Sammy. What was the age of each of the eight individuals?
Read Answer
THE BAG OF NUTS. | {"url":"https://www.mathpuzzle.ca/Puzzle/Next-door-Neighbours.html","timestamp":"2024-11-10T05:57:29Z","content_type":"text/html","content_length":"14598","record_id":"<urn:uuid:bd45e19d-be48-459b-8c18-989aeb6c8901>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00329.warc.gz"} |
Preemptive scheduling in a two-stage multiprocessor flow shop is NP-hard
Johnson [1954] gave an efficient algorithm for minimizing makespan in a two-rnachine flow shop; there is no advantage to preemption in this case. McNaughton's wrap-around rule [1959] finds a shortest
preemptive schedule on identical parallel machines in linear time. A similarly efficient algorithm is unlikely to exist for the simplest common generalization of these prblems. We show that
preemptive scheduling in a two-stage flow shop with at least two identical parallel machines in oneofthe stages so as to minimize makespan is NP-hard in the strong sense. Key Words & Phrases: flow
shop, parallel machines, complexity, NP-hardness.
Publication series
Name Memorandum COSOR
Volume 9320
ISSN (Print) 0926-4493
Dive into the research topics of 'Preemptive scheduling in a two-stage multiprocessor flow shop is NP-hard'. Together they form a unique fingerprint. | {"url":"https://research.tue.nl/en/publications/preemptive-scheduling-in-a-two-stage-multiprocessor-flow-shop-is--2","timestamp":"2024-11-05T09:57:11Z","content_type":"text/html","content_length":"45577","record_id":"<urn:uuid:c31bf5cc-ab43-496f-8d8d-ebedbbc37feb>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00889.warc.gz"} |
The manager of a school radio station is conducting a survey. She wants to find out which type of music students prefer. How would the manager most likely get a random sample that represents the school population? ask every fifth student who attended the last school concert ask every fifth student who attends the school from the student list ask each member of the school band and the chorus ask five students from each grade in the school
1. Home
2. Algebra
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The Estimated Formation Pressure During Drilling Operations, in Drilling Wells, and an Exploration of Steps towards Efficient Drilling
The Estimated Formation Pressure During Drilling
Operations, in Drilling Wells, and an Exploration of
Steps towards Efficient Drilling
Mahdi Imanian Najafabadi
Department of Automation and instrumentation Petroleum University of Technology
Ahwaz , Iran
Application of estimation and monitoring in drilling industry has found a wide-spread attention among researchers . One of the main areas relates to the efficient supervision and control of drilling
operations . For instance , during drilling oil wells , a fluid is pumped into drill string . This fluid is circulated through drill pipe and drill bit to bottom of the well and then is directed to
the surface via annulus to transfer cutting material , meanwhile , cooling and lubricating the drilling devices located at the bottom of well . This causes the pressure between the drilling fluid and
formation to be varied , leading to kick phenomenon and ultimately resulting into probable blow out if it is left uncontrolled . Therefore , estimation and monitoring of bottom hole pressure is
treated as a necessary requirement during well drilling operations [2] , [3].
In this paper , we are interested to investigate the feasibility of using adaptive observer technique to realize monitoring of drilling operations in oil wells from the essential and critical
operational problem views , being considered in the work. The resulting developed estimation and monitoring systems will be implemented and evaluated in simulation environment on the basis of
accessible operational data from candidate oil wells.
Estimation , Drilling Operation , Bottomhole Pressure , Adaptive Law , Lyapunov Law
Due to the importance of the rotation system of mud drilling , it is recognized as the heart of the drilling operation. By importing hydrostatic pressure from drilling mud, the fluid formation is
prevented from flowing from the wellbore to the surface. In other words, appropriate hydrostatic pressure created by drilling fluid prevents it from reaching to the kick off point and wellbore blow
out. For creating the appropriate hydrostatic pressure, it is necessary to measure the pressure at the bottom of the formation continuously by sensors installed on the drilling bit. The pressure is
sent to the surface, and the geologist decreases or increases the mud weight based on the pressure at the bottom of the wellbore. The base on the data acquisition and the choke valve is either closed
or opened in order to provide appropriate pressure on the drilling bit .In some situations such as a pipe connection, the rotation system of mud is stopped, and the pipes are transferred from the
wellbore to the surface. In reality there are no sensors to measure the pressure of the formation at that time, therefore it is possible to have sudden flow from the formation to the wellbore, and
blow outs happen during the pipe connection
operation. Therefore, pressure sensors are not applicable in this situation and cannot be used to monitor the formation pressure .In this study, we are looking for a solution to observe and measure
the formation pressure during all drilling steps, including during the pipe connection. The problem is solvable by using the adaptive observer and estimator to estimate the pressure of the formation
when the sensors are not inside the wellbore. Theses estimators and observers proposed by the lyapunov law are applicable in both automation and control engineering. After determining the
mathematical relationships for the estimation of the formation pressure by an adaptive observer, the results provided from the adaptive observer and estimator are simulated to real data, and the
acquired data are analyzed. [7] , [8] , [9] , [10]
For the estimation of formation pressure by the comparison estimator, it is necessary to develop a mathematical model from the drilling well in dynamic mode. Then, with the presented mathematical
model , the lyapunov law is applied and the comparison raw for the observer is obtained.
2-1. The mathematical model for the
dynamic mode of the wellbore
Model [1] has recently been used as one of the nonlinear mathematical models for solving the dynamic mode of wellbore. In the model, the drilling well is assumed to be isothermal, and the drilling
fluid is the only liquid phase. According to Figure 1, the drilling fluid is pumped by a mud pump and the fluid is injected through the drillstring to the bottom of the wellbore. After passing the
fluid through the drilling bit and annular, it reaches the control choke valve. The proposed mathematical model classifies the drilling well into two parts, such as the drillstring and the annulus.
The final acquired equations of model [1] are presented as following [4] , [5] :
[3] [2] [1]
(2) 2 2 2 3 1 2 2 1 2
a a
d d
a d
Where the above (corresponding) variables are defined as following:
: The input pressure of the drilling mud in the drillstring
: The output pressure of the drilling mud in the annulus M : Equivalent mass for the drilling mud in the drillstring
: The Bulk modules of the drilling mud in the annulus : The bulk modules of the drilling mud in the drillstring
The volume of the drilling mud in the annulus
The volume of the drilling mud in the drillstring
The flow rate of the input drilling mud to wellbore
The flow rate of the drilling mud on the drilling bit
The depth of the drilling bit from the surface
The density of the drilling mud in the annulus
The density of the drilling mud in the drillstring
The frictional coefficient between the drilling mud and the annulus
The frictional coefficient between the drilling mud and the drillstring
2-2. The nonlinear observer for the
estimation of the formation pressure
Equation (4) is used due to the presented nonlinear model and regarding the estimation of the flow rate of drilling mud on the drill bit and the continuous monitoring of formation pressure. It should
be noted that the pressure formation is obtained indirectly according to Equation (4) and its relation to the flow rate of drilling mud on the bit, the input pressure of the mud in the drillstring,
and the output pressure of the
mud in the annulus. Therefore, is considered as an impossible measurement of the system and the adaptive estimator is applied to estimate the output.
(5) From the change of variable we have:
is a positive constant gain .
From the converting of equation (6) to the estimated form
, and deriving the error equation and the derivative of the equation we have:
(7) 3 1 1 3 1 1 3 1 1 1
(9) Equations (1) and (3) are substituted in equation (9)
(10) 1 1 2 1 2 2 2 3 1 3 1 1 1 3
(11) 2 2 2 3 1 2 3 1 2 3 1 2 3 1 3 1
[1] [3]
, and
(15) With writing the lyapunov law, the following equations are derived ,
(17) With the substitution of the equation (14) to equation (17) , equation (18) is yielded ,
(18) With considering the law below and it as a positive definite matrix we have:
The third part of equation (18) included the term,
second part of equation (18) included the term,
where condition (21) is canceled or becomes positive . Finally, with the consideration of
, which it is a positive definite matrix under the estimator matrix , equation (18) is converted to equation (22).
(22) Because of the impossible measurement of the variable,
, the variable,
, is used as the following:
With substitution of equation (1) and the derivative of equation (7) in equation (24) we have:
Since the is a turbulence signal is possible to measure, and is a state variable and is recognized as the impossible to measure output ,we have
With the conversion of equation (23) to the estimated parameter,
, we have:
Therefore we have
With substitution of the equations (25) and (26) in equation (31), and substitution of equation (19) and its results in equation (30) the following equation is obtained:
(32) With integration of the equation (32) we have:
(33) Briefly the derived comparison adaptive law is given as following:
In this part, the simulation of the obtained estimator in part 2 is performed based on the data in [4] and [5] that is obtained from a real wellbore in the MATLAB environment and the Simulink. The
estimated results and the variables are compared with the real data. In the specified well, we don’t have the formation pressure at t = 3600 for 10 min because of the operation of the pipe connection
and during that time period, the pressure sensors are inactive.
Table 1 : Required data for the simulation of wellbore for the Grane field of the North Sea.
As mentioned in the introduction, one of the problems during the pipe connection is the lack of monitoring and observation of the formation pressure caused by the pressure sensors installed on the
drill bit being out of the wells during the pipe connection. Therefore, it is necessary to utilize adaptive estimators in order to estimate the formation pressure during drilling operations, which
enables it to monitor the formation pressure at all times during the operation; not only during the drilling formation but the pipe connection as well. One of the reasons for the monitoring of the
formation pressure is to know when the kick off and the blow out happens in the wellbore because, in this situation, the pressure in the wellbore changes suddenly. With knowledge of these pressure
changes, it is possible to perform some actions in order to recognize the kick off and to neutralize it.
In part 2 of this paper, the appropriate model is presented for the estimated observer, and in part 3 the base on the presented model, the comparison estimator and the observer are obtained. Finally,
in part 4, the estimator in the MATLAB environment and Simulink is applied to utilize the data in [4], and [5] is obtained from a well located in the Grane field in the North Sea.
The data acquisition is observed in figures 2, 3, and 4. As seen in the figures, at the time, t = 3600s, because of the pipe connection operation for 10 min, the mud pump is turned down and the pipes
and drilling bit are transferred to the surface. During those 10 min, the formation pressure is obtained by the comparison estimator. According to the figure 3, after 1 hour and 10 min, the formation
pressure obtained from the sensors installed on the bit becomes equal to the pressure of the compression estimator. And the error yielded from the difference of pressure between the estimator and the
sensors become zero.
According to the data acquisition, it can be concluded that the estimator can be used in real systems and drilling rigs because of rapid reaction in approaching of the estimated pressure to the real
[1] G .O . Kassa ,2007,”A Simple Dynamic Model of Drilling “,Technical Report ,Statoil , Norway.
[2] H .K. Khalil ,2002,"Nonlinear Systems” , Englewood Cliffs .
[3] E .Jahnshahi , K. Salahshoor , R. Kharrat ,2007,"A Modified Distributed Delay Model for Void Wave Dynamics in Gas Lift Oil Wells" ,JPST.
[4] J. Zhou , O . N. Stamnes ,O.M. Aamo ,G.O. Kassa ,2010 "Pressure Regulation with Kick Attenuation in a Managed Pressure Drilling System," The 48th[ IEEE ]
Conference in Decision and Control ,Shanghai ,P.R. China.
[5] G. Nygaard ,2006 "Multivariable Process Control in High Temperature and High Pressure Environment Using Non-Intrusive Multi Sensor Data Fusion", Phd Thesis, NTNU.
[6] D. Hargreaves, S. Jardine , and B. Jeffryes ,2001 “Early kick detection for deepwater drilling: New probabilistic methods applied in SPE Annual Technical Conference and Exhibition.
[7] H. Santos, E. Catak, J. Kinder, and P. Sonnemann ,2007, “Kick detection and control in oil-based mud: Real well-test results using micro control equipment”, in SPE/IADC Drilling Conference,
Proceedings, vol. 1. Society of Petroleum Engineers (SPE), Richardson , United States, pp. 429-438.
[8] C. L. Helio Santos and S. Shayegi ,2003, “ Micro- ux control :the next generation in drilling process for ultra-deepwater” , in SPE Latin American and Caribbean Petroleum Engineering Conference,
ser. SPE8113, Port-of-Spain, Trinildad, West Indies.
[9] M. Doria and C. Morooka , 1997, “Kick detection in coating drilling rigs, in SPE/IADC 39004. IADC/SPE Drilling Conference, Amsterdam.
[10]J .J. Azar and G.R.Samuel,2007,” Drilling Engineering”. Penwell Corporation.
Figure 3.The pressure of the drilling mud at the bottom of the wellbore: 1 – obtained from the pressure sensors on the drilling bit, Grane field located in the North Sea, 2- The estimation of the
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Waterfall Charts Using Measures in Power BI
Power BI waterfall charts provide a great way for users to visualise how pieces of an overall plan (or results) are combined to contribute to an outcome. For example, you could use a waterfall chart
to show how sales have increased by year and category as illustrated below using the standard waterfall chart in Power BI.
One of the features of the waterfall chart about is that “Year” and “Category” are both columns in the data model. The columns are added to the X Axis and the results “build” from left to right.
Measures, not Columns?
But what if you want to build a waterfall chart that “builds” on a concept such as the components of revenue e.g. Cost, Profit, Tax and Sales? See example below.ย If you have these components as
columns in your data model (eg as with General Ledger data), then you are fine.
But what if you don’t have such a data model, and each of the components has to be calculated as its own measure?ย The standard waterfall chart does not support using measures on the x-axis, and
hence you cannot create such a chart just using measures.
Disconnected Tables and Switch Measures
One solution to this problem is to turn the measures into columns in a table by using SWITCH measures and disconnected tables.ย The steps to do this are as follows:
• Create a disconnected table containing the measure name you want to see on the waterfall chart axis
• write a SWITCH measure to display the measure values for each row in the table
• use the column in the disconnected table as a category in the Waterfall chart
• use the SWITCH measure as values in the Waterfall chart.
Disconnected Table with Measure Names
The first step is to create a table with measure names using Enter Data in Power BI Desktop. I like to add an ID column to use in the SWITCH measure as it is easier to write, plus you can use the ID
column to sort the Waterfall chart.
After loading the table, sort the Measure Desc by ID column. I have called my table SwitchTable, but you could call it DriversOfRevenue or something similar if you wanted something more descriptive.
SWITCH Measure to Display the Values
Write the following switch measure to identify what measure values need to be displayed.
Display =
SWITCH (
SELECTEDVALUE ( SwitchTable[ID] ),
1, [Total Cost],
2, [Total Profit],
3, [Total Tax]
Create the Waterfall Chart
From here you can create the Waterfall chart.
• Place the column (mine is called Measure Desc) on the X Axis
• Place the measure onto values
I then did some tweaking to the chart such as changing the title, hiding the legend, etc.ย Also, I couldn’t find a way to change the final bar of the chart – it simply is called “Total” with no
seeming way to rename this (if you know of a way, please let me know in the comments below).ย I added a text box over the top of this label so I could rename it.
Or You Could Use Calculation Groups
You could also use Calculation Groups instead of SWITCH Measures to solve this problem. You can read more about Calculation Groups here
29 thoughts on “Waterfall Charts Using Measures in Power BI”
Hi Matt,
Thanks a lot. It helped me in a big way.
Hey Matt,
I have a project to analyze MoM Customer Growth with categories for โ Starting Customersโ , โ Customers Churnedโ , and โ Customers Gainedโ . I have to present this data on a waterfall chart
in power bi. My x axis is very crowded because it is concatenating the Months and the categories. The waterfall chart does not give me this option. Is there any way I can use Dax to remove the
Hi Matt, Thanks for this article.
However, I have a question, how can I show one of my category (measure) as result column that starts with 0 on axis?
Matt, can you provide a reference of how calculation tables can be used in a waterfall chart. I have a calculation table inside a matrix format, but unable to transition to a waterfall chart.
Thank you
Sorry, I don’t really understand what you are asking. This article shows how to create a table and use that in a waterfall chart. That is what you seem to be asking, which is what the article
describes. As long as you have a column in your table to use on the x axis of your chart, it will work
Hi, I was needing to show the variance of LY vs CY by using a Waterfall Chart. Also I was needing the total of Ly and CY as you are having in the first picture. It worked with your function but I
can’t sort the buckets alphabetically (they can only be sorted with their correspondent values). Did you encountered the same issue?
Hi, Thank you for the post, I am not sure I am following Disconnected Table with Measure Names. Can you please explain it to me? Tahnk you so much
Hi Danzi,
A disconnected table is a table in your model that doesn’t have a relationship with any other table in the model.
The disconnected table is required so you can use switch to control the measures being returned to the chart.
Hii Matt,
Thanks for your way forward. It helped me yo sort my problem.
Can we use waterfall chart without measure. Because I ready calculate data in table . if we use data do a measure for waterfall the value is not correct.
Amazing solution! Thank you!
Hey, thank you for your help!
I would like to get some help with power bi. i need to show period over period cycle, so Iโ m using the waterfall chart and use the โ Typeโ as breakdown and the visual makes the difference
between the periods by itself. But, what if i want to make a calculate measure on top of this difference?
For example, case when last_period_sale > 0 and diff< 1000 then โ New_Basicโ end as โ Typeโ . Thank you ๐ ๐ ป
Hi Matt,
Could you please explain how to add a waterfall chart using Calculation Groups?
At the heart, a calculation group is just like a disconnected table like the one shown in this demo. So create the calculation group instead of the disconnected table, and it should work.
There is a link on how to create a calculation group above.
Chris Many Many thanks for this forum
Please advice me to hide Total Column from waterfall chart.
Thank you
So clear. Thanks.
I love this and see how it should work. But, mine produces no numbers. It looks like it not even calculating. I proved the measures work and are correct.
Here is my call for help on the community.Powerbi.com.
Hi, I have a problem. First my date data is redundant in my dataset. I have different sets of breakdowns (both positive and negative numbers) for different categories in my dataset. I have tried
to use the Dax examples to solve this problem, it doesn’t work. It replicates the breakdown for all the fields in all the category fields just like your first pictogram. can u pls help? thanks.
It’s difficult to provide support here. I suggest you ask a question at community.powerbi.com and provide as much detail as you can
You are a great expert in Excel. Thanks
Great post Matt. I did it differently to make the chart reveal real business insights. I structured it as follows:
PAT Drivers Display =
1, [Revenue Actual],
2, [CoS Actual],
3, [Operating Expenses Actual],
4, [D & A Actual],
5, [Net Interest Expense/Income Actual],
6, [Taxes Actual],
7, [PAT Actual]
It is actually drivers of Profit After Tax (PAT). The Total bar chart is meaningless and formatted its color to white so that it does not display. if there a way of not showing Totals, i would
have disabled it.
Unfortunately, i cannot not attach an image of the chart
Thanks for sharing Chris
Hi Chris, pls have u tried writing DAX for a waterfall chart that has different breakdowns (both + & – numbers) for different category fields in the same data set? Your formula only works if
the breakdown is the same for all the category fields..
Matt, Chris, thanks for you ideas! Didnt yiu find a way to disable “Total” field?
Anther new DAX function that is not supported in Power Pivot
Agreed. And I donโ t see it changing actually.
Would it be possible to use someting else instead of Selectedvalue in this case?
Yes, you can use MAX() instead
Thanks for the input Matt!
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RBSE Solutions for Class 11 Physics Chapter 6 GravitationRBSE Solutions for Class 11 Physics Chapter 6 Gravitation
Rajasthan Board RBSE Class 11 Physics Chapter 6 Gravitation
RBSE Class 11 Physics Chapter 6 Textbook Exercises with Solutions
RBSE Class 11 Physics Chapter 6 Very Short Answer Type Questions
Question 1.
Weightlessness is experienced in artificial satellite but why not on the Moon?
Weightlessness is not experienced on the Moon because due to its greater mass it has its own gravitational acceleration where as gravitational acceleration on artificial satellite is negligible.
Question 2.
How much energy is required by a satellite to revolve in its orbit?
Since, work done by the centripetal force is zero. Hence, no energy is required.
Question 3.
If Earth is a hollow sphere; then what would be the mass of an object at a depth 10 km below the Earth’s surface?
The value of g is equal to zero in a hollow sphere, hence, the weight (mg) will also be zero.
Question 4.
Why are high tides and low tides in the sea?
Due to gravitational effect of Moon there is high tides and low tides in the sea.
Question 5.
Why is the Earth fiat at the poles?
Due to rotation of Earth about is own axis, the Earth is flat at the poles.
Question 6.
If the Earth revolves around the Sun in circular orbit then what would be the work done by the gravitational force?
Gravitational force provides the necessary centripetal force, which at right angles to motion of the Earth, so by the relation W = Ed cos θ
∴ W = Fd cos 90 = 0
hence no work is done by gravitational force.
Question 7.
How many Newtons are there in 1 kg weight?
∵ W = mg, ∴ 1 kg weight = 1 × g = 1 × 9.8 = 9.8 N
Question 8.
The value of escape velocity for any object from the surface of the Earth is 11.2 km/s. If the object is thrown at an angle of 30° then what would be the value of escape velocity?
Since the escape velocity does not depend upon the angle of projection, hence the escape velocity will be same.
Question 9.
Moon is very light in comparison to the Earth. Then why does not it fall down due to Earth’s gravitation?
Earth’s gravitational force is normal to motion of Moon, therefore work done is zero. Hence the Moon does not fall due to Earth’s gravitation.
Question 10.
The weight of 10 g gold is more at the poles in comparison to the equator. Why?
Weight of an object, W = mg. The value of g is maximum at poles and minimum at equator by the relation g = \(\frac{G M}{R^{2}}\) This is why the weight of 10 gram gold is greater at poles than at
Question 11.
Name the first satellite launched by India.
Aryabhatta; 19^th April, 1975.
Question 12.
Give the dimensional formula of gravitational potential.
Intensity of gravitational field,
E[g] = \(\frac{F}{m}\)
Dimensional formula of E[g] = \(\frac{\left[\mathrm{M}^{1} \mathrm{L}^{1} \mathrm{T}^{-2}\right]}{\left[\mathrm{M}^{1}\right]}\)
= [M^0LT^-2].
RBSE Class 11 Physics Chapter 6 Short Answer Type Questions
Question 1.
Give differences between weight and mass.
Mass : The mass of an object tells, how much quantity of substance is there in the object. Measurement of mass is done with the help of physical balance in unit of kg.
Weight : The weight of an object is the force by which Earth attracts the object towards its centre. It is force and its value is obtained by mg where m is the mass of the object and g is the
acceleration due to gravity at the location of the object. The measurement of the weight is done by spring balance.
On changing the place on the Earth, the mass remains unchanged while the weight varies according to change in the value of ‘g’.
Question 2.
If during revolving around the Earth’s orbit, the mass of a satellite is doubled due to any reason, then what would be the effect on time period?
The time period of the satellite,
T = \(2 \pi \sqrt{\frac{r^{3}}{G M}}\)
where, r = radius of the orbit,
M mass of Earth.
In above formula the mass of the satellite (m) is not used. Therefore the time period T will not depend on m. Hence, on doubling the mass of the satellite, its time period will remain unchanged.
Question 3.
Can any artificial satellite be placed at any orbit so that it seen always above Rajasthan’s capital Jaipur? Explain clearly.
A satellite can be seen always only in the orbit above the equator and Jaipur does not lie in equatorial plane. Therefore such an artificial satellite is not possible which may always be seen above
Question 4.
Generally why are rockets launched at the equator surface from west to east?
The direction of rotation of Earth is from west to east. Therefore the rockets are also launched in this direction so that they may take the place of geostationary satellite. The rockets are launched
on equator surface because they may be geo-station only in this plane. The reason of launching the rockets from west to east is to minimise the consumption of fuel because by this action, the
relative velocity of the rocket = velocity of rocket + velocity of earth.
Thus, the velocity of rocket increases resulting the less consumption of fuel.
Question 5.
If a watch based on the simple pendulum is kept at the center of the Earth, then what would be its time period? Will the watch work?
The value of acceleration due to gravity at the depth of d below Earth’s surface,
g’ = \(g\left(1-\frac{d}{R}\right)\)
where g is the acceleration due to gravity at Earth’s surface and R is it’s radius.
For the centre of the earth,
d = R
∴ g’ = \(g\left(1-\frac{R}{R}\right)\) = g (1 – 1)
or g’ = 0
∴ The time period of simple pendulum.
T = \(2 \pi \sqrt{\frac{l}{g^{\prime}}}=2 \pi \sqrt{\frac{l}{0}}\) = ∞ (infinity)
Thus the time period of the watch based on simple pendulum will be infinite i.e., the watch will not work.
Question 6.
Due to the greenhouse effect if the ice at the poles will melt, then what would be the effect on the duration of the day on Earth?
On melting the ice will convert into water which moves away from axis of rotation. Therefore the moment of inertia increases. As a result, the angular velocity of rotation of Earth decrease and hence
the Earth will take more time to complete the rotation. Hence the duration of the day will increase.
Question 7.
Different countries establish different communication satellites in the orbits for their communication system. Are these orbits different? Explain.
The time period of each geo-stationary satellite revolving the Earth is 24 hrs and the height of this orbit from the surface of Earth is 3600 km. Therefore each country requires its own
geo-stationary satellite for its communication system. Thus it is clear that the communication satellites of each country are in the’same orbit but in different situations.
Question 8.
If the Earth stops rotation on its axis, then what would be the change in the value of gravitational acceleration at equator and poles?
The value of gravitational acceleration (in latitude),
g’ = g – ω^2Rcos^2 λ
(i) For poles, λ = 90°, ∴ cos^2 λ = 0
Therefore g’ = g
Therefore, even the Earth stops its rotation, i.e, ω = 0, there will be no change in gravitational acceleration.
(ii) For equator, λ = 0. ∴ cos λ = 1
If ω = 0 i.e., Earth stops rotation, then
g’ = g
Thus the value of gravitational acceleration increase on the equator when Earth stops it’s rotation.
Question 9.
Are there some celestial bodies for which the value of acceleration due to gravity is infinity?
Acceleration due to gravity for celestial body
where g = \(\frac{G M}{R^{2}}\)
M = the mass of the body
R = the radius of the body
Therefore, for g to be infinity,
(i) The value of M should be infinite
(ii) The radius R should be zero.
Above both the conditions are not possible for a real body. Therefore such a celestial body is not possible on which the value of gbe infinite.
Question 10.
A satellite revolves around a planet in elliptical orbit. If F is gravitational force then what would be the central force?
For satellite,
Required centripetal force = Gravitational attraction between planet and satellite.
or F[c] – F[g]
This gravitational force will be the central force.
Question 11.
Define gravitational potential and give its units in S.I.
Gravitational potential : Gravitational potential at any point in the gravitational field is equal to work done in bringing unit mass from infinity to that point. It is represented by V[G] .
∴ V[G] = \(\frac{W}{M}\)
Where W = the work done in bringing mass M from infinity to that point
∴ S.I. unit of V[G] = \(\frac{J}{\mathrm{kg}}\) = J. kg^-1
Question 12.
If due to any reason the kinetic energy of a satellite increases 100% then what would be its behaviour?
The kinetic energy of the satellite,
K = \(\frac{1}{2} \frac{G M m}{r}\)
If the kinetic energy is increased by 100%, then
Therefore, the binding energy of the satellite in this new situation
E[t] = K’ + U = \(\frac{G M m}{r}-\frac{G M m}{r}\) = 0
or E[t] = 0
Therefore the satellite will escape.
RBSE Class 11 Physics Chapter 6 Long Answer Type Questions
Question 1.
Derive the formulae for kinetic energy and binding energy of a satellite.
Energy of a Satellite
When the satellite is revoling in its orbit it has kinetic energy due to the orbital velocity and due to its position it has potential energy. The sum of these two types of energies in an orbit is
the total energy of the satellite.
It is clear that the total energy is negative which means that the satellite is an attached body to the planet. To free it we require external energy.
Binding Energy of a Satellite
“The minimum amount of energy required for a satellite by which it can leave out its orbit or escape from the gravitational influence is called the binding energy of a satellite”. It is represented
by E[b].
Total energy of the satellite, E[t] = \(-\frac{1}{2} \frac{G M m}{r}\)
When the satellite escapes at infinity its total energy will be zero. Hence binding energy of the satellite
E[b] = E[b] – E[t] = 0 – \(\left[-\frac{1}{2} \frac{G M m}{r}\right]\)
or E[b] = \(\frac{1}{2} \frac{G M m}{r}\)
Question 2.
Write about the contributions of Indian astronomers.
Dr. Vikram Sarabhai is known as the father of Indian Space Programme. ISRO (Indian Space Research Organisation) is for higher level space research programmes. Indian Space Research Organisation was
established in the year 1969 on 15th August. Indian space research journey is very long and significant; which can be described from first satellite Aryabhata (1975) to the present Mangalyaan (2013).
Indian Astrophysicist Dr. Subrahmanyam Chandrashekhar was awarded Nobel prize in the year 1983 for Chandrashekhar Limit.
According to this, mass of all the White Dwarfs stars is in the limit given by Chandrashekhar Limit. If the mass of a star becomes 1.44 times the mass of the Sun then it will become a Black hole. In
his honour, NASA (National Aeronautics and Space Administration) established an X-ray laboratory dedicated to the study of stars in the year 1999.
Principles of Dr. C.V. Raman and Shri Meghnad were also useful in spaceworks. Shri Vikram Sarabhai established Physical Research Laboratory in Ahemedabad.
India’s first satellite was made by ISRO, which was launched by Soviet Union on 19th April 1975. Rohini satellite was launched by Indian launching I vehicle SLV-3 in the year 1980 from Sriharikota.
This was first launch by India. In the year 2014 a heavy satellite GSAT-14 was projected in its orbit by GSLV-D5. ISRO constructed Indian cryogenic engine for this. After this first successful flight
India has become the sixth country to have this technique in the world. After this GSAT-6 vehicle was successfully launched on 27th August 2015 by GSLV-D6. On July 2012, former President of India and
‘Missile Man’ Dr. A.P.J. Abdul Kalam said that ISRO and DRDO work for making the space techniques less expensive and more efficient. This is the reason that Mangalyaan’s successful launching was less
expensive than the hollywood film ‘Gravity.’
Indian Space Research achievements are as following :
1. In the year 1963, India launched its first rocket from Thumba.
2. In the year 1967, an Experimental Satellite Communication Earth Station was established at Ahmedabad.
3. For various space researches Indian Space Commission and Department of Space was established in the year 1972.
4. On 19th April, 1975 first satellite was launched in the name of great scientist Aryabhatta.
5. Bhaskar-1 was launched in the year 1989.
6. In the year 1981 communication satellite named Apple was launched.
7. INSAT-lA (Indian National Satellite) was launched in the year 1982.
8. INSAT-1B was launched in the year 1983.
9. Shri Rakesh Sharma was the first Astronaut of India in the year 1984.
10. India’s first operational remote sensing satellite was launched in the year 1988.
11. INSAT-ID was launched in the year 1990.
12. INSAT-2C was launched in the year 1995.
13. INSAT-2D was not launched successfully in June 1997, but in September, 1997 IRS-1D was successfully launched.
14. GSLV-D1 was partly successful in the year 2001.
15. In the year 2004, GSLV-EDUSAT was successfully launched.
16. On 22nd October, 2008 Chandrayaan-1 was successfully launched.
17. On 5th November, 2013 Mangalyaan was successfully launched.
18. On 24th September, 2014 Mangalyaan (298 days after launching) successfully established itself in the orbit of Mars.
19. On 5th January, 2014 GSLV-D5 was successfully launched.
20. On 18th December, 2014 the flight of GSLV-MKIII was successful.
21. On 28th September, 2015 India’s first dedicated multi-wavelength space observatory was launched named ‘Astrosat.’
Main Achievements of the Century
Chandrayaan-l made an important discovery by discovering the presence of water molecules on the Lunar surface.
Chandrayaan-1 was sent on 22nd October, 2008 on the Moon. It was launched by PSLV-Cll from Satish Dhawan Space Center.
Its main objective was to discover water and He gas. For this achievement India has become the sixth country in the world. After its disconnection from the orbit it was closed. According to ISRO’s
Head Shri Madhwan Nayar on 30th August 2009 ISRO closed Chandrayaan-1 officially.
According to Times Magazine Mangalyaan was selected in the list of Best Inventions in the year 2014. Mangalyaan’s successful launch was the showcase of Indian scientists talent which establishes
themselves in the whole world.
Mangalyaan, Mass Orbitter Mission (MOM) was launched on 5^th November, 2013 by ISRO from Shriharikota (Andhra Pradesh).
On 24^th September, 2014 India was the first country to reach on Mars in its first attempt. This is a great achievement of Indian scientists and reseaschers.
Question 3.
What do you mean by orbital velocity and escape velocity? Give formulae and relationship for these.
Escape Velocity
“Escape velocity is the minimum speed needed for an object to escape from the gravitational influence of a massive body.” It does not have a fixed direction. Hence, it would be more right to say it
as escape speed rather than escape velocity.
Formula for escape velocity from the Earth’ surface
We know that when an object is thrown away from the Earth’s surface then its kinetic energy changes into its gravitational potential energy. And when the complete kinetic energy changes into the
gravitational potential energy from that place the object returns back.
Escape velocity = Value of gravitational potential energy
It is clear that escape velocity does not depend upon the mass of the object.
∵ g = 9.8m/s^2 and R = 6.4 × 10^6 m
∴ Escape velocity from the surface of the Earth
Escape velocity for any object at height h from the surface of the Earth at point
Keeping the values g= 9.8 m/s^2 and R = 6.4 × 10^6 m in the above equation;
v[e] = \(\sqrt{9.8 \times 6.4 \times 10^{6}}\) = 7.92 km/s
In this situation the escape velocity will be equal to the orbital velocity of the satellite revolving close to the Earth.
Relation between Orbital Velocity and Escape Velocity
In the article (6.10.1) we will study that the orbital velocity of the satellite revolving close to the Earth is,
v[0] = \(\sqrt{g R}\) …………. (1)
Escape velocity of the body at the surface of the Earth
Hence, it is clear that if due to any reason the orbital velocity is done \(\sqrt{2}\) times, the satellite would escape from its orbit.
Question 4.
Derive relationship between g and G.
Relation between Acceleration due to Gravity (g) and Universal Gravitational Constant (G)
The acceleration produced by the gravitational force of the Earth is called acceleration due to gravity (g)
“The acceleration gained by an object freely falling towards the Earth because of gravitational force is called acceleration due to gravity.”
It is represented by ‘ g’. This value is independent of the shape and mass of the body. Its S.I. unit is m/s^2.
Suppose, that the Earth’s radius is R and its mass is M. The value of the gravitational force acting on a particle of mass ‘ m’ at a distance ‘ r’ from the center;
F = \(\frac{G M m}{r^{2}}\) ……………. (1)
By, Newton’s second law of motion
F = ma
∴ If the acceleration due to gravity is g at point P
F = mg ………………. (2)
from the equation (1) and (2) we get
Where, ‘ h’ is the height of the particle above the Earth’s surface. If point P is close to the surface of the Earth then the value of h will be negligible with reference to the radius R of the
Earth, i.e.
h << R’;
∴ h + R ≈ R
from the equation (3)
∴ g = \(\frac{G M}{R^{2}}\) ………….. (4)
If Earth’s average density is ρ then;
Mass of the Earth, M = Volume × Density
M = \(\frac{4}{3}\) πR^3ρ
∴ From the equation (4)
g = \(\frac{G \frac{4}{3} \pi R^{3} \rho}{R^{2}}\)
or g = \(\frac{4}{3}\) πGRρ ……………… (5)
This equation (5) shows the relationship between (g) and density (ρ).
Question 5.
Prove that a satellite revolving close to the Earth’s surface has orbital velocity approximately 8 km/s.
Orbital Velocity of a Satellite
The velocity of object by which is revolving in the orbit, is called orbital velocity.
Suppose a satellite of mass M is revolving around the Earth (mass M) at a distance h from the surface of earth with orbital velocity v[0]. The radius of the orbit of the satellite; r = (R + h)
The gravitational force between the satellite and the Earth provides the necessary centripetal force.
∵ Acceleration due to gravity at the surface of the Earth
Question 6.
Calculate the height of Geo-stationary satellite above the Earth’s surface. How can it be used for communication?
Geo-stationary Satellite
A satellite placed at a definite height directly above the Earth’s equator and revolves in the same direction as the Earth rotates; so that its orbital time period is same as the Earth’s rotation
period (24 hours), is called a Geo-stationary satellite. The observer at the equator views the satellite as stationary, hence such types of satellites are also called geo-synchronous satellites.
These are used for communication, radio broadcasting, universe related studies & researches and gathering weather information. Communication satellite is also a Geo-stationary satellite.
We know the time period of a satellite,
T = \(\frac{2 \pi r^{3 / 2}}{\sqrt{G M}}\)
∴ r = \(\left[\frac{G M T^{2}}{4 \pi^{2}}\right]^{1 / 3}\) ……………. (1)
Putting G = 6.67 × 10^-11 Nm^2/kg^2, mass of Earth M = 6 × 10^24 kg, time period T = 24 hours in above equation.
Orbital radius of the satellite from the equation (1) by putting all values;
r = 4.2 × 10^4 = 42000 km
but r = R + h
Hence, height of the satellite from the surface of the Earth;
h = r – R
h = 42000 km – 6400 km
or h ≈ 36000 km
Hence, the height of a geo-stationary satellite from the Earth’s surface is approximately 36000 km.
Angular Velocity of Geo-stationary Satellite
Since, angular velocity ω = \(\frac{2 \pi}{T}=\frac{2 \times 3.14}{24 \times 60 \times 60}\)
ω = 7.3 × 10^-5 rad/s
Orbital velocity v = ωr
∴ v = 3.1 km/s.
For communication system, a satellite transmits signals to only \(\frac{1}{3}\) part of the Earth. Hence, to broadcast information throughout the Earth minimum three satellites are necessary.
Any communication satellite (geo-stationary) can not be placed over India’s capital New Delhi because it will not be in equatorial line.
Question 7.
Give reasons for the change in the intensity of the gravitational field of the Earth with respect to;
(1) Height, (2) Depth, (3) Due to rotation of the Earth.
Variation in Acceleration due to Gravity (g) with Height, Depth Shape of the Earth and Its Rotation
The variation in the value of (g) above the surface of earth : Assume that M is the mass of the Earth and R is the radius. Hence, acceleration due to gravity on the surface of the Earth;
Value of acceleration due to gravity at point P at a height h above the Earth’s surface.
It is clear that as we move above the Earth’s surface, the value of g reduces.
From equation (3);
It is clear from the above the earth’s surface, the value of g reduces
g’ = \(g\left[1+\frac{h}{R}\right]^{-2}\)
If h << R then according to binomial theorem
[(1 + x)^n = (1 + nx + ………. )] ≈ (1 + nx)
It is clear from the above figure that for h << R there would be a simple straight line change in the value of the acceleration due to gravity with h but for higher values of h it would be curved.
Percentage error in the variation of ‘g’
from equation (4)
The variation in the value of g due to the shape of earth : In reality the shape of the Earth is not completely a sphere but it is flat at the poles. Assuming that R[e] and R[p] are the equatorial
radius and the polar radius respectively. Then, it is clear from the figure 6.9(a) that;
It is clear that acceleration due to gravity at poles is greater than acceleration due to gravity at the ’ equator. This means that on moving towards the poles from the equator, the value of
acceleration due to gravity increases.
The variation in the value of ‘g’ due to the rotation of the Earth on its axis : Let R be the radius of the Earth and it is rotating on its axis with a definite angular velocity ω. Hence, it is clear
that every object (body) on the surface of the Earth will move with the same constant angular velocity.
The necessary centripetal force required for this circular motion is obtained from the one component of the attraction force and the rest is obtained in the form of (mg’ )
Suppose that a particle of mass m is at a point P whose degree of latitude is λ . If there is no rotation of the Earth and the acceleration due to gravity is g then the weight of the particle (body)
at P,
mg’ = mg – mrω^2 cos λ
or g’ = g – rω^2 cosλ,
∴ From ΔOPO’
\(\frac{r}{R}\) = cos λ ⇒ r = R cos λ
∴ From equation (12);
g’ = g – Rω^2 cos^2 λ
Question 8.
Define gravitational potential energy. Calculate the change in the potential energy in sending a particle to height h from the Earth’s surface. When h << R discuss about the changes in potential
Gravitational Potential Energy
The gravitational potential energy is equal to the amount of work done to bringing a mass from infinity to a point inside the gravitational field without any acceleration. It is a scalar quantity. It
is always negative and consider to be zero at infinity.
Suppose, M is the mass of the Earth and R is the radius. A point T is at a height h above the Earth’s surface. The gravitational potential energy at this point is to be calculated for a body of mass
M. For this we will have to calculate the work done to bringing the object from infinity to the point T. To calculate this in the direction OT there is another point S at a distance x from the
center. The value of applied force on the body mass m;
F = \(\frac{G M m}{x^{2}}\)
Work done in displacing a body by dx in the direction of the force
The distance of T from the centre of the Earth r = (R + h)
Hence, work done to bringing the body from infinity to point T or the gravitational potential energy of the body at T;
Situation 1 : For the surface of the Earth, h = 0,
∴ r = R
∴ U = \(\frac{-G M}{R}\) ………….. (2)
Situation 2 : Work done in taking the object to a height h above the Earth surface will be equal to the change in potential energy i. e.:
ΔU = W
∴ W = ΔU = U[2] – U[1]
ΔU = Potential energy at height h – Potential energy at Earth’s surface
Question 9.
Explain Kepler’s laws.
Kepler (1571-1630) studied the researches of Tycho Brahe for many years and gave the following three laws :
First Law or Law of Orbit : Every planet revolves around the Sun in elliptical orbit and with the Sun at one of the focus.
Second Law or Law of Area’s: An imaginary line drawn from the centre of the Sun to the centre of the planet will sweep out equal areas in equal intervals of time.
Suppose, a planet covers dA area in dt time, areal speed \(\frac{d A}{d t}\) = constant. As shown in the figure (6.19) the time taken by the planet to cover the distance from B to A is same as the
time taken to cover the distance from B’ to A’. Hence, area SAB= area SA’ B’. Since, both the areas are same hence the orbital speed of the planet changes. When the planet moves away from the Sun,
the speed decreases and when it comes closer it increases. Hence, writing areal speed in terms of angular momentum, the angular momentum of the moving planet remains constant; i.e; .
\(\frac{d A}{d t}=\frac{L}{2 m}\)
where L is angular momentum and m is the mass of the planet.
Third Law or law of periods : The ratio of the squares of the periods of any two planets is equal to the ratio of the cubes of their average distances from the Sun.
T^2 ∝ r^3
or T^2 = Kr^3
where K is a constant and r is the average distance between the planet and the Sun.
Therefore, it is clear that farther the planet from the Sun its time period will be more. This is the reason that the time period of Mercury is very less and that of Neptune maximum.
Question 10.
Explain weightlessness in space.
The weight of an object (or body) is felt due to the attraction force acting on it. If the attraction force is zero then the state of weightlessness is felt. For example, if a person jumps freely
downwards, then there is no attraction force applied to the person. Hence, weightlessness is felt.
In artificial satellites the objects are in a state of weightlessness. This truth can be understood as follows :
Suppose the mass of the Earth is M and M’ is the mass of any satellite which is revolving with an orbital speed v[0] in an orbit of radius r. The necessary centripetal force is obtained from the
gravitational force.
Suppose inside the satellite at its surface an object of mass m is kept on which the reaction force is R opposite in direction to its weight W. Hence, equation of motion of the object :
This means that the weight of the body is zero. In other words it can be said that in artificial satellites objects are in a state of weightlessness.
Question 11.
Derive formula for velocity of projection.
Projection Velocity
The minimum velocity to throw an object vertically upwards to a definite height is called the projection velocity”.
Suppose, if an object having mass (m) is projected with v velocity to a height (h) from the Earth’s surface, when the body is thrown upwards then the body’s kinetic energy is changed into its
gravitational potential energy. Hence, from the law of conservation of energy :
Total energy of the body at the surface of the Earth (S) = Total energy of body at point P at height h
This is the formula for the projection velocity of a body going to a height h from the surface of the Earth.
The maximum height (h) gained by the object from equation (1) is;
Question 12.
Write Newton’s gravitation law. Derive it in vector form an explain that it follows action-reaction law.
Gravitational Law in Vector Form
As shown in the figure (6.3) the position vectors of particles M[1] and M[2] are \(\overrightarrow{r_{1}}\) and \(\overrightarrow{r_{2}}\) respectively.
Negative sign tells about the attractive nature of force
Similarly force on M[1] and M[2]
Hence, it is clear that the applied forces are equal and opposite. Hence, the gravitational force follows Newton’s third law.
RBSE Class 11 Physics Chapter 6 Numerical Questions
Question 1.
Calculate the gravitational force between two metal spheres of masses 50 kg and 100 kg respectively and the separation between their centres is 50 cm.
Given; m[1] = 50kg; m[2] = 100kg;
r = 50cm = 50 × 10^-2 m ; F = ?;
G = 6.67 × 10^-11 Nm^2kg^-2
The magnitude of gravitational force,
F = \(G \frac{m_{1} m_{2}}{r^{2}}\)
= 6.67 × 10^-11 × \(\frac{50 \times 100}{50 \times 50 \times 10^{-4}}\)
= 13.34 × 10^-7 n
or F = 1.334 × 10^-6N
Question 2.
If due to any reason, the orbital speed of a satellite increases 41.4%. Then in this situation would the satellite escape? Explain.
Orbital velocity of the satellite = v[0]
When any hour, its orbital velocity is increases by 41.4% then the new value of orbital velocity
v’[0] = v[0] + \(\frac{41.4}{100} v_{0}=v_{0}\left[1+\frac{41.4}{100}\right]\)
= v[0] [1 + 0.414] .
= v[0] × 1.414 = v[0]\(\sqrt{2}\)
or v’[0] = v[e] (escape velocity)
Therefore, the satellite will escape.
Question 3.
A body is thrown with a velocity of 10 km/s from the Earth’s surface, which is only a little less from the value of escape velocity. Calculate the height up to which the body will be reached.
Maximum height achieved by the projectile,
= 0.228 × 10^8
= 2.28 × 10^7 m
= 2.28 × 10^4 km
Question 4.
On the surface of the Earth a body’s weight is 72 N. How much would be the gravitational force at a height of half the radius of the Earth?
Weight of a body on Earth’s surface, W[s] = mg = 72N
or mg = 72N ………… (i)
Gravitational acceleration at height h,
Question 5.
Escape velocity on the surface of the Earth is 11.2 km/s. If any object is thrown with double the escape velocity, what would be its speed at maximum distance? Neglect the presence of Sun and other
celestial bodies.
Escape velocity from the Earth’s surface,
v[e] = 11.2 km-s^-1
Projection velocity from the Earth,
v[1] = 2v[e]
∴ Kinetic energy of projection
Now applying the principle” of conservation of energy,
Total energy on Earth’s surface = Total energy at infinity
Question 6.
Three bodies of same masses are kept at the top of an equilateral triangle with side ‘ a’. What speed would the three bodies be moved in a circle so that the triangle moves in the circular orbit and
there should be no change in the side of the triangle?
The gravitational force acting between each couple of masses
F = \(\frac{G M M}{a^{2}}=\frac{G M^{2}}{a^{2}}\)
Each internal angle of equilateral triangle is of 60°. Therefore the resultant of forces acting on each mass will be directed towards the centre O of the circle and the same will the role of
centripetal force for circular motion of the whole triangle.
Centripetal force = Resultant force
Question 7.
The potential energy of 3 kg body is -54 J. Calculate its escape velocity.
Given; potential energy on the Earth’s surface U[i] = – 54 J; mass of the body m = 3 kg
If the body escapes, then its potential energy at infinity U[f] = 0
Escape energy = U[f] – U[i] = 0 – (-54) = 54J
or \(\frac{1}{2} m v_{e}^{2}\) = 54
or v[e] = \(\sqrt{\frac{2 \times 54}{m}}=\sqrt{\frac{2 \times 54}{3}}=\sqrt{36}\)
or v[e] = 6 ms^-1
Question 8.
At how much height above the Earth’s surface, the value of gravitational acceleration would be 10% less than that on the surface?
The gravitational acceleration at height h from the Earth’s surface,
Question 9.
‘Lagrangian Point’ is a location in space between the Earth and the Sun; where the net gravitational force on any body due to both the Earth and the Sun is zero. Calculate the distance of this point
from Earth. The distance between Earth and Sun is approximately 10^8 km, and the mass of the Sun is 3.24 × 10^5 times the mass of the Earth.
Given; distance between Earth and Sun,
r = 10^8 km = 10^11 m
Mass of Sun, M[s] = 3.24 × 10^5M[e]
The distance of Lagrangian Point from the Earth x =?
A Lagrangian point P,
or x (568.8 + 1) = 10^11
or x × 569.8 = 10^11
∴ x = \(\frac{10^{11}}{569.8}=\frac{1000}{569.8} \times 10^{8}\)
or x = 1.75 × 10^8 m
Question 10.
Assume that any object is revolving around a big star in a circular orbit of radius R. Its time period is T. If the gravitational force between the object and the star is directly proportional to (R^
-5/2) then how is its time period dependent on radius?
If the mass of the object and the star be m and M respectively, then according to question,
Question 11.
Calculate the escape velocity on the Moon. Given Earth’s radius is 4 times the radius of Moon and mass of Earth is 80 times of the Moon.
Given M[e] = 80 M[m]; R[e] = 4 R[m]
Question 12.
A space laboratory whose mass is 2 × 10^3 kg is transferred from an orbit of radius 2R to .an orbit of radius 31?. Calculate the work done. Here, R = 6400 km (Earth’s radius).
Mass of Earth = M; mass of laboratory m = 2 × 10^3 kg;
r[1] = 2R; r[2] = 3R?; W = ?;
R = 6400km = 6.4 × 10^36 m
The work done in transferring the laboratory
Question 13.
If radius of the Earth is 6400 km. Then what would be the linear velocity of any object at the equator?
R = 6400 km; Time period of Earth’s rotation T = 24 hr
Question 14.
How much would be the increase in the potential energy if a body of mass m is taken to a height R equal to the radius of the Earth?
Question 15.
A satellite is revolving at a distance x from the centre of the Earth. How much would be the increase in its speed if the radius of the circular orbit reduces by 1%?
Orbital speed of the satellite,
∴ The percentage increase in orbital velocity = 0.5%
Question 16.
How much would be the radius of the Sun so that it becomes a Black Hole? Take the mass of the Sun constant (10^30 kg) and the maximum limit of velocity of projection should be equal to speed of the
light because as given by Einstein the velocity of any object cannot be more than the speed of high.
’’ Hint: Black Hole are those celestial bodies from which no object can escape because their gravitational acceleration is very much.]
Given; mass of the Sun M = 1030 kg
G = 6.67 × 10^11 Nm^2 kg^-2
v[max] = 3 × 10^8 ms^-1; R[min] =?
Gravitational potential energy on the surface of the Sun
Question 17.
If an object is placed in a tunnel running through the Earth’s centre, it does simple harmonic motion. Calculate the time period for this. If radius of Earth is 6.4 × 10^6 m and mass is 6 × 10^24 kg.
Time period of simple harmonic motion of an object placed on the tunnel running through the centre of the Earth,
or T = 84.7 min = 85 min
Question 18.
What would be the change in escape velocity, if the radius of the Earth reduces 4% and mass remains constant?
Escape velocity,
v[e] = \(\sqrt{\frac{2 G M}{R}}=\sqrt{2 G M} R^{-\frac{1}{2}}\)
From this formula, it is clear that on decreasing the value of R[e] escape velocity v[e] will increase.
∴ For maximum percentage error in v[e]
∴ Percentage increase in escape velocity = 2%
Question 19.
Two bodies of masses M[1] and M[2] are kept at a distance d from each other. Prove that the point at which the intensity of the gravitational field is zero, the gravitational potential will be :
V = \(\frac{-G}{d}\left[M_{1}+M_{2}+2 \sqrt{M_{1} M_{2}}\right]\)
Suppose the intensity of gravitational field intensity at point P is zero. Therefore, the gravitational field intensity is a vector quantity. So, | {"url":"https://www.rbsesolutions.com/class-11-physics-chapter-6-english-medium/","timestamp":"2024-11-03T04:09:57Z","content_type":"text/html","content_length":"225719","record_id":"<urn:uuid:54502d67-b7e0-47f5-9603-417a0738455e>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00551.warc.gz"} |
What Is the Probability of a Recession? The Message from Yield Spreads
Congress has given the Federal Open Market Committee (FOMC) a dual mandate: to maintain stable prices and maximum sustainable employment.Technically, the mandate also includes “moderate long-term
interest rates,” but it is generally thought that such moderation in interest rates would be the natural outgrowth of achieving price stability. The mandate to maintain stable prices is generally
thought to imply low and stable inflation, which the FOMC has specifically interpreted as a 2% annual change in the personal consumption expenditure (PCE) price index. After a recent burst of
inflation to a peak of almost 7% in June 2022, PCE inflation has been declining back toward the FOMC’s 2% inflation target, while expected inflation—implied by bond prices and surveys—has remained
consistent with the FOMC’s target.
Starting in March 2022, the FOMC responded to this burst of inflation by hiking the federal funds target range from 0%-0.25% to its current level of 5.25%-5.50%. Although inflation has declined and
that decline will probably continue, it is uncertain what the shift from accommodative to this moderately restrictive monetary policy will do to real economic activity. To assess the likelihood of
recession in the next year or so, this blog post explains and reviews recession forecasts from the yield curve, i.e., nominal and real term spreads.
A yield curve is just a picture of interest rates or yields on bank deposits and bonds with similar risk characteristics at a point in time. That is, a yield curve describes how short, medium, and
long rates relate to each other at a single time. The figure below shows that short rates are usually lower than long rates. In other words, one-year yields are usually lower than five-year yields,
which are usually lower than 10-year yields. That is, the yield curve usually slopes upward if you graph these yields by maturity. When short rates are about equal to long rates, that is called a
flat yield curve. An inverted yield curve is one in which short rates are higher than long yields. In other words, an inverted yield curve means that the yield curve is sloping down instead of up.
Yield Curve Inversion as a Predictor of Recessions
Since late 2022, several prominent measures of the yield spread—the short rates less long rates—have been very low or negative. That is, short rates are now higher than long rates and they have been
for most of the past year. This is concerning because past yield curve inversions have reliably predicted recessions, that is, sustained downturns in economic activity, as defined by the National
Bureau of Economic Research (NBER). There is nothing special about a yield curve that happens to be slightly downward sloping as opposed to flat. Both predict reduced economic activity in the future.
There are multiple stories about why an inverted yield curve predicts recession. Two of the most common are the following:
• Fed tightening raises short-term interest rates, and this action makes it more expensive to borrow for investment and consumption and thereby slows the economy.
• Low medium- and long-term interest rates indicate relatively low desired investment and lower growth in the future.
There are many measures of the slope of the yield curve, and economists and forecasters have different views as to the best one to use. Eric Engstrom and Steven Sharpe, for example, argue for the use
of the near-term forward spread, which is the difference in the 18-month and three-month interest rate.See Engstrom, Eric C.; and Sharpe, Steven A. “The Near-Term Forward Yield Spread as a Leading
Indicator: A Less Distorted Mirror.” Financial Analysts Journal, 2019, Vol. 75, No. 4. DOI: 10.1080/0015198X.2019.1625617. The authors argue that this spread is more closely related to near-term
monetary policy expectations.
Nominal vs. Real Yield Spreads
There is a longer history, however, of using spreads between long-term and short-term interest rates. I follow the practice from a regular forecast produced by the Federal Reserve Bank of New York,
which uses the spread between the 10-year and three-month Treasury rates.The Federal Reserve Bank of New York succinctly explains the procedures for the recession prediction. For a deeper dive, see
Arturo Estrella and Gikas A. Hardouvelis’s 1991 article, Arturo Estrella and Frederic S. Mishkin’s 1996 article, and Arturo Estrella and Mary R. Trubin’s 2006 article. The next figure shows two
interesting features of the 10-year-three-month term spread.
10-Year-Three-Month Yield Spread in Nominal and Real Terms
SOURCES: FRED (Federal Reserve Economic Data), Federal Reserve Bank of New York, Survey of Professional Forecasters, Livingston Survey and author’s calculations.
NOTE: Shaded areas depict NBER recessions.
First, like other term spreads, it is usually positive. That is, long-term rates are almost always higher than short-term rates. Second, when the 10-year-three-month spread declines, a recession
becomes statistically more likely. A negative spread—i.e., an inverted yield curve—has preceded each recession since the 1950s. The model is not perfect, however; there were false positives—that is,
high probabilities of recession when no recession occurred—in the late 1960s and late 1990s.
A statistical model called a “probit” translates the level of the spread into a formal probability of recession. The spread does predict recessions well, in the sense that implied recession
probabilities rise before each recession. The next figure illustrates the time series of estimated 12-month ahead recession probabilities from the nominal and real 10-year-three-month term spreads.
12-Month Ahead Recession Probabilities Derived from Yield Spreads
SOURCES: FRED (Federal Reserve Economic Data), Federal Reserve Bank of New York, Survey of Professional Forecasters, Livingston Survey and author’s calculations.
NOTE: Shaded areas depict NBER recessions.
The nominal yield spread is currently negative—quite low by historical standards—and predicts a 65% probability of recession in 12 months.The recession forecasts quoted here can vary modestly for a
number of reasons, including different estimation samples or use of different final data on which to condition current forecasts. For example, at the time of this writing, the New York Fed’s website
reports a 66% probability of recession based on the nominal 10-year-three-month yield spread using July 2023 data as the latest predictive information, with the model’s parameters estimated using
data from January 1959 to December 2009. This recession probability would be unprecedentedly high for a false positive. The near-term forward spread from the Board of Governors currently implies a
50% chance of recession 12 months from now.The near-term forward spread is the difference between the forward rate on a three-month Treasury bill six quarters in the future and the yield on a
three-month Treasury bill.
The Value of Examining Real Spreads
There are good arguments, however, for looking at real—that is, inflation adjusted—spread measures, rather than nominal measures. Because borrowers, lenders, consumers, and investors make decisions
based on inflation-adjusted interest rates, real interest rates should be more closely related than nominal interest rates to both the stance of monetary policy and expected future economic growth.
For example, a nominal federal funds rate of 5% is a very restrictive rate if expected inflation is 1%, but very accommodative if expected inflation is 10%. Recently, however, 10-year and one-year
inflation expectations differed more than usual, meaning that the current 10-year-three-month nominal spread could be less informative than it has been in the past.Inflation expectations can be
estimated in several ways. The most common are “break-even” expected inflation from real and nominal bonds and survey measures. Some surveys are of the general public; other surveys are of
specialized groups, such as economic forecasters.
To measure the impact of a similar “real” spread, I correct the 10-year-three-month spread for expected inflation using forward-looking 10-year inflation expectations from the Survey of Professional
Forecasters and one-year inflation expectations from the Livingston Survey. I use one-year expected inflation to deflate the three-month yield, rather than shorter horizon expectations, such as a
six-month yield, because one-year expectations are slightly more stable than six-month data (and both are very highly correlated).
The third figure shows implied probabilities of recession over time using probit models of real and nominal spreads. The “real” interest rate spread implies a still elevated, but lower probability of
recession in 12 months, of about 40%.I have used survey data on 10-year inflation expectations from the Survey of Professional Forecasters and data from the Livingston Survey on one-year inflation
expectations. This 40% probability is the highest probability in the history of the series, exceeding that even in any actual recession.
One interpretation of this discrepancy between the implied probabilities from nominal and real yield spreads is that the probability of recession in 12 months is somewhat lower than usually claimed.
Unfortunately, the real spread does not fit the recession data as well as the nominal spread by conventional statistical measures. It isn’t obvious why it should not. This poorer fit might reduce the
confidence in the forecast from the real spread.
The difference between the nominal and real yield spreads is equal to the difference between 10-year and one-year expected inflation. So, this difference in expected inflation over long-short
horizons must drive the difference in how the spread data predict recessions.
The figure below shows that the difference between the one-year and 10-year expected inflation has usually been pretty small, less than one percentage point, since the early 1960s. But this
difference did increase significantly in about 1975—during the first oil shock and coincident inflation—and in about 1981—during the great disinflation.
During these two periods, which each coincided with recessions, 10-year expected inflation was much lower than one-year expected inflation, and the real 10-year-three-month spread rose relative to
the nominal version, which made the real spread a weaker predictor of those recessions. In other words, the nominal spread predicted the 1975 and 1981 recessions better than the real spread. The
reason for this disparity is not clear and it could just be “bad luck.”
This difference in historical data means that the real spread’s predictions are not as sensitive to fluctuations in the real spread. That is, the model implied by the real spread just doesn’t predict
as well as that of the nominal spread.
10-Year Minus One-Year Expected Inflation
SOURCES: Survey of Professional Forecasters, Livingston Survey and author’s calculations.
NOTE: Shaded areas depict NBER recessions.
One should take recession forecasts with a grain of salt and look at a variety of indicators, because even if a model were literally correct instead of an approximation, these models predict 40%,
50%, 60% probabilities of recessions, which means there is a fair likelihood of a soft landing. In addition, one should remember that forecasting relations can and do break down. Past success doesn’t
guarantee future results. Information from other sources can give us greater confidence in, or suggest skepticism about, the yield curve forecasts.
In summary, the inverted yield curve is consistent with the claim that currently monetary policy is moderately restrictive and that there is a relatively high probability of recession in the next 12
months. But no forecast is certain.
This post was updated Oct. 6 to add the author’s interview on Timely Topics podcast series.
1. Technically, the mandate also includes “moderate long-term interest rates,” but it is generally thought that such moderation in interest rates would be the natural outgrowth of achieving price
2. See Engstrom, Eric C.; and Sharpe, Steven A. “The Near-Term Forward Yield Spread as a Leading Indicator: A Less Distorted Mirror.” Financial Analysts Journal, 2019, Vol. 75, No. 4. DOI: 10.1080/
3. The Federal Reserve Bank of New York succinctly explains the procedures for the recession prediction. For a deeper dive, see Arturo Estrella and Gikas A. Hardouvelis’s 1991 article, Arturo
Estrella and Frederic S. Mishkin’s 1996 article, and Arturo Estrella and Mary R. Trubin’s 2006 article.
4. The recession forecasts quoted here can vary modestly for a number of reasons, including different estimation samples or use of different final data on which to condition current forecasts. For
example, at the time of this writing, the New York Fed’s website reports a 66% probability of recession based on the nominal 10-year-three-month yield spread using July 2023 data as the latest
predictive information, with the model’s parameters estimated using data from January 1959 to December 2009.
5. The near-term forward spread is the difference between the forward rate on a three-month Treasury bill six quarters in the future and the yield on a three-month Treasury bill.
6. Inflation expectations can be estimated in several ways. The most common are “break-even” expected inflation from real and nominal bonds and survey measures. Some surveys are of the general
public; other surveys are of specialized groups, such as economic forecasters.
7. I have used survey data on 10-year inflation expectations from the Survey of Professional Forecasters and data from the Livingston Survey on one-year inflation expectations.
This blog offers commentary, analysis and data from our economists and experts. Views expressed are not necessarily those of the St. Louis Fed or Federal Reserve System.
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Awesome Math Girls
For the past 50 years, there has been a significant interest in the research community regarding why male students perform better than female students at higher levels of mathematics. When Dr.
Gunderson and others were at the University of Chicago, they published a great report on the impact of parents’ and teachers’ gender related stereotypes on the performance of girls in math. They
stated these stereotypes impact girls’ math course selections and pursuit of math related career paths.
This report mainly focuses on parents and teachers and suggests that they have significant impact on children’s academic attitudes and performance.
The report focuses on three main arguments relating to students in America:
1. Adults (parents and teachers) have their own anxiety about math and thinking that math ability is a fixed trait and not a growth trait.
2. Children get influenced as early as preschool. Early negative math attitudes in girls have long lasting impact on girls’ selection of math courses as well as their pursuit of STEM related careers.
3. The impact of parents’ and teachers’ attitude towards math and how it translates to children’s attitude and stereotypes.
Overall this report does a nice job of showing how negative math attitude impacts girls, causing most not to pursue advanced studies in math and math related careers. The report also clearly provides
a lot of evidence and examples on how stereotypes and mindsets of parents and teachers damage girls’ math mindsets more than they damage boys’ mindsets.
The Role of Parents and Teachers in the Development of Gender-Related Math Attitudes by Elizabeth A. Gunderson, Gerardo Ramirez, Susan C. Levine, and Sian L. Beilock | {"url":"http://awesomemathgirls.org/posts/2015/","timestamp":"2024-11-02T21:28:35Z","content_type":"text/html","content_length":"37612","record_id":"<urn:uuid:dc7535b2-cf54-48c8-8532-8019dd3955dc>","cc-path":"CC-MAIN-2024-46/segments/1730477027730.21/warc/CC-MAIN-20241102200033-20241102230033-00645.warc.gz"} |
Korean middle school math Question #29 : JoJo Teacher - Korean mathematics
Korean middle school math Question.
Hello. I am Teacher JoJo, teaching mathematics in South Korea.
Today, I have brought a math problem related to the incenter of a triangle.
This problem can be solved by 2nd-year middle school students in Korea.
Why don’t you give it a try as well?
[Korean middle school math question]
Question Description)
In triangle ABC, point $I$ is the incenter.
$\angle AEI = x, \; \; \angle BDI = y, \; \; \angle ACB = 40^\circ$
$\angle x + \angle y = ?$
If you are familiar with the properties of the incenter of a circle, it’s definitely worth a challenge.
Try to solve it on your own first, and then compare your solution with mine below.
Have you found the answer yet?
If you’re a diligent Korean student, you’ll probably solve it in under a minute.
Let’s quickly solve it using the approach of a Korean student.
$\angle ACD = 40^\circ \longrightarrow \angle AIB = 110^\circ$
$\angle AIB = \angle DIE = 110^\circ$
$\angle BEC = 180 -x, \; \; \angle ADC = 180 -y$
The sum of the angles in $\Box IDCE = 360^\circ$
so, $40 + 180 -x + 180 -y + 110 = 360$
$150 = x + y$
The answer is ${\color{red} 150^\circ}$
Did you understand my solution process?
If you are familiar with the properties of the incenter, you could have solved it quickly using this method.
if you are not aware of the properties of the incenter, this part might have been confusing for you.
${\color{blue} \angle ACD = 40^\circ \longrightarrow \angle AIB = 110^\circ}$
Why does it suddenly become like this? Let me explain.
According to the properties of the incenter,
If $\angle A = x^\circ$, $\angle BIC = \cfrac{x}{2} + 90^\circ$
The incenter is the point of intersection of the angle bisectors.
Extend segment AI, and consider triangles ABI and ACI.
Let’s denote the exterior angles of each triangle.
Please refer to the diagram below.
$\angle IAB + \angle IBA = \angle BID$
$\angle IAC + \angle ICA = \angle CID$
$\angle a + \angle b + \angle c = 90^\circ$
$\angle BIC = {\color{blue} a + b + c} + {\color{green}a} = {\color{blue} 90^\circ} + {\color{green} \cfrac {\angle A}{2}}$
$\therefore \angle BIC = \cfrac{\angle A}{2} + 90^\circ$
Did my explanation make sense to you?
Even middle school students in Korea find this concept challenging.
If it’s difficult, please read it again.
If you take your time to examine it, anyone can understand it.
I have other Korean math problems on my website, and I also have a YouTube channel.
A Korean solving American high school math questions. #28
[Korean ver]
안녕하세요. 저는 한국에서 수학을 가르치고있는 조조쌤입니다.
오늘은 중학교 2학년 수준의 문제를 가지고왔습니다.
내심을 잘 이해하고있다면 쉽게 풀수있으리라 생각됩니다.
함께 문제 보실까요?
천천히 풀어보시고 아래의 제 풀이와 비교해 보세요.
아마 공부를 좀 하는 한국 학생이라면, 1분안에 풀었을 것같네요.
그럼 한국 학생처럼 빠르게 풀어볼까요?
$\angle ACD = 40^\circ \longrightarrow \angle AIB = 110^\circ$
$\angle AIB = \angle DIE = 110^\circ$
$\angle BEC = 180 -x, \; \; \angle ADC = 180 -y$
사각형IDCE의 내각의 합은 360입니다.
$40 + 180 -x + 180 -y + 110 = 360$
$150 = x + y$
${\color{red} 150^\circ}$입니다.
제 풀이 이해가 되셨나요?
이해가 되었다면 내심의 성질에 대해서 잘 알고계신것입니다.
만일 그게 아니라면, 좀 더 자세한 풀이를 보여드리겠습니다.
아마 이부분이 이상했을 것입니다.
${\color{blue} \angle ACD = 40^\circ \longrightarrow \angle AIB = 110^\circ}$
이것에 대해서 자세히 설명해드리겠습니다.
내심의 성질에 의하여,
만일 $\angle A = x^\circ$라면, $\angle BIC = \cfrac{x}{2} + 90^\circ$입니다.
내심의 성질 기억나나요? 증명까지 해보죠.
내심은 각의 이등분선의 교점입니다.
선분AI에 연장선을 긋습니다.
그리고 삼각형ABI, ACI에서 각각의 외심을 봅니다.
아래 그림을 보세요.
$\angle IAB + \angle IBA = \angle BID$
$\angle IAC + \angle ICA = \angle CID$
$\angle a + \angle b + \angle c = 90^\circ$
$\angle BIC = {\color{blue} a + b + c} + {\color{green}a} = {\color{blue} 90^\circ} + {\color{green} \cfrac {\angle A}{2}}$
$\therefore \angle BIC = \cfrac{\angle A}{2} + 90^\circ$
증명이 이해가되나요? 아마 학교에서도 배웠을 것입니다.
보통 배워도 까먹죠.
한번보고 이해가 안될수도있습니다.
천천히 다시한번 읽어본다면 더 잘 이해가 될것입니다.
그럼 오늘은 여기까지 하도록 하겠습니다.
다음에 또 재미있는 문제로 찾아뵙겠습니다.
“Korean middle school math Question #29”에 대한 6개의 응답
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[…] Korean middle school math Question #29 […]
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Truly Stealthy PGP
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Truly Stealthy PGP
Eric points out the difficulty of making a "stealth PGP" which is 100%
indistinguishable from a string of random bits. The problem is that we
have to encode the RSA encrypted number, m, which is less than n, the
RSA modulus.
PGP first puts out two bytes of bit length, then m. This obviously won't
do, since the bit length is generally much less than 2^16 and so these
two bytes are a dead giveaway. However, we could leave these two bytes
off and just output m as raw bits, padded to the length of n. The
recipient knows n so he would be able to extract m.
The problem here, as Eric points out, is that m is less than n, so the
high bits of m will look non-random. If the high two bytes of n are,
say, 0x0C12, then m's high two bytes will never be bigger than this. This
will allow the opponent to do much better than 50% on guessing which files
have embedded messages.
This was discussed some time back on the pgp developers' list, and at that
time the suggestion was made to add a multiple of n to m so that it covered
a fuller range of values. The recipient would then just take the exponent
mod n and try that.
Mathematically, call L the next multiple of 256 above n. (0x10000... in the
example above.) We want to choose k so that M = m + k*n is randomly distributed
between 0 and L-1 if m is randomly distributed between 0 and n-1. This may
not be possible in this form. Perhaps there is another deterministic and
reversible transformation would accomplish it, though. In that case we
would have M = f(m,n) such that f can be reversed given M and n (we can
recover m).
As a trivial example of this problem, given n=2 and L=3, try to come up with
a way to turn a random 0/1 value into a random 0/1/2 value which is both
reversible and produces each of 0/1/2 with 33% probability. Seems pretty | {"url":"https://cypherpunks.venona.com/date/1994/03/msg00273.html","timestamp":"2024-11-13T18:28:36Z","content_type":"text/html","content_length":"5874","record_id":"<urn:uuid:9ebea665-3934-440d-aeba-d3f7fc10cb5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00808.warc.gz"} |
Newton’s Laws of Motion
Before listing Newton’s laws, it will be worthwhile to clarify what is meant by the word “law.” In science today, the words law, theory, and hypothesis are roughly synonymous. A theory is a
descriptive or explanatory account of something. The word hypothesis usually refers to a theory that has not yet been extensively tested and confirmed by observations or experiments. The word law
typically refers to a well-confirmed theory that describes some regularity in nature. However, these terms are not always used in a precise or consistent way (for example, “string theory” is an
entirely untested hypothesis), so don’t rely on these distinctions too much. For now, just bear in mind that the words theory, hypothesis, and law are essentially synonyms: they all refer to
descriptive or explanatory accounts (though they connote different degrees of observational support).
On the other hand, when Newton used the word “law” to describe the mathematical regularities he had discovered in nature, he wasn’t using the word merely as a synonym for “theory.” Like his
predecessors Kepler and Galileo, Newton was a devout Christian, and he regarded natural regularities as the result of divinely-instituted principles that govern creation. At the end of his magnum
opus, The Mathematical Principles of Natural Philosophy (1846), Newton explicitly credits God for the order that is found in nature:
This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other
like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One… This Being governs all things, not as the soul of the world, but as Lord over all… And from
his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite,
omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and knows all things that are or can be done.The
entire text is available here.
Newton’s laws of motion concern the effects of forces on physical objects. A force is simply a push or a pull. Forces are vector quantities: they have both magnitude and direction. The net force
(total force) exerted on an object is the sum of all the forces acting on it. (See the previous page for an explanation of how vectors can be added together.)
Newton’s first law of motion says that an object’s velocity (speed and direction of motion) does not change so long as the net force on the object is zero. Newton’s first law is also known as the law
of inertia.
Newton’s first law is logically entailed by the second law, since the latter implies that acceleration is zero when the net force is zero. So the first law can be regarded as merely a special case of
the second law: the case where F = 0.
Newton’s second law of motion says that the net force on an object is equal to the object’s mass times its acceleration: F = ma In the above equation, F is the magnitude (strength) of the net force,
m is the mass of the object, and a is the magnitude (rate) of its acceleration.
The standard unit of force, called the newton (symbolized with an uppercase “N”), is the amount of force needed to accelerate one kilogram of mass at a rate of one meter per second per second (i.e.,
its velocity increases by 1 m/s every second): 1 N = 1 kg m/s^2
Suppose a net force of 6 N is exerted on a 3 kg object that was initially at rest. How fast will the object be moving after 10 seconds? In order to answer this question, we first have to determine
the acceleration using Newton’s second law of motion:
force = mass × acceleration
6 N = 3 kg × a
6 N / 3kg = a
2 m/s^2 = a
So, the object will accelerate at a rate of 2 m/s
, which means it will go 2 m/s faster every second. Since it was initially at rest and accelerated at that rate for 10 seconds, its final velocity will be 20 m/s in whatever direction the force is
pushing it.
Newton’s third law of motion says that whenever one object exerts a force on another, the second object simultaneously exerts a force on the first object. These two forces are equal in magnitude but
opposite in direction.
For example, the earth pulls on a falling apple with a downward gravitational force; the apple simultaneously pulls the earth upward with a force of equal magnitude. (Of course, this upward pull
doesn’t budge the earth noticeably, because the earth has so much mass.)
Although the forces are equal and opposite, they do not cancel each other out, since they act on different objects.
Newton’s three laws together imply the law of conservation of momentum, which says that the total momentum of a physical system (any collection of physical objects) is conserved (doesn’t change), so
long as no external forces act upon the system. In other words, the total momentum of the system (the sum of the momentum vectors for each object in the system) always stays the same unless an object
outside the system exerts a force on one or more objects within the system. This is true for the total linear momentum and for the total angular momentum of the system.
Here’s a simple example to illustrate how the conservation of momentum follows from Newton’s laws. Suppose a 2 kg rock is moving to the right with a velocity of 5 m/s, and a 3 kg rock is moving to
the left with a velocity of 4 m/s. The momentum of the smaller rock (its mass × velocity) is 2 kg × 5 m/s = 10 kg m/s to the right. The momentum of the larger rock is 3 kg × 4 m/s = 12 kg m/s to the
left. Since the two momentum vectors are pointing in opposite directions, adding those vectors together yields a vector of length 2, pointing to the left. (See the previous page for an explanation of
vector addition.) So, the total momentum of the two-rock system, initially, is 2 kg m/s to the left.
Now suppose the rocks exert forces on each other. By Newton’s third law, those forces must be equal in magnitude but opposite in direction. For example, if the large rock exerts 30 newtons of force
pushing the small rock to the left, the small rock must exert 30 newtons pushing the large rock to the right. Let’s see what happens if the rocks exert that much force on each other for a duration of
1 second. First, consider what happens to the small rock, which has a mass of 2 kg. According to Newton’s second law:
force = mass × acceleration
30 N = 2 kg × acceleration
15 m/s^2 = acceleration
The small rock will accelerate at a rate of 15 m/s^2 to the left—the direction in which it is being pushed. In other words, its leftward velocity will increase by 15 m/s every second. Since it was
initially traveling to the right at 5 m/s, after 1 second it will be going 10 m/s to the left. Therefore, its final momentum will be 2 kg × 10 m/s = 20 kg m/s to the left.
Next, let’s see what happens to the big rock, which has a mass of 3 kg:
force = mass × acceleration
30 N = 3 kg × acceleration
10 m/s^2 = acceleration
The big rock will accelerate at a rate of 10 m/s^2 to the right. Since it was initially moving to the left at 4 m/s, after 1 second it will be going 6 m/s to the right. Therefore, its final momentum
will be 3 kg × 6 m/s = 18 kg m/s to the right.
So, the final momentum of the little rock is 20 kg m/s to the left, and the final momentum of the big rock is 18 kg m/s to the right. What is the total momentum of the two-rock system now, after the
rocks have exerted forces on each other? Adding these final momentum vectors together yields a vector of length 2, pointing to the left. In other words, the final momentum of the two-rock system is 2
kg m/s to the left—the same as it was at the beginning! Although both rocks are moving in different directions and at different speeds than they were initially, and their individual momenta have
changed, the total momentum of the two-rock system hasn’t changed at all. | {"url":"https://www.faithfulscience.com/classical-physics/laws-of-motion.html","timestamp":"2024-11-12T10:15:37Z","content_type":"text/html","content_length":"10882","record_id":"<urn:uuid:9539c4e9-e8a6-4c14-bb72-ea7a21e8d116>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00715.warc.gz"} |
If a simple random sample of size \(n\) is selected without replacement from a finite population of size \(N\), and the sample size is more than \(5 \%\) of the population size \((n>0.05 N)\), better
results can be obtained by using the finite population correction factor, which involves multiplying the margin of error \(E\) by \(\sqrt{(N-n) /(N-1)}\) For the sample of 100 weights of M\&M candics
in Data Set 27 "M\&M Weights" in Appendix \(\mathrm{B},\) we \(\operatorname{get} \bar{x}=0.8565 \mathrm{g}\) and \(s=0.0518 \mathrm{g}\). First construct a \(95 \%\) confidence interval cstimate of
\(\mu\), assuming that the population is large; then construct a \(95 \%\) confidence interval cstimate of the mean weight of M\&Ms in the full bag from which the sample was taken. The full bag has
465 M\&Ms. Compare the results. | {"url":"https://www.vaia.com/en-us/textbooks/math/elementary-statistics-13-edition/chapter-7/?page=6","timestamp":"2024-11-08T01:38:39Z","content_type":"text/html","content_length":"231754","record_id":"<urn:uuid:1ee51516-a74a-4281-aae7-429e6c082748>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00494.warc.gz"} |
The Excel FIND function
If you use Excel often, you’re already aware of its numerous useful functions. Many of these formulas perform calculations or analyze data. However, if you want to make life easier when working with
large data sets, you should know about the FIND function. What is the syntactical structure? How do you use the function? And what’s the difference between FIND and FINDB? You can find this out and
more in the following article.
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What is the Excel FIND formula used for?
It’s easy to lose track of things when working in large spreadsheets with hundreds of rows. Like all other Office products, Excel has a built-in search function. But that isn’t always what you
actually need. The search function automatically searches the entire document, meaning you can’t narrow down the range. More importantly, you can’t use the returned values in other functions. You
can’t forward the search result because the search function operates only on the interface.
But if you want to search within specific cells and integrate the search directly into your worksheet, you can use the FIND function. Simply enter the word or phrase and the function to find the
location of the first occurrence of your search term in the text string (meaning the text in the cell).
The FIND function in Excel is a building block with a range of further applications. Combine FIND with other functions to unlock its full potential. For example, you can use it to see whether a
certain term appears in your worksheet or to extract specific parts of a text string.
Excel FIND function – the syntax explained
The syntax of FIND isn’t very complex. In the standard version, you only have to specify two arguments: What do you want to find? And where do you want to find it?
You can also customize your search to start at a specific character:
You can use the parameters to specify various information:
• Find_text: This is the text string you want to find. You have to enclose the text in quotes. You can also specify the cell containing the text. This parameter is always case-sensitive.
• Within_text: This parameter specifies the text within which you want to search. You typically specify a cell containing the text. You could also type the text directly into the formula. In that
case, you also have to enclose it in quotes.
• Start_num: You use a numerical value to specify the character where the search is to start. This parameter is optional. If you don’t enter this value, the search will start at position 1.
The FIND function is case-sensitive and does not support wildcards. To get around this restriction, you can use the SEARCH function.
Excel returns a number as the result. This value tells you where the search string begins, meaning the first occurrence from left to right. If the term occurs again in the cell, the FIND function
will disregard it by itself. To find further positions, you have to use nesting. The returned value counts every single character, including spaces. The number indicates the position of the first
character of the returned string. This means that the first letter or number of the search is counted in the result.
Besides FIND, Excel also offers the FINDB function. Both functions achieve the same results and have identical syntax. They only differ in the character sets they support. FIND only works with
single-byte character sets (SBCS). This includes the Latin alphabet. If you use Asian characters from Chinese, Japanese and Korean (CJK), use FINDB because it supports double-byte character sets
(DBCS) for these languages. Each character is encoded in two bytes, so counting is adjusted accordingly.
The Excel FIND function in practice
For many users, the purpose of this function is not always immediately apparent. Finding the position of a search term within a text doesn’t seem all that useful at first. But the true power of this
function is revealed when used in combination with other functions.
FIND & FIND: Nested groups
Suppose we want to find the second, third or nth occurrence of the search term rather than the first.
This formula shows how you can use the optional third parameter. In the start_num position of this formula, we once again insert the formula that returns the position of the first occurrence. This
value plus one indicates the position at which you want the search to begin. If you then want to find a third position, nest the function again, and so on.
FIND & ISNUMBER: True or false statements
The FIND function in Excel allows you to form a true or false statement from the specified position: Does the text contain the search term or not?
The ISNUMBER function returns the value TRUE if the result of FIND is a number, otherwise it will return FALSE. Since the FIND function in Excel specifies the position of the term as a whole number,
the ISNUMBER function can respond to it. If the text does not contain the search term, FIND returns an error message, which of course is not a number, and ISNUMBER responds accordingly with FALSE.
You may also be interested in seeing where search terms appear. You can do this if you’ve entered data in multiple cells, such as a list of products sold. You can add this formula (like any other
formula) to the conditional formatting rules. This allows you to select any sales entry related to teddy bears, for example.
FIND & MID: Extracting characters
Product codes can be very long and confusing, so you may want to extract a certain number of characters from a string. Excel provides three functions for this purpose: LEFT, RIGHT and MID. These are
very useful on their own, but the formulas are even more powerful when combined with FIND. Suppose your product codes always follow a specific pattern consisting of letters, numbers, and hyphens:
ABCDE-A-12345-T. You want to extract the numerical part in the middle.
However, because the string does not have a fixed length, you can’t use the basic extract functions. These functions require a specific number of characters, which you can’t easily provide in this
case. But thanks to the hyphens, you can use the FIND function. It gives you the position information you need.
Since there are multiple hyphens in the string, you have to nest the FIND function. In this example, we’re assuming that the number part always contains five characters.
FIND & IF: If, then, otherwise
Product codes can be very long and confusing, so you may want to extract a certain number of characters from a string. Excel provides three functions for this purpose: LEFT, RIGHT and MID. These are
very useful on their own, but the formulas are even more powerful when combined with FIND. Suppose your product codes always follow a specific pattern consisting of letters, numbers, and hyphens:
ABCDE-A-12345-T. You want to extract the numerical part in the middle.
However, because the string does not have a fixed length, you can’t use the basic extract functions. These functions require a specific number of characters, which you can’t easily provide in this
case. But thanks to the hyphens, you can use the FIND function. It gives you the position information you need.
Since there are multiple hyphens in the string, you have to nest the FIND function. In this example, we’re assuming that the number part always contains five characters.
=PART(A2;FIND("-"; A2;FIND("-"; A2;FIND("-";A2)+1))+1;5)
However, if the length is not defined, further nesting of FIND functions can be helpful. Since the string you want to find ends in a hyphen, you can search for it to determine the length.
=PART(A2;FIND("-";A2;FIND("-";A2;FIND("-";A2)+1))+1; FIND("-";A2;FIND("-";A2;FIND("-";A2)+1)+1)-FIND("-";A2;FIND("-";A2;FIND("-";A2)-1))-3)
Admittedly, this formula is very confusing, but it actually manages to achieve the goal. No matter how many characters you place between the two hyphens, Excel will always extract the correct
characters using the FIND function.
FIND & IF: If, then, otherwise
You can also easily combine FIND with IF. For example, you may want a certain action to happen if a specific string occurs in the cell. You can do this by combining IF and FIND: If the string
appears, this will happen, otherwise that will happen. A problem may arise if FIND returns an error if the string does not appear. Therefore, you also have to use the ISERROR function.
If the FIND function doesn’t find the search term (“bear” in this example), it returns an error message. For ISERROR, this means the condition is met and IF returns the first option: No, the search
term does not appear. However, if the FIND function finds the string, it returns a number, which does not meet the condition for ISERROR. The alternative is returned: Yes, the search term appears.
FIND is a helpful function, especially when combined with other functions. It has a wide range of combination options and different applications. This small but useful function can help you solve
many problems you encounter when creating formulas in Excel.
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Moment distribution
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross and published in 1930 in an ASCE journal. The method only
accounts for flexural effects and ignores axial and shear effects. In the moment distribution method, every joint of the structure to be analyzed is fixed so as to develop the fixed-end moments. Then
each fixed joint is sequentially released and the fixed-end moments (which by the time of release are not in equilibrium) are distributed to adjacent members until equilibrium is achieved. The moment
distribution method in mathematical terms can be demonstrated as the process of solving a set of simultaneous equations by means of iteration. The moment distribution method falls into the category
of displacement method of structural analysis. | {"url":"https://www.thestructuralengineer.info/education/structural-analysis/moment-distribution","timestamp":"2024-11-05T10:07:34Z","content_type":"text/html","content_length":"51651","record_id":"<urn:uuid:b5bf4c4d-6afd-41ff-b312-0eb1868e59fe>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00614.warc.gz"} |
Exploring the Importance of Precision: How Many Decimals of Pi Do We Actually Need?
Precision is a crucial aspect of mathematics and scientific calculations. It allows us to obtain accurate results and make reliable predictions. In the realm of mathematics, one of the most
well-known and important numbers is pi (π). Pi is an irrational number that represents the ratio of a circle's circumference to its diameter. It has fascinated mathematicians for centuries, and the
search for more digits of pi has been an ongoing endeavor. In this article, we will explore the concept of precision, the value of pi, the quest for more digits, the applications of pi in various
fields, and the practical considerations of how many decimals of pi we actually need.
Key Takeaways
• Precision is crucial for obtaining accurate results in mathematics and scientific calculations.
• Pi (π) is an irrational number that represents the ratio of a circle's circumference to its diameter.
• The search for more digits of pi has been an ongoing endeavor for mathematicians.
• Pi has various applications in geometry, trigonometry, physics, and engineering.
• There is a trade-off between precision and efficiency when considering the number of decimals of pi needed.
The Concept of Precision
Defining Precision in Mathematics
Precision in mathematics refers to the level of detail or accuracy in a calculation or measurement. It is the degree to which a value is expressed in terms of significant figures or decimal places.
In mathematical terms, precision can be defined as the number of digits used to represent a value.
In order to understand the concept of precision, it is important to distinguish it from accuracy. While precision refers to the level of detail, accuracy refers to how close a measured or calculated
value is to the true or accepted value.
To illustrate the importance of precision, consider the following example:
In this table, the value of pi is represented with different levels of precision. As the number of digits used to represent pi increases, the precision also increases. This demonstrates how precision
can affect the level of detail and accuracy in mathematical calculations.
The Role of Precision in Scientific Calculations
Precision plays a crucial role in scientific calculations. It refers to how close the agreement is between repeated measurements. The precision of a measurement system is determined by the number of
significant figures it can provide. Significant figures are the digits in a measurement that are known with certainty, plus one digit that is estimated. The more significant figures a measurement
has, the more precise it is. For example, a measurement with three significant figures is more precise than a measurement with two significant figures. It allows scientists to make more accurate
calculations and draw more reliable conclusions.
The Value of Pi
A Brief History of Pi
Pi, symbolized by the Greek letter π, is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It has been studied and used in mathematics for thousands of
years, with its earliest known approximations dating back to ancient civilizations such as the Egyptians and Babylonians. One of the most notable contributions to the history of pi was made by an
eighteenth-century French mathematician named Georges Buffon. He devised a way to calculate π based on probability, which is still used today in various mathematical and statistical applications.
Buffon's method involves dropping needles onto a lined surface and analyzing the probability of the needle crossing a line. This experiment provides an estimation of π and demonstrates the connection
between probability and geometry.
The Importance of Pi in Mathematics
Pi is a fundamental mathematical constant that plays a crucial role in various mathematical calculations. It is a transcendental number, meaning that it is not the root of any non-zero polynomial
equation with rational coefficients. The value of pi, approximately 3.14159, is used in a wide range of mathematical formulas and equations, including those in geometry, trigonometry, and calculus.
Pi is also essential in the field of statistics, where it is used in probability distributions and statistical analysis. Its significance in mathematics cannot be overstated.
The Search for More Digits
The Quest for the Most Accurate Value of Pi
Throughout history, mathematicians and scientists have been on a relentless quest to calculate the most accurate value of pi. This pursuit has led to numerous approximations and formulas that have
pushed the boundaries of precision. One notable approximation is the value of pi accurate within 0.04% of the true value. This achievement, reached before the beginning of the modern era, showcases
the dedication and ingenuity of early mathematicians. Despite the advancements, calculating pi with absolute precision remains an elusive goal.
The Limitations of Calculating Pi
Calculating the value of pi is a challenging task due to its infinite and non-repeating decimal representation. While there are formulas and algorithms for calculating pi, achieving a high level of
precision requires significant computational resources and time. The search for more digits of pi has led to the development of powerful supercomputers and advanced mathematical techniques. However,
there is a limit to how accurately pi can be calculated, as the decimal expansion of pi is believed to be infinitely long and non-repeating. This inherent limitation poses a challenge for
mathematicians and scientists striving for absolute precision in their calculations.
Applications of Pi
Pi in Geometry and Trigonometry
In geometry and trigonometry, the value of pi plays a crucial role in various calculations and formulas. It is used to determine the relationships between angles, sides, and areas of geometric
shapes. For example, the sine of pi is equal to the y-coordinate of the point with polar coordinates (r, theta) = (1, pi). This relationship is fundamental in trigonometry and is used to calculate
the values of trigonometric functions for different angles. Additionally, pi is also used in formulas for calculating the circumference, area, and volume of circles and spheres. The precise value of
pi is essential for accurate calculations in geometry and trigonometry.
Pi in Physics and Engineering
Pi plays a crucial role in the fields of physics and engineering. It is used in various calculations and formulas to solve complex problems and understand the fundamental laws of nature. For example,
the Pi theorem, introduced by the American physicist Edgar Buckingham in 1914, is one of the principal methods of dimensional analysis. It states that if ...
Practical Considerations
How Many Decimals of Pi Do We Actually Need?
The precision of the value of pi is a topic of great interest and debate among mathematicians and scientists. While pi is an irrational number that goes on forever and never ends, the question
arises: how many decimals of pi do we actually need for practical purposes?
For most practical calculations, a precision of 7 significant decimal digits is sufficient. This level of precision is commonly used in everyday applications, such as in electronic calculators and
ordinary computer software. In fact, most calculators and software provide even more decimal places for pi.
However, there are certain specialized fields, such as theoretical physics and advanced engineering, where a higher precision of pi may be required. In these cases, calculations may involve millions
of digits of pi to ensure accuracy and reliability.
It is important to note that increasing the precision of pi comes at a cost. Calculating and storing a large number of decimal places requires more computational resources and can significantly
impact the efficiency of calculations. Therefore, there is a trade-off between precision and efficiency when it comes to using pi in practical applications.
The Trade-off Between Precision and Efficiency
When it comes to the precision of Pi, there is a trade-off between accuracy and efficiency. The more decimals of Pi we calculate, the more accurate our calculations become. However, this increased
precision comes at the cost of computational resources and time. It is important to find the right balance between precision and efficiency, depending on the specific application.
In some cases, a high level of precision is necessary. For example, in scientific research or engineering calculations, where small errors can have significant consequences, it is crucial to use a
large number of decimals of Pi. On the other hand, in everyday calculations or practical applications, a few decimals of Pi are usually sufficient.
Finding the optimal number of decimals of Pi requires considering the specific requirements of the problem at hand. It is a balance between achieving the desired level of accuracy and minimizing
computational costs. By carefully evaluating the trade-off between precision and efficiency, we can make informed decisions and optimize our calculations.
In conclusion, the precision of Pi is a topic that has fascinated mathematicians and scientists for centuries. While the value of Pi is infinite and its decimal representation continues indefinitely,
the question of how many decimals of Pi we actually need is a practical one. Through our exploration, we have discovered that the number of decimals required depends on the specific application. For
everyday calculations, a few decimal places of Pi are sufficient. However, in fields such as astronomy and physics, where extreme precision is necessary, a higher number of decimals may be required.
It is important to strike a balance between precision and practicality, as excessive precision can lead to unnecessary complexity and computational burden. Overall, understanding the importance of
precision in the context of Pi allows us to make informed decisions about the level of accuracy required for different applications.
Frequently Asked Questions
1. Why is precision important in mathematics?
Precision is important in mathematics because it allows for accurate calculations and reliable results. It ensures that the values obtained are as close to the true values as possible, minimizing
errors and uncertainties.
2. How is precision defined in mathematics?
Precision in mathematics refers to the level of detail or accuracy to which a value or calculation is expressed. It is often measured by the number of significant digits or decimal places used to
represent a number.
3. What is the role of precision in scientific calculations?
Precision plays a crucial role in scientific calculations as it determines the accuracy and reliability of the results. Scientists need to use precise values and calculations to ensure the validity
of their experiments and theories.
4. Why is pi important in mathematics?
Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is a fundamental constant in mathematics and appears in various mathematical formulas and
equations. Pi is important for geometry, trigonometry, and many other areas of mathematics.
5. What is the quest for the most accurate value of pi?
The quest for the most accurate value of pi involves using advanced mathematical algorithms and supercomputers to calculate pi to as many decimal places as possible. This pursuit is driven by the
desire to understand the mathematical properties of pi and to test the limits of computational precision.
6. How many decimals of pi do we actually need?
The number of decimals of pi needed depends on the specific application. For most everyday calculations, a few decimal places of pi are sufficient. However, in certain scientific and engineering
fields, more decimals may be required to ensure the accuracy and precision of the calculations. | {"url":"https://www.iancollmceachern.com/single-post/exploring-the-importance-of-precision-how-many-decimals-of-pi-do-we-actually-need","timestamp":"2024-11-13T19:02:29Z","content_type":"text/html","content_length":"1050626","record_id":"<urn:uuid:f906fc6d-376f-4238-824f-1147b0a2d901>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00271.warc.gz"} |
Measurement - The Lesson Study Group
In this resource, we discuss the teaching and learning of measurement. We highlight important aspects of teaching measurement topics to children and address the main stages of understanding that
students go through when learning what it means to measure a specific attribute (e.g. length, volume, etc.). Along the way, we examine areas that students often struggle with and make suggestions for
how to support students in understanding them.
Important Aspects of Measurement
Measurement is one of the most practical topics in the elementary school mathematics curriculum. In the K-5 CCSS, students are expected to learn the following measurement attributes: length, liquid
volume, elapsed time, mass (weight), area, volume, and angle. When discussing teaching and learning of measurement, we need to keep in mind there are three different (yet clearly related) aspects
that students must learn. They are:
• Understanding the attribute being measured (e.g. what is length?)
• Understanding the process of measurement (e.g. how do you describe how long something is?)
• Learning how to use measuring instruments (e.g., how do you use a ruler?)
The Four Stages of Measurement
There are typically four stages to the teaching and learning of each measurement attribute: (1) direct comparison; (2) indirect comparison; (3) measuring with arbitrary (non-standard) units; and (4)
measuring with standard units. In the CCSS, these stages are explicit in teaching and learning of linear measurements in Kindergarten through Grade 2. However, students need to experience these
stages with all measurable attributes even though they may spend less time on some stages as they gain more experiences in measurement. For example, when students learn about volume of solids in
fifth grade they will likely spend much less time on the first three stages compared to what they did when they learned about length. Still though, it is still important for students to go through
these stages so that they can build their understanding of what volume is, the process of measuring it, and how to use the tools to do so.
Stage 1: Direct Comparison
So, why is it important to follow these four stages as we begin our instruction on measurement? The major focus of the first two stages is to help students understand the attributes that are being
measured. After all, before we can measure anything, we really need to understand what it is that we want to measure. For example, before we can measure length, we need to understand what length is.
By putting two objects next to each other (direct comparison), students can determine which is longer and which is shorter. Through such experiences, students gain the understanding that length is
the amount of space between the two ends of an object. (Although we may use different words, “height” is not really an attribute. It can be thought of as length in the vertical orientation.) Of
course, through direct comparison activities, students are gaining some fundamental understanding about how to measure an object as well. For example, when comparing the lengths of two objects, it
is important that one end of the objects must be lined up.
You cannot say the segment on top in the figure below is longer just because it “sticks out” farther to the right.
Students will also learn that the “amount of space” we are interested in is along a straight path. Thus, we cannot simply compare the positions of the end points as shown in the figure below.
Before we can compare these two objects, the end points of the second object must be arranged in a straight line. These learnings are fundamental to a child’s understanding of measurement.
Stage 2: Indirect Comparison
Unfortunately, it is not always possible to directly compare two objects. In situations where it is not, it is sometimes useful to use a third object that can be compared directly to each of the two
objects and act as a reference between them. For example, if one doorway is wider than your arm span but another doorway is narrower than your arm span, you know that the first one is wider than the
second one. Indirect comparison provides more flexibility as you compare two objects. It also provides opportunities for children to experience an important mathematical property of relationships
called transitivity. This property states that if a > b and b > c, then a > c. Of course, the formal study of such property will not take place until much later.
Stage 3: Measuring with Arbitrary (Non-Standard) Units
Perhaps a more important reason to teach and learn indirect comparison is that it sets the stage for the most fundamental idea about the measurement process: the use of a unit. When using a third
object to indirectly compare, its quantity may not always be in between that of the original two objects. For example, a wooden stick may be much shorter than both of the two doorways. In those
cases, however, it may be possible to determine that one doorway is taller than three (of the same) wood sticks put end to end while the other one is shorter than three wood sticks. Now, we can say
that the first doorway is taller (longer) than the second one.
You can easily see that such experiences become the foundation for the idea of expressing an attribute in terms of the number of a third object, unit, necessary to “cover” it. When we move into this
stage, we are now indeed “measuring” in the sense that we are assigning a number to an object in terms of how much of the attribute it has. There are many merits for expressing the amount of an
attribute using numbers. Clearly, it simplifies the process of comparison as we no longer need to find different objects to use as the reference — comparison of multiple objects can be easily done
by simply comparing numbers of one established unit. Another important advantage to assigning numbers to attributes is that we can answer not only “which is longer?” but also “by how much?” In
general, once we express the amount of an attribute with numbers, arithmetic operations may be used to answer more complex questions. Although the CCSS does not explicitly state these merits, I hope
teachers help students experience and understand them.
Why Teach Students to Use Non-Standard Units?
Some people may argue that, once we get to this stage, we should just use standard units. This argument perhaps makes sense later in the elementary grades after students have learned about measuring
three or four different attributes. However, at the primary grade level, it is important to keep in mind that students are still learning about the process of measurement: pick a unit, then
determine how many iterations of the unit is necessary to equal the object’s attribute that you are measuring. For us, this is so obvious, but not so with children. Introducing standard units at
this stage will require children to deal with two new ideas simultaneously — new units and a new process. There are also other considerations. First, some units may be too small or too large so
that the size of the resulting numbers may not be appropriate for children at this particular time. By using non-standard units, teachers can control the range of numbers students might obtain.
Also, it is important to note that measuring with standard units typically means measuring with various instruments. For example, if you are measuring with inches, you are most likely to be
measuring with a ruler. However, learning to use a ruler, or any measurement instrument, is also a challenging task in and of itself — this might be a third new idea students have to grapple with if
we are introduce standard units without having taught non-standard units.
Stage Four: Measuring with Standard Units
Although it may sound a bit paradoxical, the use of non-standard units is a useful experience for children to understand the need for having standard units. For example, if two students measure the
width of the same doorway using their pencils, they may get different results. This type of experience will allow them to see that they cannot compare numbers unless their units are the same. This
is when we can introduce the idea of common units, then finally the idea of standard units such as inches, feet, centimeters and meters.
Teaching Measurement Tools
Finally, learning how to measure with common instruments such as rulers is not as simple as adults might think. Because of this, it may be useful if children had some experiences using their own
measurement tools. For example, during the third stage (measuring with non-standard unit), students can tape together index cards to form their own measuring “tape.” Initially, students may count the
number of index cards every time they measure an object, but eventually they may realize labeling the cards 1, 2, 3,… would make it easier. The figure below shows the evolution of a
student-constructed “ruler.”
Such experiences will allow students to understand that what we are counting on a measurement tape is the number of spaces between the tick marks, and the numerical label at a given tick mark
indicates the total number of units up to that mark. Furthermore, as we learned in the first stage, the end (actually the starting point) of the measuring tape must be lined up with the end of the
object, rather than the tick mark on the measuring tape labeled “1.” A variety of home-made measuring instruments can be made to measure length, capacity/volume, weight, and even angles. Making
and measuring with student-made instrument may be a very fruitful experiences as students learn to measure with standard units.
Discussion Questions
After reading this resource, it may be helpful to discuss the following questions as a team.
1. What ideas about teaching-learning measurement discussed in this essay strike you as important in your setting?
2. How does your curriculum handle the topic of measurement? What about it do you like and/or notice that has opportunity for improvement?
3. What experiences have your students had that currently inform their understandings of measurement?
4. What makes it difficult for students to progress through each of the four stages of measurement?
1. From direct to indirect comparison
2. From indirect comparison to non-standard units
3. From non-standard to standard units
5. What do you want to learn more about?
Next Steps
This concludes our overview of teaching and learning measurement. Our resources below include more specific information on the measurement attributes covered in the CCSS, and are meant to support a
deeper look at your area of focus.
See the Resource tiles below for expanded discussions on a selection of sub-topics related to measurement. | {"url":"https://lessonresearch.net/content-resource/measurement-2/","timestamp":"2024-11-04T08:14:16Z","content_type":"text/html","content_length":"280945","record_id":"<urn:uuid:a8fd388c-bf70-46ab-a3cb-45a7a8aacbd8>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00694.warc.gz"} |
watts_strogatz_graph(n, k, p, seed=None)[source]¶
Return a Watts-Strogatz small-world graph.
n : int
The number of nodes
k : int
Each node is connected to k nearest neighbors in ring topology
Parameters :
p : float
The probability of rewiring each edge
seed : int, optional
Seed for random number generator (default=None)
First create a ring over n nodes. Then each node in the ring is connected with its k nearest neighbors (k-1 neighbors if k is odd). Then shortcuts are created by replacing some edges as follows:
for each edge u-v in the underlying “n-ring with k nearest neighbors” with probability p replace it with a new edge u-w with uniformly random choice of existing node w.
In contrast with newman_watts_strogatz_graph(), the random rewiring does not increase the number of edges. The rewired graph is not guaranteed to be connected as in connected_watts_strogatz_graph
[R313] Duncan J. Watts and Steven H. Strogatz, Collective dynamics of small-world networks, Nature, 393, pp. 440–442, 1998. | {"url":"https://networkx.org/documentation/networkx-1.9/reference/generated/networkx.generators.random_graphs.watts_strogatz_graph.html","timestamp":"2024-11-11T13:58:24Z","content_type":"text/html","content_length":"17133","record_id":"<urn:uuid:1b534a77-db7d-40df-887e-cd4eb0c06f02>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00597.warc.gz"} |
Euler Circle Spring Paper: Čebotarev Density Theorem
Creative Commons CC BY 4.0
In this paper, we do exactly what the title implies: prove the Čebotarev Density Theorem. This is an extremely valuable theorem because it is a vast generalization of Dirichlet's Theorem on primes in
an arithmetic progression. Our theorem goes even further to the case of other number fields; we will show that the prime ideals in an imaginary quadratic field K are virtually equidistributed among
the conjugacy classes of Artin symbols in the Galois group of a Galois extension L over K. Note that L need not be abelian over K! | {"url":"https://tr.overleaf.com/latex/templates/euler-circle-spring-paper-cebotarev-density-theorem/qzgxffhsqmzy","timestamp":"2024-11-14T20:11:37Z","content_type":"text/html","content_length":"95051","record_id":"<urn:uuid:b967ddb2-1c5e-4a3c-9fd7-b622e3eb4977>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00683.warc.gz"} |
Cantor’s Diagonal Argument
Cantor’s Diagonal Argument
“Diagonalization seems to show that there is an inexhaustibility phenomenon for definability similar to that for provability”—Franzén…
Georg Cantor (1845-1918)’s correspondence with mathematician Richard Dedekind (1831-1916) in the years 1873-74 has been narrated in a previous newsletter. Their letters detail the process by which
Cantor arrived at the discovery that the set of the real numbers is uncountably infinite. That is, Cantor showed that although both the natural numbers and the real numbers are infinite in number and
so go on forever, there “aren’t enough” natural numbers to create a one-to-one correspondence between them and the real numbers. Cantor’s brilliant discovery, in other words, showed rigorously and
undeniably that infinity comes in different sizes, some of which are larger than others. One of his methods of proving this assertion, Cantor’s diagonal argument, was later employed by both Gödel and
Turing in their most renowned results.
The Uncountability of Real Numbers (Cantor, 1891)
Cantor first demonstrated the uncountability of real numbers using a topological proof based on the Bolzano-Weierstrass theorem in the 1874 paper Ueber eine Eigenschaft des Inbegriffes aller reellen
algebraischen Zahlen (“On a Property of the Collection of All Real Algebraic Numbers“) in Crelle’s Journal. Here, Cantor showed:
The Uncountability of Real Numbers
Given any sequence of real numbers and any interval [α ... β], one can determine a number η in [α ... β] that does not belong to the sequence. Hence, one can determine infinitely many such
numbers η in [α ... β].
A real number ℝ is a value of a continuous quantity that can represent a distance along a line. Any real number can be determined by a possibly infinite decimal representation, such as that of e.g.
8.632, 0.00001, 10.1 and so on, where each consecutive digit is measured in units one tenth the size of the previous one. The statement that the real numbers are uncountable is equivalent to the
|ℝ| ≠ |ℕ|.
Although Cantor had already shown the statement to be true in 1874, he reproved it again seven years later using what would later be known as the diagonal argument.
His proof was published in the 1891 paper Ueber eine elementare Frage der Mannigfaltigkeitslehre (“On an Elementary Question of Manifold Theory”). In his paper, Cantor presents the argument as
Proof that the set of all infinite binary sequences are not denumerable. Consider the set M of all infinite sequences of the binary numbers m and w. Sequences such as:
E₁ = (m, m, m, m, m, ...),
E₂ = (w, w, w, w, w, ...),
E₃ = (m, w, m, w, m, ...),
E₄ = (w, m, w, m, w, ...),
E₅ = (m, m, w, w, m, ...)
Cantor asserts that there exists a set M that does not have the “breath” of the series E₁, E₂, E₃ … , meaning M is of a different size than the sum of each sequence En, i.e. that even though M is
constructed of all the infinite sequences of the binary numbers m and w, he can always construct a new sequence E₀ which “is both an element of M and is not an element of M.”
The new sequence E₀ is constructed using the complements of one digit from each sequence E₁, E₂, … En. A complement of a binary number is defined as the value obtained by inverting the bits in
the representation of the number (swapping m for w and visa versa). So, the new sequence is made up of the complement of the first digit from the sequence E₁ (m), the complement of the second
digit from the sequence E₂ (w), the complement of the third digit from the sequence E₃ (m) and so on to finally the complement of the nth digit from the sequence En. From the example sequences
above, the new sequence E₀ would then be:
E₀ = (w, m, w, w, w, ...)
By its construction, E₀ differs from each sequence En since their nth digits differ. Hence, E₀ cannot be one of the infinite sequences in the set M.
The proof provided below applies the argument specifically to prove the uncountability of the real numbers ℝ. It is an excerpt from the book Real Analysis and Applications* by Frank Morgan (2005).
Proof of the Uncountability of the Real numbers ℝ (Morgan, 2005)
Assume that the following proposition is false:
The set of reals ℝ is uncountable.
We will derive a contradiction. In particular, we will assume that the reals can be listed, and then exhibit a real number missing from the list, the desired contradiction. Since the argument
applies to any list, that will complete the proof.
Suppose that the reals were countable. Then the positive reals would also be countable and could be listed, for example as such:
1. 6 5 7 . 8 5 3 2 6 0 ...
2. 2 . 3 1 3 3 3 3 ...
3. 3 . 1 4 1 5 9 2 ...
4. . 0 0 0 3 0 7 ...
5. 4 9 . 4 9 4 9 4 9 ...
6. . 8 7 3 2 5 7 ...
To obtain a contradiction, it suffices to show that there exists some real α that is missing from the list. The construction of such an α works by making its first decimal place different from
the first decimal place of the first number of the list, by making the second decimal place different from the second decimal place of the second number, and in general by making the nth decimal
place different from the nth decimal place of the nth number on the list.
Even simpler, for our α we'll make the nth decimal place 1 unless it is already 1, in which case we'll make it 2. By this process, for our example list of numbers, we obtain:
α = . 1 2 2 1 1 1 ...
Which, by construction cannot be a member of the list we created. And so, by contradiction, our list of all reals cannot contain every number, and so must be uncountable.
Summarized, to prove the uncountability of the reals we first list all such numbers and assume this list is exhaustive. Next, starting from the top, we construct a new real number using one decimal
place from each of the numbers in the list, altering each digit to not match the digits from the list we use in the construction, to infinity. We observe that this new number, the nth number on the
list will always differ from all of those already the list. Thus, our list of the real numbers is not exhaustive, and so they must be uncountable.
Another way of thinking about it is provided in Richard Hammock’s Book of Proof*(2013):
Imagine making a table for the function f(n): ℕ → ℝ where values of n in ℕ are the left-hand column and the corresponding values of f(n) are on the right. The first few entries might look
something as follows:
n | f(n)
1 | 0 . 4 0 0 0 0 0 0 0 0 0 ...
2 | 8 . 5 0 0 6 0 7 0 8 6 6 ...
3 | 7 . 5 0 5 0 0 9 4 0 0 4 ...
4 | 5 . 5 0 7 0 4 0 0 8 0 4 ...
5 | 6 . 9 0 0 2 6 0 0 0 0 0 ...
6 | 6 . 8 2 8 0 9 5 8 2 0 0 ...
7 | 6 . 5 0 5 0 5 5 5 0 6 5 ...
8 | 8 . 7 2 0 8 0 6 4 0 0 0 ...
9 | 0 . 5 5 0 0 0 0 8 8 8 8 ...
10 | 0 . 5 0 0 2 0 7 2 2 0 7 ...
In this table, the real numbers f(n) are written with all their decimal places trailing off to the right. Thus, even though f(1) happens to be the real number 0.4, we write it as 0.4000000....,
etc. There is a diagonal bolded band in the table. For each n ∈ ℕ, this diagonal covers the nth decimal place of f(n):
The 1st decimal place of f(1) is the 1st entry
The 2nd decimal place of f(2) is the 2nd entry
The 3rd decimal place of f(3) is the 3rd entry etc
The diagonal helps us construct a number b ∈ ℝ that is unequal to any f(n). Just let the nth decimal place of b differ from the nth entry of the diagonal. Then the nth decimal place of b differs
from the nth decimal place of f(n).
- Excerpt, Book of Proof* by Richard Hammock (2013)
The Incompleteness of First-Order Logic (Gödel, 1931)
Forty years later, logician Kurt Gödel (1906-78) utilized Cantor’s diagonal argument to show that any system of logic is inherently incomplete, in his famous paper:
Colourized photograph of Kurt Gödel and the first page of his 1931 paper utilizing Cantor’s diagonal argument
Biographer John W. Dawson describes Gödel’s result as follows:
"Gödel's achievement was to show how to construct a statement whose undecidability could be (informally) demonstrated. He did so by showing in very detailed fashion that a large number of basic
syntactic notions are representable, through their code numbers, by formulas of formalized arithmetic.
Specifically, he showed that there is a binary, primitive recursive predicate B(x,y) that formalizes the notion "x is the code number of a proof of the formula whose code number is y", so that if
n and m are natural numbers and n and m are the symbols denoting those numbers within the formal theory, then B(n,m) is disprovable otherwise.
It follows that the monadic predicate Bew(y) defined as an abbreviation for the formula (∃x)B(x,y), formalizes the notion "y is provable". The negation of Bew(y) then formalizes the notion "y is
not provable"; and that notion, Gödel realized, could be exploited by resort to a diagonal argument reminiscent of Cantor's."
- Excerpt, Logical Dilemmas* by John W. Dawson (2006)
Complicated as Gödel’s proof by contradiction is, its construction may be superficially described as follows:
• Statements in the system (which may be true of false) can be represented by natural numbers, later known as Gödel numbers. Statements which are true have certain properties which are different
than statements which are false. The truth and falsehood of statements in the system in other words may be demonstrated by examining their Gödel numbers.
• As a consequence, one can construct a mathematical formula which expresses the idea that “statement S is provable in the system”. This formula can applied to any statement S in the system in
order to investigate its provability.
• Using Cantor’s diagonal argument, in all formal systems which are complete, we must thus be able to construct a Gödel number whose matching statement, when interpreted, is self-referential.
• The meaning of one such statement is equivalent to the English statement “I am unprovable” (read: “The Liar Paradox”). Such a self-referential statement demonstrates that there must exist
statements in the system which are neither provable nor disprovable.
The fact that there exists a statement in the system which is neither provable nor disprovable demonstrates that that the system cannot be consistent if it is both complete (every formula may be
derived from its axioms) and decidable (there exists an effective method for deriving the correct answer to any question).
The Undecidability of the Entscheidungsproblem (Turing, 1937)
Colourized photograph of Alan Turing and the first page of his 1937 paper utilizing Cantor’s diagonal argument
The Entscheidungsproblem (so-called “decision problem”) was a mathematical challenge originally posed by David Hilbert (1862–1943) and his doctoral student Wilhelm Ackermann (1896–1962) in 1928. The
challenge asked for an algorithm which takes a logical statement in a formal language and outputs “True” or “False” depending on the truth value of the statement. It was published in Turing’s
By introducing his theoretical “universal computing machine” (now called a Turing machine), defined as:
A Turing machine (a-machine) is a mathematical model of computation that defines an abstract machine which manipulates symbols on a strip of tape according to a table of rules.
Turing in the paper proved that:
• A Turing machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm (a finite sequence of well-defined instructions). Moreover:
• There cannot be a solution to the Entscheidungsproblem because the halting problem for Turing machines is undecidable, meaning that it is not possible to decide algorithmically whether a Turing
machine will ever halt (read: Turing Uncomputability).
Briefly, sets of numerical strings have two properties that are of particular relevance to Turing’s paper. They are enumerability and decidability:
A set E of strings is computably enumerable if it is possible to program a computer to compute and print out the members of E.
A set E of strings is computably decidable if it is possible to program a computer to decide, given any string s of symbols, whether or not is in E.
Turing’s proof of the so-called undecidability theorem is structured as follows (Franzén, 2004):
Create a computable enumerable list of all possible programs:
P₀, P₁, P₂, ..., Pᵢ
Define i to be the index of the program Pᵢ. Now let K be the set of all i such that Pᵢ terminates and outputs a value when given the index i of the program as an input. The set K is computably
enumerable by definition.
However, the set K is not decidable. To show this, suppose that K was decidable. We can then define a procedure which given any input i first checks whether i is in K. If not, we give 0 as
output. If i is in K so that Pᵢ does terminate with i as input, we let Pᵢ compute its result and then give as output that result with a further symbol added as the end.
Since this defines a program P that given any string computes another string as output, P must be identical with Pm for some m. But P and Pm do not agree on m, so they are not identical. Hence, K
is not decidable.
Here we see the diagonal argument appear in the sentence “If i is in K so that Pᵢ does terminate with i as input, we let Pᵢ compute its result and then give as output that result with a further
symbol added as the end.”
The properties and implications of Cantor’s diagonal argument and their later uses by Gödel, Turing and Kleene are outlined more technically in the paper:
* This story contains Amazon Affiliate links | {"url":"https://www.cantorsparadise.org/cantors-diagonal-argument-c594eb1cf68f/","timestamp":"2024-11-05T10:20:16Z","content_type":"text/html","content_length":"50273","record_id":"<urn:uuid:3d683834-5fcb-475d-80c6-ad3549838907>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00239.warc.gz"} |
Gauge transform
Many fields arising in PDE can be viewed as a section or connection on a gauge bundle, which is typically a principal G-bundle over a domain ${\displaystyle \Omega }$, where G is the gauge group. To
(locally) coordinatize these sections and connections, one chooses a (local) trivialization of the gauge bundle, which identifies the bundle with the trivial bundle ${\displaystyle \Omega \times G}$.
This converts sections into G-valued fields ${\displaystyle \sigma }$, and connections D into ${\displaystyle {\mathfrak {g}}}$-valued one-form ${\displaystyle A}$, thus ${\displaystyle D_{\alpha }=\
partial _{\alpha }+A_{\alpha }}$. Such a trivialization is known as a gauge.
Given any G-valued field U, one can transform the trivialization by applying the group element U(x) to the fiber of the trivial bundle at x. This is a gauge transform; it maps ${\displaystyle \sigma
}$ to ${\displaystyle U\sigma }$ and ${\displaystyle A_{\alpha }}$ to ${\displaystyle UA_{\alpha }U^{-1}-\partial _{\alpha }UU^{-1}}$.
One reason for applying a gauge transform is to convert a connection ${\displaystyle A_{\alpha }}$ into a better form. However, there is an obstruction to flattening a connection entirely, namely the
curvature of the connection. Nevertheless, there are a number of gauges which seek to make the connection as mild as possible given its curvature.
Examples of gauges
Equations in which gauge transforms arise | {"url":"https://dispersivewiki.org/DispersiveWiki/index.php?title=Gauge_transform","timestamp":"2024-11-03T16:03:45Z","content_type":"text/html","content_length":"28835","record_id":"<urn:uuid:1f5e4520-730b-479f-a85c-7be93c0ce2e7>","cc-path":"CC-MAIN-2024-46/segments/1730477027779.22/warc/CC-MAIN-20241103145859-20241103175859-00520.warc.gz"} |
0009741: Simplified Arbitrary Grid - MantisBT
View Issue Details
ID Project Category View Status Date Submitted Last Update
0009741 ardour features public 2024-07-10 22:23 2024-07-10 22:23
Reporter erdavis7 Assigned To
Priority normal Severity minor Reproducibility N/A
Status new Resolution open
Platform Apple Macintosh OS MacOS OS Version 10.12 or later
Product 8.6
Summary 0009741: Simplified Arbitrary Grid
Grid Mode and its associated menus should be simplified and enhanced to support fractional divisions of any denominator.
Currently, the Grid Mode menu (on the toolbar) has three separate sections: one for powers of 2, one for triplets, and one for
quintuplets. Additionally, the "Edit/Snap & Grid" menu has a section for septuplets. Instead, there should be a single menu which adapts to whatever "base" fraction is currently selected,
with the base fraction being any fraction with a numerator of 1, and an arbitrary denominator with the powers of 2 removed.
The denominator of the base fraction should be selectable with a text input box, where the user can type any number to divide the duration of a whole note. If the requested number contains
a factor of a power of 2, this should be removed to set the base fraction internally.
For a given base fraction, the grid menu should be filled with the base fraction, and below that, its multiples by powers of two. For
example, when the base fraction is 1/3, then the grid menu should read:
No Grid
Description ...
If the user had requested a base fraction of 1/6, this would select a base fraction of 1/3 (which would not necessarily be externally
visable), but the active fraction of the grid would remain 1/6. Regardelss of the base fraction, the commands "Previous Quantize Grid
Choice" and "Next Quantize Grid Choice" (hotkeys "5" and "6", respectively) should navigate up and down this grid menu.
The default base fraction should be 1/1, which would give the standard durations of powers of 2:
No Grid
The behavior described here would result in a simpler, unified menu, and it would also be more flexible.
If a swing feature is ever added to the Grid, then the swing could be applied to the current power of 2, shifting every other grid line.
Tags fraction, grid
Date Modified Username Field Change
2024-07-10 22:23 erdavis7 New Issue
2024-07-10 22:23 erdavis7 Tag Attached: fraction
2024-07-10 22:23 erdavis7 Tag Attached: grid | {"url":"https://tracker.ardour.org/view.php?id=9741","timestamp":"2024-11-12T12:28:16Z","content_type":"text/html","content_length":"16264","record_id":"<urn:uuid:e0f48eb3-f46c-4932-b18b-869360c76aaa>","cc-path":"CC-MAIN-2024-46/segments/1730477028273.45/warc/CC-MAIN-20241112113320-20241112143320-00739.warc.gz"} |
How to add a legend to a graph in Matlab?
To add a legend to a graph in MATLAB, you can use the "legend" function. Here is an example of how to add a legend to a simple line plot:
1 x = 0:0.1:2*pi;
2 y1 = sin(x);
3 y2 = cos(x);
5 plot(x, y1, 'r-', x, y2, 'b--');
6 legend('sin(x)', 'cos(x)');
In the example above, the "plot" function is used to plot two lines, one for the sine function and one for the cosine function. The color and line style for each line is specified using the 'r-' and
'b--' arguments. Then, the "legend" function is used to add a legend to the plot, with the labels 'sin(x)' and 'cos(x)' corresponding to the two lines.
You can also customize the location, font size, and other properties of the legend using additional arguments to the "legend" function. For example:
1 legend('sin(x)', 'cos(x)', 'Location', 'northwest', 'FontSize', 12);
This sets the legend location to the northwest corner of the plot and the font size to 12. | {"url":"https://devhubby.com/thread/how-to-add-a-legend-to-a-graph-in-matlab","timestamp":"2024-11-12T03:13:29Z","content_type":"text/html","content_length":"130118","record_id":"<urn:uuid:869a4935-337e-492d-82df-fb196017efe1>","cc-path":"CC-MAIN-2024-46/segments/1730477028242.50/warc/CC-MAIN-20241112014152-20241112044152-00263.warc.gz"} |
Using an oscilloscope to measure R, X, VSWR and power
The oscilloscope is a very powerful tool that has many functions. A very interesting one, at least for educational purposes, is the measurement of complex loads: R, X, VSWR, Return Loss, Phase,
capacitance, inductance and real power.
The measurement method
The measurement of an unknown load, and the power that is being delivered to it, is conceptually very simple thanks to the Ohm’s law:
Z? = V1 / I1
Power at Z? = V1 * I1
The only difference between these measurements and those made at DC is that here all involved values are complex numbers. Beside the magnitude of V1 and I1, we need also to know φ, i.e. the phase
difference between voltage and current.
As we know, an oscilloscope can measure voltage, but how can it measure current? The answer is very simple, use one of these configurations:
We measure V1 by simply connecting the CH1 probe across the load. Instead we measure I1 by adding a known resistor R in series with Z and measuring the voltage drop across it with the CH2 probe. This
measurement gives us the current I along the entire circuit (I = V2/R). Thus, the phase difference between CH1 and CH2 gives the φ between voltage and current.
Measurement example #1
To easen the measurements I prepared a simple fixture made of two SMA female panel connectors soldered back to back. The central pins are soldered together, while the ground flanges are connected by
mean of a 12Ω resistor, which is my known “R”.
Fixture used to take complex loads measurements.
Form my example #1, I prepared a SMA male connector with soldered a 33Ω SMD resistor in parallel with a 330pF SMD capacitor. This test load will later allow me accurate and repeatable measurements on
a Vector Network Analyzer for comparison.
I can proceed now to connect my oscilloscope:
The channel 1 probe (yellow on my scope) is connected to the central pin and to the ground on the generator side.
The channel 2 probe (cyan on my scope) is connected to the 12Ω “R” resistor towards the load. Its ground clip is connected to the ground on the generator side together with the ground of channel 1.
The oscilloscope probes must be set to 10X to reduce their parasitic capacitance, thus reducing the measurement error.
The generator in my case is lab RF signal generator set to 13dBm output at 10 MHz, but any sine wave generator can be used. For example, it can be a regular ham RTX, maybe with an attenuator to
reduce the mismatch seen by the transceiver.
This is what I see on my scope with this setup:
The values needed to perform the calculations are:
1. V1=0.880V[rms], in my setup, the yellow trace;
2. V2=0.282V[rms], in my setup, the cyan trace;
3. WL=100ns, i.e. the wavelength; this value is taken by measuring the length of a full wave;
4. PD=-5.84ns: this the horizontal shift of the two curves;
Note that the “PD” (Phase Difference) value is negative: by convention of the online calculator we will use, we have that:
• the value must be given as negative if we are measuring the distance from the yellow trace (CH1) to the cyan trace (CH2) going leftwards, like in this example;
• the value must be given as positive if we are going to measure rightwards;
• the measurement unit of WL and PD does not matter as long as it is the same; you can measure the distances on the screen with a ruler and enter the data as mm or inches.
To measure the phase difference with the best precision, it is useful to zoom in the trace by increasing the horizontal scale:
At this point we are ready for the calculations. Instead of going through al the complex number calculations, I developed a dedicated online calculator which I will use for this test. This is the
input data:
The calculator reports the usual information normally available from VNAs: R, X, Return Loss, etc. Note that, unlinke many cheap antenna analyzers, this measurement method is able to correctly detect
the reactance sign.
We shall now compare these results with those obtained by other instruments.
VNA analysis
In order to verify these results, I measured the same load using a calibrated Vector Network Analyzer, so we can compare them with a more accurate measurement.
This is the result:
Let’s compare the results:
Label Scope VNA Error
R 22.954Ω 23.224Ω 1.17%
X -13.434Ω -14.356Ω 6.64%
RL -6.247dB -7.804dB 1.56dB
Phase -143.151° -140.7° 2.45°
Par.R 30.817Ω 32.098Ω 4.08%
Par.C 302.268pF 306.504pF 1.39%
As we can see, the impedance measurement made by the oscilloscope looks quite accurate.
Power measurement
As we have seen, besides of the load impedance, this method measures also the real power delivered to the load.
From the previous chapter, we already know the impedance of the load as measured by the VNA: R=23.224 X=-14.356. This impedance is the equivalent of a 32.098Ω resistor in parallel with a 306.504pF
The impdedance of the fixture with the 12Ω resistor reads R=12.018 X=1.021, which is the equivalent of a 12.018Ω resistor in series with a tiny 16nH inductor.
We can now measure the RMS voltage using a Rohde & Schwarz RF millivoltmeter:
We can now create a SPICE circuit with a 10MHz generator providing 0.870Vrms (as read by the millivoltmenter) on the discrete components calculated by the VNA. This is the SPICE circuit with the RMS
power levels calculated by it:
These are the power values calculated by SPICE compared to those calculated by the online calculator from the oscilloscope data:
Label Scope SPICE Difference
Power on Z 12.676mW (11.03dBm) 12.376mW (10.93dBm) 0.10db
Power on R 6.627mW (8.21dBm) 6.3959mW (8.06dBm) 0.15db
Overall power 19.303mW (12.86dBm) 18.777mW (12.74dBm) 0.12db
The power measurements, with a difference of only 0.10/0.15dB with the VNA+RF millivoltmeter method, appear to be quite accurate.
Measurement example #2
In this example, we shall measure an inductor made of 36 turns of AWG#20 wire on an Amidon T80-2 toroid at 5 MHz. I have chosen this configuration because it is one of the examples documented by
From Amidon tables we see that its inductance should be 7.8μH.
We can now prepare the fixture and take our measurement:
The oscilloscope gives us this reading:
To have a better reading of the phase difference, once again I have zoomed the oscilloscope reading:
Unlike the example #1, this time the blue trace is shifted to the right, showing that the load is inductive instead of capacitive. According once again to the convention, since we are measuring
starting from CH1 and going rightwards, the phase difference (45.6ns) must be positive.
We have now all the data we need and we can feed the online calculator:
Results validation
Let us as usual take the measurement of the same inductor with a calibrated VNA to compare the results:
The result of the VNA is the following:
The VNA measures an inductance of 7.8314μH, which matches Amidon’s declared 7.8μH. The the scope reports a little bit higher 8.154μH (4% difference).
To be fair I have to say that to obtain this measurement, the VNA reference plane has been moved at the inductor plane, thus cancelling the impedance transformations due to the fixture. This function
is quite advanced and it is not available on most cheap VNA/Antenna Analyzers. If I repeat the measurement without this correction, as most of the cheap VNAs do, I obtain this:
The series inductance now “appears” as 8.0435μH, which is very close to the 8.154μH measured by the scope (+1.3%).
• With a simple fixture, a two channels oscilloscope can be used to measure voltage, current and phase difference among them (φ).
• A RF source is required to do the measurements: it can be a signal generator, a normal transceiver or anything else able to generate enough power for the scope to measure the voltage.
• With the data above, the Ohm’s law and some complex numbers calculation we can derive all the data normally produced by vector network analayzers (R, X, RL, phase, VSWR, capacitance, inductance,
• With the same data, the real power delivered on the load under test can be calculated thus making the oscilloscope act as a power meter.
• Unlike common power meters, that can measure power on a given load (for example, 50Ω), this method allows the measurement of power actually dissipated by load of any impedance.
• The stray capacitance of the probes and the low precision in voltage measurement ad high frequencies limit the maximum frequency at which this method gives reliable results; with a common
50-100MHz oscilloscope, probably the safe limit is around 30 MHz.
8 Comments
1. Nice article.
Would it be possible to send the formulas used by your calculator?
Thank you.
2. If it’s possilble , pleas send me too,
good work!
thanks adam
3. excellent..! tnx.
de SV1COX
4. Good documentation, but I´m missing the equation of the calculator.
Would it be possible to send the formulae? Thank you for your good work. Greeting from Salzburg in Austria.
Siegfried Georg Burger
5. Hello, This is excellent. I was wondering though on the output side:
is X the same as Xc (capacitive reactance),
is R the same as ESR,
and what is Qz (part of the output of the calculator but not this example).
6. Measurement example #1 shows a table to compare the VNA and O-Scope measurements.
I see that the RL of the O-Scope is -6.247dB and the VNA is -7.804dB.
Yet I cannot find where you got -6.247dB, did I overlook something. I see the calculation on the website that you used shows 7.806dB in RL
Did you do some other calculations to get -6.247dB that I overlooked?
7. Hi, Thank you, can you share the equations which you used to calculate values?
□ You can find the equations at this link: | {"url":"https://www.iz2uuf.net/wp/index.php/2018/02/14/oscilloscope-as-vna/","timestamp":"2024-11-02T11:13:45Z","content_type":"text/html","content_length":"77493","record_id":"<urn:uuid:ffb7b508-96bb-4b0f-b765-ea4dfd16e97e>","cc-path":"CC-MAIN-2024-46/segments/1730477027710.33/warc/CC-MAIN-20241102102832-20241102132832-00467.warc.gz"} |
Quant Question of the Day Chat
Hey guys, wondering if it’s more effective to dedicate study sessions to one/a few topics at a time (e.g. doing combinatorics & probabilities-related questions only in the session) or do a drill of
questions with mixed topics?
You might find below useful:
3 tips on how to study effectively
Explore how the brain learns and stores information, and find out how to apply this for more effective study techniques.
"Forgetting is the friend of learning." - Robert Bjok.
1. How?
📖 Testing with flashcards and quizzes is an effective study technique. It actively retrieves and strengthens memory, providing a more accurate assessment of what you know.
🤔 Making mistakes while trying to recall information can improve long-term learning, as it activates relevant knowledge and aids in better integration.
2. What?
🃏 Mixing different subjects in study sessions, known as interleaving, enhances retention and strengthens memory by forcing the brain to temporarily forget and retrieve information.
3. When?
📆 Spacing review sessions across multiple days with rest and sleep in between is more effective than cramming, as it allows the brain to actively integrate knowledge in the neocortex for
long-term retention. | {"url":"https://gmatclub.com/forum/quant-chat-group-join-the-discussion-400376-3380.html","timestamp":"2024-11-04T05:02:39Z","content_type":"application/xhtml+xml","content_length":"823381","record_id":"<urn:uuid:2a5b2808-8b82-4385-a80d-b6d478c85101>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00842.warc.gz"} |
Counting Algebraic Structures
Posted by John Baez
The number of groups with $n$ elements goes like this, starting with $n = 0$:
0, 1, 1, 1, 2, 1, 2, 1, 5, …
The number of semigroups with $n$ elements goes like this:
1, 1, 5, 24, 188, 1915, 28634, 1627672, 3684030417, 105978177936292, …
Here I’m counting isomorphic guys as the same.
But how much do we know about such sequences in general? For example, is there any sort of algebraic gadget where the number of gadgets with $n$ elements goes like this:
1, 1, 2, 1, 1, 1, 1, 1, … ?
No! Not if by “algebraic gadget” we mean something described by a bunch of operations obeying equational laws — that is, an algebra of a Lawvere theory.
This follows from a result of László Lovász in 1967:
On Mastodon, Omar Antolín sketched a proof that greases the wheels with more category theory. It relies on a rather shocking lemma:
Super-Yoneda Lemma. Let $\mathsf{C}$ be the category of algebras of some Lawvere theory, and let $A, B \in \mathsf{C}$ be two algebras whose underlying sets are finite. If the functors $\mathrm{hom}
(-,A)$ and $\mathrm{hom}(-,B)$ are unnaturally isomorphic, then $A \cong B$.
Here we say the functors $\mathrm{hom}(-,A)$ and $\mathrm{hom}(-,B)$ are unnaturally isomorphic if
$\mathrm{hom}(X,A) \cong \mathrm{hom}(X,B)$
for all $X \in \mathsf{C}$. We’re not imposing the usual commuting naturality square — indeed we can’t, since we’re not even giving any specific choice of isomorphism!
If $\mathrm{hom}(-,A)$ and $\mathrm{hom}(-,B)$ are naturally isomorphic, you can easily show $A \cong B$ using the Yoneda Lemma. But when they’re unnaturally isomorphic, you have to break the glass
and pull out the Super-Yoneda Lemma.
Given this shocking lemma, it’s easy to show this:
Theorem. Let $A, B$ be two algebras of a Lawvere theory whose underlying sets are finite. If $A^k \cong B^k$ for some natural number $k$ then $A \cong B$.
Here’s how. Since $A^k \cong B^k$, we have natural isomorphisms
$\mathrm{hom}(-,A)^k \cong \mathrm{hom}(-, A^k) \cong \mathrm{hom}(-, B^k) \cong \mathrm{hom}(-,B)^k$
so for any $X \in \mathsf{C}$ the sets $\mathrm{hom}(X,A)^k$ and $\mathrm{hom}(X,B)^k$ have the same cardinality. This means we have an unnatural isomorphism
$\mathrm{hom}(-,A) \cong \mathrm{hom}(-,B)$
The lemma magically lets us conclude that
$A \cong B$
Now, how do we use this to solve our puzzle? Let $a(n)$ be the number of isomorphism classes of algebras whose underlying set has $n$ elements. We must have
$a(n^k) \ge a(n)$
since we’ve just seen that nonisomorphic algebras with $n$ elements give nonisomorphic algebras with $n^k$ elements. So, for example, we can never have $a(4) \lt a(2)$, since $4 = 2^2$. Thus, the
sequence can’t look like the one I showed you, with
$a(0) = 1, \; a(1) = 1, \; a(2) = 2, \; a(3) = 1,\; a(4) = 1, ...$
Nice! So let’s turn to the lemma, which is the really interesting part.
I’ll just quote Omar Antolín’s proof, since I can’t improve on it. I believe the ideas go back to Lovász, but a bit of category theory really helps. Remember, $A$ and $B$ are algebras of some Lawvere
theory whose underlying sets are finite:
Let $\mathrm{mon}(X, A)$ be the set of monomorphisms, which here are just homomorphisms that are injective functions. I claim you can compute the cardinality of $\mathrm{mon}(X, A)$ using the
inclusion-exclusion principle in terms of the cardinalities of $\mathrm{hom}(Q, A)$ for various quotients of $X$.
Indeed, for any pair of elements $x, y \in X$, let $S(x, y)$ be the set for homomorphisms $f \colon X \to A$ such that $f(x) = f(y)$. The monomorphisms are just the homomorphisms that belong to
none of the sets $S(x, y)$, so you can compute how many there are via the inclusion-exclusion formula: you’ll just need the cardinality of intersections of several $S(x_i, y_i)$.
Now, the intersection of some $S(x_i, y_i)$ is the set of homorphisms $f$ such that for all $i$, $f(x_i) = f(y_i)$. Those are in bijection with the homorphisms $Q \to A$ where $Q$ is the quotient
of $X$ obtained by adding the relations $x_i=y_i$ for each $i$.
So far I hope I’ve convinced you that if $\mathrm{hom}(-, A)$ and $\mathrm{hom}(-, B)$ are unnaturally isomorphic, so are $\mathrm{mon}(-, A)$ and $\mathrm{mon}(-, B)$. Now it’s easy to finish:
since $\mathrm{mon}(A, A)$ is non-empty, so is $\mathrm{mon}(A, B)$, so $A$ is isomorphic to a subobject of $B$. Similarly $B$ is isomorphic to a subobject of $A$, and since they are finite, they
must be isomorphic.
But if you look at this argument you’ll see we didn’t use the full force of the assumptions. We didn’t need $A$ and $B$ to be algebras of a Lawvere theory. They could have been topological spaces, or
posets, or simple graphs (which you can think of as reflexive symmetric relations), or various other things. It seems all we really need is a category $\mathsf{C}$ of gadgets with a forgetful functor
$U \colon \mathsf{C} \to \mathsf{FinSet}$
that is faithful and has some extra property… roughly, that we can take an object in $\mathsf{C}$ and take a quotient of it where we impose a bunch of extra relations $x_i = y_i$, and maps out of
this quotient will behave as you’d expect. More precisely, I think the extra property is this:
Given any $X \in \mathsf{C}$ and any surjection $p \colon U(X) \to S$, there is a morphism $j \colon X \to Q$ such that the morphisms $f \colon X \to Y$ that factor through $j$ are precisely
those for which $U(f)$ factors through $p$.
Can anyone here shed some light on this property, and which faithful functors $U \colon \mathsf{C} \to \mathsf{FinSet}$ have it? These papers should help:
but I haven’t had time to absorb them yet.
By the way, there’s a name for categories where the super-Yoneda Lemma holds: they’re called right combinatorial.
And there’s a name for the sequences I’m talking about. If $T$ is a Lawvere theory, the sequence whose $n$th term is the number of isomorphism classes of $T$-algebras with $n$ elements is called the
fine spectrum of $T$. The idea was introduced here:
• Walter Taylor, The fine spectrum of a variety, Algebra Universalis 5 (1975), 263–303.
though Taylor used not Lawvere theories but an equivalent framework: ‘varieties’ in the sense of universal algebra. For a bit more on this, go here.
I’m interested in which sequences are the fine spectrum of some Lawvere theory. You could call this an ‘inverse problem’. The direct problem — computing the fine spectrum of a given Lawvere theory —
is already extremely difficult in many cases. But the case where there aren’t any equational laws (except trivial ones) is manageable:
Some errors in Harrison’s paper were corrected here:
I suspect Harrison and Tureček’s formulas could be nicely derived using species, since they’re connected to the tree-like structures discussed here:
• François Bergeron, Gilbert Labelle and Pierre Leroux, Combinatorial Species and Tree-Like Structures, Cambridge U. Press, Cambridge, 1998.
For all I know these authors point this out! It’s been a while since I’ve read this book.
Posted at September 17, 2023 1:00 AM UTC
Re: Counting Algebraic Structures
“If the functors hom(−,A) and hom(−,A) are unnaturally isomorphic” Pretty sure they’re naturally isomorphic.
Posted by: Allen Knutson on September 17, 2023 3:15 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Posted by: John Baez on September 17, 2023 4:10 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Neat! I think the following hypothetical variant of the theorem on powers could give even stronger constraints on such “number of algebras” sequences:
Theorem? Let $A$, $B$, $C$ be algebras of a Lawvere theory whose underlying sets are finite. If $A \times C \cong B \times C$, then $A \cong B$.
The problem with the analogous proof is that trying to cancel $- \times \mathrm{hom}(-,C)$ from an isomorphism may not be possible because $\mathrm{hom}(X, C)$ could be empty for some $X$.
But perhaps one can get this to work with some additional cleverness? Or are there obvious counterexamples?
If it’s true, then one could conclude that a “number of algebras” sequence must increase (non-strictly) not only along powers, but along arbitrary multiples, provided that there is at least one
algebra of every positive finite cardinality.
Posted by: Tobias Fritz on September 17, 2023 4:52 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
You might look at
If you have a single constant in the signature and the constant always gives a subalgebra then you never have empty hom sets and you can do this.
Posted by: Benjamin Steinberg on September 18, 2023 3:58 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
This works if every nonempty finite algebra contains a copy of the 1 element algebra. So groups, rings, monoids and semi groups are fine.
Posted by: Benjamin Steinberg on September 18, 2023 4:07 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Alas, Tobias’s conjectured theorem isn’t true in full generality. Here’s a counterexample.
Take the theory consisting of three constants and no equations. Thus, an algebra is a set $A$ together with distinguished elements $a_1$, $a_2$ and $a_3$, in order, which may or may not be equal.
Let $A = B = C = \{1, 2, 3\}$. Make $A$, $B$ and $C$ into algebras by taking the following constants:
• $a_1 = 2$, $a_2 = 2$, $a_3 = 3$
• $b_1 = 1$, $b_2 = 2$, $b_3 = 3$
• $c_1 = 2$, $c_2 = 3$, $c_3 = 3$.
Then $A \times C$ is the 9-element set $\{1, 2, 3\} \times \{1, 2, 3\}$ with distinguished elements
$(2, 2), (2, 3), (3, 3),$
and $B \times C$ is the same 9-element set with distinguished elements
$(1, 2), (2, 3), (3, 3).$
Since both $A \times C$ and $B \times C$ are 9-element sets whose three distinguished elements are all distinct, they are isomorphic as algebras. Concretely, the self-map of $\{1, 2, 3\}^2$ that
exchanges $(2, 2)$ and $(1, 2)$ while leaving everything else fixed defines an isomorphism of algebras $A \times C \cong B \times C$.
On the other hand, $A$ is not isomorphic to $B$, because the three distinguished elements of $B$ are distinct but those of $A$ are not.
Posted by: Tom Leinster on September 19, 2023 8:21 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Fiendishly clever, Tom!
Your trick reminds me of how geometric realization of simplicial sets preserves products thanks to the miracle of degeneracies. The simplicial 1-simplex (the simplicial set that looks like an
interval) has vertices $0$ and $1$. It has a degenerate 2-simplex with vertices $0, 0, 1$ and another with vertices $0, 1, 1$. These give the product of two simplicial 1-simplexes a nondegenerate
2-simplex with vertices $(0,0), (0,1)$ and $(1,1)$. As we move along from $(0,0)$ to $(0,1)$ to $(1,1)$, at each stage the first coordinate may stay the same, or the second coordinate may stay the
same, but they don’t both stay the same.
Was this on your mind?
Posted by: John Baez on September 19, 2023 10:53 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
No, I just guessed the general conjecture would be false and looked for counterexamples in the simplest possible algebraic theories.
Posted by: Tom Leinster on September 20, 2023 4:04 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
That’s very nice indeed! It’s a shame that my conjecture is wrong in general.
In fact, I realize now that also my conjectured corollary about every “number of algebras” sequence being non-strictly increasing along multiples is wrong. For example, the number of Boolean algebras
on $2$ elements is one, but the number of Boolean algebras on $2 \cdot 3$ elements is zero. In this case, the problem is not that $\mathrm{hom}(-,C)$ can be empty, which is how my envisioned argument
fails in your example, but rather the complete absence of algebras in cardinalities that are not powers of two.
Posted by: Tobias Fritz on September 20, 2023 10:32 AM | Permalink | Reply to this
Re: Counting Algebraic Structures
Oh, that’s a very simple counterexample, Tobias!
Given a set of natural numbers closed under multiplication, is there a Lawvere theory whose finite algebras only have cardinalities in this set?
Posted by: John Baez on September 20, 2023 10:45 AM | Permalink | Reply to this
Re: Counting Algebraic Structures
I take it that what you’re asking is this: given a multiplicative submonoid $S \subseteq \mathbb{N} \setminus \{0\}$, does there exist a Lawvere theory which has algebras precisely in cardinalities
in $S$?
Here’s a positive answer in some special cases. The set $S = \{1, p^k, p^{2k}, \ldots\}$ for a prime $p$ and exponent $k \in \mathbb{N}$, can be realized by the theory of vector spaces over the
finite field $\mathbb{F}_{p^k}$. Given sets $S_1, \ldots, S_m$ which are of this form for different primes, we can also realize the set of all products $S_1 \cdots S_m$ by using products of Lawvere
theories, so that an algebra of such a structure is simply the product of $m$ vector spaces over the respective fields. More generally, this shows that the set of realizable sets is itself a monoid
under multiplication!
Looking at other cases seems difficult. Here are two of them which should be at roughly the next level of difficulty:
• Does there exist a Lawvere theory which realizes the powers of two, except for two itself?
A natural idea would be to try and extend the structure of Boolean algebra or vector space over $\mathbb{F}_2$ in such a way that this extra structure exists on all algebras except on the two-element
one, but it’s unclear how to do this.
• What about a Lawvere theory realizing all the powers of $6$?
This one seems easier, and perhaps one can consider pairs consisting of a vector space over $\mathbb{F}_2$ and one over $\mathbb{F}_3$ and somehow force them to have the same dimension.
Posted by: Tobias Fritz on September 25, 2023 5:26 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Here’s a positive example on my first bullet point, namely what should be a Lawvere theory which has finite algebras in cardinalities exactly the powers of two except for two itself. Consider first
the Lawvere theory whose algebras are product sets $V \times W$ where $V$ is a vector space over $\mathbb{F}_2$ and $W$ is a vector space over $\mathbb{F}_4$; this is the product of the individual
Lawvere theories for these kinds of vector spaces. Now throw in two additional unary operations corresponding to maps $s : V \to W$ and $r : W \to V$ and impose the equation $rs = 1_V$; you can also
impose $\mathbb{F}_2$-linearity if you like, but it’s not needed. The idea is that $s$ embeds $V$ into $W$, and that such maps exist if and only if $|V| \le |W|$.
In this way, the possible cardinalities of $V \times W$ are clearly of the form $2^m 4^n$ with $2^m \le 4^n$. It’s easy to see that these numbers are exactly the powers of two except for two itself.
(It’s enough to use $m \in \{0,1\}$ to get these.)
There should be a pretty general idea here on how to build Lawvere theories without models in specific cardinalities, but I’m not quite sure about how to formulate it.
Posted by: Tobias Fritz on September 28, 2023 7:25 AM | Permalink | Reply to this
Re: Counting Algebraic Structures
The powers of 2 except for 2 are of the form $8^m 4^n$, and I’m convinced you can tackle general powers of 2 by just adding more powers into the product.
For powers of 2 and 3, there are uncountably many multiplicative submonoids. (Consider $\{2^m 3^n\,|\,m \le \alpha n\}$, which encodes any positive real $\alpha$.)
Somewhere in between, we can ask what this gadget can produce.
I’m assuming you can impose inequalities of the form $|V_1V_2| \le |V_3V_4|$, which means we can abstract it down to the following.
Let $I$ be a (finite?) set and consider functions $q:I \to \mathbb{N}, m:I \to \mathbb{N}$ and a collection of inequality conditions $C \subseteq P(I)^2$. Then we are asking what sets are of the form
$S_{I,q,C} = \mathbb{N} \cap \left\{\prod_{i \in I} {q_i}^{m_i} \,|\, \prod_{c \in C_1} {q_c}^{m_c} \le \prod_{c \in C_2} {q_c}^{m_c} \quad\forall (C_1,C_2) \in C\right\}$
(We also have to impose the condition that each $q_i$ is a prime power. This is the main restriction we are trying to remove in the second puzzle.)
Posted by: unekdoud on October 4, 2023 4:20 AM | Permalink | Reply to this
Re: Counting Algebraic Structures
Posted by: James Sheppard on September 19, 2023 2:28 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Posted by: John Baez on September 19, 2023 10:58 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
I didn’t find that the politest of responses. Of course I mis-wrote: my point was that the automorphism group of $S_n$ is not what one “expects” (ie $S_n$ again) precisely when $n=2,6$, and that set
doesn’t have the property of algebraic structures which is under discussion.
Posted by: James Sheppard on September 20, 2023 8:54 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Sorry, I was trying to joke around, which is always risky without facial expressions to indicate it. I suspected many people would skim over the sequence and not notice that the first “2” was wrong.
If you expand the scope a bit and look for epimomorphisms from $S_n$ to $S_m$ with $m \le n$ then you’ll also find our friend the sign homomorphism from $S_n$ to $S_2$ and one more exotic delight:
the nontrivial homomorphism from $S_4$ to $S_3$, which arises from the 3 ways to partition a 4-element set into 2-element subsets.
Posted by: John Baez on September 20, 2023 9:49 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
It is almost surely unrelated, but talk of the nontrivial homomorphism $S_4 \to S_3$ makes me think of the unique non-null-homotopic map $S^4\to S^3$.
Posted by: David Roberts on September 21, 2023 12:58 AM | Permalink | Reply to this
Re: Counting Algebraic Structures
Actually I think the nontrivial homomorphism $S_4 \to S_3$ is related to the existence of a nontrivial homomorphism $\mathrm{SO}(4) \to \mathrm{SO}(3)$.
If we take the 4-element set $\{1,2,3,4\}$, there are 3 ways to split it:
$\{1,2\} + \{3,4\}$$\{1,3\} + \{4,2\}$$\{1,4\} + \{2,3\}$
and $S_4$ permutes these 3 splittings. This gives a nontrivial homomorphism $S_4 \to S_3$.
Similarly, if we take $\mathbb{R}^4$ with its usual basis $e_1, e_2, e_3, e_4$, the space of self-dual elements of $\Lambda^2 \mathbb{R}^4$ is 3-dimensional with basis
$e_1 \wedge e_2 + e_3 \wedge e_4$$e_1 \wedge e_3 + e_4 \wedge e_2$$e_1 \wedge e_4 + e_2 \wedge e_3$
and $\mathrm{SO}(4)$ preserves this 3-dimensional space. This gives a nontrivial homomorphism $\mathrm{SO}(4) \to \mathrm{SO}(3)$.
We can think of these two phenomena as ‘the same thing’ over the field with one element and the field $\mathbb{R}$. But the real case is different because we also have a 3-dimensional space of
anti-self-dual elements of $\Lambda^2 \mathbb{R}^4$, with basis
$e_1 \wedge e_2 - e_3 \wedge e_4$$e_1 \wedge e_3 - e_4 \wedge e_2$$e_1 \wedge e_4 - e_2 \wedge e_3$
This gives a different nontrivial homomorphism $\mathrm{SO}(4) \to \mathrm{SO}(3)$. But arguably this is because $\mathbb{R}$ lets us use minus signs, which we can’t do in the field with one element.
I should really see how this story plays out in all the finite fields $\mathbb{F}_p$. I get a bit confused about orthogonal groups and exterior algebras in characteristic $p$, especially when $p = 2$
Posted by: John Baez on September 21, 2023 1:11 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
This stuff is so cool, John! It’s so nice to see categorical tools being useful in such a concrete, ‘hard math’ question.
Re the condition on the functor $U:C \to \mathrm{FinSet}$ to prove super-Yoneda, it looks very much like exponentiability for a functor.
Quoting from the nLab:
A functor $p\colon E\to B$ is exponentiable if for any morphism $\alpha\colon a\to b$ in $E$ and any factorization $p a \overset{\beta}{\to} c \overset{\gamma}{\to} p b$ of $p \alpha$ in $B$, we
1. there exists a factorization $a \overset{\tilde{\beta}}{\to} d \overset{\tilde{\gamma}}{\to} b$ of $\alpha$ in $E$ such that $p \tilde{\beta} = \beta$ and $p \tilde{\gamma} = \gamma$, and
2. any two such factorizations in $E$ are connected by a zigzag of commuting morphisms which map to $id_c$ in $B$.
You see, the first condition is almost an exact match of yours.
Posted by: Matteo Capucci on September 21, 2023 2:24 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Thanks! This could be a wiser way of formulating my condition. It will take me a while to understand the precise relation, but I can see they’re similar.
Posted by: John Baez on September 23, 2023 3:32 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
If it can help, I like to express the exponentiability condition (also called Conduché property) for a functor in a different way (I used the other one to stress the similarity to yours):
Let $F:C \to D$ be a functor. $F$ is exponentiable iff the displayed category of its fibers, which is a unitary lax functor $F^{-1}:D \to \mathbf{Prof}$, is actually a pseudofunctor.
So exponentiability gives you ‘half’ of the structure of a fibration, namely that composition of cartesian lifts is still cartesian, and vice versa. What it doesn’t guarantee is that all cartesian
lifts exists, only that when they do, they compose nicely.
This other guarantee holds for prefibrations, and indeed a Grothendieck fibration is exactly an exponentiable prefibration!
This is all stuff more or less explicitly treated by Benaboù in ‘Distributors at work’.
Posted by: Matteo Capucci on September 25, 2023 12:15 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
Coming to this a bit late, but I don’t see how exponentiability says something about cartesian morphisms if they don’t exist.
If all weakly cartesian lifts do exist, so that the normal lax functor $D\to Prof$ lands in $Cat$, then the weakly cartesian arrows are closed under composition if and only if the lax functor is
pseudo, hence if and only if the functor $C\to D$ is exponentiable. But if weakly cartesian lifts don’t exist, I don’t see how assuming that they are closed under composition is relatable to
Posted by: Mike Shulman on December 11, 2023 10:53 PM | Permalink | Reply to this
Re: Counting Algebraic Structures
(Sorry to come to this a bit late; this semester was kind of overwhelming for me.)
Given any $X\in C$ and any surjection $p:U(X)\to S$, there is a morphism $j:X\to Q$ such that the morphisms $f:X\to Y$ that factor through $j$ are precisely those for which $U(f)$ factors through
That looks to me like saying exactly that $p$ has a semi-final lift. In particular, since $j$ factors through itself, $U(j)$ factors through $p$, and this condition together with faithfulness of $U$
implies that it is the initial such factorization.
In particular, therefore, this condition holds whenever $U$ is a solid functor. This holds if:
1. $U$ is topological concrete, or
2. $U$ is the restriction of a monadic functor into $Set$ with the property that quotients of finite algebras are finite.
However, it seems to me that the argument is using a bit more than this assumption. Don’t you need at the end that $U$ is conservative or something?
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Stacks Project Blog
Theorem Tag 04I7 now has a complete proof. It is the case of schemes for the result I mentioned in this post. It says that given two schemes X, Y any morphism of locally ringed topoi
(Sh(X_{etale}), O_X) —> (Sh(Y_{etale}), O_Y)
comes from a morphism of schemes X —> Y. To prove it you use that an affine scheme V etale over Y can be embedded into A^n_Y for some n (and that it is cut out by polynomial equations in there).
Of course, it would perhaps be quicker to try and directly prove the corresponding result for algebraic spaces or Deligne-Mumford stacks (haven’t worked out the details yet), but I want mostly to
stick with the philosophy that each result is proved in various levels of generality: commutative algebra, schemes, algebraic spaces, algebraic stacks, higher topos theory, etc, etc.
In a related discussion Brian Conrad pointed me to Theorem A.4.1 of the preprint by Conrad-Lieblich-Olsson entitled “Nagata compactification for algebraic spaces”. This theorem states that the
category of all first order thickenings of algebraic spaces is equivalent to the category of pairs (X, A —> O_X) where X is an algebraic space and A –> O_X is a surjection of sheaves of rings on X_
{etale} with quasi-coherent square zero kernel.
It seems to me that it is useful to think about the locally ringed small etale topos of an algebraic space in order to formulate and prove such results, even though it will not necessarily simplify,
or shorten the proofs. Namely, in that language Theorem A.4.1 can be reformulated as follows:
• if X —> X’ is a first order thickening of algebraic spaces, then X_{etale} = X’_{etale}, i.e., the topos doesn’t change,
• define a locally ringed topos (Sh(C), A) to be an algebraic space if it is equivalent to (Sh(X_{etale}), O_X) for some algebraic space X, and
• if (Sh(C), A) is an algebraic space and A’ –> A is a surjection of rings with square zero quasi-coherent kernel then (Sh(C), A’) is an algebraic space.
The functoriality takes care of itself by the result discussed higher up.
One thought on “Update”
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Codeforces 109 1B / 2D - Colliders
Link to statement
I kind of solved this during the contest, I just made the absurdly wrong assumption that I only needed prime factors up to sqrt(100000), this is completely wrong and stupid. A prime factor can be up
to (N/2). I also did something complicated to get the number of the conflicting collider (I knew that only a collider can use a prime factor, but I forgot about that when making this operation). I
have no idea why this happened.
I find this problem to share a lot of traits to
the topcoder problem
from the round just after. The same logic to ensure that the numbers are co-prime is very useful in both problems.
n and m can be sort of large, up to 100000. It is best to make the algorithms better than O(n*m), just for example. The following solution will work in O(m * sqrt(n)) time, so that's quite fine.
First of all, co-prime numbers. Although gcd(a,b) = 1 is the definition used in the statement, it really pays to see the problem as (No two colliders can share a prime factor). You can be sure that
the maximum number of
prime factors a number can have is O(log(n)) (It is actually better than that). For example, try 2*3*5*7*11*13*17 , that is already larger than 100000. Thus you can assume a number will never have
more than 7 distinct prime factors.
So, the algorithm is rather simple, when considering that. Assume that you have a list of all the prime factors of each of the numbers from 1 to 100000. That is an array [6][100001]. Which is
perfectly fine for the memory limit. Then, let's say that you have an array that for each prime factor, sets whether it is currently in use by a collider (A collider that is a multiple of that prime
exists) or not. The solution becomes an easy simulation: When turning on a collider, set the array to true for all of its prime factors. When verifying if you can turn a collider on, see for each of
its prime factors if the array is already true or not. When turning a collider off, set the boolean values of its prime factors to false (This works because only one collider at the same time can be
a multiple of a prime factor).
We forgot about something, the conflict message requires you to say the number of a collider that conflicts with the request. That is easy, instead of just setting leaving true or false to the array
that determines used prime factors. Leave the number of the collider that uses it or -1 if no collider uses it.
Prime factors quickly
The previous solution is correct and works in O(m * log(n) ) time but it assumes we already have a list of the prime factors for each number. We need to get this list quickly.
Looping through each of the n numbers, and factorizing them, may work in time. Maybe 100000 * sqrt(100000) is fine.
. If you want something faster, here is a trick that uses a slight modification to the
Sieve of Eratosthenes
. In the normal sieve, you mark the numbers you find to be composite as such. However, think of it, whenever you mark a composite number as false for the first time, you are doing so because you have
just found the first prime factor of the number. Thus, you can modify the Sieve of Eratosthenes slightly to make it return a list of the first prime factor for each number from 2 to n (For a prime
number, its first and only prime factor is itself). You can then use this list to quickly factorize any number from 2 to n:
- Extract first prime factor : q
- Divide x by q until x is no longer a multiple of x.
- Use the new value of x to find the next prime factor, and repeat until x is 1.
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Interfacially adsorbed bubbles determin
SciPost Submission Page
Interfacially adsorbed bubbles determine the shape of droplets
by Alessio Squarcini, Antonio Tinti
This is not the latest submitted version.
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users): Alessio Squarcini
Submission information
Preprint Link: scipost_202302_00024v1 (pdf)
Date submitted: 2023-02-13 12:54
Submitted by: Squarcini, Alessio
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
• Condensed Matter Physics - Theory
Specialties: • Condensed Matter Physics - Computational
• Statistical and Soft Matter Physics
Approaches: Theoretical, Computational
The characterization of density correlations in the presence of strongly fluctuating interfaces has always been considered a difficult problem in statistical mechanics. Here we study -- by using
recently developed exact field-theoretical techniques -- density correlations for an interface with endpoints on a wall forming a droplet in 2D. Our framework applies to interfaces entropically
repelled by a hard wall as well as to wetting transitions. In the former case bubbles adsorbed on the interface are taken into account by the theory which yields a systematic treatment of finite-size
corrections to one- and two-point functions and show how these are related to Brownian excursions. Our analytical predictions are confirmed by Monte Carlo simulations without free parameters. We also
determine one- and two-point functions at wetting by using integrable boundary field theory. We show that correlations are long ranged for entropic repulsion and at wetting. For both regimes we
investigate correlations in momentum space by generalizing the notion of interface structure factor to semi-confined systems. Distinctive signatures of the two regimes manifest in the structure
factor through a term that we identify on top of the capillary-wave one.
Current status:
Has been resubmitted
Reports on this Submission
Report #2 by Anonymous (Referee 2) on 2023-6-12 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202302_00024v1, delivered 2023-06-12, doi: 10.21468/SciPost.Report.7335
A very well written pedagogical account of new developed field theory approaches to fluid interfacial phenomena that tests more phenomenological ideas.
None other than this is not an easy topic and it is likely only specialists that will learn from it. Nevertheless there is no doubt this is very good theoretical physics.
The authors should be congratulated on producing a very well written pedagogical account of recent developments that allow the calculation of correlation functions at interfacial phase transitions
using field theoretic methods applied to microscopic models. These allow one to test the quantitive accuracy of more phenomenological approaches based on interfacial models and determine
systematically the corrections to these when one is away from the scaling limit ( when all lengths are small to the interfacial roughness etc). The article is well written and mathematically sound. I
found the glossary particularly useful where they show the correspondence between the quantum field and statistical physics interpretations. Indeed I would place this more prominently at the
beginning or end conclusions and even add another column where they provide the definitions of some of the more obscure terms eg wedge covariance = a mapping between observables for wetting at planar
walls and wedge corners, with opening angle (\pi/2 - 2\alpha), involving the simple change \theta \to \theta -\alpha.
REPLY TO REPORT 2
The authors should be congratulated on producing a very well written pedagogical account of recent developments that allow the calculation of correlation functions at interfacial phase transitions
using field theoretic methods applied to microscopic models. These allow one to test the quantitive accuracy of more phenomenological approaches based on interfacial models and determine
systematically the corrections to these when one is away from the scaling limit ( when all lengths are small to the interfacial roughness etc). The article is well written and mathematically sound. I
found the glossary particularly useful where they show the correspondence between the quantum field and statistical physics interpretations. Indeed I would place this more prominently at the
beginning or end conclusions and even add another column where they provide the definitions of some of the more obscure terms eg wedge covariance = a mapping between observables for wetting at planar
walls and wedge corners, with opening angle (\pi/2 - 2\alpha), involving the simple change \theta \to \theta -\alpha.
We are particularly grateful to the referee for having expressed enthusiastic appreciation of our work, its scientific validity and our efforts in describing the results in a pedagogical fashion. As
also pointed out by Referee 1, we agree with Referee 2 that the glossary/dictionary deserves a more prominent position in the paper, and we thank the referee for this constructive suggestion. Indeed,
the shifting of the glossary right after the introduction enhances the readability. In addition, we have sharpened the content of the last line in the dictionary signaling how a wedge with opening
angle $\pi-2\alpha$ can be implemented through a double boost of the vertical walls. In order to further detail this point, we have added an explanatory paragraph above equation (6).
Report #1 by Anonymous (Referee 1) on 2023-5-31 (Invited Report)
• Cite as: Anonymous, Report on arXiv:scipost_202302_00024v1, delivered 2023-05-31, doi: 10.21468/SciPost.Report.7284
The work presents interesting and important results in the field of near-critical interfacial phenomena.
The authors consider the two-dimensional Ising model on the half-plane x > 0 with boundary conditions enforcing a droplet shape. They consider two different scenarios for the droplet shape: entropic
repulsion where the droplet is partially covering the absorbing surface, and the wetting transition where the droplet is covering the complete surface. For both scenarios they use the equivalence
between the near-critical Ising model and a relativistic quantum field theory to obtain expressions for the density profile (Eqs. (1) and (7)) and correlation functions (Eqs. (5) and (11)). For the
former scenario the analytical results are compared with MC simulations. Finally, in Sec. (5) the authors provide a technique to calculate the interface structure factor for semi-confined systems.
The work presents interesting and important results in the field of near-critical interfacial phenomena. However, some issues must be addressed and clarified before the paper can be accepted; please
see my remarks below.
Specific remarks:
- Introduction: ”From the theoretical side, boundary-induced effects on near-critical systems have been intensively studied by means of several techniques ranging from mean field theory, perturbative
field theory [9–11], and numerical simulations [12].”(p2)
References are missing for “mean field theory”
1) Introduction:
- ”...the exact analytic form of correlations in the presence of strongly fluctuating interfaces is largely unknown...” (p3).
This statement seems to contradict the result of Onsager mentioned on the top of page 3:
- ” The celebrated Onsager’s solution yields the exact result for the decay of scaled truncated two-point function, … [23]” (p3).
The authors probably mean that the exact analytic form of correlations in the presence of a boundary is largely unknown. If this is so, the authors should make this statement more precise.
2) Entropic repulsion
- ”For the Ising model with a = − and b = +1 this protocol introduces a droplet of negative magnetization enclosed in the Peierls contour of Fig. 1 (a) [47].” (p5)
I suggest to briefly explain what a Peierls contour is. Moreover, a = − should be written as a = −1.
- ”The mapping between variables in the lattice gas and Ising model is n i = (1 + s i )/2, where n i ∈ 0, 1 stands for an empty/filled site [6,24,48]. (p5)
An explanation/elaboration of s_i is seemingly missing. It is obvious that these are the spin variables s_i = ±1, but this does not seem to be stated anywhere.
- The Ising Hamiltonian is never defined. Although many readers will be familiar with the Ising Hamiltonian, for completeness it should at least be stated once. This way it will also be immediately
clear how the coupling J and J_b enter the Hamiltonian.
- The authors assign the variable $a$ to two different quantities. First they write ”For the Ising model with a = − ...” (p5)
and shortly after
”...and those in the adjacent layer is taken to be of the form J_b = aJ, where 0 ≤ a ≤ 1, meaning that J_b is a weakened bond;” (p5)
This is confusing. It seems that the first a = −1 does not have to be introduced if the authors introduce the spin variables s_i = ±1.
- ”... there exist a wetting temperature T w (a) < T c such that T w (a → 0 + ) = T c − and ...” (p5)
Here both T_c and T^-_c are not defined. Most likely T_c is the critical temperature (which the referee will henceforth assume), but this should be stated explicitly.
- ” To be definite, in our simulations for the case of entropic repulsion we simply take J_b = J, corresponding to a = 1, meaning that for all subcritical temperatures the system is non-wet.”
Should subcritical temperatures not be positive temperatures, since T_w (a \to 1^− ) = 0^+ ?
- ”... for the Ising model ⟨σ⟩ = ±M where M \sim (T_c − T )^{1/8} ...” (p6)
To be consistent with previous notation, and to avoid the need for introducing useless variables, they should change ⟨σ⟩ to ⟨s⟩ or change the s i on page 5 to σ_i . Furthermore, the relation M \sim
(T_c − T )^{1/8} only applies close to the critical temperature. It would be more correct if the authors state the exact result M = [1 − \sinh^{−4}(2J/k_B T)]^{1/8}
- One of the main results of this manuscript is shown in Eq. (1). Since the authors always write scaling relations close to the critical temperature, i.e. M \sim (T_c − T )^{1/8} and \xi_b \sim (T_c
− T )^{-\nu} , it is not clear whether Eq. (1) only applies close to the critical temperature T_c , or for arbitrary temperatures. If the former is the case (which is suggested according to Sec. 6)
then this must be explicitly stated.
If the latter is the case, then the fully general result for M and \xi_b in terms of the coupling J and temperature T should be stated accordingly.
- ”The correction term \mathcal{A}P (x, y)\propto R^{−1/2} , which we have identified, turns out to resolve the aforementioned mismatch and eventually yields – if included in the profile – an
accurate comparison between theory and numerics, as shown in Fig. 2. The agreement is perfect due to such a term and without taking it into account the MC data fall systematically away from
the analytic prediction.” (p7)
To be able to faithfully make such a statement the authors must include the analytical profile without the correction terms in Fig. 2. Without this the reader simply cannot see the importance of the
correction term, since it can very well be that without the correction term the density profiles already match quite nicely.
- ”In general, it is possible to regard the interface as the result of an exploration process which starts from one pinning point and ends at the other; see points denoted “in” and “out” in Fig. 3.”
The idea of regarding the interface as a Brownian bridge is not new and has been pursued in quite a few previous works. See for example:
- M. E. Fisher, Walks, walls, wetting, and melting, J. Stat. Phys. 34, 667 (1984).
- Hryniv, Ostap. On local behaviour of the phase separation line in the 2D Ising model. Probability theory and related fields. 110.1 (1998): 91-107.
- Ganguly, Shirshendu, and Reza Gheissari, Local and global geometry of the 2D Ising interface in critical prewetting. (2021): 2076-2140.
- K. Blom, N. Ziethen, D. Zwicker, A. Godec, Thermodynamically consistent phase-field theory including nearest-neighbor pair correlations, Phys. Rev. Res. 5, 1 (2023).
It therefore seems to be appropriate to cite these works in the above mentioned sentence.
- Figures 2, 4, and 5 show a comparison between the analytical theory and MC simulations. However, many details about the MC simulations are missing. For example, the number of trajectories that were
considered, the uncertainty of the data points, and the conversion from the discrete lattice spacing to a continuous length variable x. These details must be mentioned/stated
in the manuscript.
3) Wetting transition
- The variable m is introduced for the first time on page 11, whereas its explanation only comes much later on page 12 under equation (9): ” … where m is the surface tension of the interface”. This
explanation should come directly after the first mentioning of m on page 11.
- In Eq. (8) we find P_2 has 3 variables, whereas in Eq. (10) we find it has 4. This is inconsistent and needs to be resolved.
- Compared with the previous section the referee was wondering why Eqs. (7) and (11) – which provide analytical results for the density profile and correlation function – are not compared with MC
simulations (given that these were carried out)? For completeness and scientific correctness, the authors should provide a comparison with MC simulations here. Considering the fact that they have
done this for the entropic repulsion case, this should not be too difficult.
4) Theoretical framework in a nutshell
- This section is very well-written and provides a clear analogy between the relativistic quantum field theory and the interface theory. In fact, this section (or a “diluted” version thereof) would
suit much better at the beginning of the manuscript before the section Entropic repulsion.
This will also immediately resolve an issue mentioned above, since this section clearly shows that the applied techniques are only valid close to the critical temperature T_c.
5) Conclusion:
- ”Field theory yields exact results for order parameter profiles and correlations in the regime of subcritical temperatures with R >> ξ b .” (p19)
This is confusing w.r.t. to following statement in Sec. 6:
”The near-critical behavior in 2D can be described by analytic continuation of a (1 + 1)- relativistic quantum field theory to a 2D Euclidean field theory in the plane (y = −it).” (p15)
So, do the results apply to near-critical or to sub-critical temperatures? This needs to be clarified.
- The analytical progress along mentioned \to The analytical progress mentioned
- page 6: and P(x; y) are super-universal, \to and P (x, y) are super-universal
- page 14: and yields the solid black arc of ellipse shown in Fig. 1 (c)\to and yields the solid black arc of ellipse shown in Fig. 1 (b)
- page 20: The results presented in this paper and show how to \to The results presented in this paper show how to ...
REPLY TO REPORT 1
REFEREE’S COMMENT: The authors consider the two-dimensional Ising model on the half-plane x > 0 with boundary conditions enforcing a droplet shape. They consider two different scenarios for the
droplet shape: entropic repulsion where the droplet is partially covering the absorbing surface, and the wetting transition where the droplet is covering the complete surface. For both scenarios they
use the equivalence between the near-critical Ising model and a relativistic quantum field theory to obtain expressions for the density profile (Eqs. (1) and (7)) and correlation functions (Eqs. (5)
and (11)). For the former scenario the analytical results are compared with MC simulations. Finally, in Sec. (5) the authors provide a technique to calculate the interface structure factor for
semi-confined systems. The work presents interesting and important results in the field of near-critical interfacial phenomena. However, some issues must be addressed and clarified before the paper
can be accepted; please see my remarks below.
REPLY: We thank the referee for appreciating the scientific importance of our work, and for the constructive remarks. Below we provide a collection of responses to all points raised by the referee.
We have revised our manuscript accordingly.
REFEREE’S COMMENT: Specific remarks: Introduction: ”From the theoretical side, boundary-induced effects on near-critical systems have been intensively studied by means of several techniques ranging
from mean field theory, perturbative field theory [9–11], and numerical simulations [12].”(p2). References are missing for “mean field theory”
REPLY: The mean field theory of wetting transition is actually discussed in Sec. C of Ref. 9 at p. 98 (“Landau Theory”). Since this reference is already present in the paper, we have added a footnote
and linked it appropriately.
REFEREE’S COMMENT: 1) Introduction: - ”...the exact analytic form of correlations in the presence of strongly fluctuating interfaces is largely unknown...” (p3). This statement seems to contradict
the result of Onsager mentioned on the top of page 3: - ” The celebrated Onsager’s solution yields the exact result for the decay of scaled truncated two-point function, … [23]” (p3). The authors
probably mean that the exact analytic form of correlations in the presence of a boundary is largely unknown. If this is so, the authors should make this statement more precise.
REPLY: As correctly pointed out by the referee, the claim in the text is the following: the exact analytic form of spin-spin correlations in the 2D Ising model in the presence of a boundary enforcing
an interface is largely unknown (for the reasons mentioned in the text), while the spin-spin correlation function for the Ising model in the unbounded (i.e. infinite) plane is known from Onsager’s
solution. The fact that the exact analytic form of spin-spin correlation functions for a system in a strictly bounded geometry with boundaries enforcing an interface is largely unknown does not
contradict the fact that Onsager’s result for the spin-spin correlation function in the 2D Ising model on the infinite plane is a well established result. To avoid misconceptions, we added a footnote
in which we clarify that we are referring to interfaces pinned at boundaries and we refer the interested reader to Chapter 21 of book in Ref. [26] for the calculation of correlation functions in the
case of a uniform boundary in the massive (i.e., off-critical) Ising model.
REFEREE’S COMMENT: 2) Entropic repulsion - ”For the Ising model with a = − and b = +1 this protocol introduces a droplet of negative magnetization enclosed in the Peierls contour of Fig. 1 (a) [47].”
(p5) I suggest to briefly explain what a Peierls contour is.
REPLY: The notion of Peierls contour has been explained and linked to literature; Ref. [52] has been added. The newly added Figure 4 explains the symbols entering in the Ising Hamiltonian and also
illustrates Peierls contour without referring to Figure 6.
REFEREE’S COMMENT: Moreover, a = − should be written as a = −1.
REPLY: We replaced $a=-$ with $a=-1$ in the text. However, when the labels $a$ and $b$ are used as subscripts we prefer to keep the shorthand notation by writing, e.g., $<\sigma(x,y)>_{-+}$ instead
of $<\sigma(x,y)>_{-1,+1}$. We believe that this notation is clear enough and no ambiguities can possibly arise.
REFEREE’S COMMENT: ”The mapping between variables in the lattice gas and Ising model is n i = (1 + s i )/2, where n i ∈ 0, 1 stands for an empty/filled site [6,24,48]. (p5). An explanation/
elaboration of s_i is seemingly missing. It is obvious that these are the spin variables s_i = ±1, but this does not seem to be stated anywhere. Explain that sigma is the field and s is the spin
variable. The Ising Hamiltonian is never defined. Although many readers will be familiar with the Ising Hamiltonian, for completeness it should at least be stated once. This way it will also be
immediately clear how the coupling J and J_b enter the Hamiltonian. Provide the Ising Hamiltonian
REPLY: In Eq. (10) we have provided the explicit expression of the Hamiltonian explaining the role of the various terms. The previous section 2 is now section 3. We have added a paragraph in Section
3 introducing the observables of our interest and explaining that $\sigma(x,y)$ is the field variable while $s_{i,j}$ is the spin variable entering the Hamiltonian (10).
REFEREE’S COMMENT: - The authors assign the variable a a to two different quantities. First they write ”For the Ising model with a = − ...” (p5) and shortly after ”...and those in the adjacent layer
is taken to be of the form J_b = aJ, where 0 ≤ a ≤ 1, meaning that J_b is a weakened bond;” (p5). This is confusing. It seems that the first a = −1 does not have to be introduced if the authors
introduce the spin variables s_i = ±1.
REPLY: For the sake of clarity we have replaced $a$ with $\alpha$.
REFEREE’S COMMENT: ”... there exist a wetting temperature T w (a) < T c such that T w (a → 0 + ) = T c − and ...” (p5) Here both T_c and T^-_c are not defined. Most likely T_c is the critical
temperature (which the referee will henceforth assume), but this should be stated explicitly.
REPLY: We have stated explicitly in Sec. 2 that $T_{c}$ is the critical temperature.
REFEREE’S COMMENT: ” To be definite, in our simulations for the case of entropic repulsion we simply take J_b = J, corresponding to a = 1, meaning that for all subcritical temperatures the system is
non-wet.” Should subcritical temperatures not be positive temperatures, since T_w (a \to 1^− ) = 0^+ ?-
REPLY: Possibly there is a misunderstanding that here we clarify. The aim of the paragraph is to describe some features of the wetting phase diagram. For $0<\alpha<1$ the wetting temperature is
strictly positive and bounded from above by the bulk critical temperature. The wetting temperature tends to zero from above as $\alpha$ tends to one from below. By taking $\alpha=1$ it follows that
any positive temperature is such that $T_{w}<T<T_{c}$ and strictly speaking the wetting temperature vanishes.
REFEREE’S COMMENT: ”... for the Ising model ⟨σ⟩ = ±M where M \sim (T_c − T )^{1/8} ...” (p6) To be consistent with previous notation, and to avoid the need for introducing useless variables, they
should change ⟨σ⟩ to ⟨s⟩ or change the s i on page 5 to σ_i . Furthermore, the relation M \sim (T_c − T )^{1/8} only applies close to the critical temperature. It would be more correct if the authors
state the exact result M = [1 − \sinh^{−4}(2J/k_B T)]^{1/8}
REPLY: We agree with the referee that the spontaneous magnetization is $<s>=M$. However, we keep the notation separate because $\sigma$ refers to the field and $s$ to the spin variable. After having
introduced the Ising Hamiltonian we introduce the field variable $\sigma(x,y)$. Since we use field theory to obtain one- and two-point correlation functions, these results are expressed in terms of
the field $\sigma$; hence, we use the notation $<\sigma(x,y)>_{-+}$ for the one-point function. Within the field-theoretical language the spontaneous magnetization is the so-called vacuum expectation
value and the standard notation for that is $<\sigma>=M$. The suggestion proposed by the referee to replace $\sigma$ with $<s>$ would alter all equations for one- and two-point functions, not only
the vacuum expectation value $\sigma$. Since this paper is about field-theoretical results, we believe that it is appropriate to write results in terms of field variables. Of course, it remains
obvious that $<s>=M$. The exact expressions for the spontaneous magnetization and the bulk correlation length are provided in the last paragraph at page 12.
REFEREE’S COMMENT: One of the main results of this manuscript is shown in Eq. (1). Since the authors always write scaling relations close to the critical temperature, i.e. M \sim (T_c − T )^{1/8} and
\xi_b \sim (T_c − T )^{-\nu} , it is not clear whether Eq. (1) only applies close to the critical temperature T_c , or for arbitrary temperatures. If the former is the case (which is suggested
according to Sec. 6) then this must be explicitly stated. If the latter is the case, then the fully general result for M and \xi_b in terms of the coupling J and temperature T should be stated
REPLY: The fact that the theory is valid close to $T_{c}$ should not to be intended as a point of weakness but rather as a strong point of the approach since it allows to find exact results for a
broad range of universality classes within a unified language. For example, Eq. (9) and Eq. (13) do not apply only to the Ising model although this is the system we considered in simulations. We
stated clearly that our results apply in the regime of temperatures close to the critical one with $T$ and $R$ satisfying the double-sided inequality (11), that we have added. The bulk correlation
length has to be large with respect to the lattice spacing in order to allow for a continuum description in terms of fields. Then, the system size $R$ has to be large with respect to the bulk
correlation length in order for phase separation to emerge. This explains the domain of applicability of the framework and the corresponding results.
REFEREE’S COMMENT: ”The correction term \mathcal{A}P (x, y)\propto R^{−1/2} , which we have identified, turns out to resolve the aforementioned mismatch and eventually yields – if included in the
profile – an accurate comparison between theory and numerics, as shown in Fig. 2. The agreement is perfect due to such a term and without taking it into account the MC data fall systematically away
from the analytic prediction.” (p7) To be able to faithfully make such a statement the authors must include the analytical profile without the correction terms in Fig. 2. Without this the reader
simply cannot see the importance of the correction term, since it can very well be that without the correction term the density profiles already match quite nicely.
REPLY: We have updated the plot as requested by the referee. In the new version of the figure (Fig. 5) we have plotted the density profile including the subleading term proportional to R^{-1/2} and,
in addition, we have shown also the density profile without such a subleading term. We have indicated them with solid and dashed lines, respectively. The systematic deviation mentioned in the text is
clearly visible in the plot.
REFEREE’S COMMENT: ”In general, it is possible to regard the interface as the result of an exploration process which starts from one pinning point and ends at the other; see points denoted “in” and
“out” in Fig. 3.” (p8) The idea of regarding the interface as a Brownian bridge is not new and has been pursued in quite a few previous works. See for example: - M. E. Fisher, Walks, walls, wetting,
and melting, J. Stat. Phys. 34, 667 (1984). - Hryniv, Ostap. On local behaviour of the phase separation line in the 2D Ising model. Probability theory and related fields. 110.1 (1998): 91-107. -
Ganguly, Shirshendu, and Reza Gheissari, Local and global geometry of the 2D Ising interface in critical prewetting. (2021): 2076-2140. - K. Blom, N. Ziethen, D. Zwicker, A. Godec, Thermodynamically
consistent phase-field theory including nearest-neighbor pair correlations, Phys. Rev. Res. 5, 1 (2023). It therefore seems to be appropriate to cite these works in the above mentioned sentence.
REPLY: We are aware that the idea of regarding the interface as a Brownian bridge is not new since this idea goes back (at least to) 1984 (M. E. Fisher, J. Stat. Phys. 34, 667 (1984), current Ref.
[54]), a reference that we have already cited in the section on entropic repulsion. In fact, we believe it is profitable for the reader to stress that exact results for strongly fluctuating (i.e.,
off-critical) interfaces in the 2D Ising model (see, e.g., the review article [13]) were used to test the reliability of heuristic description in terms of random walks [54]. Moreover, random curves
describing fluctuating cluster boundaries emerging in lattice models are often studied in the context of critical interfaces — i.e., interfaces arising from systems right at the critical point by
tracking the interface on the lattice — as shown in this paper. The interpretation in terms of a random exploration process is still possible but, contrary to the off-critical case, critical curves
are self-similar and can be described rigorously in terms of SLE. We believe that the above remarks help the reader to better contextualize the idea of regarding the interface as an exploration
process and its mathematical description in the different scenarios: i.e., random walks for off-critical interfaces and SLE for critical ones. We therefore added three paragraphs and referred to the
appropriate literature comprising heuristic approaches [54], exact solution for 2D Ising Interfaces from lattice calculations [13], rigorous results in mathematics [75-82], as well as the suggested
REFEREE’S COMMENT: Figures 2, 4, and 5 show a comparison between the analytical theory and MC simulations. However, many details about the MC simulations are missing. For example, the number of
trajectories that were considered, the uncertainty of the data points, and the conversion from the discrete lattice spacing to a continuous length variable x. These details must be mentioned/stated
in the manuscript.
REPLY: We have added a paragraph at the end of section 4.1 in which the requested simulation details are carefully provided.
REFEREE’S COMMENT: 3) Wetting transition - The variable m is introduced for the first time on page 11, whereas its explanation only comes much later on page 12 under equation (9): ” … where m is the
surface tension of the interface”. This explanation should come directly after the first mentioning of m on page 11.
REPLY: The definition of $m$ is indicated at its first occurrence.
REFEREE’S COMMENT: In Eq. (8) we find P_2 has 3 variables, whereas in Eq. (10) we find it has 4. This is inconsistent and needs to be resolved. this has been corrected
REPLY: We have corrected the arguments of $P_2$ everywhere.
REFEREE’S COMMENT: Compared with the previous section the referee was wondering why Eqs. (7) and (11) – which provide analytical results for the density profile and correlation function – are not
compared with MC simulations (given that these were carried out)? For completeness and scientific correctness, the authors should provide a comparison with MC simulations here. Considering the fact
that they have done this for the entropic repulsion case, this should not be too difficult.
REPLY: We concur with the referee that the comparison between theory and simulations for one- and two-point correlation functions at wetting deserves further investigations that can hardly fit into
an already lengthy paper. We have already planned to complete this comparison in a followup paper.
REFEREE’S COMMENT: 4) Theoretical framework in a nutshell - This section is very well-written and provides a clear analogy between the relativistic quantum field theory and the interface theory. In
fact, this section (or a “diluted” version thereof) would suit much better at the beginning of the manuscript before the section Entropic repulsion. This will also immediately resolve an issue
mentioned above, since this section clearly shows that the applied techniques are only valid close to the critical temperature T_c.
REPLY: We are glad that the referee expresses appreciation for this section. We followed the referee’s suggestion, therefore we moved the “Theoretical framework in a nutshell” in Sec. 2 and moved the
former section “Models” in Sec. 3 and rename it “Models, geometry, and observables”.
REFEREE’S COMMENT: 5) Conclusion: - ”Field theory yields exact results for order parameter profiles and correlations in the regime of subcritical temperatures with R >> ξ b .” (p19) This is confusing
w.r.t. to following statement in Sec. 6: ”The near-critical behavior in 2D can be described by analytic continuation of a (1 + 1)- relativistic quantum field theory to a 2D Euclidean field theory in
the plane (y = −it).” (p15). So, do the results apply to near-critical or to sub-critical temperatures? This needs to be clarified.
REPLY: We emphasize once more that we are studying phase separation for temperatures close to the critical one but slightly less to it. Therefore we are dealing with sub-critical temperatures but of
course sub-critical temperatures can be near-critical provided $T \rightarrow T_{c}$ with $T<T_{c}$. More precisely, we consider the regime of temperatures $T$ and system size $R$ such that $a_{0} <<
\xi_{b} << R$, as explained in a reply to a previous comment.
REFEREE’S COMMENT: Typos: - The analytical progress along mentioned \to The analytical progress mentioned - page 6: and P(x; y) are super-universal, \to and P (x, y) are super-universal - page 14:
and yields the solid black arc of ellipse shown in Fig. 1 (c)\to and yields the solid black arc of ellipse shown in Fig. 1 (b) - page 20: The results presented in this paper and show how to \to The
results presented in this paper show how to …a
REPLY: We thank the referee for having spotted the above typos. We have corrected all of them. | {"url":"https://scipost.org/submissions/scipost_202302_00024v1/","timestamp":"2024-11-09T06:55:14Z","content_type":"text/html","content_length":"69909","record_id":"<urn:uuid:5309b005-5fdb-47ad-a2e5-017097c4b457>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00886.warc.gz"} |
Teaching Resources
I have taught a wide range of classes at both the graduate and undergraduate level. I will collect various resources I have created over the years on this page, including lecture notes. Please
contact me if you wish to use these resources or would like any homework sets associated with these classes (lavrentm @ gmail.com).
Graduate Quantum Mechanics I
Lecture notes based off of
Quantum Physics
by Michel Le Bellac, Cambridge University Press (2011) (ISBN: 978-1107602762):
Lecture Notes
Graduate Quantum Mechanics II
Lecture notes based off of
Quantum Physics
by Michel Le Bellac, Cambridge University Press (2011) (ISBN: 978-1107602762) and also
Modern Quantum Mechanics, 3rd ed.
by J. J. Sakurai and Jim Napolitano, Cambridge University Press (2020) (ISBN: 978-1108473224):
Lecture Notes
Undergraduate Thermal Physics
Graduate Statistical Mechanics
Graduate Biophysics
Undergraduate Biophysics | {"url":"https://www.maxlavrentovich.com/p/teaching.html","timestamp":"2024-11-11T04:35:50Z","content_type":"application/xhtml+xml","content_length":"42442","record_id":"<urn:uuid:29a63c5f-9570-4e37-b82f-80e69e45eace>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00359.warc.gz"} |
Antecedent | Lexique de mathématique
That which precedes or premises in reasoning.
In the statement “if a quadrilateral has one pair of parallel sides, then this quadrilateral is a trapezoid“, the antecedent is the proposition “a quadrilateral has one pair of parallel sides“, and
the consequence of the proposition “this quadrilateral is a trapezoid“. | {"url":"https://lexique.netmath.ca/en/antecedent/","timestamp":"2024-11-06T18:00:05Z","content_type":"text/html","content_length":"63199","record_id":"<urn:uuid:10afe1aa-7ec1-449f-8105-a668736137c9>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00859.warc.gz"} |
What is Statistics? Why we use it ? & How we use statistics?
Statistics is a branch of mathematics in which we study the information in a meaningful way.
In statistics, we discuss various terms like mean, mode, median, classmark, Histogram, bar graph etc.
1.) Why we use statistics?
In numerical data or theoretical data, we can't understand the clear meaning.
In statistics, we show this data by a bar graph, pie chart, etc. Statistics make it easy to read this information for us.
2.) How to use statistics?
We use statistics as graphical representation & to measure central tendency in various ways to show the information-
1) Bar graph
The bar graph is used to compare among two quantities.
2) Pie graph (Circle graph)
The pie graph is a whole circle in which we show the part of the whole.
3) Histogram
The histogram is used for grouped data & it shows the variation of quantity in a particular class interval.
4) Line graph
The line graph shows how data continues to changes.
1) Mean
Mean is the average of whole data.
2) Mode
A mode is the number in given data which have high frequency.
A number which comes more time than other numbers is known as Mode.
3) Median
Median is a number which divides the whole data into two parts. | {"url":"https://www.tirlaacademy.com/2020/10/what-is-statistics-why-how-we-use-statistics%20.html","timestamp":"2024-11-02T23:29:16Z","content_type":"application/xhtml+xml","content_length":"318749","record_id":"<urn:uuid:acd601b5-b6b5-40ff-927d-be71e596351e>","cc-path":"CC-MAIN-2024-46/segments/1730477027768.43/warc/CC-MAIN-20241102231001-20241103021001-00816.warc.gz"} |
Julia 中文社区
We held a two day Julia tutorial at MIT in January 2013, which included 10 sessions. MIT Open Courseware and MIT-X graciously provided support for recording of these lectures, so that the wider Julia
community can benefit from these sessions.
This session is a rapid introduction to julia, using a number of lightning rounds. It uses a number of short examples to demonstrate syntax and features, and gives a quick feel for the language.
The rationale and vision behind julia, and its design principles are discussed in this session.
DataFrames is one of the most widely used Julia packages. This session is an introduction to data analysis with Julia using DataFrames.
This session demonstrates Julia's statistics capabilities, which are provided by these packages: Distributions, GLM, and LM.
Julia provides a built-in interface to the FFTW library. This session demonstrates the Julia's signal processing capabilities, such as FFTs and DCTs. Also see the Hadamard package.
This session focuses largely on using Julia for solving linear programming problems. The algebraic modeling language discussed was later released as JuMP. Benchmarks are shown evaluating the
performance of Julia for implementing low-level optimization code. Optimization software in Julia has been grouped under the JuliaOpt project.
Julia is homoiconic: it represents its own code as a data structure of the language itself. Since code is represented by objects that can be created and manipulated from within the language, it is
possible for a program to transform and generate its own code. Metaprogramming is described in detail in the Julia manual.
Parallel and distributed computing have been an integral part of Julia's capabilities from an early stage. This session describes existing basic capabilities, which can be used as building blocks for
higher level parallel libraries.
Julia provides asynchronous networking I/O using the libuv library. Libuv is a portable networking library created as part of the Node.js project.
The Grid of Resistors is a classic numerical problem to compute the voltages and the effective resistance of a 2n+1 by 2n+2 grid of 1 ohm resistors if a battery is connected to the two center points.
As part of this lab, the problem is solved in Julia in a number of different ways such as a vectorized implementation, a devectorized implementation, and using comprehensions, in order to study the
performance characteristics of various methods. | {"url":"https://cn.julialang.org/blog/2013/03/julia-tutorial-MIT/index.html","timestamp":"2024-11-03T04:12:22Z","content_type":"text/html","content_length":"16634","record_id":"<urn:uuid:b6ff7b6e-4ce4-4dbf-bede-d97c0e902f29>","cc-path":"CC-MAIN-2024-46/segments/1730477027770.74/warc/CC-MAIN-20241103022018-20241103052018-00648.warc.gz"} |
No More Learning
He was prevented from
succeeding by respect for the authority of Aristotle, whom he could
not believe guilty of definite, formal fallacies; but the subject
which he desired to create now exists, in spite of the patronising
contempt with which his schemes have been treated by all superior
Mysticism and Logic and Other Essays by Bertrand Russell
In what way does it contribute to the beauty of human
existence? As respects those pursuits which contribute only remotely,
by providing the mechanism of life, it is well to be reminded that not
the mere fact of living is to be desired, but the art of living in the
contemplation of great things. Still more in regard to those
avocations which have no end outside themselves, which are to be
justified, if at all, as actually adding to the sum of the world's
permanent possessions, it is necessary to keep alive a knowledge of
their aims, a clear prefiguring vision of the temple in which creative
imagination is to be embodied.
The fulfilment of this need, in what concerns the studies forming the
material upon which custom has decided to train the youthful mind, is
indeed sadly remote--so remote as to make the mere statement of such a
claim appear preposterous. Great men, fully alive to the beauty of the
contemplations to whose service their lives are devoted, desiring that
others may share in their joys, persuade mankind to impart to the
successive generations the mechanical knowledge without which it is
impossible to cross the threshold. Dry pedants possess themselves of
the privilege of instilling this knowledge: they forget that it is to
serve but as a key to open the doors of the temple; though they spend
their lives on the steps leading up to those sacred doors, they turn
their backs upon the temple so resolutely that its very existence is
forgotten, and the eager youth, who would press forward to be
initiated to its domes and arches, is bidden to turn back and count
the steps.
Mathematics, perhaps more even than the study of Greece and Rome, has
suffered from this oblivion of its due place in civilisation. Although
tradition has decreed that the great bulk of educated men shall know
at least the elements of the subject, the reasons for which the
tradition arose are forgotten, buried beneath a great rubbish-heap of
pedantries and trivialities. To those who inquire as to the purpose of
mathematics, the usual answer will be that it facilitates the making
of machines, the travelling from place to place, and the victory over
foreign nations, whether in war or commerce. If it be objected that
these ends--all of which are of doubtful value--are not furthered by
the merely elementary study imposed upon those who do not become
expert mathematicians, the reply, it is true, will probably be that
mathematics trains the reasoning faculties. Yet the very men who make
this reply are, for the most part, unwilling to abandon the teaching
of definite fallacies, known to be such, and instinctively rejected by
the unsophisticated mind of every intelligent learner. And the
reasoning faculty itself is generally conceived, by those who urge its
cultivation, as merely a means for the avoidance of pitfalls and a
help in the discovery of rules for the guidance of practical life. All
these are undeniably important achievements to the credit of
mathematics; yet it is none of these that entitles mathematics to a
place in every liberal education. Plato, we know, regarded the
contemplation of mathematical truths as worthy of the Deity; and
Plato realised, more perhaps than any other single man, what those
elements are in human life which merit a place in heaven. There is in
mathematics, he says, "something which is _necessary_ and cannot be
set aside . . . and, if I mistake not, of divine necessity; for as to
the human necessities of which the Many talk in this connection,
nothing can be more ridiculous than such an application of the words.
_Cleinias. _ And what are these necessities of knowledge, Stranger,
which are divine and not human? _Athenian. _ Those things without some
use or knowledge of which a man cannot become a God to the world, nor
a spirit, nor yet a hero, nor able earnestly to think and care for
man" (_Laws_, p. 818). [10] Such was Plato's judgment of mathematics;
but the mathematicians do not read Plato, while those who read him
know no mathematics, and regard his opinion upon this question as
merely a curious aberration.
Mathematics, rightly viewed, possesses not only truth, but supreme
beauty--a beauty cold and austere, like that of sculpture, without
appeal to any part of our weaker nature, without the gorgeous
trappings of painting or music, yet sublimely pure, and capable of a
stern perfection such as only the greatest art can show. The true
spirit of delight, the exaltation, the sense of being more than man,
which is the touchstone of the highest excellence, is to be found in
mathematics as surely as in poetry. What is best in mathematics
deserves not merely to be learnt as a task, but to be assimilated as a
part of daily thought, and brought again and again before the mind
with ever-renewed encouragement. Real life is, to most men, a long
second-best, a perpetual compromise between the ideal and the
possible; but the world of pure reason knows no compromise, no
practical limitations, no barrier to the creative activity embodying
in splendid edifices the passionate aspiration after the perfect from
which all great work springs. Remote from human passions, remote even
from the pitiful facts of nature, the generations have gradually
created an ordered cosmos, where pure thought can dwell as in its
natural home, and where one, at least, of our nobler impulses can
escape from the dreary exile of the actual world.
So little, however, have mathematicians aimed at beauty, that hardly
anything in their work has had this conscious purpose. Much, owing to
irrepressible instincts, which were better than avowed beliefs, has
been moulded by an unconscious taste; but much also has been spoilt by
false notions of what was fitting. The characteristic excellence of
mathematics is only to be found where the reasoning is rigidly
logical: the rules of logic are to mathematics what those of structure
are to architecture. In the most beautiful work, a chain of argument
is presented in which every link is important on its own account, in
which there is an air of ease and lucidity throughout, and the
premises achieve more than would have been thought possible, by means
which appear natural and inevitable. Literature embodies what is
general in particular circumstances whose universal significance
shines through their individual dress; but mathematics endeavours to
present whatever is most general in its purity, without any irrelevant
How should the teaching of mathematics be conducted so as to
communicate to the learner as much as possible of this high ideal?
Here experience must, in a great measure, be our guide; but some
maxims may result from our consideration of the ultimate purpose to be
One of the chief ends served by mathematics, when rightly taught, is
to awaken the learner's belief in reason, his confidence in the truth
of what has been demonstrated, and in the value of demonstration. This
purpose is not served by existing instruction; but it is easy to see
ways in which it might be served. At present, in what concerns
arithmetic, the boy or girl is given a set of rules, which present
themselves as neither true nor false, but as merely the will of the
teacher, the way in which, for some unfathomable reason, the teacher
prefers to have the game played. To some degree, in a study of such
definite practical utility, this is no doubt unavoidable; but as soon
as possible, the reasons of rules should be set forth by whatever
means most readily appeal to the childish mind. In geometry, instead
of the tedious apparatus of fallacious proofs for obvious truisms
which constitutes the beginning of Euclid, the learner should be
allowed at first to assume the truth of everything obvious, and should
be instructed in the demonstrations of theorems which are at once
startling and easily verifiable by actual drawing, such as those in
which it is shown that three or more lines meet in a point. In this
way belief is generated; it is seen that reasoning may lead to
startling conclusions, which nevertheless the facts will verify; and
thus the instinctive distrust of whatever is abstract or rational is
gradually overcome. Where theorems are difficult, they should be first
taught as exercises in geometrical drawing, until the figure has
become thoroughly familiar; it will then be an agreeable advance to be
taught the logical connections of the various lines or circles that
occur. It is desirable also that the figure illustrating a theorem
should be drawn in all possible cases and shapes, that so the abstract
relations with which geometry is concerned may of themselves emerge
as the residue of similarity amid such great apparent diversity. In
this way the abstract demonstrations should form but a small part of
the instruction, and should be given when, by familiarity with
concrete illustrations, they have come to be felt as the natural
embodiment of visible fact. In this early stage proofs should not be
given with pedantic fullness; definitely fallacious methods, such as
that of superposition, should be rigidly excluded from the first, but
where, without such methods, the proof would be very difficult, the
result should be rendered acceptable by arguments and illustrations
which are explicitly contrasted with demonstrations.
In the beginning of algebra, even the most intelligent child finds, as
a rule, very great difficulty. The use of letters is a mystery, which
seems to have no purpose except mystification. It is almost
impossible, at first, not to think that every letter stands for some
particular number, if only the teacher would reveal _what_ number it
stands for. The fact is, that in algebra the mind is first taught to
consider general truths, truths which are not asserted to hold only of
this or that particular thing, but of any one of a whole group of
things. It is in the power of understanding and discovering such
truths that the mastery of the intellect over the whole world of
things actual and possible resides; and ability to deal with the
general as such is one of the gifts that a mathematical education
should bestow. But how little, as a rule, is the teacher of algebra
able to explain the chasm which divides it from arithmetic, and how
little is the learner assisted in his groping efforts at
comprehension! Usually the method that has been adopted in arithmetic
is continued: rules are set forth, with no adequate explanation of
their grounds; the pupil learns to use the rules blindly, and
presently, when he is able to obtain the answer that the teacher
desires, he feels that he has mastered the difficulties of the
subject. But of inner comprehension of the processes employed he has
probably acquired almost nothing.
When algebra has been learnt, all goes smoothly until we reach those
studies in which the notion of infinity is employed--the infinitesimal
calculus and the whole of higher mathematics. The solution of the
difficulties which formerly surrounded the mathematical infinite is
probably the greatest achievement of which our own age has to boast.
Since the beginnings of Greek thought these difficulties have been
known; in every age the finest intellects have vainly endeavoured to
answer the apparently unanswerable questions that had been asked by
Zeno the Eleatic. At last Georg Cantor has found the answer, and has
conquered for the intellect a new and vast province which had been
given over to Chaos and old Night. It was assumed as self-evident,
until Cantor and Dedekind established the opposite, that if, from any
collection of things, some were taken away, the number of things left
must always be less than the original number of things. This
assumption, as a matter of fact, holds only of finite collections; and
the rejection of it, where the infinite is concerned, has been shown
to remove all the difficulties that had hitherto baffled human reason
in this matter, and to render possible the creation of an exact
science of the infinite. This stupendous fact ought to produce a
revolution in the higher teaching of mathematics; it has itself added
immeasurably to the educational value of the subject, and it has at
last given the means of treating with logical precision many studies
which, until lately, were wrapped in fallacy and obscurity. By those
who were educated on the old lines, the new work is considered to be
appallingly difficult, abstruse, and obscure; and it must be confessed
that the discoverer, as is so often the case, has hardly himself
emerged from the mists which the light of his intellect is dispelling.
But inherently, the new doctrine of the infinite, to all candid and
inquiring minds, has facilitated the mastery of higher mathematics;
for hitherto, it has been necessary to learn, by a long process of
sophistication, to give assent to arguments which, on first
acquaintance, were rightly judged to be confused and erroneous. So far
from producing a fearless belief in reason, a bold rejection of
whatever failed to fulfil the strictest requirements of logic, a
mathematical training, during the past two centuries, encouraged the
belief that many things, which a rigid inquiry would reject as
fallacious, must yet be accepted because they work in what the
mathematician calls "practice. " By this means, a timid, compromising
spirit, or else a sacerdotal belief in mysteries not intelligible to
the profane, has been bred where reason alone should have ruled. All
this it is now time to sweep away; let those who wish to penetrate
into the arcana of mathematics be taught at once the true theory in
all its logical purity, and in the concatenation established by the
very essence of the entities concerned.
If we are considering mathematics as an end in itself, and not as a
technical training for engineers, it is very desirable to preserve the
purity and strictness of its reasoning. Accordingly those who have
attained a sufficient familiarity with its easier portions should be
led backward from propositions to which they have assented as
self-evident to more and more fundamental principles from which what
had previously appeared as premises can be deduced. They should be
taught--what the theory of infinity very aptly illustrates--that many
propositions seem self-evident to the untrained mind which,
nevertheless, a nearer scrutiny shows to be false. By this means they
will be led to a sceptical inquiry into first principles, an
examination of the foundations upon which the whole edifice of
reasoning is built, or, to take perhaps a more fitting metaphor, the
great trunk from which the spreading branches spring. At this stage,
it is well to study afresh the elementary portions of mathematics,
asking no longer merely whether a given proposition is true, but also
how it grows out of the central principles of logic. Questions of this
nature can now be answered with a precision and certainty which were
formerly quite impossible; and in the chains of reasoning that the
answer requires the unity of all mathematical studies at last unfolds
In the great majority of mathematical text-books there is a total lack
of unity in method and of systematic development of a central theme.
Propositions of very diverse kinds are proved by whatever means are
thought most easily intelligible, and much space is devoted to mere
curiosities which in no way contribute to the main argument. But in
the greatest works, unity and inevitability are felt as in the
unfolding of a drama; in the premisses a subject is proposed for
consideration, and in every subsequent step some definite advance is
made towards mastery of its nature. The love of system, of
interconnection, which is perhaps the inmost essence of the
intellectual impulse, can find free play in mathematics as nowhere
else. The learner who feels this impulse must not be repelled by an
array of meaningless examples or distracted by amusing oddities, but
must be encouraged to dwell upon central principles, to become
familiar with the structure of the various subjects which are put
before him, to travel easily over the steps of the more important
deductions. In this way a good tone of mind is cultivated, and
selective attention is taught to dwell by preference upon what is
weighty and essential.
When the separate studies into which mathematics is divided have each
been viewed as a logical whole, as a natural growth from the
propositions which constitute their principles, the learner will be
able to understand the fundamental science which unifies and
systematises the whole of deductive reasoning. This is symbolic
logic--a study which, though it owes its inception to Aristotle, is
yet, in its wider developments, a product, almost wholly, of the
nineteenth century, and is indeed, in the present day, still growing
with great rapidity. The true method of discovery in symbolic logic,
and probably also the best method for introducing the study to a
learner acquainted with other parts of mathematics, is the analysis of
actual examples of deductive reasoning, with a view to the discovery
of the principles employed. These principles, for the most part, are
so embedded in our ratiocinative instincts, that they are employed
quite unconsciously, and can be dragged to light only by much patient
effort. But when at last they have been found, they are seen to be few
in number, and to be the sole source of everything in pure
mathematics. The discovery that all mathematics follows inevitably
from a small collection of fundamental laws is one which immeasurably
enhances the intellectual beauty of the whole; to those who have been
oppressed by the fragmentary and incomplete nature of most existing
chains of deduction this discovery comes with all the overwhelming
force of a revelation; like a palace emerging from the autumn mist as
the traveller ascends an Italian hill-side, the stately storeys of the
mathematical edifice appear in their due order and proportion, with a
new perfection in every part.
Until symbolic logic had acquired its present development, the
principles upon which mathematics depends were always supposed to be
philosophical, and discoverable only by the uncertain, unprogressive
methods hitherto employed by philosophers. So long as this was
thought, mathematics seemed to be not autonomous, but dependent upon a
study which had quite other methods than its own. Moreover, since the
nature of the postulates from which arithmetic, analysis, and geometry
are to be deduced was wrapped in all the traditional obscurities of
metaphysical discussion, the edifice built upon such dubious
foundations began to be viewed as no better than a castle in the air.
In this respect, the discovery that the true principles are as much a
part of mathematics as any of their consequences has very greatly
increased the intellectual satisfaction to be obtained. This
satisfaction ought not to be refused to learners capable of enjoying
it, for it is of a kind to increase our respect for human powers and
our knowledge of the beauties belonging to the abstract world.
Philosophers have commonly held that the laws of logic, which underlie
mathematics, are laws of thought, laws regulating the operations of
our minds. By this opinion the true dignity of reason is very greatly
lowered: it ceases to be an investigation into the very heart and
immutable essence of all things actual and possible, becoming,
instead, an inquiry into something more or less human and subject to
our limitations. The contemplation of what is non-human, the discovery
that our minds are capable of dealing with material not created by
them, above all, the realisation that beauty belongs to the outer
world as to the inner, are the chief means of overcoming the terrible
sense of impotence, of weakness, of exile amid hostile powers, which
is too apt to result from acknowledging the all-but omnipotence of
alien forces. To reconcile us, by the exhibition of its awful beauty,
to the reign of Fate--which is merely the literary personification of
these forces--is the task of tragedy. But mathematics takes us still
further from what is human, into the region of absolute necessity, to
which not only the actual world, but every possible world, must
conform; and even here it builds a habitation, or rather finds a
habitation eternally standing, where our ideals are fully satisfied
and our best hopes are not thwarted. It is only when we thoroughly
understand the entire independence of ourselves, which belongs to this
world that reason finds, that we can adequately realise the profound
importance of its beauty.
Not only is mathematics independent of us and our thoughts, but in
another sense we and the whole universe of existing things are
independent of mathematics. The apprehension of this purely ideal
character is indispensable, if we are to understand rightly the place
of mathematics as one among the arts. It was formerly supposed that
pure reason could decide, in some respects, as to the nature of the
actual world: geometry, at least, was thought to deal with the space
in which we live. But we now know that pure mathematics can never
pronounce upon questions of actual existence: the world of reason, in
a sense, controls the world of fact, but it is not at any point
creative of fact, and in the application of its results to the world
in time and space, its certainty and precision are lost among
approximations and working hypotheses. The objects considered by
mathematicians have, in the past, been mainly of a kind suggested by
phenomena; but from such restrictions the abstract imagination should
be wholly free. A reciprocal liberty must thus be accorded: reason
cannot dictate to the world of facts, but the facts cannot restrict
reason's privilege of dealing with whatever objects its love of beauty
may cause to seem worthy of consideration. Here, as elsewhere, we
build up our own ideals out of the fragments to be found in the world;
and in the end it is hard to say whether the result is a creation or a
It is very desirable, in instruction, not merely to persuade the
student of the accuracy of important theorems, but to persuade him in
the way which itself has, of all possible ways, the most beauty. The
true interest of a demonstration is not, as traditional modes of
exposition suggest, concentrated wholly in the result; where this does
occur, it must be viewed as a defect, to be remedied, if possible, by
so generalising the steps of the proof that each becomes important in
and for itself. An argument which serves only to prove a conclusion is
like a story subordinated to some moral which it is meant to teach:
for aesthetic perfection no part of the whole should be merely a means.
A certain practical spirit, a desire for rapid progress, for conquest
of new realms, is responsible for the undue emphasis upon results
which prevails in mathematical instruction. The better way is to
propose some theme for consideration--in geometry, a figure having
important properties; in analysis, a function of which the study is
illuminating, and so on. Whenever proofs depend upon some only of the
marks by which we define the object to be studied, these marks should
be isolated and investigated on their own account. For it is a defect,
in an argument, to employ more premisses than the conclusion demands:
what mathematicians call elegance results from employing only the
essential principles in virtue of which the thesis is true. It is a
merit in Euclid that he advances as far as he is able to go without
employing the axiom of parallels--not, as is often said, because this
axiom is inherently objectionable, but because, in mathematics, every
new axiom diminishes the generality of the resulting theorems, and the
greatest possible generality is before all things to be sought.
Of the effects of mathematics outside its own sphere more has been
written than on the subject of its own proper ideal. The effect upon
philosophy has, in the past, been most notable, but most varied; in
the seventeenth century, idealism and rationalism, in the eighteenth,
materialism and sensationalism, seemed equally its offspring. Of the
effect which it is likely to have in the future it would be very rash
to say much; but in one respect a good result appears probable.
Against that kind of scepticism which abandons the pursuit of ideals
because the road is arduous and the goal not certainly attainable,
mathematics, within its own sphere, is a complete answer. Too often it
is said that there is no absolute truth, but only opinion and private
judgment; that each of us is conditioned, in his view of the world, by
his own peculiarities, his own taste and bias; that there is no
external kingdom of truth to which, by patience and discipline, we may
at last obtain admittance, but only truth for me, for you, for every
separate person. By this habit of mind one of the chief ends of human
effort is denied, and the supreme virtue of candour, of fearless
acknowledgment of what is, disappears from our moral vision. Of such
scepticism mathematics is a perpetual reproof; for its edifice of
truths stands unshakable and inexpungable to all the weapons of
doubting cynicism.
The effects of mathematics upon practical life, though they should not
be regarded as the motive of our studies, may be used to answer a
doubt to which the solitary student must always be liable. In a world
so full of evil and suffering, retirement into the cloister of
contemplation, to the enjoyment of delights which, however noble, must
always be for the few only, cannot but appear as a somewhat selfish
refusal to share the burden imposed upon others by accidents in which
justice plays no part. Have any of us the right, we ask, to withdraw
from present evils, to leave our fellow-men unaided, while we live a
life which, though arduous and austere, is yet plainly good in its own
nature? When these questions arise, the true answer is, no doubt, that
some must keep alive the sacred fire, some must preserve, in every
generation, the haunting vision which shadows forth the goal of so
much striving. But when, as must sometimes occur, this answer seems
too cold, when we are almost maddened by the spectacle of sorrows to
which we bring no help, then we may reflect that indirectly the
mathematician often does more for human happiness than any of his more
practically active contemporaries. The history of science abundantly
proves that a body of abstract propositions--even if, as in the case
of conic sections, it remains two thousand years without effect upon
daily life--may yet, at any moment, be used to cause a revolution in
the habitual thoughts and occupations of every citizen. The use of
steam and electricity--to take striking instances--is rendered
possible only by mathematics. In the results of abstract thought the
world possesses a capital of which the employment in enriching the
common round has no hitherto discoverable limits. Nor does experience
give any means of deciding what parts of mathematics will be found
useful. Utility, therefore, can be only a consolation in moments of
discouragement, not a guide in directing our studies.
For the health of the moral life, for ennobling the tone of an age or
a nation, the austerer virtues have a strange power, exceeding the
power of those not informed and purified by thought. Of these austerer
virtues the love of truth is the chief, and in mathematics, more than
elsewhere, the love of truth may find encouragement for waning faith.
Every great study is not only an end in itself, but also a means of
creating and sustaining a lofty habit of mind; and this purpose should
be kept always in view throughout the teaching and learning of
[10] This passage was pointed out to me by Professor Gilbert Murray.
The nineteenth century, which prided itself upon the invention of
steam and evolution, might have derived a more legitimate title to
fame from the discovery of pure mathematics. This science, like most
others, was baptised long before it was born; and thus we find writers
before the nineteenth century alluding to what they called pure
mathematics. But if they had been asked what this subject was, they
would only have been able to say that it consisted of Arithmetic,
Algebra, Geometry, and so on. As to what these studies had in common,
and as to what distinguished them from applied mathematics, our
ancestors were completely in the dark.
Pure mathematics was discovered by Boole, in a work which he called
the _Laws of Thought_ (1854). This work abounds in asseverations that
it is not mathematical, the fact being that Boole was too modest to
suppose his book the first ever written on mathematics. He was also
mistaken in supposing that he was dealing with the laws of thought:
the question how people actually think was quite irrelevant to him,
and if his book had really contained the laws of thought, it was
curious that no one should ever have thought in such a way before. His
book was in fact concerned with formal logic, and this is the same
thing as mathematics.
Pure mathematics consists entirely of assertions to the effect that,
if such and such a proposition is true of _anything_, then such and
such another proposition is true of that thing. It is essential not to
discuss whether the first proposition is really true, and not to
mention what the anything is, of which it is supposed to be true. Both
these points would belong to applied mathematics. We start, in pure
mathematics, from certain rules of inference, by which we can infer
that _if_ one proposition is true, then so is some other proposition.
These rules of inference constitute the major part of the principles
of formal logic. We then take any hypothesis that seems amusing, and
deduce its consequences. _If_ our hypothesis is about _anything_, and
not about some one or more particular things, then our deductions
constitute mathematics. Thus mathematics may be defined as the subject
in which we never know what we are talking about, nor whether what we
are saying is true. People who have been puzzled by the beginnings of
mathematics will, I hope, find comfort in this definition, and will
probably agree that it is accurate.
As one of the chief triumphs of modern mathematics consists in having
discovered what mathematics really is, a few more words on this
subject may not be amiss. It is common to start any branch of
mathematics--for instance, Geometry--with a certain number of
primitive ideas, supposed incapable of definition, and a certain
number of primitive propositions or axioms, supposed incapable of
proof. Now the fact is that, though there are indefinables and
indemonstrables in every branch of applied mathematics, there are none
in pure mathematics except such as belong to general logic. Logic,
broadly speaking, is distinguished by the fact that its propositions
can be put into a form in which they apply to anything whatever. All
pure mathematics--Arithmetic, Analysis, and Geometry--is built up by
combinations of the primitive ideas of logic, and its propositions are
deduced from the general axioms of logic, such as the syllogism and
the other rules of inference. And this is no longer a dream or an
aspiration. On the contrary, over the greater and more difficult part
of the domain of mathematics, it has been already accomplished; in the
few remaining cases, there is no special difficulty, and it is now
being rapidly achieved. Philosophers have disputed for ages whether
such deduction was possible; mathematicians have sat down and made the
deduction. For the philosophers there is now nothing left but graceful
The subject of formal logic, which has thus at last shown itself to be
identical with mathematics, was, as every one knows, invented by
Aristotle, and formed the chief study (other than theology) of the
Middle Ages. But Aristotle never got beyond the syllogism, which is a
very small part of the subject, and the schoolmen never got beyond
Aristotle. If any proof were required of our superiority to the
mediaeval doctors, it might be found in this. Throughout the Middle
Ages, almost all the best intellects devoted themselves to formal
logic, whereas in the nineteenth century only an infinitesimal
proportion of the world's thought went into this subject.
Nevertheless, in each decade since 1850 more has been done to advance
the subject than in the whole period from Aristotle to Leibniz. People
have discovered how to make reasoning symbolic, as it is in Algebra,
so that deductions are effected by mathematical rules. They have
discovered many rules besides the syllogism, and a new branch of
logic, called the Logic of Relatives,[11] has been invented to deal
with topics that wholly surpassed the powers of the old logic, though
they form the chief contents of mathematics.
It is not easy for the lay mind to realise the importance of symbolism
in discussing the foundations of mathematics, and the explanation may
perhaps seem strangely paradoxical. The fact is that symbolism is
useful because it makes things difficult. (This is not true of the
advanced parts of mathematics, but only of the beginnings. ) What we
wish to know is, what can be deduced from what. Now, in the
beginnings, everything is self-evident; and it is very hard to see
whether one self-evident proposition follows from another or not.
Obviousness is always the enemy to correctness. Hence we invent some
new and difficult symbolism, in which nothing seems obvious. Then we
set up certain rules for operating on the symbols, and the whole thing
becomes mechanical. In this way we find out what must be taken as
premiss and what can be demonstrated or defined. For instance, the
whole of Arithmetic and Algebra has been shown to require three
indefinable notions and five indemonstrable propositions. But without
a symbolism it would have been very hard to find this out. It is so
obvious that two and two are four, that we can hardly make ourselves
sufficiently sceptical to doubt whether it can be proved. And the same
holds in other cases where self-evident things are to be proved.
But the proof of self-evident propositions may seem, to the
uninitiated, a somewhat frivolous occupation. To this we might reply
that it is often by no means self-evident that one obvious proposition
follows from another obvious proposition; so that we are really
discovering new truths when we prove what is evident by a method which
is not evident. But a more interesting retort is, that since people
have tried to prove obvious propositions, they have found that many of
them are false. Self-evidence is often a mere will-o'-the-wisp, which
is sure to lead us astray if we take it as our guide. For instance,
nothing is plainer than that a whole always has more terms than a
part, or that a number is increased by adding one to it. But these
propositions are now known to be usually false. Most numbers are
infinite, and if a number is infinite you may add ones to it as long
as you like without disturbing it in the least. One of the merits of a
proof is that it instils a certain doubt as to the result proved; and
when what is obvious can be proved in some cases, but not in others,
it becomes possible to suppose that in these other cases it is false.
The great master of the art of formal reasoning, among the men of our
own day, is an Italian, Professor Peano, of the University of
Turin. [12] He has reduced the greater part of mathematics (and he or
his followers will, in time, have reduced the whole) to strict
symbolic form, in which there are no words at all. In the ordinary
mathematical books, there are no doubt fewer words than most readers
would wish. Still, little phrases occur, such as _therefore, let us
assume, consider_, or _hence it follows_. All these, however, are a
concession, and are swept away by Professor Peano. For instance, if we
wish to learn the whole of Arithmetic, Algebra, the Calculus, and
indeed all that is usually called pure mathematics (except Geometry),
we must start with a dictionary of three words. One symbol stands for
_zero_, another for _number_, and a third for _next after_. What these
ideas mean, it is necessary to know if you wish to become an
arithmetician. But after symbols have been invented for these three
ideas, not another word is required in the whole development. All
future symbols are symbolically explained by means of these three.
Even these three can be explained by means of the notions of
_relation_ and _class_; but this requires the Logic of Relations,
which Professor Peano has never taken up. It must be admitted that
what a mathematician has to know to begin with is not much. There are
at most a dozen notions out of which all the notions in all pure
mathematics (including Geometry) are compounded. Professor Peano, who
is assisted by a very able school of young Italian disciples, has
shown how this may be done; and although the method which he has
invented is capable of being carried a good deal further than he has
carried it, the honour of the pioneer must belong to him.
Two hundred years ago, Leibniz foresaw the science which Peano has
perfected, and endeavoured to create it.
He was prevented from
succeeding by respect for the authority of Aristotle, whom he could
not believe guilty of definite, formal fallacies; but the subject
which he desired to create now exists, in spite of the patronising
contempt with which his schemes have been treated by all superior
persons. From this "Universal Characteristic," as he called it, he
hoped for a solution of all problems, and an end to all disputes. "If
controversies were to arise," he says, "there would be no more need of
disputation between two philosophers than between two accountants. For
it would suffice to take their pens in their hands, to sit down to
their desks, and to say to each other (with a friend as witness, if
they liked), 'Let us calculate. '" This optimism has now appeared to be
somewhat excessive; there still are problems whose solution is
doubtful, and disputes which calculation cannot decide. But over an
enormous field of what was formerly controversial, Leibniz's dream has
become sober fact. In the whole philosophy of mathematics, which used
to be at least as full of doubt as any other part of philosophy, order
and certainty have replaced the confusion and hesitation which
formerly reigned. Philosophers, of course, have not yet discovered
this fact, and continue to write on such subjects in the old way. But
mathematicians, at least in Italy, have now the power of treating the
principles of mathematics in an exact and masterly manner, by means of
which the certainty of mathematics extends also to mathematical
philosophy. Hence many of the topics which used to be placed among the
great mysteries--for example, the natures of infinity, of continuity,
of space, time and motion--are now no longer in any degree open to
doubt or discussion. Those who wish to know the nature of these things
need only read the works of such men as Peano or Georg Cantor; they
will there find exact and indubitable expositions of all these quondam
In this capricious world, nothing is more capricious than posthumous
fame. One of the most notable examples of posterity's lack of judgment
is the Eleatic Zeno. This man, who may be regarded as the founder of
the philosophy of infinity, appears in Plato's Parmenides in the
privileged position of instructor to Socrates. He invented four
arguments, all immeasurably subtle and profound, to prove that motion
is impossible, that Achilles can never overtake the tortoise, and that
an arrow in flight is really at rest. After being refuted by
Aristotle, and by every subsequent philosopher from that day to our
own, these arguments were reinstated, and made the basis of a
mathematical renaissance, by a German professor, who probably never
dreamed of any connection between himself and Zeno. Weierstrass,[13]
by strictly banishing from mathematics the use of infinitesimals, has
at last shown that we live in an unchanging world, and that the arrow
in its flight is truly at rest. Zeno's only error lay in inferring (if
he did infer) that, because there is no such thing as a state of
change, therefore the world is in the same state at any one time as at
any other. This is a consequence which by no means follows; and in
this respect, the German mathematician is more constructive than the
ingenious Greek. Weierstrass has been able, by embodying his views in
mathematics, where familiarity with truth eliminates the vulgar
prejudices of common sense, to invest Zeno's paradoxes with the
respectable air of platitudes; and if the result is less delightful to
the lover of reason than Zeno's bold defiance, it is at any rate more
calculated to appease the mass of academic mankind.
Zeno was concerned, as a matter of fact, with three problems, each
presented by motion, but each more abstract than motion, and capable
of a purely arithmetical treatment. These are the problems of the
infinitesimal, the infinite, and continuity. To state clearly the
difficulties involved, was to accomplish perhaps the hardest part of
the philosopher's task. This was done by Zeno. From him to our own
day, the finest intellects of each generation in turn attacked the
problems, but achieved, broadly speaking, nothing. In our own time,
however, three men--Weierstrass, Dedekind, and Cantor--have not merely
advanced the three problems, but have completely solved them. The
solutions, for those acquainted with mathematics, are so clear as to
leave no longer the slightest doubt or difficulty. This achievement is
probably the greatest of which our age has to boast; and I know of no
age (except perhaps the golden age of Greece) which has a more
convincing proof to offer of the transcendent genius of its great men.
Of the three problems, that of the infinitesimal was solved by
Weierstrass; the solution of the other two was begun by Dedekind, and
definitively accomplished by Cantor.
The infinitesimal played formerly a great part in mathematics. It was
introduced by the Greeks, who regarded a circle as differing
infinitesimally from a polygon with a very large number of very small
equal sides. It gradually grew in importance, until, when Leibniz
invented the Infinitesimal Calculus, it seemed to become the
fundamental notion of all higher mathematics. Carlyle tells, in his
_Frederick the Great_, how Leibniz used to discourse to Queen Sophia
Charlotte of Prussia concerning the infinitely little, and how she
would reply that on that subject she needed no instruction--the
behaviour of courtiers had made her thoroughly familiar with it. But
philosophers and mathematicians--who for the most part had less
acquaintance with courts--continued to discuss this topic, though
without making any advance. The Calculus required continuity, and
continuity was supposed to require the infinitely little; but nobody
could discover what the infinitely little might be. It was plainly not
quite zero, because a sufficiently large number of infinitesimals,
added together, were seen to make up a finite whole. But nobody could
point out any fraction which was not zero, and yet not finite. Thus
there was a deadlock. But at last Weierstrass discovered that the
infinitesimal was not needed at all, and that everything could be
accomplished without it. Thus there was no longer any need to suppose
that there was such a thing. Nowadays, therefore, mathematicians are
more dignified than Leibniz: instead of talking about the infinitely
small, they talk about the infinitely great--a subject which, however
appropriate to monarchs, seems, unfortunately, to interest them even
less than the infinitely little interested the monarchs to whom
Leibniz discoursed.
The banishment of the infinitesimal has all sorts of odd consequences,
to which one has to become gradually accustomed. For example, there is
no such thing as the next moment. The interval between one moment and
the next would have to be infinitesimal, since, if we take two moments
with a finite interval between them, there are always other moments in
the interval. Thus if there are to be no infinitesimals, no two
moments are quite consecutive, but there are always other moments
between any two. Hence there must be an infinite number of moments
between any two; because if there were a finite number one would be
nearest the first of the two moments, and therefore next to it. This
might be thought to be a difficulty; but, as a matter of fact, it is
here that the philosophy of the infinite comes in, and makes all
The same sort of thing happens in space. If any piece of matter be cut
in two, and then each part be halved, and so on, the bits will become
smaller and smaller, and can theoretically be made as small as we
please. However small they may be, they can still be cut up and made
smaller still. But they will always have _some_ finite size, however
small they may be. We never reach the infinitesimal in this way, and
no finite number of divisions will bring us to points. Nevertheless
there _are_ points, only these are not to be reached by successive
divisions. Here again, the philosophy of the infinite shows us how
this is possible, and why points are not infinitesimal lengths.
As regards motion and change, we get similarly curious results. People
used to think that when a thing changes, it must be in a state of
change, and that when a thing moves, it is in a state of motion. This
is now known to be a mistake. When a body moves, all that can be said
is that it is in one place at one time and in another at another. We
must not say that it will be in a neighbouring place at the next
instant, since there is no next instant. Philosophers often tell us
that when a body is in motion, it changes its position within the
instant. To this view Zeno long ago made the fatal retort that every
body always is where it is; but a retort so simple and brief was not
of the kind to which philosophers are accustomed to give weight, and
they have continued down to our own day to repeat the same phrases
which roused the Eleatic's destructive ardour. It was only recently
that it became possible to explain motion in detail in accordance with
Zeno's platitude, and in opposition to the philosopher's paradox. We
may now at last indulge the comfortable belief that a body in motion
is just as truly where it is as a body at rest. Motion consists merely
in the fact that bodies are sometimes in one place and sometimes in
another, and that they are at intermediate places at intermediate
times. Only those who have waded through the quagmire of philosophic
speculation on this subject can realise what a liberation from antique
prejudices is involved in this simple and straightforward commonplace.
The philosophy of the infinitesimal, as we have just seen, is mainly
negative. People used to believe in it, and now they have found out
their mistake. The philosophy of the infinite, on the other hand, is
wholly positive. It was formerly supposed that infinite numbers, and
the mathematical infinite generally, were self-contradictory. But as
it was obvious that there were infinities--for example, the number of
numbers--the contradictions of infinity seemed unavoidable, and
philosophy seemed to have wandered into a "cul-de-sac. " This
difficulty led to Kant's antinomies, and hence, more or less
indirectly, to much of Hegel's dialectic method. Almost all current
philosophy is upset by the fact (of which very few philosophers are as
yet aware) that all the ancient and respectable contradictions in the
notion of the infinite have been once for all disposed of. The method
by which this has been done is most interesting and instructive. In
the first place, though people had talked glibly about infinity ever
since the beginnings of Greek thought, nobody had ever thought of
asking, What is infinity? If any philosopher had been asked for a
definition of infinity, he might have produced some unintelligible
rigmarole, but he would certainly not have been able to give a
definition that had any meaning at all. Twenty years ago, roughly
speaking, Dedekind and Cantor asked this question, and, what is more
remarkable, they answered it. They found, that is to say, a perfectly
precise definition of an infinite number or an infinite collection of
things. This was the first and perhaps the greatest step. It then
remained to examine the supposed contradictions in this notion. Here
Cantor proceeded in the only proper way. He took pairs of
contradictory propositions, in which both sides of the contradiction
would be usually regarded as demonstrable, and he strictly examined
the supposed proofs. He found that all proofs adverse to infinity
involved a certain principle, at first sight obviously true, but
destructive, in its consequences, of almost all mathematics. The
proofs favourable to infinity, on the other hand, involved no
principle that had evil consequences. It thus appeared that common
sense had allowed itself to be taken in by a specious maxim, and that,
when once this maxim was rejected, all went well.
The maxim in question is, that if one collection is part of another,
the one which is a part has fewer terms than the one of which it is a
part. This maxim is true of finite numbers. For example, Englishmen
are only some among Europeans, and there are fewer Englishmen than
Europeans. But when we come to infinite numbers, this is no longer
true. This breakdown of the maxim gives us the precise definition of
infinity. A collection of terms is infinite when it contains as parts
other collections which have just as many terms as it has. If you can
take away some of the terms of a collection, without diminishing the
number of terms, then there are an infinite number of terms in the
collection. For example, there are just as many even numbers as there
are numbers altogether, since every number can be doubled. This may be
seen by putting odd and even numbers together in one row, and even
numbers alone in a row below:--
1, 2, 3, 4, 5, _ad infinitum_.
2, 4, 6, 8, 10, _ad infinitum_.
There are obviously just as many numbers in the row below as in the
row above, because there is one below for each one above. This
property, which was formerly thought to be a contradiction, is now
transformed into a harmless definition of infinity, and shows, in the
above case, that the number of finite numbers is infinite.
But the uninitiated may wonder how it is possible to deal with a
number which cannot be counted. It is impossible to count up _all_ the
numbers, one by one, because, however many we may count, there are
always more to follow. The fact is that counting is a very vulgar and
elementary way of finding out how many terms there are in a
collection. And in any case, counting gives us what mathematicians
call the _ordinal_ number of our terms; that is to say, it arranges
our terms in an order or series, and its result tells us what type of
series results from this arrangement. In other words, it is impossible
to count things without counting some first and others afterwards, so
that counting always has to do with order. Now when there are only a
finite number of terms, we can count them in any order we like; but
when there are an infinite number, what corresponds to counting will
give us quite different results according to the way in which we carry
out the operation. Thus the ordinal number, which results from what,
in a general sense may be called counting, depends not only upon how
many terms we have, but also (where the number of terms is infinite)
upon the way in which the terms are arranged.
The fundamental infinite numbers are not ordinal, but are what is
called _cardinal_. They are not obtained by putting our terms in order
and counting them, but by a different method, which tells us, to begin
with, whether two collections have the same number of terms, or, if
not, which is the greater. [14] It does not tell us, in the way in
which counting does, _what_ number of terms a collection has; but if
we define a number as the number of terms in such and such a
collection, then this method enables us to discover whether some other
collection that may be mentioned has more or fewer terms. An
illustration will show how this is done. If there existed some country
in which, for one reason or another, it was impossible to take a
census, but in which it was known that every man had a wife and every
woman a husband, then (provided polygamy was not a national
institution) we should know, without counting, that there were exactly
as many men as there were women in that country, neither more nor
less. This method can be applied generally. If there is some relation
which, like marriage, connects the things in one collection each with
one of the things in another collection, and vice versa, then the two
collections have the same number of terms. This was the way in which
we found that there are as many even numbers as there are numbers.
Every number can be doubled, and every even number can be halved, and
each process gives just one number corresponding to the one that is
doubled or halved. And in this way we can find any number of
collections each of which has just as many terms as there are finite
numbers. If every term of a collection can be hooked on to a number,
and all the finite numbers are used once, and only once, in the
process, then our collection must have just as many terms as there are
finite numbers. This is the general method by which the numbers of
infinite collections are defined.
But it must not be supposed that all infinite numbers are equal. On
the contrary, there are infinitely more infinite numbers than finite
ones. There are more ways of arranging the finite numbers in different
types of series than there are finite numbers. There are probably more
points in space and more moments in time than there are finite
numbers. There are exactly as many fractions as whole numbers,
although there are an infinite number of fractions between any two
whole numbers. But there are more irrational numbers than there are
whole numbers or fractions. There are probably exactly as many points
in space as there are irrational numbers, and exactly as many points
on a line a millionth of an inch long as in the whole of infinite
space. There is a greatest of all infinite numbers, which is the
number of things altogether, of every sort and kind. It is obvious
that there cannot be a greater number than this, because, if
everything has been taken, there is nothing left to add. Cantor has a
proof that there is no greatest number, and if this proof were valid,
the contradictions of infinity would reappear in a sublimated form.
But in this one point, the master has been guilty of a very subtle
fallacy, which I hope to explain in some future work. [15]
We can now understand why Zeno believed that Achilles cannot overtake
the tortoise and why as a matter of fact he can overtake it. We shall
see that all the people who disagreed with Zeno had no right to do so,
because they all accepted premises from which his conclusion followed.
The argument is this: Let Achilles and the tortoise start along a road
at the same time, the tortoise (as is only fair) being allowed a
handicap. Let Achilles go twice as fast as the tortoise, or ten times
or a hundred times as fast. Then he will never reach the tortoise. For
at every moment the tortoise is somewhere and Achilles is somewhere;
and neither is ever twice in the same place while the race is going
on. Thus the tortoise goes to just as many places as Achilles does,
because each is in one place at one moment, and in another at any
other moment. But if Achilles were to catch up with the tortoise, the
places where the tortoise would have been would be only part of the
places where Achilles would have been. Here, we must suppose, Zeno
appealed to the maxim that the whole has more terms than the part. [16]
Thus if Achilles were to overtake the tortoise, he would have been in
more places than the tortoise; but we saw that he must, in any period,
be in exactly as many places as the tortoise. Hence we infer that he
can never catch the tortoise. This argument is strictly correct, if we
allow the axiom that the whole has more terms than the part. As the
conclusion is absurd, the axiom must be rejected, and then all goes
well. But there is no good word to be said for the philosophers of the
past two thousand years and more, who have all allowed the axiom and
denied the conclusion.
The retention of this axiom leads to absolute contradictions, while
its rejection leads only to oddities. Some of these oddities, it must
be confessed, are very odd. One of them, which I call the paradox of
Tristram Shandy, is the converse of the Achilles, and shows that the
tortoise, if you give him time, will go just as far as Achilles.
Tristram Shandy, as we know, employed two years in chronicling the
first two days of his life, and lamented that, at this rate, material
would accumulate faster than he could deal with it, so that, as years
went by, he would be farther and farther from the end of his history.
Now I maintain that, if he had lived for ever, and had not wearied of
his task, then, even if his life had continued as event fully as it
began, no part of his biography would have remained unwritten. For
consider: the hundredth day will be described in the hundredth year,
the thousandth in the thousandth year, and so on. Whatever day we may
choose as so far on that he cannot hope to reach it, that day will be
described in the corresponding year. Thus any day that may be
mentioned will be written up sooner or later, and therefore no part of
the biography will remain permanently unwritten. This paradoxical but
perfectly true proposition depends upon the fact that the number of
days in all time is no greater than the number of years.
Thus on the subject of infinity it is impossible to avoid conclusions
which at first sight appear paradoxical, and this is the reason why so
many philosophers have supposed that there were inherent
contradictions in the infinite. But a little practice enables one to
grasp the true principles of Cantor's doctrine, and to acquire new and
better instincts as to the true and the false. The oddities then
become no odder than the people at the antipodes, who used to be
thought impossible because they would find it so inconvenient to stand
on their heads.
The solution of the problems concerning infinity has enabled Cantor to
solve also the problems of continuity. Of this, as of infinity, he has
given a perfectly precise definition, and has shown that there are no
contradictions in the notion so defined. But this subject is so
technical that it is impossible to give any account of it here.
The notion of continuity depends upon that of _order_, since
continuity is merely a particular type of order. Mathematics has, in
modern times, brought order into greater and greater prominence. In
former days, it was supposed (and philosophers are still apt to
suppose) that quantity was the fundamental notion of mathematics. But
nowadays, quantity is banished altogether, except from one little
corner of Geometry, while order more and more reigns supreme. The
investigation of different kinds of series and their relations is now
a very large part of mathematics, and it has been found that this
investigation can be conducted without any reference to quantity, and,
for the most part, without any reference to number. All types of
series are capable of formal definition, and their properties can be
deduced from the principles of symbolic logic by means of the Algebra
of Relatives. The notion of a limit, which is fundamental in the
greater part of higher mathematics, used to be defined by means of
quantity, as a term to which the terms of some series approximate as
nearly as we please. But nowadays the limit is defined quite
differently, and the series which it limits may not approximate to it
at all. This improvement also is due to Cantor, and it is one which
has revolutionised mathematics. Only order is now relevant to limits.
Thus, for instance, the smallest of the infinite integers is the limit
of the finite integers, though all finite integers are at an infinite
distance from it. The study of different types of series is a general
subject of which the study of ordinal numbers (mentioned above) is a
special and very interesting branch. But the unavoidable
technicalities of this subject render it impossible to explain to any
but professed mathematicians.
Geometry, like Arithmetic, has been subsumed, in recent times, under
the general study of order. It was formerly supposed that Geometry was
the study of the nature of the space in which we live, and accordingly
it was urged, by those who held that what exists can only be known
empirically, that Geometry should really be regarded as belonging to
applied mathematics. But it has gradually appeared, by the increase of
non-Euclidean systems, that Geometry throws no more light upon the
nature of space than Arithmetic throws upon the population of the
United States. Geometry is a whole collection of deductive sciences
based on a corresponding collection of sets of axioms. One set of
axioms is Euclid's; other equally good sets of axioms lead to other
results. Whether Euclid's axioms are true, is a question as to which
the pure mathematician is indifferent; and, what is more, it is a
question which it is theoretically impossible to answer with certainty
in the affirmative. It might possibly be shown, by very careful
measurements, that Euclid's axioms are false; but no measurements
could ever assure us (owing to the errors of observation) that they
are exactly true. Thus the geometer leaves to the man of science to
decide, as best he may, what axioms are most nearly true in the actual
world. The geometer takes any set of axioms that seem interesting, and
deduces their consequences. What defines Geometry, in this sense, is
that the axioms must give rise to a series of more than one dimension.
And it is thus that Geometry becomes a department in the study of
In Geometry, as in other parts of mathematics, Peano and his disciples
have done work of the very greatest merit as regards principles.
Formerly, it was held by philosophers and mathematicians alike that
the proofs in Geometry depended on the figure; nowadays, this is known
to be false. In the best books there are no figures at all. The
reasoning proceeds by the strict rules of formal logic from a set of
axioms laid down to begin with. If a figure is used, all sorts of
things seem obviously to follow, which no formal reasoning can prove
from the explicit axioms, and which, as a matter of fact, are only
accepted because they are obvious. By banishing the figure, it becomes
possible to discover _all_ the axioms that are needed; and in this way
all sorts of possibilities, which would have otherwise remained
undetected, are brought to light.
One great advance, from the point of view of correctness, has been
made by introducing points as they are required, and not starting, as
was formerly done, by assuming the whole of space. This method is due
partly to Peano, partly to another Italian named Fano. To those
unaccustomed to it, it has an air of somewhat wilful pedantry. In this
way, we begin with the following axioms: (1) There is a class of
entities called _points_. (2) There is at least one point. (3) If _a_
be a point, there is at least one other point besides _a_. Then we
bring in the straight line joining two points, and begin again with
(4), namely, on the straight line joining _a_ and _b_, there is at
least one other point besides _a_ and _b_. (5) There is at least one
point not on the line _ab_. And so we go on, till we have the means of
obtaining as many points as we require. But the word _space_, as Peano
humorously remarks, is one for which Geometry has no use at all.
The rigid methods employed by modern geometers have deposed Euclid
from his pinnacle of correctness. It was thought, until recent times,
that, as Sir Henry Savile remarked in 1621, there were only two
blemishes in Euclid, the theory of parallels and the theory of
proportion. It is now known that these are almost the only points in
which Euclid is free from blemish. Countless errors are involved in
his first eight propositions. That is to say, not only is it doubtful
whether his axioms are true, which is a comparatively trivial matter,
but it is certain that his propositions do not follow from the axioms
which he enunciates. A vastly greater number of axioms, which Euclid
unconsciously employs, are required for the proof of his propositions. | {"url":"https://nomorelearning.com/gem/195535","timestamp":"2024-11-09T23:24:14Z","content_type":"text/html","content_length":"217857","record_id":"<urn:uuid:3dafc826-bf71-4b28-8da0-de4f68d3c02c>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00706.warc.gz"} |
[OLD] Teen learn - Throwback to Teen Challenge 2019 | Reply Challenges
Practice makes you perfect, but learning the best practices from some experts may boost your coding performances.
This page contains the first three problems from the 2019 Reply Code Challenge - Teen Edition. Let’s take a look at them, but before...
Train with the 2019 Reply Code Challenge problems!
Get your own hands dirty and test your skills, trying out the problems of past editions in sandbox mode.
problem 1 _ Quantum Teleportation
Uncover the solution beyond the statement
The solution of the first problem is simple as it only requires implementing the simulation explained in the problem statement.
Given the initial position and the list of teleportation portals, we have to check the nearest portal according to the procedure detailed at each step. That is, the sum of the absolute differences
between the two coordinates.
The only two things to watch out for are to implement the correct tie-breaker, in case distances are the same distances, and to remove a portal once it’s used.
Source code of a test case solution:
def solve_testcase(tc):
(X, Y, X0, Y0, N, points) = tc
MOD = 100003
IN = [points[i][:2] for i in range(N)] # Qubit position
OUT = [points[i][-2:] for i in range(N)] # Qubit destination
seen = [False for _ in range(N)] # Qubit remained
# Manhattan distance
def dist(a, b):
return abs(a[0]-b[0]) + abs(a[1]-b[1])
# Nearest point and equal distance
def less(pos, a, b):
if dist(pos,a) < dist(pos, b):
return True
elif dist(pos,a) == dist(pos, b):
return a < b
return False
# total distance and actual position
tot = 0
pos = [X0, Y0]
for i in range(N):
# find nearest point
best = None
for p in range(N):
# If already seen, skip
if seen[p]:
# If nearest, select this point
if best == None or less(pos, IN[p], IN[best]):
best = p
tot += dist(pos, IN[best]) # add travelled distance
pos = OUT[best] # teleport to position
seen[best] = True # mark as seen
return tot % MOD
problem 2 _ Riceboard
A little deeper mathematical knowledge for 3 different solutions of increasing difficulty
The second problem is more complicated and can be described simply by calculating the value of the sum:
S = (R0 + R1 + R2 + ... + RX-2 + RX-1) % M
Where R, M and X are given in input and, in particular, X = N2 is the number of cells in a chessboard of size N.
Case R = 2
This is the simplest case, if you know how binary numbers work. According to the binary system we know this important property:
2^0 + 2^1 + 2^2 + ... + 2^(N-1) + 2^N = 2^(N+1) - 1
Given this formula, it’s easy to find the value of S.
Source code to solve the first three levels:
def solve(io):
# read input
[R, N, M] = [int(x) for x in io.readline().strip().split()]
X = pow(N, 2)
return (pow(2,X)-1) % M
Case M prime and R any value
If you’re a good mathematician, you’ll have noticed the problem is actually the calculation of a geometric series. For this problem, a closed formula exists that can be used to find the solution:
S = (1 - R^X)/(1 - R)
Before this using simple formula, we have to pay attention because we’re looking for the values modulo M. As we’re working in modular arithmetic it’s not possible to compute the division in the
standard way, but we need to find the multiplicative inverse of (1-R). In python, this can be done using the pow function.
Once we’ve found the multiplicative inverse, computing the value of S is straightforward.
Source code to solve also the 4th level:
def solve_testcase(t):
(R, N, M) = t
X = N**2
num = (pow(R, X, M) - 1) % M # numerator
den = (1-R)%M # denominator
den_inv = pow(den, -1, M) # inverse of denominator
return (num * den_inv) % M
Case M and R any value
For the general case where N, M and R can be any value, we need to work a little to arrive at an efficient solution. The value of S can be found by constructing its value step by step, using some
simple mathematical properties.
In particular, the procedure is:
1. Find the values of R0, R2, R4, R8, R16 until not reaching the power of X. This operation requires only log(X) steps as we can calculate successive powers using previous values, like R16 = (R8)2
2. Find the values of (R0), (R0+R1), (R0+...+R3), (R0+...+R7), ... until not reaching the power of X. Even this operation requires only log(X) steps and can be calculated using previous values and
the value of 1).
Example: (R0+...+R15) = (R0+...+R7) + R8(R0+...+R7) = (R0+...+R7)+(R8+...+R15)
3. Find the final value of S = (R0 + ... + RX-1) by using the values calculated in 1) and 2).
Example: S = (R0 + ... + R24) = (R0+...+R15) + R16(R0+...+R7) + R16R8(R0)
For the last steps, the values to sum can be found using the binary exponentiation techniques based on the binary decomposition of (X-1).
In this case, 25 = 11001 in base 2 = 2^0 + 2^3 + 2^4 = 16 + 8 + 1
Source code of a test-case solution:
def solve_testcase(t):
(R, N, M) = t
X = (N**2)
powr = [1, R] # 1, R^2, R^4, R^8, ...
sumr = [1, 1+R] # 1, 1+R, 1+R+R^2+R^3, ...
i = 1
# Compute values of powr and sumr
while i < X:
i *= 2
powr.append(powr[-1]*powr[-1] % M)
sumr.append((((1+powr[-1]) % M)*sumr[-1]) % M)
mul, S = 1, 0
b = bin(X)[2:]
# Exponentiation by squaring
for i in range(len(b)):
if b[i] == '0':
i = len(b)-i-1
S = (S + ((mul*sumr[i]) % M)) % M
mul = (mul*powr[i+1]) % M
return S
problem 3 _ T9
Find the most common K passwords according to the probability matrix given by the input data
This problem can be solved using a dynamic programming approach: we can construct our solution starting from the right-most digit, adding a new character step by step.
We’ll start with an empty string. Each time a button is pressed we’ll going to push all its corresponding characters to the last available solution. At each step, we’ll use the probability matrix to
find the most likely passwords by adding to the last passwords the new character’s probabilities.
Source code of a test case solution:
class Bucket:
def __init__(self, d, K):
self.K = K
self.buckets = [[] for _ in range(len(MAP[d]))]
def prepend_key(self, d, prob):
b = Bucket(d, self.K)
for i in range(len(MAP[d])):
c = MAP[d][i]
self.fill_bucket(b.buckets[i], c, prob)
return b
def fill_bucket(self, b, c, prob):
indicies = [0 for _ in range(len(self.buckets))]
while len(b) < self.K:
best_index = -1
best_value = -1
for i in range(len(self.buckets)):
if indicies[i] >= len(self.buckets[i]):
suffix = self.buckets[i][indicies[i]]
value = prob[ord(c) - ord('A')][ord(suffix[1]
[0]) - ord('A')] + suffix[0]
if value > best_value:
best_index = i
best_value = value
if best_index < 0:
new_suffix = c + self.buckets[best_index][indicies[best_index]][1]
b.append([best_value, new_suffix])
indicies[best_index] += 1
def solve_testcase(t):
[L, K, T, S] = t
T2 = [[0 for _ in range(27)] for _ in range(27)]
for i in range(26):
for j in range(26):
T2[i][j] = T[i][j]
T = T2
keys = []
for s in S:
keys.append(ord(s) - ord('0'))
b = Bucket(keys[L-1], K)
for i in range(len(MAP[keys[L-1]])):
b.buckets[i].append([0, MAP[keys[L - 1]][i:i+1]])
for i in range(L-2, -1, -1):
b = b.prepend_key(keys[i], T)
b = b.prepend_key(0, T)
result = []
for s in b.buckets[0]:
return str(result[K-1]) | {"url":"https://challenges.reply.com/challenges/coding-teen/learn-train/teen-challenge-2019-solution-1ilhpa/","timestamp":"2024-11-05T14:58:30Z","content_type":"text/html","content_length":"44611","record_id":"<urn:uuid:bebd1ee2-3bf8-4421-8c46-d1bcab3bff8e>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00489.warc.gz"} |
Numerical Modelling of Rock Fracture Using Polygonal Finite Elements
Polygonal finite elements have been drawing increasing attention during the last 20 years in computational mechanics due to some of their superior features over the traditional finite elements. They
offer, for example, better flexibility in meshing complex geometries and better accuracy in the numerical solution of some problems. However, the main negative aspect of the polygonal finite elements
is the more involved numerical integration since the interpolation functions are usually rational functions.
In this paper, we present some results on a research project aiming at the simulation of rock fracture with a mesoscopic model based on polygonal finite elements. As the mineral texture of many rocks
is polygonal, the polygonal finite element method is a natural choice. Here, the rock meso-structure is described as a Voronoi diagram where the Voronoi cells are the physical polygonal finite
elements. Then, the minerals constituting the rock are represented by random clusters of polygonal finite elements.
In order to account for rock fracture, the meso-scopic rock material description is equipped with a damage-viscoplasticity model based on the Hoek-Brown criterion. Due to the asymmetry of the tension
and compression behavior of rocks, separate scalar damage variables, driven by viscoplastic strain, are employed in tension and compression. The final aim is to study problems with transient impact
loadings, e.g. percussive drilling. For this reason, the system equations of motion are solved by explicit time marching.
In the numerical examples, the capabilities of the present rock material description are demonstrated. Namely, uniaxial tension and compression tests of a numerical rock sample are simulated under
plane strain conditions. Finally, the dynamic Brazilian disc test simulations are carried out as a dynamic example. These simulations demonstrate that the present method can capture the salient
features, including the stress-strain response and the failure modes, of typical rock behavior in these applications.
Conference The 44th Israel Symposium on Computational Mechanics
Lyhennettä ISCM-44
Maa/Alue Israel
Kaupunki Beer Sheva
Ajanjakso 22/03/18 → …
Sukella tutkimusaiheisiin 'Numerical Modelling of Rock Fracture Using Polygonal Finite Elements'. Ne muodostavat yhdessä ainutlaatuisen sormenjäljen. | {"url":"https://researchportal.tuni.fi/fi/publications/numerical-modelling-of-rock-fracture-using-polygonal-finite-eleme","timestamp":"2024-11-04T04:37:28Z","content_type":"text/html","content_length":"55182","record_id":"<urn:uuid:67e33537-6417-467f-ac0f-7c6be6f2cda9>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00518.warc.gz"} |
Electrical Technology Fundamentals (PDF Notes) - Gate Knowledge
Electrical Technology Fundamentals
Below is the syllabus for Electrical Technology Fundamentals:-
D. C. circuit excited by independent voltage/current source (steady state): Ohm’s Law, junction & node, circuit elements classification: Linear & nonlinear, active & passive, lumped and distributed,
unilateral & bilateral with examples, KVL, KCL, Loop analysis of resistive circuit in the context of dc voltage & current, Node-voltage analysis of resistive circuit in the context of dc voltages &
currents. Star-Delta transformation for set of pure resistors. Relevant D. C. circuit analytical problems for quantitative analysis.
Network Theorems: Superposition, Thevenin’s and Norton’s theorem all in the context of dc voltage and current sources acting in a resistive network, maximum power transfer theorem, Relevant D. C.
circuit analytical problems for quantitative analysis.
AC Fundamentals: Mathematical representation of various wave functions, Sinusoidal periodic signal, instantaneous & peak values, polar & rectangular form representation of impedances & phasor
quantities, Adition & subtraction of two or more phasor sinusoidal quantities using component resolution method. RMS & average values of various waveforms including clipped, clapped, half wave
rectified & full wave rectified sinusoidal periodic waveforms etc. Generation of alternating emf (dynamo). Relevant analytical problems for quantitative analysis.
AC Circuits: Behaviour of various compoents fed by AC source (steady state respose of pure R, pure L, pure C, RL, RC, RLC series with waveforms of instantaneous voltage, current & power on
simultaneous real axis scale and corresponding phasor diagrams), P. F. active, reactive $ apparent power. Frequency response of Series & Parallel RLC circuit including resonance, Q factor, cut-off
frequency & bandwidth. Relevant AC circuit analytical problems solutions using ‘j-omega’ operator method.
Balanced Three Phase Systems: Necessity & advantage of three phase system, mode of generation of 3 phase supply. Phase and line voltages & currents, power. Measurement of 3-phase power by two
wattmeter method for various types of star & delta connected balanced resistive, inductive & capacitive loads including phasor diagrams at various power factors. Phase sequence significance. Relevant
problems for quantitative analysis.
Electromagnetism & Magnetic circuits (Quanlitative analysis only): Laws of EMI, statically & dynamically induced emf, self & mutual induction, dot notation, RH Screw rule, Fleming’s RH & LH rules.
MMF, Relation between magnetic flux, mmf and reluctance, magnetic fringing, Hysteresis & Eddy current losses & their minimization.
Single Phase Transformer (Qualitative analysis only): Principle, construction & emf equation. Phasor diagram for ideal case and at no load. Winding resistance & leakage reactance. Actual transformer
at resistive, inductive & capacitive loads with phasor diagram. Losses & Efficiency, condition of maximum efficiency, regulation. OC & SC test, direct load test, equivalent circuit, concept of auto
Electrical Machines (Qualitative analysis only):
Prime mover, Stator-Rotor, Field-Armature, necessity of a starter.
D. C. Machines: Principle, general construction & working, Split ring/Commutator working in DC generator & motor, generated emf equation, Torque Equation. Types of DC Machines, speed control of DC
Shunt motor.
A. C. Machines: 3-phase Induction motor: Concept of rotating magnetic field, principle, types, general construction and working. Concept of slip & its significance.
Synchronous Generator (alternator): Principle, general construction & working.
Synchronous motor: Principle, general construction & working
General comparison amongst squirrel cage I. M., phase wound rotor type I. M. & DC motor.
General comparison between alternator & DC generator.
1. Vijay Kumar Garg, Basic Electrical Engg: A Complete Solution, Wiley India Ltd.
2. Rajendra Prasad, Electrical Engg. Fundamentals, PHI Pub.
1. S. K. Sahdev, Basic Electrical Engg., Pearson Education
2. PV Prasad, Basic Electrical Engg., Sivangaraju, Cengage Learning Pub.
3. Bobrow, Electrical Engg. Fundamentals, Oxford Univ. Press
4. Kulshreshtha, Basic Electrical Engg., McGraw Hill Pub
Below is the link to download Electrical Technology Fundamentals notes. | {"url":"https://gateknowledge.in/electrical-technology-fundamentals/","timestamp":"2024-11-04T04:13:07Z","content_type":"text/html","content_length":"109997","record_id":"<urn:uuid:e94e1fc8-dfe6-4194-928d-ec11558334b9>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00522.warc.gz"} |
State Bank Officials Training Scheme (SBOTS)Sample Exam Paper-2017
State Bank Officials Training Scheme SBOTS Sample Exam Paper-2017 (PDF)
♦ Mathematics MCQs Questions With Answers, Sample Paper For Preparation
♦ General Knowledge MCQs With Answers, Sample Paper Solved for (SBOTS)
♦ English MCQs Questions With Answers, Sample Papers Solved for (SBOTS)
Mathematics MCQs Questions With Answers, Sample Paper For Preparation
1.In a city 90% of the population own a car, 15% own a motorcycle, and everybody own one or the other or both. What is the percentage of
A) 15%B) 5%
C) 75%
D) 33 1/3%
Answer: D
2. If 15 workers can paint a certain number of houses in 24 days, how many days will 40 workers take, working at the same rate, to do the
same job?
A) 12 Days
B) 18 Days
C) 15 Days
D) 9 Days
Answer: D
3. Jafer drew a square, He then erased it and drew a second square whose sides were 3 times the side of the first square. By what percent
was the area of the square increased?
A) 300%B) 800%
C) 400%
D) 200%
Answer: B
4. The price of a can of acid was increased by 20%. How many cans can be purchased for the amount of money that used to buy 300
A) 250
B) 320
C) 150
D) 240
Answer: A
5. At bilal’s Discount Hardware everything is sold for 30% less than the price market. If Bilal’s buys tool bits for Rs. 96, What price should
he marked them if he wants to make a 20% profit on his cost?
A) 117
B) 146
C) 96
D) 127
Answer: B
6.A man borrows Rs.360. If he pays it back in 12 monthly installments of Rs.31.50, what is his interest rate?
A. 1.5%
B. 4.5%
C. 10%
D. 5%
Answer: D
7. If 1 inch = 2.54 centimeters, centimeter equals which of the following in inches?
A. 6.77
B. 0.95
C. 0.39
D. 0.38
E. 0.15
Answer: E
8. One number is 5 times another number and their sum is -60. What is lesser of the two numbers?
A. -5
B. -10
C. -48
D. -50
Answer: D
9. The average of x numbers is 15. If two of the numbers are each increased by y, the new average will be increased by how much?
A. 2y
B. y
C. y/x
D. 2y/x
Answer: D
10. A fair coin is tossed three times. What is the probability that at least one head appears?
A. 1/8
B. 7/8
C. 6/8
D. 4/8
Answer: D
11. A square and an equilateral triangle have the same perimeter. The area of the square is 225 cm. What is the length of a side of the
A. 18.5 cm
B. 20 cm
C. 20.5 cm
D. 22 cm
Answer: B
12. A German class has 12 boys and 18 girls. What fraction of the class are boys?
A) 1/6B) 3/5
C) 2/3
D) 4/15
Answer: B
13. The least number which when divided by 35, leaves remainder of 25; when divided with 45 leaves a remainder of 35 and when divided
by 55 leaves 45 as remainder, is
A) 3455
B) 3465
C) 3475
D) 10
Answer: A
14. It takes Riaz 30 minutes to mark a paper. Razi only need 25 minutes to mark a paper. If they both start marking paper at 11:00 AM,
What is the first time they will finish marking a paper at the same time?
A) 12:30
B) 12:45
C) 1:30
D) 12:25
Answer: C
15. A rectangular field which is twice as long as it is broad, has an are of 14450 M2, What is it perimeter?
A) 85 m
B) 510 m
C) 165 m
D) 170 m
Answer: B
16. The difference between the first two perfect squares that end with a 9 is :
A) 11
B) 40
C) 30
D) 120
Answer: B
17. Asim has an average of 60 on his four math tests. After taking the next test, his average dropped to 58. Find his most recent test grade.
A) 40
B) 50
C) 48
D) 32
Answer: B
18. If A gets 25% more than B and B gets 20% more than C. The share of C out of a sum of Rs. 740 is:
A) 150
B) 200
C) 250
D) 300
Answer: B
19. Honda Atlas issued 100,000 shares of stock. In 1990, each share of stock was worth Rs. 152.50. In 1993, each share of the stock was
worth Rs. 21.20. How much less the 100,000 shares worth in 1993 than in 1990?
A) 2052459
B) 3130000
C) 15250000
D) 12120000
Answer: B
20. If 15 boys working independently and at the same rate can assemble 30 machines in an hour, How many machines would 48 boys
working independently and at the same rate assemble in 40 minutes?
A) 54
B) 64
C) 96
D) 68
Answer: B
21. An office has a staff of 6 people. A certain project will take the regular staff 24 hours to complete. Assuming that all the workers will
perform at this rate, How many additional workers must be employed to complete the job in 8 hours?
A) 18
B) 06
C) 12
D) 10
Answer: C
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16 Nov 2020
escr - m0lecon 2020 - crypto
The chall
We are given the code for a custom hash function and we have to create 10 collisions in a row, let’s have a look at the actual code.
def rotl(x, n):
return ((x << n) & 0xffffffffffffffff) | x >> (64 - n)
def rotr(x,n):
return rotl(x, 64 - n)
class ToyHash(object):
def __init__(self):
self.state = [ 0x6A09E667F3BCC908, 0xBB67AE8584CAA73B, 0x3C6EF372FE94F82B,
0x510E527FADE682D1, 0x9B05688C2B3E6C1F, 0x1F83D9ABFB41BD6B,
0x243F6A8885A308D3, 0x13198A2E03707344, 0xA4093822299F31D0]
self.rounds = 91
self.mod = 2**64
def R(self, a, b, c, m1, m2, m3):
self.state[a] = (self.state[a] + self.state[b] + m1) % self.mod
self.state[b] = rotl(self.state[b]^self.state[c]^m2,16)
self.state[c] = (self.state[b] + self.state[c] + m3) % self.mod
self.state[a] = (self.state[a] + self.state[b] + m1) % self.mod
self.state[b] = rotr(self.state[b]^self.state[c]^m2,48)
self.state[c] = (self.state[b] + self.state[c] + m3) % self.mod
def compress(self, block):
mini_blocks = [int(block[64*i:64*i+64], 2) for i in range(9)]
for _ in range(self.rounds):
self.R(0, 3, 6, mini_blocks[0],mini_blocks[1],mini_blocks[2])
self.R(1, 4, 7, mini_blocks[3],mini_blocks[4],mini_blocks[5])
self.R(2, 5, 8, mini_blocks[6],mini_blocks[7],mini_blocks[8])
def hash(self, m):
bm = bin(bytes_to_long(m))[2:]
l = len(bm) % 0x7ff
bm = bm + '0'*((576-len(bm))%576) + '0'*564 + '1' + bin(l)[2:].rjust(11, '0')
blocks = [bm[576*i:576*i+576] for i in range(len(bm)//576)]
for b in blocks:
h = [self.state[i]^self.state[i+3]^self.state[i+6] for i in range(3)]
return ''.join(hex(n)[2:].ljust(16, 'f') for n in h).encode()
Basically the message is splitted in blocks of length 576 bits (padded with 0’s if necessay ) and another block with the length of the message is appended at the end, then each block is passed to the
compress function that updates the internal state. The final hash is calculated from the internal state.
The idea is to find a block b' different from b but that satisfies compress(b') = compress(b), so that we can create a second message different from the first, but that updates the inernal state in
the same way and therefore has the same hash.
The exploit
Since rotl(x, 16) and rotr(x, 48) are actually the same we can simplify the function R like so (and we need to double the number of rounds):
def R(self, a, b, c, m1, m2, m3):
self.state[a] = (self.state[a] + self.state[b] + m1) % self.mod
self.state[b] = rotl(self.state[b]^self.state[c]^m2,16)
self.state[c] = (self.state[b] + self.state[c] + m3) % self.mod
We can observe that a change in m1 has effect ONLY on state[a], in particular if we call 2*rounds times the function R the state[a] will change like so: newState[a] = (
something_that_dosent_depend_on_m1 + 2*rounds*m1 ) % mod.
Since gcd(2*rounds, mod) = 2 there exists another value m1' different from m1 that satisfies 2*rounds*m1' = 2*rounds*m1 (mod n). In particular that value is m1' = ( m1 + mod/2 ) % mod. ( in this case
is the same as flipping the most significant bit: m1' = m1 ^ (1<<63))
Putting everything together
So we want to modify the value of either mini_blocks[0], mini_blocks[3], mini_blocks[6] as described above. If we study how the message is splitted into mini_blocks we see that mini_blocks[0]
corresponds to the first 64 bits of the message, that means that the most significant bit will always be one, and it also means that if we flip it we will obtain a new message that is shorter than
the original, no good. Thats not a big deal, we can flip the first bit of mini_blocks[3].
Since each block is 64 bits long we’ll have to flip the 193-th bit of the message (counting from one), that is easy enough and can be done for example like so:
def gencoll(plain):
plain = bytes_to_long(plain)
coll = plain ^ (1<<(plain.bit_length() - 64*3 - 1))
return long_to_bytes(coll) | {"url":"https://fibonhack.it/2020/m0lecon2020/escr","timestamp":"2024-11-09T01:22:46Z","content_type":"text/html","content_length":"32184","record_id":"<urn:uuid:bf9d676c-7787-43e9-991f-bdc1d8263ab1>","cc-path":"CC-MAIN-2024-46/segments/1730477028106.80/warc/CC-MAIN-20241108231327-20241109021327-00528.warc.gz"} |
Audiogon Discussion Forum
You can always send it back to BAT for a refresh if things ever get uninspiring. Being what it is, I can't see how that can happen.
Whoever bought this was probably happy camper. Seems like decent price. The current P12 is $13k!
I got the VK-P5 for 1150, which I think was a great deal.
My main ppre is a Sonneteer Sedley which has been great.
Excited to do some testing and listening.
$1150 is a fair price.
My phonostage has 60db, and I was about to try a .25mv. cart(AT ART9XA) I played it safe with a .4mv instead.
My phonostage is tubed as well. All tubed-rectified/active gain-no step up. Things get extra hissy in quiet passages when the volume knob gets cranked,so getting more juice out of the cart is
paramount. Enjoy.
I received the step up transformer and tried it on my Sonneteer Sedley to begin with.
It was hard to figure out what it did and if the playback had improved any. The difference became apparent when I used my db meter to level match the test. I set the volume at 77 db gain average and
heard the following:
1) I was able to discern a ringing sound in the mids and highs that was not apparent to me before. It evidences itself as a smidge of harshness.
2) With the ringing diminished, more detail became apparent, and at the same average spl, the min and max spl had a wider range
3) mid bass also improved. highs were less bright.
The overall all effect is not really an improvement of the frequency response, but a refinement of the sound. Its that much more pleasing and sweet with the base being more pronounced. The refinement
also allowed a little more detail and some new sounds to come through. Not a whole lot, but observable. I have some screechy and tinny sounding vinyl records that were really quite acceptable after
running it through the SUT. Interesting.
I'll be trying the new preamp next week.
Also notice a dramatic reduction groove / tracking noise between tracks. Also, the click from switching off power to the motor disappeared.
Interesting results.
I once took a Zesto Allaso for a test drive. Plugged into the MM input of my Fosgate .I was curious to hear the difference (if any) between passive and the active gain.
While I did appreciate a smidge more of quiet when blasting something, I had to concentrate to hear what type of difference in SQ there was. Very subtle level of overall reduction of "organic/real"
(best way I can describe it) presentation? Maybe a cartridge specific thing. I didn't have an exotic LOMC-just used a AT ART9 .5mv.
Maybe a different story with another cart? Perhaps one of those real low ->.3mv, fancy builds?
So my Sonneteer Sedley (SS) preamp is making a come back. In trying a lot of configurations, the Sonneteer Sedley with the DV cart combinations and permutations I'm rating below.
BEST - DV on SS set for MC 50 ohm load and 10pf capacitance
VERY GOOD - DV on SS set for MM with 47K on load and 147pf capacitance. Transformer in between.
GOOD - DV on SS set for MC at 100 ohm and 10 pf capacitance (great but has some slightly perceptible ringing)
When Im done refreshing the BAT preamp, going through the tubes, I am putting in a 50 ohm setting (user option with open pins inside) and a 150 pf option.
I will be comparing the DV on the BAT on MC, 50 ohm/100ohm at 100pf. And Also the DV on the BAT on MM with the transformer set at 47Kohm and 150pf with the transformer.
I am also looking at purchasing a Hana SL to try out. The DV has been with me for about 5 years and could use a substitute. The Hana will be a perfect match for the transformer, at least on paper.
Thanks for keying me onto this. I think I have figured out a path to try.
First, I believe I have found a resource that agrees with my findings. My Dynavector 20x2, 5 ohm 2.58mv cart, doesnt sound as good with a 280 ohm load with a transformer vs. being loaded at 100 and
50 ohm load using my phono preamp. I hear positive things, but over all, its not the best setting.
Here is the resource - "A typical MC cartridge will sound somewhat “lightweight” when it is loaded too lightly (load impedance is too high) and will sound somewhat dull when the load impedance is too
That said, I reached out to Andrew at Rothwell and he confirmed my computations and gave me some information that allowed me to understand the MC-1 much better. I used three primary resources to
figure out my next move.
I need to get the SUT to present the right load to the cartridge.
I studied up on the science and formulas using Rothwell's site - http://www.rothwellaudioproducts.co.uk/html/mc_step-up_transformers_explai.html
K&K provides a schematic how to load the SUT so that it presents my desired load to the cartridge.
I reverse calculated the way my MC-1 is built so that I can compute the load resistor I am going to use in the schematic from K&K using MH-Audio's SUT Guide - http://mh-audio.nl/Calculators/
I arrived at the following conclusion. The MC-1 presents a 280 ohm load with a 20 and 258 turns of wire on the primary and the secondary. Or thereabouts, the exact ratio is 12.9.
The nonlinearity of the impedance transformation gets a little complicated. The impedance differs by the square of the turns ratio. For a 1:12.9 (22dB) step-up, the impedance is transformed by a
factor of 167. That is, with the secondary loaded by 47k ohms, the primary reflects what appears to be 280 ohms to the cartridge.
The question now is what resistor should you put in parallel with the 47k phono stage to change the loading to 50 ohms? I used the following formula:
First get the SUT impedance presented to the phono preamp with the MC on the SUT - (Transformer Ratio^2 * Desired Loading) = (12.9^2 * 50) = 8320 ohms
From there, you can get the Resistor load using this formula 1 / (1 / Desired Loading - 1 / Phonostage) whereby 1 / ( 1/8320 ohms - 1/47Kohms) = 10.1 Kohms
I verified my calculations using this site - http://www.hagtech.com/loading.html
Soldering a 10.1 Kohm resistor to the output of my SUT, with the phono preamp set at 47Kohm will present a 50 ohm load to my cartridge.
You can also put the parallel resistor on the primary side, but then the value is the above divided by the turns ratio squared.
Ive got a few resistors incoming to modify my SUT. I am hoping I am able to keep the sweetness of the SUT's sound, while retaining the brilliance of the cartridge.
Man, I havent done this much analysis and math since school | {"url":"https://d2dve11u4nyc18.cloudfront.net/discussions/trying-a-bat-vk-p5-with-a-step-up-transformer","timestamp":"2024-11-07T19:42:24Z","content_type":"text/html","content_length":"97262","record_id":"<urn:uuid:1ee069bb-9042-47c1-aa56-772cbf329eac>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00742.warc.gz"} |
1A - 150 Essential Exam Practice Questions - Tim Gan Math Online Courses
1A – 150 Essential Exam Practice Questions
Secondary 1 Express 150 Essential Practice Questions is the first of a two-book series specially written for Secondary 1 students to prepare for their various continual assessments and semestral
assessments in Mathematics. The materials in this book follow closely to the latest Mathematics syllabus implemented by the Ministry of Education, containing the first 7 chapters of the Secondary 1
curriculum. These materials are relevant for the students in both the ‘O’ Level and Integrated Programme (IP) tracks and serve to provide a concise yet complete essential collection of practice
questions that a student needs to fully comprehend each given topic bounded by the syllabus. | {"url":"https://online.timganmath.edu.sg/courses/150-essential-questions-vol-a/","timestamp":"2024-11-05T16:17:52Z","content_type":"text/html","content_length":"258079","record_id":"<urn:uuid:2270f836-74c1-4fe3-86bc-3d0863014db3>","cc-path":"CC-MAIN-2024-46/segments/1730477027884.62/warc/CC-MAIN-20241105145721-20241105175721-00443.warc.gz"} |
Python Function: Find Biggest Product of Numbers
Oops, something went wrong. Please try again in a few moments.
def find_biggest_product(numbers):
Function to find the biggest product of a list of numbers.
- numbers: list
A list of numbers.
- int:
The biggest product of the numbers.
- ValueError:
Raises an error if the input list is empty or contains non-numeric values.
# Checking if the input list is empty
if not numbers:
raise ValueError("Input list is empty.")
# Checking if the input list contains non-numeric values
if not all(isinstance(num, (int, float)) for num in numbers):
raise ValueError("Input list contains non-numeric values.")
# Initializing the biggest product as the first number in the list
biggest_product = numbers[0]
# Iterating through the remaining numbers in the list
for num in numbers[1:]:
# Updating the biggest product if the current number multiplied by the previous biggest product is greater
if num * biggest_product > biggest_product:
biggest_product = num * biggest_product
return biggest_product
# Example usage of the find_biggest_product function
numbers = [8, 9, 5, 7, 6, 7]
result = find_biggest_product(numbers)
print(f"The biggest product of the numbers {numbers} is {result}.") | {"url":"https://codepal.ai/code-generator/query/br2vi1lJ/python-function-find-biggest-product","timestamp":"2024-11-06T04:43:35Z","content_type":"text/html","content_length":"111595","record_id":"<urn:uuid:8713bed7-199c-4a89-8880-ddbcda903baa>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00098.warc.gz"} |
Simpson's Paradox
Applet Instructions
: Simpson's Paradox is the name given to the phenomenon in which relationships observed between groups reverse when the groups are divided into subgroups based on a lurking variable.
Illustrated example
Data Table
: The data are displayed in the table at the top of the applet. The greater percentages are boxed (in orange if Simpson's Paradox is observed and in green otherwise).
: For each of the comparison groups the plot shows the percentage of observations in the specified category of the outcome variable as a function of the percentage in a category of the lurking
variable. Colored dots on the lines indicate the percentages in the lurking variable category for each of the comparison groups. The left extreme of each line indicates the percentage that would be
in the specified outcome category if 0% of the observations were in the indicated lurking variable category and the right extreme corresponds to 100% in the lurking variable category.
The Sliders
: The sliders allow the user to adjust the percentage in the lurking variable category for each of the comparison groups and to see how this affects the observed relationships. As a slider is
adjusted, a circle on the corresponding line (circle color the same as slider dot color) moves. Dashed lines from the circles to the axes highlight the relationship between the variable values. The
data in the table is updated as the sliders are adjusted. The combined counts for the comparison groups and the percentages in the outcome category for the subgroups are fixed.
Points to Ponder
• Describe what you see in the table when Simpson's Paradox is observed.
• Describe what you see in the plot when Simpson's Paradox is observed.
• When adjusting the sliders, at what point does Simpson's Paradox appear?
• What do the conditions you observe in the applet tell you about what's going on in the data when Simpson’s Paradox occurs?
Baker-Kramer Data: Source: Wainer, H. (2002) The BK plot: Making Simpson's Paradox Clear to the Masses.
• Comparison groups: Treatments A and B
• Outcome variable: Percent Surviving
• Lurking variable: Percent women
Berkeley Admissions Data: Source: Hammel, W., Bickel, P., and O'Connell, J.W. (1975) Is There a Sex Bias in Graduate Admissions?
. 187.
• Comparison groups: Male and Female Applicants
• Outcome variable: Percent admitted
• Lurking variable: Acceptance rate of desired major.
Florida Death Penalty Data: Source: Radelet, M. L. and Pierce, G. L. (1991).
Florida Law Review
• Comparison groups: Black and White defendants in murder trials
• Outcome variable: Percent receiving death penalty
• Lurking variable: Race of victim
Airlines Data: Source: Moore, McCabe, Craig.
• Comparison groups: Alaska and America West Airlines
• Outcome variable: Percent of flights delayed
• Lurking variable: Flight origination
1964 Civil Rights Act Data: Source: Simpson's Paradox. Wikipedia.
• Comparison groups: Democrats and Republicans
• Outcome variable: Percent in favor
• Lurking variable: Origin of representative (Northern v. southern)
20 Year Smoker Survival: Source: Vanderpump, M.P.J., Tunbridge, W.M.G., French, J.M., Appleton, D., Bates, D., Clark, F., Grimley Evans, J. Rodgers, H. Tunbridge F., and Young, E.T. (1996) The
Development of Ischemic Heart Disease in Relation to Autoimmune Thyroid Disease in a 20-Year Follow-up Study of an English Community
• Comparison groups: Smokers and non-smokers
• Outcome variable: Percent alive at 20 year follow-up
• Lurking variable: Age of subject (under 65 v. 65 and older)
House pet data: Source: Schneiter (2012) Hypothetical study data.
• Comparison groups: Dogs and cats
• Outcome variable: Percent kept in the house
• Lurking variable: Size of pet (small v. large) | {"url":"https://www.usu.edu/math/schneit/CTIS/SP/","timestamp":"2024-11-09T12:37:05Z","content_type":"text/html","content_length":"8878","record_id":"<urn:uuid:9a9479cf-ea04-4583-83e9-d24d4268a4e8>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00235.warc.gz"} |
measure signal power
8 years ago
●19 replies●
latest reply 8 years ago
753 views
I have implemented a filter and passing composite signal with multiple frequencies.
I would like to measure the power of signal before and after filtering. This is for me to verify how signal has been affected in terms of power after filtering. Can someone guide me in this regard?
PS: I tried rms function in octave but it looks like this is not yet implemented. Next, I was thinking I could implement custom function myself.
[ - ]
Reply by ●March 7, 2017
rms = square root of mean of squared values.
so square each sample, get the mean of squares (this is mean power)
get square root of that if you wish
[ - ]
Reply by ●March 7, 2017
Hi. There are two types of "power" that we talk about regarding discrete sequence: (i) the "instantaneous power", and the (ii) "N-point average power".
The instantaneous power of an x[n] sequence is merely the sequence (x[n])^2.
The N-point average power of an x[n] sequence is merely summing (x[n])^2 + (x[n-1])^2 + (x[n-2])^2 + (x[n-3])^2 + ... + (x[n-(N-1)])^2. And then divide that sum by N to produce a single "N-point
average power" sample. You repeat the summing and division operations as each new sample that arrives to your system. This "power" sequence could rightly be called the "N-point moving average
[ - ]
Reply by ●March 7, 2017
Thanks a lot for your inputs.
Dear Rick,
Thanks. Probably, this is what I am looking for.
I would like to come back to my main intention of computing the power.
Input is made of 4 sine signals. Output is expected to filter the 3rd and 4th sine signals as they are above the filter cut off.
Now, I want to do the following,
1) measure the power of each input sine signal
2) measure the power of composite input signal
3) measure the power of signal that is output from filter
Just compare to see if the values co-relate. Ideally, I expect to see power from 3 and 4th sine to be filtered out.
Next, I want to play around with amplitudes of sine1, sine2, sine3 and sine4 and see its impact.
Am I going in the right direction in order to analyze the filter function?
[ - ]
Reply by ●March 7, 2017
I think your purpose is to measure effect of filter on power rather than measure power per se.
The best way (in software) is to inject an impulse input of unity power then look at filter output spectrum. It tells you what happens to power at every frequency.
[ - ]
Reply by ●March 7, 2017
Thanks. I could do that but what can I expect assuming the filter is a low pass filter?
[ - ]
Reply by ●March 7, 2017
As explained by Rick you look at input/output spectrum.
For unit impulse input, the spectrum will be a nice flat top all over frequencies. The filtered output of LPF should be 0dB at dc and as flat as 0dB as you approach cutoff where it starts to descend
towards stop band. This is the case of dc unity filter. If it is above or below 0dB then you have lost unity gain though filter shaping may still be correct(but shifted up or down)
[ - ]
Reply by ●March 7, 2017
I hope I have done it correctly. The following is how the plots look like:
[ - ]
Reply by ●March 7, 2017
There are some funny scaling issues.
you can avoid all by just this command:
[ - ]
Reply by ●March 7, 2017
Kaz, Rick,
I have used the approach given by Fernandoorg to measure power of various signals and it looks as below:
1 power of input signal 1.99495
2 power of output signal 0.963378
3 expected power of output signal 0.998553
4 power of sine 1 signal 0.499276
5 power of sine 2 signal 0.499276
6 power of sine 3 signal 0.499276
7 power of sine 4 signal 0.499276
Input signal is composed of the 4 sine signals whose power is given on line 4-7. Actual output power is given in line 2. I computed expected power by adding power of sine 1 and 2.
I am a little surprised that actual output power is less than expected output power. In reality I actual output power to be > expected output power due to the fact that beyond cut-off filter still
allows frequencies although with certain degree of attenuation.
Can you please throw some light?
[ - ]
Reply by ●March 7, 2017
Well have you considered correlation issues. To explain that what is the power of two sine waves at same frequency but opposite in phase (zero isn't it)
[ - ]
Reply by ●March 7, 2017
in reality, such signals would cancel each other anyway even before they appear at the filter input. If I just compute power of 2 sine waves individually then it would have some finite power but when
you add 2 of such sine signals and compute power then it would be 0.
[ - ]
Reply by ●March 7, 2017
I mean that when it comes to power addition you can add power of two or more signals if they are not correlated else the sum will vary depending on how much they cancel out each other. So doing power
additions is not a good idea. Just use freqz(h) and it will tell you all.
A filter also changes phase (though linearly if symmetric fir) and this needs be taken into account.
[ - ]
Reply by ●March 7, 2017
[ - ]
Reply by ●March 7, 2017
Hi. kaz's March 4th comment told you how to measure the frequency behavior of your filter. You can verify the results of kaz's test by comparing the spectral magnitude (or spectral power) of your
filter's input sequence and the spectral magnitude (or spectral power) the filter's output sequence. At the filter's output you should see greatly reduced spectral magnitudes for your 3rd and 4th
sine waves.
[ - ]
Reply by ●March 7, 2017
I have another observation. It has been my experience that an arbitrary "long term average" may not be enough for an accurate measurement if the window length is less than the period of the lowest
frequency of the signal's modulation due to not averaging an integral number of those low-frequencies. This "noise" is not due to the SNR of the signal!
This can be proactively corrected for signals modulated with fixed frequencies (ignoring Gaussian noise, which an "average of averages" can reduce) rather than pseudo-random frequencies like voice.
My experience has been with VOR and ILS signals that have periods of 30 Hz and 15 Hz, respectively, In my case, I was trying to create an instrument to accurately measure these signals to ensure safe
landings and enroute guidance.
As an example, if the period of averaging is not harmonic with the "characteristic" frequency, then there will be not only variations due to noise, but perhaps more significantly, due to cyclic
Note that a simple average is the DC part of a Fourier transform and those too are subject to Gibbs ripple phenomenon, which is cyclic truncation. The measurement variation due to noncyclic
(nonharmonic) sampling can be quite large.
[ - ]
Reply by ●March 7, 2017
Hi artmez. I think your comments are very sensible.
[ - ]
Reply by ●March 7, 2017
You can measure the energy of the signal as the squared sumation of all its samples. It could be something like
energy = sum(x.^2)
where x is the signal to be measured.
The power of a signal is a time-average value, then you should take the calculated energy and normalize it in time.
If N is the number of samples in x you could do something like
power = (1/N)*energy
Finally the RMS value of the signal is obtained by taking the square root of the power, so
rms = sqrt(power)
Hope it helps! My apologies for my poor english
[ - ]
Reply by ●March 7, 2017
My apologies for my poor english
It seemed perfect to me; no need to apologize. :-)
[ - ]
Reply by ●March 7, 2017
The only thing that I'd add to this is that if you're RMS-ing noise, it can take an astonishingly long averaging interval to get a good-looking output. I've never sat down and done the math, but
it's like tens or hundreds times longer than 1/bandwidth. | {"url":"https://dsprelated.com/thread/2178/measure-signal-power","timestamp":"2024-11-07T00:20:46Z","content_type":"text/html","content_length":"67575","record_id":"<urn:uuid:abc4b1b1-74bd-4e54-a416-cdc92812525a>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.54/warc/CC-MAIN-20241106230027-20241107020027-00295.warc.gz"} |
No 6 (2023)
Hydrogen, one of the most abundant elements in nature, is potentially suitable to produce, store and consume clean energy, namely for various industrial uses. The safest technique for hydrogen
storage is metal hydride, e.g. as MgH2, with safe facilities to store and transport. A critical indicator of a metal-hydrogen system is the kinetic of hydrogen sorption, which is known particularly
low for magnesium. However, under given thermodynamic conditions, the kinetics of magnesium/hydrogen reaction is especially sensitive to the particle size i.e. specific surface, crystallite/grain
size i.e. boundary extension and nature of potential additives as catalysts. The present calculations aimed at determining the occurrence of hydride nucleation are based on a new energy ratio, as
proposed at first time, which takes into account both physicschemical and mechanical factors. The calculations of the energy ratio are based on the minimum total energy of the system, which correctly
reflects the processes occurring during hydride formation in magnesium. The critical size of a nucleus at the phase formation (hydride) in magnesium is controlled by the ratio of the volume and
surface area of the emerging component, similarly to a crystallization process from a solution. The influence of the mechanical response of the system to the formation of hydride allows one to
propose an interpretation of some phenomena regularly recorded during such experiments, i.e. the influence of special additives and mechanical texture, which lead to the acceleration of hydride
formation. The results obtained suggest a mechanism favoring oriented nucleation of the hydride in a textured magnesium matrix thanks to the anisotropy of the elastic characteristics of the newly
formed phase.
PNRPU Mechanics Bulletin. 2023;(6):5-17
The conditions of occurrence of acoustic stress resonances at the boundaries of an anisotropic layer are investigated. In general, under the action of an incident elastic wave, six elastic waves are
formed in an anisotropic layer. The total effect of these waves determines the stressstrain state of the layer and is displayed in the spectra of waves scattered by the layer into the environment.
The scattering spectra and acoustic stresses were modeled by solving the equations of motion of a continuous medium and the generalized Hooke's law. This system of differential equations is solved
with respect to the components of the displacement vector and the stress tensor in the Cartesian coordinate system. The Peano-Becker method of solving a system of differential equations by means of a
matrix exponential is used. The components of the displacement vector and the stress tensor at two opposite boundaries of the layer with thickness d are expressed through each other using a
sixth-order transfer matrix T = exp(Wd), where matrix W is determined by the parameters of the layer under study. The method of scaling and multiple squaring is used. According to this approach, T =
(exp(Wd/m))m. A method for selecting the scaling parameter m is proposed to estimate the errors of truncation and rounding when calculating exp(Wd/m). A guaranteed accuracy and the best efficiency of
calculations of all elements of the matrix exponential of the sixth order, in comparison with other known methods, is provided by the use of the method of polynomials of the principal minors of
matrix W. The modeling of elastic wave scattering spectra (conversion coefficients) and stress dependences on the angles of incidence for cubic crystal layers is given using the example of indium.
The interpretation of resonances of acoustic stresses arising in the crystal layer under the action of a shear wave incident on the crystal is given.
PNRPU Mechanics Bulletin. 2023;(6):18-28
The constructions made of composite materials are subjected to cyclic, dynamic, vibration and other loads in operations related to damage accumulation and degradation of progressive mechanical
characteristics. Thus it is important to conduct experimental and theoretical studies of the combined impact effects on the change in the mechanical characteristics of the material. In this case, it
is necessary to take into account the complex stress state realized in the structures. This work deals with an experimental study of the degradation patterns of the stiffness characteristics of
fiberglass tubular specimens obtained by continuous winding as fatigue damage accumulates due to biaxial proportional cyclic loading. The methodological aspects of realization of biaxial loading are
considered. Quasi-static and fatigue tests were performed on specimens with different winding angles under uniaxial tension, torsion, and proportional tension, i.e. torsion with three different
ratios of the normal and shear stress tensor components. The presence of a decreasing region in the torsional load diagrams has been revealed. Strength surfaces are constructed. By using the
approximation of the fatigue sensitivity curves previously developed by the authors, we process the experimental data on the decrease in the dynamic modulus of elasticity as the number of cycles of
exposure increases. The high descriptive ability of the developed model and the low values of the variation coefficients of the calculated parameters were noted. Nonmonotonic dependences of model
parameters on the type of stress state are revealed. A significant influence of the winding angle on the fatigue sensitivity of the composite has been found. We have made the conclusion about
necessity of taking into account the decrease of mechanical characteristics of materials in calculations of constructions and rationality of further experimental researches for verification of
earlier developed models.
PNRPU Mechanics Bulletin. 2023;(6):29-40
In this paper, an experimental study of the strain fields at the fatigue crack tip was carried out. The strain fields were measured by an optical camera based on the digital image using the
correlation method. Images were recorded using a Basler acA2440-75uc optical camera with a TC23007 OptoEngineering lens to achieve a spatial resolution of at least 3 μm. Recording frequency was 100
Hz. The possibility of using the solution of the linear singularity problem of elasticity theory to estimate the distribution of plastic strain at the fatigue crack tip was shown. Mechanical tests of
uniaxial cyclic deformation with simultaneous registration of the strain field at the crack tip of different lengths were carried out on flat specimens of titanium alloys Ti Grade 2, Ti-1.1Al- 0.9Mn,
Ti Grade 9. The specimens were loosened by means of a lateral semicircular notch in order to localize the crack. The solution of the problem of a specimen with a notch in the elastic formulation was
carried out numerically in the finite element modelling package Comsol Myltiphysics. The peculiarity of the work is the use of the hypothesis of the functional relationship between real deformations
and the elastic solution and the value of the secant modulus of the material to estimate the plastic deformation at the crack tip. The size of the zone of intense plastic deformations at the fatigue
crack tip for different crack lengths was determined experimentally and numerically. By comparing the calculated and experimental data we showed the possibility of using the proposed dependence to
estimate the distribution of the plastic strain field at the crack tip. The results obtained allow the analysis of the irreversible strain fields at the crack tip for mixed mode loading.
PNRPU Mechanics Bulletin. 2023;(6):41-49
Within the theory of finely dispersed nanocomposites, the dependence of the effective Young's modulus on the absolute size of the reinforcing particles is obtained. Two cases of controlling/changing
the effective Young's modulus at a constant relative volume fraction of reinforcing particles are considered. The first is the disintegration of reinforcing particles into smaller ones, followed by
diffusion throughout the volume of the matrix. In this case, the effective modulus of the nanocomposite increases. The second one is the agglomeration of reinforcing particles into larger ones. In
this case, the effective modulus of the nanocomposite decreases. These patterns seem to be universal and independent of heat treatment technology. It can be assumed that the agglomeration or
decomposition of the reinforcing particles depends on the choice of a heat treatment technology for a nanocomposite. It is important to emphasize that the selected heat treatment technology is to be
such that during the heat treatment no phase transitions occur either in the material of the reinforcing particles or in the matrix material. It is necessary to eliminate the appearance of phase
transitions, since the new phase represents a field of defects, in particular, the field of substitutional dislocations. For such processes, the gradient theory of a defect-free medium is no longer
valid. It is necessary to build models of defective environments that are more complex. Therefore, this article does not consider the criteria for choosing a heat treatment technology. The question
remains open that, along with the gradient generalization of the theory of composites, a nonlinear generalization is possible. Indeed, unlike ceramics, which retain physical linearity almost until
destruction, metal composites exhibit plasticity over a large range of deformations. However, generalization to physical nonlinearity, and even more so to plasticity, is complicated by the fact that
there is still no generally accepted theory for constructing a stress-strain curve even for homogeneous materials.
PNRPU Mechanics Bulletin. 2023;(6):50-56
The paper considers new applications of the models, algorithms, software and methods developed by the authors to study shell structures of spherical shells (domes). For this type of structures, a
method has been proposed to bypass the singularity at the top of the dome by choosing modified approximating functions. The mathematical model is geometrically nonlinear; it takes into account
transverse shears, and is presented as a functional of the total potential strain energy. To reduce the variational problem to solving a system of algebraic equations, the Ritz method was used. The
resulting system is solved by the method of continuing the solution using the best parameter with an adaptive mesh selection. The algorithm is implemented in the Maple analytical computing
environment. A steel dome was estimated using different methods of border fixing, the values of the critical buckling load and the limit stress load were obtained. A graph of the load – deflection
relationship and the deflection fields in the subcritical and supercritical stages were constructed. Fields are shown in the local and global Cartesian coordinate systems. The convergence of the Ritz
method in terms of the critical load value is demonstrated. The methodology was verified by comparing the solution to the test problem with the known solution obtained by E.I. Grigolyuk and E.A.
Lopanitsyn. The comparison results demonstrate the reliability of the data obtained. It was revealed that for the dome under consideration, the loss of strength occurs much earlier than the buckling,
and therefore it can be recommended to select a steel grade with a higher yield strength for its design. A simply support border condition in this case gives a higher value of the maximum permissible
PNRPU Mechanics Bulletin. 2023;(6):57-67
The paper considers the influence of physical and mechanical characteristics (PMC) of materials of the constructions to propagate acoustic waves in gas at a model channel. The research of the
influence of PMC materials, pipelines, in particular, on the propagation of wave processes is associated with the problem of noise that arises during the transportation of natural gas and hydrogen-
containing mixtures. The problem of noise is especially relevant given the forecasts for the development of the hydrogen energy transportation and storage industry. Modeling of acoustic processes is
often associated with the sources of occurrence and propagation in the simulated environment. In this case, the possible occurrence of the resonance phenomena or processes of attenuation of acoustic
waves in the dynamic gas-structure system are not taken into account. The boundary value problem is formulated in a bidirectional interaction statement (2-way Fluid-Structure Interaction or 2FSI)
between the deformable structure and the hydrogen flow. Predicting the behavior of the pipeline structure in a model representation under the influence of gas during transportation will make it
possible to select the optimal PMC material to reduce the acoustic impact both inside and outside the channel. The research presented in this work is carried out using the ANSYS engineering analysis
system, which allows modeling the processes under consideration in the 2FSI statement. The paper analyzes the behavior of a wave generated by a unimodal sound source interacting with a barrier
clamped in a rectangular pipe. The main results of the research are presented in the form of dependencies of the pressure amplitude versus time at characteristic points; the dependence of
displacement versus time of the model barriers made of different PMC materials; the dependencies of changes in pressure and displacement for different working fluids.
PNRPU Mechanics Bulletin. 2023;(6):68-77
The paper deals with X-ray computed tomography results of a polyetheretherketone (PEEK) and a short carbon fibre reinforced by acrylonitrile butadiene styrene (ABS+CF). Individual carbon fibres and
3D printing defects (consolidated structures of interconnected fibres and densified resin clumps) are detected on an initial microstructure of the ABS+CF composite. The carbon fibres and the
consolidated structures are densely packed with uniform sub-horizontal locations throughout in a sample volume. In the PEEK samples, the process-induced defects during the composite manufacturing
process are visualised as tubular structures of a densified resin with internal voids. Significant changes in the structure of both composites are observed after five times pulsed laser shock
peening. In case of a single pulse exposure and a surface treatment, no microstructural changes occur. In a test mode without a protective layer, a material evaporation to a depth of 0.3 mm and a
structural degradation of the PEEK samples takes place, while the process-induced interlayer voids do not close. A single consolidated area with a porous spongy structure occurs due to melting of the
carbon fibres in the ABS+CF composite. The results show that the laser shock peening has a significant effect on the surface microstructure. It is therefore necessary to carry out further experiments
to select the optimum laser shock peening parameters and a protective layer material to eliminate the process-induced defects and improve the strength properties of the composites.
PNRPU Mechanics Bulletin. 2023;(6):78-90
Autofrettage processes are designed to strengthen hollow cylindrical and spherical parts and usually consist of one load-unload cycle. At the first stage, the workpiece is loaded to cause either
partial or complete plastic deformations. During unloading, residual compressive stresses are formed in the vicinity of the inner surface of a part. The present work is devoted to a theoretical study
of the process of rotational autofrettage of a hollow cylinder with fixed ends. The formulation of the problem is based on the theory of infinitesimal elastoplastic deformations, the Tresca
plasticity condition and the flow rule associated with it. It is assumed that at the loading stage the cylinder material follows the linear-exponential law of isotropic hardening, and when unloaded
it behaves as purely elastic body. The effect of a decrease in Young's modulus during unloading as a result of preliminary plastic deformation and its influence on residual stresses caused by
rotational autofrettage of the cylinder are studied. To quantitatively describe the variation in Young's modulus, an exponential model with saturation is used. For the load stage, an exact analytical
solution is obtained based on the Lambert W-function. Calculation of residual stresses in the cylinder is performed using the Runge-Kutta method. As an example, materials with significant decrease in
Young's modulus are considered, namely aluminum alloy AA6022, steel DP980 and manganese steel. It has been established that taking into account the variable Young's modulus can lead to a significant
reduction in the calculated level of residual stresses. This effect is especially important for the calculation of thick-walled cylinders and fairly high autofrettage velocities.
PNRPU Mechanics Bulletin. 2023;(6):91-103
The paper investigates the process of local damages in the hip joint endoprosthesis (HJ) made of unidirectional carbon-carbon composite material (C/C composite) with pyrolytic carbon (PС) matrix. A
mathematical model of deformation of the endoprosthesis from the C/C composite has been developed taking into account the processes of local damage. These damages are possible due to overloads, which
may be caused by accidental circumstances during human movements. The developed model is a synthesis of an algorithmic model that takes into account the heterogeneity of the pyrocarbon matrix and
composite, and an engineering computational model of the biomechanical endoprosthesis-femur system. The matrix algorithm solves the stochastic boundary-value problem of finding mesostresses in PС
grains taking into account possible damages. The result of this algorithm is the probability distribution densities for meso-stresses in PС crystallites and the properties of the damaged matrix. The
results of calculations based on the engineering model are the fields of macrostrains and macrostresses. At each step of loading of the endoprosthesis, the state of the matrix is monitored and the
effective modules of the carbon composite are changed. This is implemented by a continuous exchange of data between the two algorithms, the recalculation of the properties of the composite, which are
the input data for the engineering model. The continuous change in the effective properties of the C/C composite during deformation is replaced by a stepwise change. To do this, the volume of the
endoprosthesis was divided into areas in which the properties become variable, starting with a certain loading step. The areas of change were determined based on the distribution patterns of
macrodeformation fields. A nonlinear loading diagram of the endoprosthesis is constructed taking into account the damage. It is shown that the destruction of the carbon part of the prosthesis begins
with local damage, which gradually engulfs the neighboring areas. Damage occurs when the standard load exceeds 1740 Newtons. The maximum force response of the prosthesis to an external load is equal
to 2004 newtons. The deformation of the prosthesis at the stage of a critical reduction in load-bearing capacity exceeds the deformation at standard load by 16 %. The high reliability of the
considered variant of the endoprosthesis was confirmed, the absence of catostrophic sharp decreases in load-bearing capacity under a significant excess of standard loads was confirmed.
PNRPU Mechanics Bulletin. 2023;(6):104-114
Plunger pumps used in oil production are made of long hollow bimetallic cylinders. These components are thermo-mechanically treated to improve strength and other physical and mechanical properties.
These operations result in residual stresses within the parts, which can lead to positive, undesirable and unacceptable changes in the geometry. In the present work, we consider the problem of
choosing the optimal machining modes. Estimations of residual stresses in the whole product take too much time, so it was decided to use small rings, which are representative for each particular
pipe. In view of complexity or impossibility of applying the existing methods, the authors have designed a novel technique to estimate the level of residual stresses. For this purpose, we formulated
and solved this problem within the theory of elasticity. An analytical solution, which makes it possible to find the level of stresses depending on the experimental measurements when cutting the
rings, has been obtained. Three different steels were chosen. i.e. 38Cr2MoAl, 15Cr5Mo, 12Cr18Ni10Ti. Based on operating conditions, four optimization criteria for the heat treatment have been
produced: the minimum level of residual stresses in the pipe; the minimum difference between stresses in the shell and liner; the minimum change in the pipe radius after the treatment; the highest
value of adhesion between the liner and the shell. The obtained results have been analyzed based on the above four criteria. We revealed the optimum and intolerable modes of thermo-mechanical
processing, which enabled undesirable changes in products. The required degree of deformation and temperature of the post-deformation heating have been found for each steel under study. As a result,
recommendations for industrial enterprises have been drawn up.
PNRPU Mechanics Bulletin. 2023;(6):115-123
Additive technologies, including laser powder deposition (a repair technology), enable a sequential deposition of powder layers. This process involves large temperature gradients and technological
residual stresses, which can lead to shape violations and change mechanical and operational characteristics of products. To control and prevent residual deformations in the hardfacing body, it makes
sense to carry out finite element modeling of the laser powder hardfacing using layer-by-layer activation technology or adding new finite elements to the surface of the hardfacing model. The Element
Birth/Death method is the most suitable method for this problem. In this case the elements for the material to be created are deactivated (so not included in the solution area), and then gradually
revived and included in the solution area. The material is built up discretely. At each sub-stage of the calculation, corresponding to the revival of the next sub-domain of dead elements, the coupled
problem of thermal conductivity and solid mechanics is solved, and thus the result of the solution of the previous sub-stage serves as the initial conditions for the next one. A mathematical model
and an algorithm for modeling warping during the deposition are developed, and calculations for the deposition of cylindrical specimens are carried out. During the calculations, the multilinear MISO
plasticity model for the sample material and the BISO bilinear plasticity model for the filler powder were used. We verified the model based on the optical control results of changes in the geometry
of the experimental samples after the deposition had been carried out. The error in warpage calculation did not exceed 5%.
PNRPU Mechanics Bulletin. 2023;(6):124-134
The use of structural superplasticity is promising in the development of production technologies with complex shapes and improved physical, mechanical and operational characteristics. Deformation in
the superplasticity regime is characterized by reduced (compared to conventional plastic processing) loads on tools and decreased number of finishing operations. It seems preferable to use the
superplasticity regime at relatively moderate homologous temperatures (less than 0.7) and high strain rates (on the order of 10–2 s–1), in which the equiaxed grain shape can be preserved with an
insignificant change in its size. Under these conditions, staged (bell-shaped) tension curves are observed in experiments on uniaxial tension with access to the structural superplasticity regime for
many alloys preliminarily prepared by severe plastic deformations. The latter is associated with the action and interaction of various physical mechanisms, the change in their roles during the
deformation and evolution of defective material structures. The above factors are influenced by the initial temperature and strain rate conditions and characteristics of material structures after the
pretreatment, in particular, grain shapes and sizes, fraction of high-angle boundaries, degree of recrystallization of the structure, presence of alloying additives that can form various phases in
materials. This review attempts to systematize experimental data on superplasticity of aluminum alloys 1420 and 1421 with a focus on the main characteristics of material structures before and during
the superplastic deformation tests, as well as its effect on the acting mechanisms. This will make it possible to form a more complete understanding the physical nature of deformation with a
transition to structural superplasticity regimes for aluminum alloys and to develop a scenario for the action and interaction of mechanisms taking the influence of the evolving material structure
into account. The above will be the concept basis for development the multilevel constitutive models of inelastic deformations of alloys to describe the material structure evolution and change in
deformation regimes, which is necessary to improve superplastic forming technologies.
PNRPU Mechanics Bulletin. 2023;(6):135-157
Within the refined theory of bending, a coupled initial-boundary value problem of thermoelastic- plastic deformation of flexible circular cylindrical shells with arbitrary reinforcement structures is
formulated. The tangential displacements of the shell points and the temperature along the thickness of the structures are approximated by high-order polynomials. This makes it possible to take into
account, with varying degrees of accuracy, the weak resistance of fibrous sheaths to transverse shear and to calculate wave processes in them. From the obtained two-dimensional equations of the
refined theory, in the first approximation, the relations of the traditional non-classical Ambartsumian theory are obtained. The geometric nonlinearity is modeled in the Karman approximation. The
inelastic deformation of the components of the composition is described by the relations of the theory of flow with isotropic hardening. In this case, the loading functions of the materials of the
composition phases depend not only on the strengthening parameter, but also on the temperature. For the numerical solution of the formulated nonlinear coupled two-dimensional thermomechanical
problem, an explicit scheme of time steps is used. We studied the axisymmetric elastic-plastic deformation of flexible long cylindrical shells, which are reinforced in the circumferential and axial
directions. Fiberglass and metal-composite structures from the inner front surface are loaded with pressure, which corresponds to the action of an air blast wave. It is shown that for an adequate
calculation of temperature fields in the structures under consideration, it is advisable to approximate the temperature over their thickness with a 7th order polynomial. It has been demonstrated that
at some points fiberglass shells can additionally heat up for a short time by only 10…11 C, so the thermal response can be disregarded in their calculations. Metal-composite structures can
additionally heat up by more than 40 C. However, for their calculation it is also possible to use the model of elastoplastic deformation of the materials of the composition components. It is shown
that when studying the dynamic inelastic behavior of both fiberglass and metalcomposite cylindrical shells, it is advisable to use the refined theory of their bending, rather than its simplest
version, the Ambartsumian theory.
PNRPU Mechanics Bulletin. 2023;(6):158-169 | {"url":"https://ered.pstu.ru/index.php/mechanics/issue/view/382","timestamp":"2024-11-11T01:22:09Z","content_type":"application/xhtml+xml","content_length":"89904","record_id":"<urn:uuid:ac60db67-3952-47f8-80ff-af6097b0e638>","cc-path":"CC-MAIN-2024-46/segments/1730477028202.29/warc/CC-MAIN-20241110233206-20241111023206-00132.warc.gz"} |
The probability that a meal at a restaurant is overcooked is 10%. Estimate the probability that exactly 1 of the next 2 meals is overcooked.
Find an answer to your question 👍 “The probability that a meal at a restaurant is overcooked is 10%. Estimate the probability that exactly 1 of the next 2 meals is overcooked. ...” in 📗 Mathematics
if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
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Aaditya Chandrasekhar
Postdoctoral Scientist, Argonne National Laboratory
I'm a postdoc at Argonne National Labs, working on physics integrated machine learning techniques for X-ray imaging at the Advanced Photon Sources.
I have a PhD in mechanical engineering from UW Madison. Advised by Krishnan Suresh, my research primarily focused on physics constrained machine learning, primarily for design optimization. Along
these lines, I've worked on algorithms for optimizing material selection during design, design representation using neural networks and imposing manufacturing constraints in design optimization.
I was a PhD resident at Google X, working on an undisclosed project related to inverse design of photonic devices.
FluTO: Graded Multiscale Fluid Topology Optimization using Neural Networks
Rahul K Padhy*, Aaditya Chandrasekhar*, Krishnan Suresh
Engineering with Computers, 2022
FRC-TOuNN: Topology Optimization of Continuous Fiber Reinforced Composites using Neural Network
Aaditya Chandrasekhar, Amir Mirzendehdel, Morad Behandish, Krishnan Suresh
Computer Aided Design, 2022
A Generalized Framework for Microstructural Optimization using Neural Networks
Aaditya Chandrasekhar*, Saketh Sridhara*, Krishnan Suresh
Materials and Design, 2022
Graded Multiscale Topology Optimization using Neural Networks
Aaditya Chandrasekhar*, Saketh Sridhara*, Krishnan Suresh
Advances in Engineering Software, 2022
Integrating Material Selection with Design Optimization via Neural Networks
Aaditya Chandrasekhar*, Saketh Sridhara*, Krishnan Suresh
Engineering with Computers, 2022
Towards Assembly-Free Methods for Additive Manufacturing Simulation
Anirudh Krishnakumar, Aaditya Chandrasekhar, Krishnan Suresh
ASME-IDETC, 2015 | {"url":"https://www.aadityacs.com/","timestamp":"2024-11-06T12:19:40Z","content_type":"text/html","content_length":"111756","record_id":"<urn:uuid:949a3703-f446-4089-aa6e-3b6a3948f203>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00192.warc.gz"} |
Operator associativity
For the mathematical concept of associativity, see
In programming languages and mathematical notation, the associativity (or fixity) of an operator is a property that determines how operators of the same precedence are grouped in the absence of
parentheses. If an operand is both preceded and followed by operators (for example, "^ 4 ^"), and those operators have equal precedence, then the operand may be used as input to two different
operations (i.e. the two operations indicated by the two operators). The choice of which operations to apply the operand to, is determined by the "associativity" of the operators. Operators may
be left-associative (meaning the operations are grouped from the left), right-associative (meaning the operations are grouped from the right) or non-associative (meaning there is no defined
grouping). The associativity and precedence of an operator is a part of the definition of the programming language; different programming languages may have different associativity and precedence
for the same operator symbol.
Consider the expression a ~ b ~ c. If the operator ~ has left associativity, this expression would be interpreted as (a ~ b) ~ c and evaluated left-to-right. If the operator has right
associativity, the expression would be interpreted as a ~ (b ~ c) and evaluated right-to-left. If the operator is non-associative, the expression might be a syntax error, or it might have some
special meaning.
Many programming language manuals provide a table of operator precedence and associativity; see, for example, the table for C and C++.
Associativity is only needed when the operators in an expression have the same precedence. Usually + and - have the same precedence. Consider the expression 7 − 4 + 2. The result could be either
(7 − 4) + 2 = 5 or 7 − (4 + 2) = 1. The former result corresponds to the case when + and − are left-associative, the latter to when + and - are right-associative.
Usually the addition, subtraction, multiplication, and division operators are left-associative, while the exponentiation, assignment and conditional operators are right-associative. To prevent
cases where operands would be associated with two operators, or no operator at all, operators with the same precedence must have the same associativity.
A detailed example
Consider the expression 5^4^3^2. A parser reading the tokens from left to right would apply the associativity rule to a branch, because of the right-associativity of ^, in the following way:
1. Term 5 is read.
2. Nonterminal ^ is read. Node: "5^".
3. Term 4 is read. Node: "5^4".
4. Nonterminal ^ is read, triggering the right-associativity rule. Associativity decides node: "5^(4^".
5. Term 3 is read. Node: "5^(4^3".
6. Nonterminal ^ is read, triggering the re-application of the right-associativity rule. Node "5^(4^(3^".
7. Term 2 is read. Node "5^(4^(3^2".
8. No tokens to read. Apply associativity to produce parse tree "5^(4^(3^2))".
This can then be evaluated depth-first, starting at the top node (the first ^):
1. The evaluator walks down the tree, from the first, over the second, to the third ^ expression.
2. It evaluates as: 3^2 = 9. The result replaces the expression branch as the second operand of the second ^.
3. Evaluation continues one level up the parse tree as: 4^9 = 262144. Again, the result replaces the expression branch as the second operand of the first ^.
4. Again, the evaluator steps up the tree to the root expression and evaluates as: 5^262144 ≈ 6.2060699 × 10^183230. The last remaining branch collapses and the result becomes the overall
result, therefore completing overall evaluation.
A left-associative evaluation would have resulted in the parse tree ((5^4)^3)^2 and the completely different results 625, 244140625 and finally ~5.9604645 × 10^16.
Right-associativity of assignment operators
Assignment operators in imperative programming languages are usually defined to be right-associative. For example, in C, the assignment a = b is an expression that returns a value (namely, b
converted to the type of a) with the side effect of setting a to this value. An assignment can be performed in the middle of an expression. (An expression can be made into a statement by
following it with a semicolon; i.e. a = b is an expression but a = b; is a statement). The right-associativity of the = operator allows expressions such as a = b = c to be interpreted as a = (b =
c), thereby setting both a and b to the value of c. The alternative (a = b) = c does not make sense because a = b is not an lvalue.
Non-associative operators
Non-associative operators are operators that have no defined behavior when used in sequence in an expression. In Prolog, the infix operator :- is non-associative because constructs such as "a :-
b :- c" constitute syntax errors.
Another possibility distinct from left- or right-associativity is that the expression is legal but has different semantics. An example is the comparison operators (such as >, ==, and <=) in
Python: a < b < c is shorthand for (a < b) and (b < c), not equivalent to either (a < b) < c or a < (b < c).^[1]
See also
1. ^ http://docs.python.org/reference/expressions.html#comparisons
□ Programming language topics
□ Operators (programming)
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JOBU is CHARACTER*1
= 'U': U must contain an orthogonal matrix U1 on entry, and
the product U1*U is returned;
[in] JOBU = 'I': U is initialized to the unit matrix, and the
orthogonal matrix U is returned;
= 'N': U is not computed.
JOBV is CHARACTER*1
= 'V': V must contain an orthogonal matrix V1 on entry, and
the product V1*V is returned;
[in] JOBV = 'I': V is initialized to the unit matrix, and the
orthogonal matrix V is returned;
= 'N': V is not computed.
JOBQ is CHARACTER*1
= 'Q': Q must contain an orthogonal matrix Q1 on entry, and
the product Q1*Q is returned;
[in] JOBQ = 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'N': Q is not computed.
M is INTEGER
[in] M The number of rows of the matrix A. M >= 0.
P is INTEGER
[in] P The number of rows of the matrix B. P >= 0.
N is INTEGER
[in] N The number of columns of the matrices A and B. N >= 0.
[in] K K is INTEGER
L is INTEGER
K and L specify the subblocks in the input matrices A and B:
[in] L A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
of A and B, whose GSVD is going to be computed by DTGSJA.
See Further Details.
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the M-by-N matrix A.
[in,out] A On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
matrix R or part of R. See Purpose for details.
LDA is INTEGER
[in] LDA The leading dimension of the array A. LDA >= max(1,M).
B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the P-by-N matrix B.
[in,out] B On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
a part of R. See Purpose for details.
LDB is INTEGER
[in] LDB The leading dimension of the array B. LDB >= max(1,P).
[in] TOLA TOLA is DOUBLE PRECISION
TOLB is DOUBLE PRECISION
TOLA and TOLB are the convergence criteria for the Jacobi-
[in] TOLB Kogbetliantz iteration procedure. Generally, they are the
same as used in the preprocessing step, say
TOLA = max(M,N)*norm(A)*MAZHEPS,
TOLB = max(P,N)*norm(B)*MAZHEPS.
[out] ALPHA ALPHA is DOUBLE PRECISION array, dimension (N)
BETA is DOUBLE PRECISION array, dimension (N)
On exit, ALPHA and BETA contain the generalized singular
value pairs of A and B;
ALPHA(1:K) = 1,
BETA(1:K) = 0,
and if M-K-L >= 0,
[out] BETA ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L) = diag(S),
or if M-K-L < 0,
ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
Furthermore, if K+L < N,
ALPHA(K+L+1:N) = 0 and
BETA(K+L+1:N) = 0.
U is DOUBLE PRECISION array, dimension (LDU,M)
On entry, if JOBU = 'U', U must contain a matrix U1 (usually
the orthogonal matrix returned by DGGSVP).
[in,out] U On exit,
if JOBU = 'I', U contains the orthogonal matrix U;
if JOBU = 'U', U contains the product U1*U.
If JOBU = 'N', U is not referenced.
LDU is INTEGER
[in] LDU The leading dimension of the array U. LDU >= max(1,M) if
JOBU = 'U'; LDU >= 1 otherwise.
V is DOUBLE PRECISION array, dimension (LDV,P)
On entry, if JOBV = 'V', V must contain a matrix V1 (usually
the orthogonal matrix returned by DGGSVP).
[in,out] V On exit,
if JOBV = 'I', V contains the orthogonal matrix V;
if JOBV = 'V', V contains the product V1*V.
If JOBV = 'N', V is not referenced.
LDV is INTEGER
[in] LDV The leading dimension of the array V. LDV >= max(1,P) if
JOBV = 'V'; LDV >= 1 otherwise.
Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
the orthogonal matrix returned by DGGSVP).
[in,out] Q On exit,
if JOBQ = 'I', Q contains the orthogonal matrix Q;
if JOBQ = 'Q', Q contains the product Q1*Q.
If JOBQ = 'N', Q is not referenced.
LDQ is INTEGER
[in] LDQ The leading dimension of the array Q. LDQ >= max(1,N) if
JOBQ = 'Q'; LDQ >= 1 otherwise.
[out] WORK WORK is DOUBLE PRECISION array, dimension (2*N)
NCYCLE is INTEGER
[out] NCYCLE The number of cycles required for convergence.
INFO is INTEGER
[out] INFO = 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
= 1: the procedure does not converge after MAXIT cycles. | {"url":"https://netlib.org/lapack/explore-html-3.4.2/d6/da0/dtgsja_8f.html","timestamp":"2024-11-06T05:58:54Z","content_type":"application/xhtml+xml","content_length":"24371","record_id":"<urn:uuid:455b1832-48ef-4004-8019-4aa8260a64c1>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00642.warc.gz"} |
Quick, hit the brakes!
A reader emailed me a fun question from a physics exam he took, along these lines:
A car driver going at some speed v suddenly finds a wide wall at a distance r. Should he apply brakes or turn the car in a circle of radius r to avoid hitting the wall?
My first thought was that surely the question wasn't doable without more information, but it turns out that we do have enough to give a straightforward answer. Let's take the "turns in a circle" and
"slams on brakes" scenarios one at a time.
Turns in a circle:
Velocity is a vector whose magnitude is the speed and whose direction is the direction of travel. If you turn, your speed remains the same but your direction of travel changes. So the velocity is
changing even if the speed isn't. A changing velocity is by definition an acceleration, and one of the key equations of first semester physics is the acceleration required to produce uniform circular
motion. It turns out to be a function of the speed and the radius of the circle:
$latex \displaystyle a = \frac{v^2}{r} &s=1$
Since we don't have any numbers to plug in or really anywhere else to go with this, we're done with this part. The required acceleration to avoid the wall is equal to the square of the speed divided
by the radius of the circle, which is just the initial distance to the wall.
Slams on brakes:
This one is a little more involved. The direction of the velocity is not changing, but the speed is. Another of the key equations of freshman physics is the formula for position in uniformly
accelerated motion. It's:
$latex \displaystyle x = \frac{1}{2}a t^2 + v_0 t + x_0 &s=1$
where a is the acceleration, v0 is the initial velocity, x0 is the initial position, and t is the elapsed time. In this case we'd like to solve for a at the point where x = r (we define our
coordinates such that x0 = 0). But we don't know how much time has elapsed by the time the car reaches the wall, so we need the formula for velocity in uniformly accelerated motion, which we might
write from memory or find by differentiating the position equation if we know calculus:
$latex \displaystyle v = at + v_0 &s=1$
Now I'll start subscripting the letter f on the specific time when the car reaches the wall. We know that we've come to a stop at at that time, so we have:
$latex \displaystyle 0 = at_f + v_0 &s=1$
Which means
$latex \displaystyle t_f = -\frac{v_0}{a} &s=1$
Don't worry about the negative sign. a is itself negative (we're decelerating), so tf will be positive as well. Now that we know how much time has elapsed when the motion is complete, we can plug
that into our position formula:
$latex \displaystyle r = \frac{1}{2}a(-\frac{v_0}{a})^2 + v_0 (-\frac{v_0}{a}) &s=1$
Remembering that at the wall, x = r and that we defined x0 = 0. You can do the algebra to solve for a, and you'll find that
$latex \displaystyle a = -\frac{v_{0}^{2}}{2r} &s=1$
Which is (ignoring the minus sign that just tells us which way the acceleration is pointed) just half the acceleration we found for the turning scenario. So purely from a standpoint of the
acceleration car tires can produce, braking works better than swerving.
After writing this post, I came across a ScienceBlogs post on Dot Physics a few years ago on the same subject. He approaches the problem in a different way, and I think it's well worth reading both
solution methods.
More like this
This is a classic problem. You are in a car heading straight towards a wall. Should you try to stop or should you try to turn to avoid the wall? Bonus question: what if the wall is not really wide so
you don't have to turn 90 degrees? Assumption: Let me assume that I can use the normal model of…
There's a question that gets posed toward the beginning of intro physics classes to gauge the students' understanding of acceleration. If you fire a bullet horizontally while at the same instant
dropping a bullet from the same height, which hits the ground first? The point is to think clearly…
I had an interesting question posed to me recently: how frequently does the sun emit photons with an energy greater than 1 TeV? All of you know about the experiments going on at the LHC, where
particles are accelerated to an energy which is equivalent to an electron being accelerated through a…
Maybe this is a little old (in internet age), but it is a great example. Here is the Loop-the-loop stunt from the show Fifth Gear. I like this. First, it is a bold stunt. But also, there is some good
physics here. Though, most importantly, the Fifth Gear producers were kind enough to include a…
The analysis assumes that your sudden maneuver (turning, braking, or some combination thereof) doesn't send you into a power slide. That may be reasonable where you live, but in snow country, or
places like California where oil buildup during the dry season makes for really slick conditions when the rains finally do come, you can't count on that. Which makes the advantage to braking that
much greater.
As someone who learned to drive in Florida and moved to snow country as an adult, "make no sudden maneuvers, especially in snow or icy conditions" is something I have internalized. Above all, try not
to get into a situation where you have to use the brakes and the steering wheel at the same time. And remember that while your all-wheel drive will help you go, it won't help you stop.
One complication is that if you can execute a 180 while braking you'll have much more crush space between you and the wall. That would probably be difficult for anyone but a seasoned stunt driver,
but given that the whole situation is hypothetical it deserves consideration.
Put another way, since turning requires twice the acceleration as braking, you are more likely to exceed the acceleration the tires can provide, making you slide.
Just notice that the magnitude of your acceleration is fixed and as soon as you turn the component pointing away from the wall goes down. That solves it without any calculation.
I wonder whether there may be some situation, perhaps facing an obstacle not quite as wide as "a wide wall", in which turning the steering wheel and applying the brakes sensibly together gives an
advantage over only using the steering wheel without breaking, or only breaking on the straight. | {"url":"https://scienceblogs.com/builtonfacts/2013/02/27/quick-hit-the-brakes","timestamp":"2024-11-04T20:27:47Z","content_type":"text/html","content_length":"48034","record_id":"<urn:uuid:d9930d62-8862-4314-afa6-5eec36f8883d>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.16/warc/CC-MAIN-20241104194528-20241104224528-00848.warc.gz"} |
This is Zygo level 5. You can try a different level:
Level 1 Level 2 Level 3 Level 4 Level 5 Level 6 Level 7 Level 8 Level 9 Level 10 Level 11 Level 12
This number arranging puzzle was devised by Les Page and adapted as a Transum Mathematics interactive numeracy puzzle. The Zygo puzzle is copyright (©2006 Les Page) but available for syndication.
This is an ideal activity for those wishing to develop their numeracy skills and logical thinking. There are 12 different levels and each puzzle is created from random numbers every time the page is
loaded so you are very unlikely to get the same puzzle twice.
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The Zygo Story
Puzzle creator, Les Page, tells the story of how Zygo came to be:
I suppose I've got to blame Zygo onto something, so it has to be that fantastic numbers puzzle Sudoku.
When Sudoku hit the media in 2004 and the craze began I, like many millions of others who love number puzzles, got caught up with the magical 9 x 9 puzzle and became totallyaddicted to it.
I am an ideas man and it wasn't long before I started to think about devising my own unique numbers puzzle having been completely inspired by Sudoku.
It was the 28th May 2006, my 59th birthday, and all the family were around for tea and birthday cake. My grandson Harry had completely worn me out after a rough and tumble on the lounge floor and I
was spending a restive five minutes scribbling some numbers on a piece of graph paper. It was then that I had that Eureka moment. I had put three parallel lines of numbers, from 1 to 9, randomly on
the graph paper and then added the numbers in each of the top two columns together and put their totals above. Likewise I added the two bottom numbers in each of the two bottom columns together and
put their totals below like this:
It was then that I realised that the middle row of numbers were common to both sums, up and down, and immediately realised I had hit on something. I had found what I was looking for, a unique, new
puzzle and shouted, I've got it!. My family took absolutely no notice of me and Harry carried on playing with his cars and Claire my daughter-in-law continued to feed my other 3 month old grandson,
Jamie. I could not blame the family
as they all knew I was in my own little private world, always dreaming up bright ideas, which never ever came to anything. They had given me the nickname of The Mad Professor, and were always pulling
my leg about it. Secretly I loved it!
But I knew I was on to something. So I decided to make a second copy of the numbers and totals, which was my puzzle solution and marked them with letters on the side like this:
I then erased all the numbers from the solution in rows A, B and C except for the first column to form the framework of my puzzle like this:
Then concentrating on the totals I tried to replace the missing numbers. It took me some time to begin with, but after a while I could see a pattern developing. In the puzzle above I could see that
there was a total of 3 at the top and this could only mean that A + B was either 1 + 2 or 2 + 1 in A and B. And as number 1 had already been used in row B the answer had to be 1 + 2 in A ad B like
I then decided to write a spreadsheet macro program on my old favourite Lotus 123. I was self taught and had trained myself to learn all the math functions I needed off by heart. It was quite a
challenge, but I managed to succeed and eventually produced a nifty little program which produced puzzles instantly at the touch of two keys and Zygo was born.
When completed I copyrighted it in July 2006. I didn't want the same thing to happen to me as Howard Garns who devised Sudoku in 1979 and died in 1989 before getting the chance to see his creation as
a worldwide phenomenon that was never given copyright © status!
At first I gave the puzzle several other names. Because of the Japanese origin of Sudoku I tried to think of a Japanese word for my puzzle too. I looked at a Japanese to English translation program
on the internet and came up with KonnyuuKu. The Japanese word Konnyuu means adding numbers and Ku means nine, which I thought appropriate.
But later I decided to formulate puzzles with ranges 1 to 5 and 1 to 7 ( mainly for kids ) and 1 to 10, 1 to 11, 1 to 12 and 1 to 13, so the Ku, 9 part of the name became inappropriate.
So I looked for a new name. I considered NouKou, which when translated means Brain Box, but I eventually came up with Zygo after stumbling across it in the Reader's Digest Oxford Complete Wordfinder.
I was looking for an obscure name. So started looking in the Z's. I was on the last page and feeling rather despondent when there it was………….ZYGO!
Zygo- /zigo, ziggo/ comb. Form joining, pairing.
It's a combination word!
The word Zygo incorporated everything involved with the puzzle I had created. You have to find pairs of combination numbers that when added together give totals that form the basis of the entire
puzzle. Amazing. No wonder I shouted, Eureka!, at the top of my voice!
What inspired me to carry on and try to eventually get ZYGO recognised by a website like TRANSUM MATHS was my grandson, Harry.
In 2010 when Harry was 10 he came to tea after school and happened to pick up one of my Zygo puzzles and started to solve it and was totally engrossed for about five minutes.
Finished it, grandad!, he shouted. It's great!
He had got it right too. I was so proud.
Later, before he went home, he told me he had really enjoyed doing Zygo and asked me if some of his school friends could try it too.
It was then that I decided to go and see the Headteacher, Mr Ball at Harry's school and to cut a long story short he arranged a ZYGO CHALLENGE with the 10 and 11 year old pupils during lunch hours.
The results of the questionnaire the children and the teachers completed were amazing. I received scores of 9's and 10's out of 10 and some lovely comments from the pupils saying, I love this puzzle.
and I wish I could have this on my DS. A lot of the kids drew smiley faces on their questionnaire forms too, which said everything.
A teacher commented, My mum would love this puzzle!
Mr Ball sent me the following letter:
27th January 2011
Re ZYGO
Dear Les,
Finally got round to doing the ZYGO challenge, the children were really impressed and thoroughly enjoyed it, as you will see from the evaluations enclosed. I would be very interested in any other
ZYGO puzzles or material, as we could probably use them in a range of activities across the school.
Thanks again.
Brian Ball
I now have a large portfolio of number puzzles which I have devised over the last 12 years since the birth of ZYGO. One called U+ has been developed professionally as an Android App and I would like
to have my own PuzzLesPage website developed with interactive puzzles on it and have more Apps
developed for the Apple Store and Google Play.
Les Page, Author of ZYGO.
If you like this puzzle you may want to try one of the other Transum Puzzles.
Sarah C,
Monday, February 5, 2024
"THIS IS SO FUN, my teacher assigned this as a fun math puzzle and it was the best! Even after the timer ran out, many of my classmates still wanted to keep playing! :3."
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The Influence of Anthropogenic and Environmental Disturbances on Parameter Estimation of a Dengue Transmission Model
School of Applied Sciences and Engineering, Universidad EAFIT, Medellín 050022, Colombia
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Submission received: 25 August 2022 / Revised: 29 October 2022 / Accepted: 14 December 2022 / Published: 22 December 2022
Some deterministic models deal with environmental conditions and use parameter estimations to obtain experimental parameters, but they do not consider anthropogenic or environmental disturbances,
e.g., chemical control or climatic conditions. Even more, they usually use theoretical or measured in-lab parameters without worrying about uncertainties in initial conditions, parameters, or changes
in control inputs. Thus, in this study, we estimate parameters (including chemical control parameters) and confidence contours under uncertainty conditions using data from the municipality of Bello
(Colombia) during 2010–2014, which includes two epidemic outbreaks. Our study shows that introducing non-periodic pulse inputs into the mathematical model allows us to: (i) perform parameter
estimation by fitting real data of consecutive dengue outbreaks, (ii) highlight the importance of chemical control as a method of vector control, and (iii) reproduce the endemic behavior of dengue.
We described a methodology for parameter and sub-contour box estimation under uncertainties and performed reliable simulations showing the behavior of dengue spread in different scenarios.
1. Introduction
Over the past 50 years, arboviral infections transmitted by mosquitoes of the
genus have emerged as a significant health problem worldwide [
] because of climatic change, human migration, and lack of control strategies [
]. Dengue is endemic in over 100 countries and affects 100 million people annually [
]. The control actions can be modeled as parameters or inputs, depending on the type of control: biological control, chemical control, mechanical control (clean-up of the mosquito breeding sites), or
vaccination [
]. Note that the World Health Organization (WHO) recommends chemical control only in emergencies to suppress an ongoing epidemic or prevent an incipient one [
]. There are different studies that motivate the study of chemical control using mathematical models to test its effectiveness under several epidemiological conditions [
Model parameters that were estimated from real data help more to fit a feasible model that follows real conditions than models with measured-in-lab parameters [
]. Previous papers have included seasonal changes in the mosquito mortality rate [
], pulse-type inputs that describe vaccination [
] or Wolbachia enhancement [
], and mosquitoes removal from the system [
]. However, in dengue application papers, the authors implemented input parameters obtained from the literature instead of estimating them by fitting the model to available epidemiological data [
]. Thereby, we identified an opportunity to study the estimation of model parameters and chemical control inputs simultaneously with data from consecutive dengue outbreaks addressing uncertainty
issues with reliable confidence intervals.
Note that nominal parameters by themselves are not informative enough about the disease dynamic because they do not address the uncertainty in epidemiological data. Hence, it is necessary to
calculate a region inside parameters space where every set of values causes the model output to stay numerically close to the real data [
], an approximation of this region is a confidence sub-contour box (CSB) [
]. We implemented global sensitivity methods to characterize the model behavior by identifying how variations in the value of its parameters affect its output. Thus, applying these methods, we
focused on pulse input estimations that represent the change in the mosquito mortality rate in specific time intervals using real data from the municipality of Bello (Colombia) during the 2010–2014
This paper is distributed as follows: first, in the Methodology section, we introduce the study case, describe the mathematical model and the methods to perform the CSB, and the uncertainty and
sensitivity analyses (UA/SA). Then, in the Results section, we present the estimated parameters and their CSB for different subsets of epidemiological data. Finally, we present a Discussion section
and Conclusions.
2. Methods
In the following subsections, we describe the methodology used to study the dengue spread and control dynamics. This is composed of (i) description of a mathematical model with parameters and CSB
estimations using real data; (ii) implementation of pulse input signals to perform estimations, in which we consider that the parameter values are not the same during a temporal window; (iii)
validation of the CSB quality using UA and SA; and (iv) test the control strategies simulating different pulse-type inputs (vaccination and chemical control).
2.1. Characteristics of the Study Area
The municipality of Bello (Colombia) is located in the Andean Mountains at 1450 MASL, with an annual mean precipitation and temperature of 1538 mm and 21.7 °C, respectively [
]. Thus, Bello has appropriate conditions for
Aedes aegypti
reproduction and endemic dengue cases with occasional outbreaks in which the four serotype could circulate at the same time [
]. In addition, the region has suffered from multiple consecutive dengue outbreaks in some years with a high ENSO effect [
], e.g., 2010–2013, 2015–2016, and 2019–2020 [
]. Therefore, the health authorities have taken actions against the spread of the disease: (i) release of mosquitoes with Wolbachia in 2015, (ii) cleaning of breeding sites, and (iii) occasional
chemical control [
Because of the high number of dengue cases reported in 2009 and 2014 in Bello, the municipality performed two aerial spraying. They applied this control action during the 29th epidemiological week in
2010 and the 32nd epidemiological week in 2014 in different locations in the city. The insecticide implemented to manage both outbreaks was Malathion, which has a residual effect for up to 12 weeks
and only affects the adult stage of mosquitoes [
]. The reported chemical controls in Bello provided us with information about an extrinsic factor that affected mosquito populations and the number of dengue cases in specific time frames. Thus, we
modeled the effect of chemical control actions over dengue cases using pulse inputs (see
Section 2.3
2.2. Experimental Data
In Colombia, hospitals report weekly the number of confirmed dengue cases to the National Public Health Surveillance System (SIVIGILA by its initials in Spanish). Pre-established protocols are
implemented to monitor classic and hemorrhagic cases that help to identify risk factors and perform control strategies [
]. This study used data from the number of infected people during 265 epidemiological weeks in Bello (Colombia), starting in the 48th epidemiological week in 2009 until the 52nd epidemiological week
in 2014; this period covers two dengue outbreaks that occurred in the city between 2009–2010 and 2014. By that time, the population of Bello was about 407,000 inhabitants who were distributed among
urban and rural zones [
2.3. Mathematical Model and Control Actions
Different models have been used to describe dengue spread; these methods depend on the scope and detail that the modeler wants to give to the analysis and control [
]. Using a model for adult and aquatic phases (eggs, larvae, pupae), we can add different actions to control the models, which allows us to check the effect of these control actions on the mosquito
population and disease spread.
In the present study, we implemented a 10-order continuous-time and nonlinear model of dengue spread based on the model given in [
], with new parameter (
) and a vaccine control input (
$u v$
). We define the model parameters in
Table 1
. All model state-space variables and parameters are non-scaled to conserve their magnitude and biological meaning:
$E ˙ = δ ( 1 − E C ) M − ( γ e + μ e + u e ) E L ˙ = γ e E − ( γ l + μ l + u l ) L P ˙ = γ l L − ( γ p + μ p + u p ) P M s ˙ = f γ p P − β m H i M s H − ( μ m + u m ) M s M e ˙ = β m H i M s H − ( θ
m + α μ m + u m ) M e M i ˙ = θ m M e − ( α μ m + u m ) M i H s ˙ = μ h H − β h M i H s M − μ h H s − u v H s H e ˙ = β h M i H s M − ( θ h + μ h ) H e H i ˙ = θ h H e − ( γ h + μ h ) H i H r ˙ = γ h
H i − μ h H r + u v H s$
We based the dengue spread model on the following assumptions: (i) $H i$ corresponds to the reported dengue cases; (ii) the total human population H is constant ($H ˙ = H s ˙ + H e ˙ + H i ˙ + H r ˙
= c o n s t$, which is given by the last four equations in the model), i.e., the human birth and death rates are equal and constant (this is valid for studies that occur in periods when the
population does not increase considerably), (iii) the total population of mosquitoes M is variable ($M = M s + M e + M i$) because of its short lifetime and the population changes that suffer in
short periods, (iv) there is a unique serotype behavior, i.e., it represents an average behavior of the dengue propagation, (v) populations are homogeneously mixed, and the mean behavior is
represented (parameters are the average for the entire population), (vi) vertical transmission of the dengue virus is not considered, and (vii) the sample time interval is one week.
We separated the mosquito mortality rate into two parts (as shown in [
]): mortality related to natural environmental conditions (
$μ m$
) and mortality related to imposed conditions such as chemical control (
$u m$
). We modeled
$u m$
as a pulse-type input (
), in which
is the number of pulses,
$t 0 c j$
is the initial time,
$Δ t c j$
is the duration and
$A c j$
is the amplitude of
-pulse input; its value can be positive or negative, in which a positive value represents a control effect and a negative value corresponds to a increase in mosquito natality due to, for example,
favorable environmental conditions:
$u m = ∑ j = 1 n u m j , u m j = A c j , t 0 c j ≤ t ≤ t 0 c j + Δ t c j 0 , o t h e r w i s e$
In model (
), the parameter
describes the effect of the virus over the mortality rate of
$M e$
$M i$
, as shown in several studies [
]. The parameter
, in
$M s$
expression, represents the sex-ratio at adult emergence [
2.4. Parameter Estimation
The parameter estimation of mathematical models is a process that assigns optimal values to the parameters to fit simulation outputs to experimental outputs [
]. In this work, we estimated the parameters of the model (
) for three cases associated with the number of pulses
$u m j$
. The pulse scenarios describe the effect on the mortality of the mosquitoes population caused by chemical control or favorable external conditions:
We fit one epidemic outbreak over 93 epidemiological weeks, without pulse-type inputs (
$u m = 0$
in Equation (
)), i.e., we assume that there were no external dynamics that could affect the mosquitoes population (no chemical control or environmental changes);
We fit one epidemic outbreak over 93 epidemiological weeks, but with the addition of one pulse-type input, which describes an external change that perturbs the mosquito population through a
chemical control (
$u m = u m 1$
in Equation (
)). Here, we estimated the parameters of the model and the pulse input together;
We fit two epidemic outbreaks over 265 epidemiological weeks (covering 93 weeks of previous cases) and four pulse-type inputs: two positive inputs for two chemical control actions ($u m 1$, $u m
4$) and two negative inputs ($u m 2$, $u m 3$) for modeling an increase in mosquito mortality due to some favorable environmental conditions. In addition, in this case, we estimate simultaneously
the four inputs and the model parameters.
To perform parameter estimation, we applied the following methods and tools in the mathematical model (
) using the Matlab programming environment: (i) We fit the
$H i$
output model with the real data (number of reported cases per week); (ii) We used numerical methods to solve the model during parameter estimation, even for model simulations; (iii) We performed the
parameter estimation using the Trust-Region-Reflective algorithm and nonlinear least-squares criterion implemented in Matlab [
], in which the parameter and functional tolerance were equal to
$1 · 10 − 7$
; (iv) We fixed two initial conditions:
$H r ( 0 ) = 0$
(the model does not consider recovered humans at the beginning of simulation ) and
$H i ( 0 ) = 8$
(the number of dengue cases reported in the first week of the outbreak); (v) We defined lower and upper bounds in parameter intervals (see column 3 in
Table 2
) from reported lab values (see column 2 in
Table 2
) and then expanded them when the parameter tends toward an interval bound in the estimation process; this makes sense since some parameters could have higher or lower values because of wildlife
conditions. All the routines and codes are available in this
The main problem encountered during parameter estimation in models such as (
) is the reliability of the data. We have only one output data set to fit the parameters, which can lead to different sets of parameters with similar cost functions. Because multiple local minima
could exist in our search surface, the estimation algorithm could converge into an unwanted stationary point in the mean squared error surface, resulting in parameter values that are biological or
physically impossible [
]. Thus, we performed several parameter estimations to obtain information about the search surface and we started from different initial points trying to avoid local minima. We performed a different
number of estimations per case because of computational costs: case A (2420 estimations), case B (1400 estimations), and case C (640 estimations). Then, from all performed estimations, we selected
the estimations whose cost function was within a range of 5% of the minimum cost function. With this method, it is possible to take out some local minima presented during estimations and identify a
certain family of parameters. For case A, we selected almost 30% of the estimations, whereas we selected 10% for case B and 80% for case C. With this filtered data; we calculated a nominal value for
each parameter for the three cases using the statistical median because the median, unlike the mean, is a robust measure of centrality, in which atypical data do not affect the median value as much [
2.5. Confidence Sub-Contour Box
After we obtained the nominal curves for each case, we aimed to compute CSB for those parameters to address the uncertainty due to misreporting of dengue cases in the data to fit. We were looking for
those feasible parameter values that cause the model output to encompass the shape of the real data up to some uncertainty threshold. Following [
], a region in the space of parameters exhibiting that property is called a confidence contour. Unfortunately, the confidence contour usually shows a complex shape for optimization problems of
nonlinear models, which, in turn, makes its estimation virtually impossible [
]. Thus, we chose a novel method to compute a rectangular region within the confidence contour starting from some nominal values for the model parameters. Such a region is named the Confidence
Sub-contour Box because of its relation to the confidence contour. The algorithm to compute it was implemented in a free Matlab toolbox [
The method for CSB computation exploits the linkages between the theory of sensitivity and the model behavior to achieve intervals for the parameters that jointly define a region where the model
output shows similarity with the data to fit. In addition, the influence of every single parameter on the model output tends to be equally relevant within this region. We refer the reader to [
] for further reading about the method. To assess the similarity between the fitted data and the model output, we chose an uncertainty threshold level of 30% when computing the CSB for the present
study. Such a threshold determines the maximum variation allowed for the model values regarding the data values. In this way, we could provide a contour box for parameters that fit the
epidemiological data and its trend.
2.6. Sensitivity and Uncertainty Analyses
We performed UAs and global SAs to quantify the uncertainty contribution of every parameter to the model output [
]. In our case, uncertainty was linked to the size of the CSB for each parameter (95% in this study). The UA is a Monte Carlo simulation that graphically assesses the spread of uncertainty from
parameters and their interactions to model outputs. The SA attempts to determine the contribution of each parameter to the model output uncertainty. For the SA, we chose two variance-based methods
from Saltelli et al. [
] and Xiao et al. [
]; these methods were implemented in a Matlab toolbox [
]. The first method (Saltelli) is particular for scalar model outputs; hence, we used it to quantify the contribution of each parameter to the mean squared error (MSE) function output, i.e., the cost
function that quantifies the fit of a given temporal response (from the estimated model parameters) to measured data (
$H i$
). The idea behind this approach was to identify those parameters that mostly determine the output behavior when all the parameters carry the same uncertainty. In this way, we can suggest that those
parameters that had the greatest contribution are important parameters for control actions. The second SA method (Xiao) that we used in particular for temporal responses; thus, we implemented it to
explore the uncertainty contribution of parameters to the model output from the estimated confidence sub-contour box.
We focused on case C to perform all the SA procedures; furthermore, for the SA methods, we estimated the indices of the contribution of each parameter alone ($S i$) and the contribution of each
parameter and its interactions ($S T i$). For our purposes, the $S T i$ indices were more relevant than the $S i$ indices; however, we used the Saltelli estimator for $S i$ as a reliability indicator
because, when estimating the $S i$ index in this way, negative values can be obtained. By definition, sensitivity indices must not be negative, i.e., obtaining negative $S i$ values suggests that an
SA with a larger sample size is necessary. We identified the minimum sample size N that gives an informative SA result through the convergence of $S i$ and $S T i$, applying the Saltelli method over
intervals built from 1% uncertainty for the nominal parameters (this percent gives outputs close to the nominal output).
3. Results
In this section, we present the results of parameter estimation for cases A (no pulse input), B (one pulse input), and C (four pulse inputs) alongside their CSBs, which are validated by the UA and
SA. Then, we performed a more in-depth analysis of case C, carrying out simulations with the disease control scenarios, including vaccination and chemical control.
3.1. Parameter Estimations with Zero, One, and Four Pulse-Type Inputs
As a first approach, we performed multiple parameter estimations using the initial intervals presented in the literature (see
Table 2
). Then, we obtained the nominal and CSB values for cases A, B, and C, as we stated in
Section 2.4
and shown in
Table 2
. For case A, we identified the following weakness: none of the estimated parameter sets reproduce the endemic behavior, which is clear from disease data (see
Figure 1
a and
Figure 2
a), regardless of the number of estimations, i.e.,
$H i → 0$
$t → ∞$
Table 2.
Estimated nominal values and confidence sub-contours for parameters and initial conditions in model (
), using zero, one and four pulse-type inputs. The biological intervals reported in the literature [
] and intervals used in parameter estimation are specified.
Case A (Zero Pulse Input) Case B (One Pulse Input) Case C (Four Pulse Inputs)
Factor Biological Estimation Nominal CSB Nominal CSB Nominal CSB
Interval Interval Value Value Value
$E ( 0 )$ - (0, 30,000) 17,000 (13,000, 23,000) 21,600 (18,800, 24,900) 9610 (9590, 12,800)
$L ( 0 )$ - (0, 30,000) 3400 (2400, 4300) 13,200 (11,000, 19,000) 26,000 (21,000, 35,000)
$P ( 0 )$ - (0, 30,000) 5400 (3900, 7500) 12,000 (9700, 16,000) 21,900 (20,900, 22,100)
$M s ( 0 )$ - (10,000, 10,000,000) 4,800,000 (4,400,000, 5,600,000) 3,400,000 (2,700,000, 3,800,000) 8,100,000 (7,700,000, 10,000,000)
$M e ( 0 )$ - (100, 1200) 1000 (980, 1200) 310 (244, 336) 320 (280, 330)
$M i ( 0 )$ - (0, 100) 0.150000 (0.110000, 0.210000) 13 (12, 16) 16 (15, 22)
$H s ( 0 )$ (0, 400,000) (0, 450,000) 358,000 (343,000, 414,000) 160,000 (150,000, 190,000) 180,000 (170,000, 200,000)
$H e ( 0 )$ - (0, 100) 0.28 (0.21, 0.41) 10 (9, 13) 22 (21, 25)
$δ$ (65, 165) (20, 180) 92 (64, 120) 46 (38, 59) 49 (42, 58)
C (6400, 95,000) (6400, 340,000) 120,000 (95,000, 180,000) 250,000 (238,000, 290,000) 231,000 (199,000, 238,000)
$γ e$ (0.6, 2.3) (0, 2.3) 0.120 (0.099, 0.170) 1.29 (1.10, 1.43) 1.48 (1.28, 1.52)
$μ e$ - (0, 1.3) 0.0078 (0.0056, 0.0101) 0.90 (0.84, 1.30) 1.23 (1.14, 1.25)
$γ l$ (0.05, 0.5) (0, 1.6) 0.42 (0.32, 0.59) 0.70 (0.68, 0.88) 0.86 (0.83, 1.07)
$μ l$ (0.07, 3.22) (0, 3.22) 2.7 (2.0, 3.9) 1.53 (1.45, 1.88) 1.43 (1.36, 1.67)
$γ p$ (0.1, 1) (0, 1.7) 0.497 (0.415, 0.696) 0.91 (0.75, 0.97) 0.905 (0.902, 1.110)
$μ p$ (0, 1.4) (0, 1.4) 1.20 (0.91, 1.75) 0.54 (0.44, 0.65) 0.61 (0.60, 0.66)
f (0.4, 0.6) (0.3, 0.7) 0.39 (0.29, 0.52) 0.49 (0.42, 0.50) 0.506 (0.449, 0.522)
$β m$ (0, 4) (0, 4) 0.040 (0.032, 0.052) 1.52 (1.50, 1.60) 2.02 (2.01, 2.03)
$μ m$ (0.06, 0.3) (0, 0.9) 0.268 (0.244, 0.270) 0.449 (0.449, 0.456) 0.5360 (0.5357, 0.5390)
$α$ (1, 1.6) (1, 1.6) 1.03 (1.03, 1.07) 1.44 (1.44, 1.46) 1.4770 (1.4668, 1.4773)
$θ m$ (0.58, 0.88) (0.4, 1.0) 0.40 (0.29, 0.51) 0.634 (0.630, 0.660) 0.642 (0.629, 0.643)
$μ h$ - (0.00001, 0.0009) 0.000021 (0.000016, 0.000029) 0.000228 (0.000190, 0.000302) 0.000748 (0.000587, 0.000788)
$β h$ (0, 4) (0, 4) 0.227 (0.216, 0.249) 1.43 (1.39, 1.43) 1.43 (1.42, 1.44)
$θ h$ (0.7, 1.75) (0.4, 1.8) 0.400 (0.290, 0.430) 0.70 (0.61, 0.72) 0.48 (0.45, 0.49)
$γ h$ (0.5, 1.75) (0.3, 2.0) 0.328 (0.322, 0.381) 1.69 (1.65, 1.69) 1.65 (1.65, 1.67)
$A m 1$ - (0, 2) - - 0.48 (0.45, 0.57) 0.69 (0.62, 0.72)
$t 0 c 1$ - (32, 38) - - 35.60 (35.00, 36.00) 35.80 (35.60, 36.80)
$Δ t c 1$ (0, 12) (0, 12) - - 9.7 (7.9, 11.5) 11.99 (11.39, 12.25)
$A m 2$ - (−1.5, 0) - - - - −0.46 (−0.54, 0.45)
$t 0 c 2$ - (120, 134) - - - - 132 (125, 145)
$Δ t c 2$ - (0, 12) - - - - 6 (5, 7)
$A m 3$ - (−1.5, 0) - - - - −0.712 (−0.919, −0.709)
$t 0 c 3$ - (210, 235) - - - - 231 (227, 233)
$Δ t c 3$ - (0, 12) - - - - 5.67 (4.86, 5.86)
$A m 4$ - (0, 2) - - - - 0.34 (0.32, 0.36)
$t 0 c 4$ - (240, 260) - - - - 243 (240, 246)
$Δ t c 4$ (0, 12) (0, 12) - - - - 11 (10, 12)
With cases B and C, we obtained parameter values farther from the estimations bounds (see
Table 2
); these can simulate and reproduce the endemic behavior of the disease (see
Figure 1
b and
Figure 2
b). Furthermore, for case C, the model can simulate other real data dynamics, including a new outbreak and chemical control scenarios. For both cases, nominal parameter values of chemical control
inputs (
$t 0 c 1$
$Δ t c 1$
$t 0 c 4$
$Δ t c 4$
) coincide with reported values from the Bello municipality and WHO; the values of
$t 0 c 1 = 36$
$t 0 c 4 = 243$
Figure 1
c correspond to the 29th epidemiological week in 2010 and the 32nd epidemiological week in 2014, respectively.
Besides the parameters mentioned above, cases B and C show more similar nominal values for parameters $α$, $γ e$, $μ e$, $δ$, $μ l$, $γ p$, $μ p$ or $γ h$, whose intervals overlap among them. Note
that the value of $α$ is greater than 1 in the CSBs for all three cases. This result suggests the mortality of mosquitoes exposed to the virus ($M e$ and $M i$) to be greater than the mortality of
the healthy ones ($M s$).
3.2. Estimation of a Sub-Contour Box for Nominal Parameters
We used the nominal parameters from
Table 2
to estimate the CSB intervals for cases A, B, and C (also in
Table 2
); then, as the first validation for the CSB intervals, we performed a UA for all cases. As expected, following the CSB estimation process, all model outputs showed defined bounds with the same trend
of the nominal curve. As shown in
Figure 2
a, for case A, no curves represented endemic behavior, as in
Figure 2
b,c (cases B and C, respectively). We also performed an SA validation with the Xiao method for CSB, focusing on case C, which is presented in
Figure 3
. Note that we obtained a uniform tendency for the pie chart, which constitutes a good sign about successfully estimated CSB intervals, in which most of the parameters have the same relative
importance in the respective interval. Furthermore, as shown in
Figure 3
b, the vectorial sensitivity indices of the parameters (
$S T i$
in each time instant of the model output) follow a smooth trend, but it is interrupted in some time intervals, in which the parameters
$t 0 c j$
$A c j$
reflect the effects of the four pulses.
For the special case of C,
Figure 4
compares the CSB intervals with the confidence intervals from the median (see
Figure 4
). We normalized all data in each case concerning the maximum and minimum bounds of the respective estimation intervals given in
Table 2
. Note that some CSB intervals are wider than the estimation intervals, e.g.,
$t 0 c 1$
$t 0 c 2$
$Δ t c 1$
, and
$M s ( 0 )$
; however, this is not an unexpected result since the CSB interval does not depend on the estimation ranges, and their intervals can exceed the normalization limits
$[ 0 , 1 ]$
. Finally, it is remarkable that none of the estimations presented values closer to the inferior bound; indeed, the estimation values were concentrated toward the superior bound.
3.3. SA: Parameters That Determine the Model Behavior
After we obtained the nominal values for case C, we performed an SA using a scalar approach to quantify how parameters contribute to MSE fit (estimated output behavior about nominal output). First,
we defined the sample size for the SA by performing a convergence of
$S T i$
$S i$
Section 2.6
for Saltelli indices).
$∑ S i$
$∑ | S i |$
are almost equal between them so, with a size of
= 6000, we can obtain a reliable SA (as stated in the Methods section).
Figure 5
a,b, we summarize the ranking of the most important parameters for the model (
) according to the Saltelli sensitivity method, which quantifies the uncertainty contribution on the scalar MSE output with 1% uncertainty in each parameter (with greater values, the UA gives
non-representative outputs). For the scalar indices in
Figure 5
a, we note that almost 50% of the uncertainty in the output model is caused by the
$μ m$
parameters, which are linked to mosquito mortality. Conversely, the transmission rates (
$β h$
$β m$
) and the human recovery rate (
$γ h$
) represent almost 40% of the output variance, and this explains the importance of using mosquito repellents. In addition, note that these parameters take part in the
$R 0$
expression (see
Appendix A
), which reinforces the hypothesis that those parameters have the greater potential to change the behavior of the model output, i.e., those are potential targets for control strategies.
The remaining parameters contribute less than 10%, which shows that model behavior is determined primarily by five of the 37 parameters. This finding does not suggest that those less influential
parameters are irrelevant to the model; instead, it suggests that we must focus on the highly influential parameters that significantly change the behavior of the nominal curve. For the vectorial
indices in
Figure 5
b, we can see that the importance ranking presented by the scalar indices is almost the same where the parameters related to the pulse-type inputs as
$t 0 c j$
appeared in specific times, affecting the importance of some parameters during the
$Δ t c j$
3.4. Simulation of Control Strategies
In this section, we simulate three control strategies for case C using nominal parameters estimated for case C, to show some descriptive results from a mathematical model with parameters estimated
from real data. These are the considered simulation cases:
• Effect of positive and negative input amplitudes over the vector populations (aquatic and adult stages);
• Variation in pulse-type chemical control input parameters ($A c j$ and $t 0 c j$, with j = {1,4});
• Human immunization (vaccination) as a pulse-type input similar to chemical control (see
Section 2.3
In the first case, we simulated only the nominal value estimated for case C, in which the vector population suffered three major effects, as shown in
Figure 6
. These effects included one related to the initial conditions and two related to the input effects: (i) the initial conditions of the adult vector population have a pronounced decline (see initial
conditions in
Table 2
); (ii) the chemical control (positive pulse amplitude) reduces all vector populations and, when the residual effect is over, the population gradually returns to their carrying capacity value; (iii)
decreasing the mosquito mortality rate (negative pulse amplitude) contributes to increasing the vector population, while the input residual effect remains; and, like the chemical control effect, the
population recovers its original value when the input effect is over. Note that the pulse-type control inputs have a large effect on the adult population that spreads to the egg, larvae, and pupae
populations with a lower influence.
In the second case, we fixed all parameters in the nominal values for case C. Then, we performed a Monte Carlo simulation by changing the pulse-type input amplitudes
$A c 1$
$A c 4$
Figure 7
a,b show the behavior of the model by increasing and decreasing the pulse amplitudes between 20% and 60%; note that high amplitude values cause significant mosquito mortality, i.e., they reduce
dengue cases, up to a value where they saturate, as expected, whereas low amplitude values are not very effective. In a similar manner, by changing
$t 0 c 1$
$t 0 c 4$
, in another Monte Carlo simulation, we can see that the number of infected people decreases when chemical control starts earlier; otherwise, the outbreak increases (see
Figure 7
Finally, for case C, we evaluated the vaccination action over the number of
$H s$
, so we eliminated the chemical control inputs and added vaccination control while keeping the mosquito mortality constant. The vaccination is represented by the parameter
$u v$
Section 2.3
), which is implemented as a pulse-type input that moves a certain proportion of people from
$H s$
$H i$
during a period (see
Figure 8
). We simulated two scenarios with immunization rates between 10–20% and 20–30% per week and per 10 weeks in the same 37th week of chemical control, as shown in
Figure 8
; we fixed all parameters that were not related to vaccination. Note that the first outbreak is larger in this case than when using chemical control, but the second outbreak is reduced or eliminated.
4. Discussion
In the present study, we adapted the mathematical model (
) and estimated its parameters considering scenarios with and without pulse-type inputs. We identified diverse dynamics among the three studied cases: we realized that case A (no pulse-type input)
provides some estimated parameters anchored to estimation limit ranges (see
Table 2
), e.g.,
$θ m$
$θ h$
, and
$μ e$
. Those estimated values suggest that the estimation algorithm still might find a better minimum outside the estimation intervals, i.e., in a region with a lesser biological sense; moreover, the
model does not provide endemic behavior, as shown in
Figure 1
a. Conversely, both cases B and C did not present anchored parameters, as presented in case A (see
Table 2
), and they did display the endemic response (see
Figure 1
The model behavior in the three scenarios makes sense from a biological perspective because Bello authorities performed fumigation to mitigate the spread of dengue. Thus, if we did not consider a
pulse input in the model, the fumigation effect would be included as an implicit component of the estimated value in $μ m$ and other parameters. We highlight that, in case A ($u m = 0$), we could not
reach an endemic behavior, regardless of the number of estimations we performed. We hypothesize that the lack in endemic behavior is caused by the real data behavior that the model itself could not
reproduce without pulse inputs, where dengue cases increased and decreased rapidly during January 2010–July 2011 outbreak because of the anthropogenic or environmental disturbances. For the case of B
($u m = u m 1$), we incorporated the effect of the exogenous action, allowing the model to reproduce an endemic scenario after the first outbreak.
For the case of C, the model had several available pulses (
$u m = u m 1 + u m 2 + u m 3 + u m 4$
). This means that the model can fit the rapid changes in real data caused by anthropogenic and environmental disturbances such as fumigation or climatic variations. Hence, we hypothesize the pulses
act as excited signals and theoretically lead to a better estimation [
] that could reproduce trends in epidemiological data overcoming limitations of the non-pulse scenario (case A). Even more, future research could study the changes in the model equilibrium caused by
the presence and absence of pulse inputs, which is a theoretical point of view to identify mathematical characteristics that we did not address in this applied study.
After we obtained nominal values for cases A, B, and C, we estimated CSB (see
Table 2
, columns 5, 7, and 9) as described in
Section 2.5
Figure 4
shows the CSB and their relation with the biological and initial estimation intervals, which allows us to identify that some parameters are outside of biological enclosures (the respective CSB are
also outside of biological enclosures). We performed both UAs and SAs to assess the relevance of the contours we obtained from the CSB; we can see graphically these analyses in
Figure 2
. In the first example, for an uncertainty level of 30%, the UA shows that the CSB intervals define a band that includes the nominal curve and real data for each case without non-representative
curves. This case implies that CSB intervals are suitable for describing the uncertainty related to several processes, such as misreporting in real data and numerical errors during parameter and
interval estimations, among others. Moreover, the bands we obtained for case A, following the same behavior of the nominal curve, never showed the endemic behavior. Conversely, the UA of cases B and
C provides both the outbreak and endemic behavior, which shows an effect of pulse-type inputs on both parameter estimation and interval estimation.
As the next step for assessing the CSB method, we performed an SA focusing on case C to quantify how much each parameter contributes to the uncertainty in the model by varying parameter values over
CSB intervals. The results are presented in
Figure 3
a,b, which correspond to the scalar and vectorial SA, respectively. The vectorial analysis is time-dependent, so the
$S T i$
indices are calculated over the set of model output for each time step in the simulation; thus, we can determine the relative contribution of the parameters at each time. From vectorial
$S T i$
, we noticed that the CSB intervals define a region where most of the parameters play an important role, at least from intervals where one could expect this behavior, i.e., the initial conditions are
relevant at the beginning of the simulation but irrelevant when the model tends to equilibrium.
Note that mathematical models are not perfect representations of a phenomenon; thus, all parameters are unlikely to achieve high relevance in SA using the CSB approach. However, we should not remove
from the model all non-sensitive parameters. Instead, we could fix them at any value for a future work in which we want to identify those dynamics. Using the vectorial SA, we can identify that the
pulse-related parameters (
$u m j$
$t 0 c j$
$Δ t c j$
) are relevant for the time intervals where they are defined, disturbing the ranking of relevance and taking the model out of a stationary tendency (see
Figure 2
). Additionally, we found that parameters related to aquatic states are relevant only at the start of the outbreak because they do not contribute to the stationary dynamics of the model. We
hypothesize that early dynamics of the aquatic states cause a pulse effect over the mosquito population at the beginning of the outbreak. The early pulse effect could generate an increase in the
vector population and, therefore, an increase in infections that led to the first outbreak. Thus, for future analyses, we suggest performing estimations with an initial pulse-type input.
The scalar
$S T i$
indices that are summarized in
Figure 3
a were obtained using the Xiao method, whose basis relies on distance components decomposition, i.e., it is a method that allows us to synthesize all the information from the vectorial SA into a
single scalar SA index. Hence,
Figure 3
b constitutes a rapid insight into the relevance of each parameter throughout the entire simulation. The goal of the CSB approach is to achieve intervals such that all the parameters have nearly the
same contribution to the model output; however, this was not possible in our study. Thus, readdressing the discussion in the previous paragraph, we attribute the non-uniformity of the SA for CSB to
the following causes: (i) the model is an approximation of reality and does not contemplate all spread components and variables; (ii) the CSB is an approximation and only considers a sub-contour box
and does not consider the confidence contour itself; and (iii) even with a significant amount of computational time, there are numerical and computational limitations to approximate the CSB.
After performing the SA to validate the CSB intervals, we implemented another SA approach, based on the modified Saltelli method for MSE outputs, to assess parameter influence. This approach allowed
us to identify the parameters with the highest potential for changing the nominal output behavior when all the parameters have the same uncertainty level. We chose a unique uncertainty level such
that it also defined a band that enclosed the real data. Thus, we determined parameters related to mosquito mortality (
$μ m$
), infection interactions (
$β h$
$β m$
), and human recovery (
$γ h$
) has the greatest influence in model outputs.We followed these non-classical methods to identify the most important parameters in the model dynamic (UA and SA), which represent, to some extent, a
practical identifiability analysis, as proposed by Lizarralde-Bejarano et al. [
]. Therefore, alongside a classical analysis, as presented in the
Appendix A
, we can reinforce the importance of mortality, recovery, and transition rates in the dengue model behavior. Furthermore,
$t 0 c i$
parameters had a considerable effect on the model output in specific time intervals. It is remarkable that the estimated amplitude for the third pulse input (
$A m 3$
) always led to a negative total mortality rate for the model (
$A m 3 + μ m$
), then it did act as a positive external flow of susceptible, exposed and infected mosquitoes. Regarding such behavior, we hypothesize the rapid increase of dengue cases at the start of the last
outbreak to be also related with migration of exposed and infected mosquitoes from neighboring areas, which makes sense, in this case, since Bello is part of a cluster of highly conurbated
municipalities. These are significant results since they allow us to propose effective control strategies and suggest that using pulse-type inputs, which change mosquito mortality, could be an
appropriate strategy to model extrinsic perturbations.
After performing the UA and SA to validate CSB and estimated nominal parameters, we implemented the pulse-type inputs into the model to determine more information about dengue spread dynamics in
three simulation scenarios. In
Figure 6
, we identify a direct effect of the extrinsic conditions on the mosquito population. Indeed, the pulse-type inputs generate knock-down effects on the vector population, where the negative and
positive pulses act as outbreak starters or finishers because of the increase or decrease in the mosquito populations, respectively.
Figure 7
a,b show that the amplitude of chemical control input is a determining factor that reduces the outbreak, but the high intensity of insecticide spraying does not produce any effect after a certain
Figure 7
c,d show that late chemical spraying increases the dengue incidence. The simulations provide information about the consequences of performing control in the wrong moment.
Figure 7
c shows that, if the Health Surveillance Office had applied the chemical control before the first outbreak that occurred in Bello (following the endemic channel criteria proposed by
Lizarralde-Bejarano et al. [
]), the dengue outbreak would not have been as serious. We show the same result in
Figure 7
d for the second outbreak.
Figure 8
, we compared the effect between chemical and vaccination control strategies modeled as pulse inputs, where a vaccination campaign is performed as a long-term strategy in a specific time interval
instead of considering it as a constant value over time. As shown in literature, the vaccination is a better option than chemical control in long-term implementation [
] because the disease incidence could disappear from the population if we inoculate enough individuals.
Finally, we implemented pulse-type inputs to estimate and simulate extrinsic conditions, e.g., a decrease in mortality rates because of chemical control. This method could simulate other control
strategies such as vaccination, cleaning up breeding sites, Wolbachia introduction into the mosquito population, or biological control. The definition and implementation of these inputs could vary
according to the environmental dynamics or by the researcher’s criteria, e.g., instead of a pulse input, we could use a ramp, stairs, or sine curves, among others. Thus, future research could
introduce and change these input signals to study the effect of the disease dynamics and the associated uncertainty.
5. Conclusions
Dengue spread is related to extrinsic phenomena, including control strategies and environmental conditions. From this premise, we implemented a dengue spread model that considers control strategies
(chemical control) and extrinsic phenomena (climatic changes) as pulse-type inputs. This model definition led us to estimate parameter values and CSB. In this process, we determined that the most
important parameters in the model are those related to mosquito mortality ($μ m$ and $α$), which validates the importance of pulse-type inputs involved in mosquito mortality and vector control
strategies. Furthermore, we recommend including pulse-type inputs to estimate parameters and simulate consecutive outbreaks because, with this approach, it is possible to separate anthropogenic or
environmental disturbances (chemical control or climatic changes) in specific time windows.
Finally, we identified interesting research questions for a future research, e.g., more research is necessary to determine (i) how to assess the control actions (chemical, mechanical, or biological)
in specific time intervals from estimated parameters and their CSB for different geographical regions, (ii) how estimated negative pulse-type control inputs are correlated with environmental
conditions, (iii) how to adapt the model to other mosquito-borne diseases (chikungunya or zika), (iv) how to test the impact of the introduction of mosquitoes with Wolbachia bacteria in specific
moments on model parameters, or (v) how to evaluate the effect of lower-order models in parameter estimation.
Author Contributions
Conceptualization, A.C.-L., D.R.-D. and C.M.V.; methodology, A.C.-L., D.R.-D. and C.M.V.; software, A.C.-L. and D.R.-D.; validation, A.C.-L. and D.R.-D.; formal analysis, A.C.-L., D.R.-D. and C.M.V.;
investigation, A.C.-L., D.R.-D. and C.M.V.; resources, C.M.V.; data curation, A.C.-L.; writing—original draft preparation, A.C.-L., D.R.-D. and C.M.V.; writing—review and editing, A.C.-L., D.R.-D.
and C.M.V.; visualization, A.C.-L.; supervision, C.M.V.; project administration, C.M.V.; funding acquisition, C.M.V. All authors have read and agreed to the published version of the manuscript.
Research of A.C.-L., D.R.-D., and C.M.V. was supported by COLCIENCIAS (grant number 111572553478, from the Colombian Administrative Department of Science, Technology and Innovation) and Universidad
EAFIT (grant number 954-000002).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
We thank Diana Paola Lizarralde-Bejarano for sharing her advice during the writing of the document.
Conflicts of Interest
The authors declare no conflict of interest. The founders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the
decision to publish the results.
Appendix A. Reproductive Number R 0 and Equilibrium
In this section, we focused on the
$R 0$
estimation using the next generation matrix as described in Lizarralde-Bejarano et al. [
], Khan and Fatmawati [
]. For the model (
), we assumed the control inputs as
$u v = 0 , u e , u l , u p , u m = 0$
because those are modeled as temporal disturbances according to the designed signal (see Equation
). Then, the disease-free equilibrium is
${ E * , L * , P * } = C ( 1 − R m ) , γ e γ L + μ L E * , γ L γ p + μ p L * { M s * , M e * , M i * } = γ p μ m P * , 0 , 0 { H s * , H e * , H i * , H r * } = H , 0 , 0 , 0$
$R m = μ m δ f γ e γ p ( γ e + μ e ) ( γ l + μ l ) 1 + μ l γ l$
We identified the infected subsystem, i.e.,
$· M e , · M i , · H e$
$· H i$
. Then, we linearized the infected subsystem and evaluated its Jacobian matrix with the infection-free steady state to identify the transmission (
) and transition (
) matrix
$F = 0 0 0 β m M s * H 0 0 0 0 0 β h H s * M 0 0 0 0 0 0 V = − ( θ m + α μ m ) 0 0 0 θ m − α μ m 0 0 0 0 − ( μ h + θ h ) 0 0 0 θ h − ( γ h + μ h )$
We computed
$R 0 = ρ ( − F V − 1 )$
, where
is the spectral radius of the matrix. Note that the parameters presented in Equation (
) correspond to the ones presented in sensitivity analysis in
Figure 5
$R 0 = β h β m θ h θ m α μ m γ h + μ h μ h + θ h θ m + α μ m$
Finally, note that if we linearize the infected subsystem and estimate the sign of the eigenvalues as described in Khan and Fatmawati [
], Gbadamosi et al. [
], we can see that the eigenvalues for mosquitoes (
$− ( θ m + α μ m ) , − α μ m$
) and for humans (
$− ( μ h + θ h ) , − ( γ h + μ h )$
) are always less than zero. Even so, for future research, it could be important to identify the effects of the pulse input on the model (
) equilibrium. When adding the pulse input as a sum on the infected systems, the eigenvalues could not always be negative because the
$u m ∈ R$
. Thus, there are negative values in which the parameter could generate bifurcations in the system.
Figure 1.
Estimated outputs (
$H i$
) using the estimated nominal values in
Table 2
for cases A, B, and C. The pulse-type input has the same units as the mortality rate (mosquitoes per week). (
) Estimated output for one dengue outbreak without pulse-type input; (
) estimated output for one dengue outbreak with one pulse-type input; (
) estimated output for two dengue outbreaks with four pulse-type inputs.
Figure 2.
Uncertainty analysis for the dengue model in cases A, B, and C using CSB intervals from
Table 2
with 1000 simulations. The CSB method guarantees that at least 95% of curves generated from random combinations of parameters produce outputs classified as ‘right’ outputs. (
) Uncertainty analysis for dengue model with non pulses input; (
) uncertainty analysis for dengue model with one pulse input; (
) uncertainty analysis for a dengue model with four pulse inputs.
Figure 3.
Sensitivity analysis for the CSB intervals. (
) The scalar (MSE) SA results for case C and graphical CSB validation using the Xiao method and CSB intervals; (
) results of the vectorial SA plotted by time for case C and graphical CSB validation using Xiao’s method with
= 6000 [
Figure 4.
Boxplot for all parameters estimated with the four pulse-type inputs model (Case C). The boxes and whiskers are filtered estimations that define the medians (nominal value), and the red stars
represent the biological intervals identified in the literature. All values are normalized according to the minimum and maximum values of the estimation intervals proposed in
Table 2
Figure 5. Sensitivity analysis for the dengue model with four pulse-type inputs for case C (with 1% uncertainty in each nominal parameter), using (a) scalar and (b) vectorial $S T i$ indices from the
Saltelli method.
Figure 6. Simulation of case C focusing on the vector population. We used nominal values to observe how the mosquito and aquatic phases are affected by pulse-type inputs. The simulation began in week
five because of the quick and significant decrease of the state variables in the vector population. The amplitude of the pulses in the plots is amplified by a factor of 10 for better visualization.
Figure 7. Monte Carlo simulations of the dengue transmission model for the number of infected humans in two scenarios: (a) increasing the value of $A c 1$ between 30–60% and 60–90% of its nominal
value; (b) increasing and decreasing the value of $A C 4$ between 20% and 60% of its nominal value; (c,d) late and early chemical control, increasing or decreasing $t 0 c 1$ and $t 0 c 4$ between
20–60% and 10–20% of their nominal values, respectively. For each scenario, we performed 1000 simulations.
Figure 8. Dengue model simulations using estimated nominal values and immunization control by vaccination. We removed chemical control pulses for immunization simulations and left those that
represented the mosquito population increase. (a) immunization of the 10 to 20% of total human population; (b) immunization of the 20 to 30% of total human population.
Table 1.
Definition of state variables, parameters, and input variables for the mathematical model (
). The input that describes mosquitoes growth (
$u m$
) is defined by three parameters (
$A c j$
$t 0 c j$
, and
$Δ t c j$
) in Equation (
); in addition,
$u m$
could be formed by multiple
pulse-type inputs as described in
Section 2.4
. Note that, for all factors, the time unit is one week, and the rate unit is [week
Factors Description Factors Description
E Number of eggs $δ$ Oviposition rate
L Number of larvae C Egg carrying capacity
P Number of pupae $γ e$ Egg to larva transition rate
$M s$ Number of susceptible mosquitoes $μ e$ Egg mortality rate
$M e$ Number of exposed mosquitoes $γ l$ Larva to pupa transition rate
$M i$ Number of infected mosquitoes $μ l$ Larvae mortality rate
$H s$ Number of susceptible humans $γ p$ Pupae to mosquito transition rate
$H e$ Number of exposed humans $μ p$ Pupae mortality rate
$H i$ Number of infected humans f Fraction of females that emerges
$H r$ Number of recovered humans $β m$ Transmission coefficient human-mosquito
M Total number of mosquitoes $μ m$ Mosquito mortality rate
H Total number of humans $α$ Change in $μ m$ due to virus infection
$u e$ Egg control input rate $θ m$ Extrinsic incubation rate
$u l$ Larvae control input rate $μ h$ Human mortality rate
$u p$ Pupae control input rate $β h$ Transmission coefficient mosquito-human
$u m$ Mosquito control input rate $θ h$ Intrinsic incubation rate
$u v$ Vaccine control input rate $γ h$ Human recovery rate
$A c j$ Mosquito control pulse amplitude
$t 0 c j$ Mosquito control initial time
$Δ t c j$ Mosquito control pulse width
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Catano-Lopez A, Rojas-Diaz D, Vélez CM. The Influence of Anthropogenic and Environmental Disturbances on Parameter Estimation of a Dengue Transmission Model. Tropical Medicine and Infectious Disease.
2023; 8(1):5. https://doi.org/10.3390/tropicalmed8010005
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Catano-Lopez, Alexandra, Daniel Rojas-Diaz, and Carlos M. Vélez. 2023. "The Influence of Anthropogenic and Environmental Disturbances on Parameter Estimation of a Dengue Transmission Model" Tropical
Medicine and Infectious Disease 8, no. 1: 5. https://doi.org/10.3390/tropicalmed8010005
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The role of pressure in an incompressible Euler singularity
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TUR - Mathematical aspects of turbulence: where do we stand?
The blowup mechanism for an inviscid wall-bounded flow is investigated from a primitive-variables point of view. The analysis focuses on the interplay between inertia and pressure, rather than on
vorticity. The incompressible Euler equations are numerically simulated in a cylindrical container. The flow is axisymmetric with the swirl. The simulations reproduce and corroborate aspects of prior
studies by Luo and Hou reporting strong evidence for a finite-time singularity. The linearity of the pressure Poisson equation is exploited to decompose the pressure field into independent
contributions. A model is presented based on a primitive-variables formulation of the Euler equations, with closure coming from how pressure is determined from velocity. The model captures key
features in the mechanics of the blowup scenario.
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Machine Learning Model Functio...
Other Models
supports other machine learning models for advanced analysis that include association rules and feedforward neural network.
Model Type: ASSOCIATION RULES
The association rules model trains itself over rows of arrays to suggest other associated values. The model returns values commonly associated in a set that are absent in the provided value set. In
practical use, this model is often used to help in retail transactions by suggesting related items or services for purchase, based on similar past transactions. When you create the model, there
should be a single input column, which is an array of any inner type.
When you execute the model, you must provide an array with the same inner type as the input column. The array represents the current state of the transaction.
You must also provide a second argument that is a positive integer that represents the association ranking of the item to return. The value of 1 indicates to add the first most-associated item to the
The third and fourth arguments are optional constant Boolean values.
If the third argument is set to true, the model adds the most-associated item to the transaction based on other transactions that have the same items in the current transaction. Otherwise, only some
of the items must be the same. The third argument defaults to true.
If the fourth argument is set to true, the model counts duplicate items in a transaction only once. Otherwise, the model counts duplicate items multiple times. To specify the fourth argument, you
must specify the third argument. The default value for the fourth argument is true, which means that duplicate items are counted once.
Model Type: FEEDFORWARD NETWORK
The feedforward network model is a neural network model where data moves from the inputs through hidden layers to the outputs. Feedforward neural networks are fully connected (sometimes named
multilayer perceptrons). The number of inputs is determined by the number of columns in the input result set. Each input must be numeric. The last column in the input result set is the target
variable. For models with one output, the column is also a numeric. For models with multiple outputs, the result must be a 1xN matrix (a row vector).
Common uses of multiple output models are:
• Multi-class classification - Multiple outputs are one-hot encoded values that represent the class of the record. The model uses results with argmax to select the highest probability class.
• Probability modeling - Multiple output values represent probabilities between 0 and 1 that sum to 1.
• Multiple numeric prediction - Multiple output values represent different numeric values to predict against.
For faster, lower quality models, reduce the popSize, initialIterations, and subsequentIterations options. Conversely, for slower, higher quality models, increase the values for these same options.
To create a neural network to perform multi-class classification for three possible classes, use the following SQL statement. y1, y2, y3 are one-hot encoded outputs. If the values are 1 and the rest
are 0, the value 1 denotes the class that the training data belongs to from the three classes.
When you execute the model later, pass N - 1 input variables and the model returns the estimate of the target variable. In the case of multiple outputs, the result is a 1xN matrix (a row vector). If
the model uses multiple outputs to perform multi-class classification, use argmax to get the integer that represents the class.
hiddenLayers - You must set this option to a positive integer that specifies how many hidden layers to use.
hiddenLayerSize - You must set this option to a positive integer that specifies the number of nodes in each hidden layer.
outputs - You must set this option to a positive integer that specifies the number of outputs.
lossFunction - This option specifies the loss function that all hidden layer nodes and all output layer nodes use. This function can be one of several predefined loss functions, or a user-defined
loss function.
The predefined loss functions are squared_error (regression), vector_squared_error (vector-valued regression), log_loss (binary classification with target values of 0 and 1), logits_loss (binary
classification with target values of 0 and 1), hinge_loss (binary classification with target values of -1 and 1), and cross_entropy_loss (multi-class classification). If the value for this required
parameter is none of these strings, the model assumes a user-defined loss function. The user-defined loss function specifies the per-sample loss. Then, the actual loss function is the sum of this
function applied to all samples. The model should use the variable y to refer to the dependent variable in the training data, and the model should use the variable f to refer to the computed estimate
for a given sample.
activationFunction - If you set this option, the values are linear, relu (rectified linear unit), leakyrelu (leaky rectified linear unit), tanh (hyperbolic tangent function), or sigmoid (fast sigmoid
approximation). This option defaults to relu. This option affects all layers except the output layer.
outputActivationFunction - If you set this option, the values are linear, relu (rectified linear unit), leakyrelu that stands for a leaky rectified linear unit, tanh (hyperbolic tangent function), or
sigmoid (fast sigmoid approximation). Different activation functions have different output ranges. The chosen activation function should match the dependent variable of your data. For example, if the
dependent variable can be anything, then choose the linear value. If the dependent variable is always positive, then choose the relu value. If your outputs range from -1 to 1 or you perform hinge
loss classification, tanh is a good option because the hyperbolic tangent function has the same range. But, if your outputs range from 0 to 1 or you perform log loss classification, sigmoid is a
better choice for the same reason. This option defaults to linear. The option only sets the activation function for the output layer.
metrics - If you set this option to true, the model calculates the average value of the loss function.
useSoftMax - If you set this option to true, the model applies a softmax function to the output of the output layer, and before computing the loss function. This option defaults to true if the
lossFunction is set to cross_entropy_loss and false otherwise.
popSize - If you set this option, the value must be a positive integer. Sets the population size for the particle swarm optimization (PSO) part of the algorithm. This option defaults to 100.
minInitParamValue - If you set this option, the value must be a floating point number. Sets the minimum for initial parameter values in the optimization algorithm. This option defaults to -1.
maxInitParamValue - If you set this option, the value must be a floating point number. Sets the maximum for initial parameter values in the optimization algorithm. This option defaults to 1.
initialIterations - If you set this option, the value must be a positive integer. Sets the number of PSO iterations for the first PSO pass. This option defaults to 50.
subsequentIterations - If you set this option, the value must be a positive integer. Sets the number of PSO iterations for subsequent PSO iterations after the initial pass. This option defaults to
momentum - If you set this option, the value must be a positive floating point number. This parameter controls how much PSO iterations move away from the local best value to explore new territory.
This option defaults to 0.1.
gravity - If you set this option, the value must be a positive floating point number. This parameter controls how much PSO iterations are drawn back towards the local best value. This option defaults
to 0.01.
lossFuncNumSamples - If you set this option, the value must be a positive integer. This parameter controls how many points the model samples when estimating the loss function. This option defaults to
numGAAttempts - If you set this option, the value must be a positive integer. This parameter controls how many GA crossover possibilities the model tries. This option defaults to 10,000.
maxLineSearchIterations - If you set this option, the value must be a positive integer. This parameter controls the maximum allowed number of iterations when running the line search part of the
algorithm. This option defaults to 20.
minLineSearchStepSize - If you set this option, the value must be a positive floating point number. This parameter controls the minimum step size of the line search algorithm. This option defaults to
samplesPerThread - If you set this option, the value must be a positive integer number. This parameter controls the target number of samples that the model sends to each thread. Each thread
independently computes a logistic regression model, and the models are all combined at the end. This option defaults to 1 million. | {"url":"https://docs.ocient.com/v23/other-models","timestamp":"2024-11-13T09:05:16Z","content_type":"text/html","content_length":"348179","record_id":"<urn:uuid:aa0a26e1-b3b5-4a88-b208-60a238b013dc>","cc-path":"CC-MAIN-2024-46/segments/1730477028342.51/warc/CC-MAIN-20241113071746-20241113101746-00350.warc.gz"} |
The n-Category Café
February 27, 2012
Prequantization in Cohesive Homotopy Type Theory
Posted by Urs Schreiber
Chris Rogers and myself are studying structures in higher (meaning: ∞-categorified) geometric (pre)quantization. Later this year I may post something about the higher structures that appear in this
context, but right now I will highlight something else.
Our constructions proceed in two steps. First we give a general abstract axiomatic definition of geometric (pre)quantization internal to any cohesive ∞-topos. In the next step we pass to suitable
models of the axioms and work out how there this reproduces the traditional notions as well as various generalizations of these that have been proposed, and usefully generalizes all this in various
In other words, we formulate geometric prequantization in cohesive homotopy type theory and then study its models. Since here in the $n$Café we had, in the last months, discussed most of the relevant
ingredients before, I thought it would be fun to highlight just this step.
See Prequantum physics in a cohesive ∞-topos for more exposition.
Technical details on what I say below are in section 2.3.24 (general definitions) and section 3.3.17 (models in smooth cohesion).
Posted at 6:43 PM UTC |
Followups (14)
February 26, 2012
Types, Homotopy and Univalent Foundations: Special Issue
Posted by Tom Leinster
Been thinking about the interaction between type theory and homotopy theory? Got a paper you want to write? Already written your masterpiece, but looking for somewhere to submit it?
Here’s news from Nicola Gambino of a special issue of Mathematical Structures in Computer Science devoted to such things. Details below the fold.
(And in case you’re wondering, that’s not an Elsevier journal: it’s published by those nice people at Cambridge University Press.)
Posted at 7:05 PM UTC |
Post a Comment
February 22, 2012
ESI Program K-Theory and Quantum Fields
Posted by Urs Schreiber
This summer in Vienna at ESI takes place a program titled
K-Theory and Quantum Fields
May 21 - July 27, 2012 (webpage)
organized by Matt Ando, Alan Carey, Harald Grosse and Jouko Mickelsson.
It starts out with
• introductory instructional lectures in the week May 28-June 1;
• advanced instructional lectures in the week June 4- June 8.
Details should appear by beginning of March on the webpage. As far as I am aware, the following speakers and topics are expected for the instructional lectures.
• John Francis: factorization algebras/homology;
• Dan Freed (probably on June 7th and 8th, topic to be announced);
• Mathai Varghese: T-duality and K-theory;
• Frederic Rochon: introduction to K-theory;
• myself: twisted differential structures in string theory;
• Danny Stevenson: twisted K-theory, higher structures;
• Bai-Ling Wang: K-theory, K-homology and twisted geometric cycles.
I’ll post an alert in the comments below as soon as there is more information available.
Posted at 8:44 PM UTC |
Followups (5)
February 21, 2012
In Göttingen
Posted by Urs Schreiber
Currently the higher order structures group in Göttingen is running their reading seminar on our article L-infinity connections with Hisham Sati and Jim Stasheff. This week I am visiting Göttingen,
by kind invitation by Chenchang Zhu, in order to answer questions and to speak about related structures.
Today we had a long session working our way towards the basics of synthetic differential infinity-groupoids, a context where $L_\infty$-algebras exist alongside genuine smooth $\infty$-groupoids as
those who have first order infinitesimal spaces of $k$-morphisms for all $k$.
Tomorrow I’ll try to explain how using this we construct, with Domenico Fiorenza, the smooth moduli 2-stack of string 2-connections and the smooth moduli 6-stack of fivebrane 6-connections by
exponentiating those big double square diagrams of $L_\infty$-algebras discussed in the article.
Today I learned from the participants that interest in the nonabelian 2-form on the 5-brane has been a motivation to pick this reading seminar topic, so I will close with a remark on the basic idea
in our recent article on that application of the machinery.
Meanwhile, I am glad to have had a chance to chat with Chris Rogers about our project of studying the higher Atiyah/Courant algebroids of circle $n$-bundles for all $n$.
So it’s busy enough. Just thought I’d drop a note on what I am up to this week.
Posted at 8:50 PM UTC |
Post a Comment
February 16, 2012
Math 2.0
Posted by John Baez
Building on the Elsevier boycott, a lot of people are working on positive steps to make expensive journals obsolete. My email is flooded with discussions, different groups making different plans.
Email is great, but not for everything. So Andrew Stacey (the technical mastermind behind the nLab, Azimuth Wiki and Azimuth Forum) and Scott Morrison (one of the brains behind MathOverflow) have
started a forum to talk about the many issues involved:
• Math 2.0.
Check it out.
Posted at 3:43 AM UTC |
Followups (1)
February 15, 2012
Workshop on Formal Topology, Higher Dimensional Algebra, Categories and Types
Posted by Urs Schreiber
This June takes place the Fourth Workshop on Formal Topology (and related topics, including constructive and computable topology, point-free topology, and other forms of non-classical topology) in
Ljubljana, Slovenia.
Keynote speakers are Per Martin-Löf, Ieke Moerdijk and Vladimir Voevodsky.
See the workshop webpage for details.
Adjoined to that, on the last day, is a Workshop on Higher Dimensional Algebra, Categories and Types, which I am helping to organize a bit.
This should consist of three talks, roughly one on type theory, one on type theory and homotopy theory and one on higher category theory.
Two confirmed of three speakers are Thorsten Altenkirch and Steve Awodey.
For more details on this see the HDACT Workshop Webpage.
Posted at 5:19 PM UTC |
Followups (1)
February 14, 2012
Modern Perspectives in Homotopy Theory
Posted by Urs Schreiber
This April (10th - 13th) takes place a school titled
Modern Perspectives in Homotopy Theory
∞-Categories, ∞-Operads and Homotopy Type Theory
in Swansea, UK.
See the school’s website.
David Gepner will speak about $\infty$-categories, Ieke Moerdijk about $\infty$-operads (and dendroidal sets), and Mike Shulman about $\infty$-type theory.
Posted at 9:07 PM UTC |
Post a Comment
February 9, 2012
The Moduli 3-Stack of the C-Field
Posted by Urs Schreiber
The Cost of Knowledge
Posted by John Baez
As of this moment, 4760 scholars have joined a boycott of the publishing company Elsevier. Of these, only 20% are mathematicians. But since the boycott was started by a mathematician, 34 of us wrote
and signed statement explaining the boycott. Here it is.
Posted at 1:51 AM UTC |
Followups (1)
February 8, 2012
Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory
Posted by Alexander Hoffnung
This past October Alistair Savage and I organized a workshop on Category Theoretic Methods in Representation Theory at the University of Ottawa. The event was generously supported by the Fields
Following the success of the October workshop, Oded Yacobi, Chris Dodd, and I decided to hold another workshop, this time with a focus on researchers who are still very early on in their careers. The
Fields Institute has again offered funding and this time will host the event as well.
We would like to draw your attention to the upcoming Young Researchers Workshop on Higher Algebraic and Geometric Structures: Modern Methods in Representation Theory to be held May 7-9, 2012 at the
Fields Institute in Toronto.
Keep reading below the fold and see the workshop website for more on the content of the workshop, registration information, and applications for financial support.
Posted at 7:28 PM UTC |
Followups (9)
February 7, 2012
Good News
Posted by John Baez
I’m in Sydney talking to Ross Street and other category theorists at Macquarie University. Once I’d expressed worries about the fate of Australian category theory now that Street has retired. I’m
happy to say that we can put those worries to rest.
First of all, while Street has formally retired, he still comes to work every day and is very active. Second, Steve Lack has moved from Sydney University to Macquarie, so he can now have graduate
students. On top of that, he’s gotten a 4-year grant that’ll let him spend most of his time on research! Richard Garner is here too, and has a 5-year research-only postdoc. Also in the math
department are Michael Batanin, Alexei Davydov and Mark Weber. And if that weren’t enough, Dominic Verity has joined Mike Johnson over in Macquarie’s computing department.
So, it’s a great place to go if you want to talk to category theorists! I apologize to anyone whom I left out.
And on a more personal note…
Posted at 12:03 AM UTC |
Followups (21) | {"url":"https://golem.ph.utexas.edu/category/2012/02/index.shtml","timestamp":"2024-11-03T20:23:02Z","content_type":"application/xhtml+xml","content_length":"91542","record_id":"<urn:uuid:6018c847-d4f5-45c8-9251-fae66a15a6a7>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00099.warc.gz"} |
JuliaActuary is an ecosystem of packages that makes Julia the easiest language to get started for actuarial workflows.
These packages are available for use in your project. Scroll down for more information and links to the associated repository for each one.
MortalityTables.jl LifeContingencies.jl
Easily work with standard tables and parametric models with common survival calculations. Insurance, annuity, premium, and reserve maths.
ActuaryUtilities.jl ExperienceAnalysis.jl
Robust and fast calculations for internal_rate_of_return, duration, convexity, present_value, breakeven, and more. Meeting your exposure calculation needs.
FinanceModels.jl EconomicScenarioGenerators.jl
Composable contracts, models, and functions that allow for modeling of both simple and complex financial instruments. Easy-to-use scenario generation that's FinanceModels.jl compatible.
For consistency, you can lock any package in its current state and not worry about breaking changes to any code that you write. Juliaโ s package manager lets you exactly recreate a set of code and
its dependencies. More information.
Adding and Using Packages
There are two ways to add packages:
• In the code itself: using Pkg; Pkg.add("MortalityTables")
• In the REPL, hit ] to enter Pkg mode and type add MortalityTables More info can be found at the Pkg manager documentation.
To use packages in your code:
using PackageName
Hassle-free mortality and other rate tables.
• Full set of SOA mort.soa.org tables included
• survival and decrement functions to calculate decrements over period of time
• Partial year mortality calculations (Uniform, Constant, Balducci)
• Friendly syntax and flexible usage
• Extensive set of parametric mortality models.
Load and see information about a particular table:
julia> vbt2001 = MortalityTables.table("2001 VBT Residual Standard Select and Ultimate - Male Nonsmoker, ANB")
MortalityTable (Insured Lives Mortality):
2001 VBT Residual Standard Select and Ultimate - Male Nonsmoker, ANB
(:select, :ultimate, :metadata)
Society of Actuaries
mort.SOA.org ID:
mort.SOA.org link:
2001 Valuation Basic Table (VBT) Residual Standard Select and Ultimate Table - Male Nonsmoker.
Basis: Age Nearest Birthday.
Minimum Select Age: 0.
Maximum Select Age: 99.
Minimum Ultimate Age: 25.
Maximum Ultimate Age: 120
The package revolves around easy-to-access vectors which are indexed by attained age:
julia> vbt2001.select[35] # vector of rates for issue age 35
โ ฎ
julia> vbt2001.select[35][35] # issue age 35, attained age 35
julia> vbt2001.select[35][50:end] # issue age 35, attained age 50 through end of table
โ ฎ
julia> vbt2001.ultimate[95] # ultimate vectors only need to be called with the attained age
Calculate the force of mortality or survival over a range of time:
julia> survival(vbt2001.ultimate,30,40) # the survival between ages 30 and 40
julia> decrement(vbt2001.ultimate,30,40) # the decrement between ages 30 and 40
Non-whole periods of time are supported when you specify the assumption (Constant(), Uniform(), or Balducci()) for fractional periods:
julia> survival(vbt2001.ultimate,30,40.5,Uniform()) # the survival between ages 30 and 40.5
Parametric Models
Over 20 different models included. Example with the Gompertz model
m = MortalityTables.Gompertz(a=0.01,b=0.2)
m[20] # the mortality rate at age 20
decrement(m,20,25) # the five year cumulative mortality rate
survival(m,20,25) # the five year survival rate
A collection of common functions/manipulations used in Actuarial Calculations.
A collection of common functions/manipulations used in Actuarial Calculations.
cfs = [5, 5, 105]
times = [1, 2, 3]
discount_rate = 0.03
present_value(discount_rate, cfs, times) # 105.65
duration(Macaulay(), discount_rate, cfs, times) # 2.86
duration(discount_rate, cfs, times) # 2.78
convexity(discount_rate, cfs, times) # 10.62
Financial Maths
• duration:
□ Calculate the Macaulay, Modified, or DV01 durations for a set of cashflows
• convexity for price sensitivity
• Flexible interest rate options via the FinanceModels.jl package.
• internal_rate_of_return or irr to calculate the IRR given cashflows (including at timepoints like Excelโ s XIRR)
• breakeven to calculate the breakeven time for a set of cashflows
• accum_offset to calculate accumulations like survivorship from a mortality vector
Options Pricing
• eurocall and europut for Black-Scholes option prices
Risk Measures
• Calculate risk measures for a given vector of risks:
□ CTE for the Conditional Tail Expectation, or
□ VaR for the percentile/Value at Risk.
Insurance mechanics
• duration:
□ Calculate the duration given an issue date and date (a.k.a. policy duration)
Common life contingent calculations with a convenient interface.
• Integration with other JuliaActuary packages such as MortalityTables.jl
• Fast calculations, with some parts utilizing parallel processing power automatically
• Use functions that look more like the math you are used to (e.g. A, aฬ ) with Unicode support
• All of the power, speed, convenience, tooling, and ecosystem of Julia
• Flexible and modular modeling approach
Package Overview
• Leverages MortalityTables.jl for the mortality calculations
• Contains common insurance calculations such as:
□ Insurance(life,yield): Whole life
□ Insurance(life,yield,n): Term life for n years
□ aฬ (life,yield): present_value of life-contingent annuity
□ aฬ (life,yield,n): present_value of life-contingent annuity due for n years
• Contains various commutation functions such as D(x),M(x),C(x), etc.
• SingleLife and JointLife capable
• Interest rate mechanics via FinanceModels.jl
• More documentation available by clicking the DOCS badges at the top of this README
Basic Functions
Calculate various items for a 30-year-old male nonsmoker using 2015 VBT base table and a 5% interest rate
using LifeContingencies
using MortalityTables
using FinanceModels
import LifeContingencies: V, aฬ # pull the shortform notation into scope
# load mortality rates from MortalityTables.jl
vbt2001 = MortalityTables.table("2001 VBT Residual Standard Select and Ultimate - Male Nonsmoker, ANB")
issue_age = 30
life = SingleLife( # The life underlying the risk
mortality = vbt2001.select[issue_age], # -- Mortality rates
yield = FinanceModels.Yield.Constant(0.05)
lc = LifeContingency(life, yield) # LifeContingency joins the risk with interest
ins = Insurance(lc) # Whole Life insurance
ins = Insurance(life, yield) # alternate way to construct
With the above life contingent data, we can calculate vectors of relevant information:
cashflows(ins) # A vector of the unit cashflows
timepoints(ins) # The timepoints associated with the cashflows
survival(ins) # The survival vector
benefit(ins) # The unit benefit vector
probability(ins) # The probability of benefit payment
Some of the above will return lazy results. For example, cashflows(ins) will return a Generator which can be efficiently used in most places youโ d use a vector of cashflows (e.g. pv(...) or sum
(...)) but has the advantage of being non-allocating (less memory used, faster computations). To get a computed vector instead of the generator, simply call collect(...) on the result: collect
Or calculate summary scalars:
present_value(ins) # The actuarial present value
premium_net(lc) # Net whole life premium
V(lc,5) # Net premium reserve for whole life insurance at time 5
Other types of life contingent benefits:
Insurance(lc,10) # 10 year term insurance
AnnuityImmediate(lc) # Whole life annuity due
AnnuityDue(lc) # Whole life annuity due
aฬ (lc) # Shortform notation
aฬ (lc, 5) # 5 year annuity due
aฬ (lc, 5, certain=5,frequency=4) # 5 year annuity due, with 5 year certain payable 4x per year
... # and more!
Constructing Lives
SingleLife(vbt2001.select[50]) # no keywords, just a mortality vector
SingleLife(vbt2001.select[50],issue_age = 60) # select at 50, but now 60
SingleLife(vbt2001.select,issue_age = 50) # use issue_age to pick the right select vector
SingleLife(mortality=vbt2001.select,issue_age = 50) # mort can also be a keyword
Flexible and composable yield curves and interest functions.
FinanceModels.jl provides a set of composable contracts, models, and functions that allow for modeling of both simple and complex financial instruments. The resulting models, such as discount rates
or term structures, can then be used across the JuliaActuary ecosystem to perform actuarial and financial analysis.
Additionally, the models can be used to project contracts through time: most basically as a series of cashflows but more complex output can be defined for contracts.
Yields Curve Fitting in FinanceModels.jl
using FinanceModels
# a set of market-observed prices we wish to calibrate the model to
# annual effective unless otherwise specified
q_rate = ZCBYield([0.01,0.02,0.03]);
q_spread = ZCBYield([0.01,0.01,0.01]);
# bootstrap a linear spline yield model
model_rate = fit(Spline.Linear(),q_rate,Fit.Bootstrap());โ
model_spread = fit(Spline.Linear(),q_spread,Fit.Bootstrap());
# the zero rate is the combination of the two underlying rates
zero(m_spread + m_rate,1) # 0.02 annual effective rate
# the discount is the same as if we added the underlying zero rates
discount(m_spread + m_rate,0,3) โ discount(0.01 + 0.03,3) # true
# compute the present value of a contract (a cashflow of 10 at time 3)
present_value(m_rate,Cashflow(10,3)) # 9.15...
Overview of FinanceModels
A conceptual sketch of FinanceModels.jl
Often we start with observed or assumed values for existing contracts. We want to then use those assumed values to extend the valuation logic to new contracts. For example, we may have a set of bond
yields which we then want to discount a series of insurance obligations.
In the language of FinanceModels, we would have a set of Quotes which are used to fit a Model. That model is then used to discount a new series of cashflows.
Thatโ s just an example, and we can use the various components in different ways depending on the objective of the analysis.
Contracts and Quotes
Contracts are a way to represent financial obligations. These can be valued using a model, projected into a future steam of values, or combined with assumed prices as a Quote.
Included are a number of primitives and convenience methods for contracts:
Existing structs:
• Cashflow
• Bond.Fixed
• Bond.Floating
• Forward (an obligation with a forward start time)
• Composite (combine two other contracts, e.g. into a swap)
• EuroCall
• CommonEquity
Commonly, we deal with conventions that imply a contract and an observed price. For example, we may talk about a treasury yield of 0.03. This is a description that implies a Quoteed price for an
underling fixed bond. In FinanceModels, we could use CMTYield(rate,tenor) which would create a Quote(price,Bond.Fixed(...)). In this way, we can conveniently create a number of Quotes which can be
used to fit models. Such convenience methods include:
• ZCBYield
• ZCBPrice
• CMTYield
• ParYield
• ParSwapYield
• ForwardYield
FinanceModels offers a way to define new contracts as well.
A Cashflows obligation are themselves a contract, but other contracts can be considered as essentially anything that can be combined with assumptions (a model) to derive a collection of cashflows.
For example, a obligation that pays 1.75 at time 2 could be represented as: Cashflow(1.75,2).
Models are objects that can be fit to observed prices and then subsequently used to make valuations of other cashflows/contracts.
Yield models include:
• Yield.Constant
• Bootstrapped Splines
• Yield.SmithWilson
• Yield.NelsonSiegel
• Yield.NelsonSiegelSvensson
Most basically, we can project a contract into a series of Cashflows:
julia> b = Bond.Fixed(0.04,Periodic(2),3)
FinanceModels.Bond.Fixed{Periodic, Float64, Int64}(0.04, Periodic(2), 3)
julia> collect(b)
6-element Vector{Cashflow{Float64, Float64}}:
Cashflow{Float64, Float64}(0.02, 0.5)
Cashflow{Float64, Float64}(0.02, 1.0)
Cashflow{Float64, Float64}(0.02, 1.5)
Cashflow{Float64, Float64}(0.02, 2.0)
Cashflow{Float64, Float64}(0.02, 2.5)
Cashflow{Float64, Float64}(1.02, 3.0)
However, Projections allow one to combine three elements which can be extended to define any desired output (such as amortization schedules, financial statement projections, or account value
rollforwards). The three elements are:
• the underlying contract of interest
• the model which includes assumptions of how the contract will behave
• a ProjectionKind which indicates the kind of output desired (cashflow stream, amortization schedule, etcโ ฆ)
Fitting Models
Model Method
| |
|------------| |---------------|
fit(Spline.Cubic(), CMTYield.([0.04,0.05,0.055,0.06,0055],[1,2,3,4,5]), Fit.Bootstrap())
• Model could be Spline.Linear(), Yield.NelsonSiegelSvensson(), Equity.BlackScholesMerton(...), etc.
• Quote could be CMTYields, ParYields, Option.Eurocall, etc.
• Method could be Fit.Loss(x->x^2), Fit.Loss(x->abs(x)), Fit.Bootstrap(), etc.
This unified way to fit models offers a much simpler way to extend functionality to new models or contract types.
Using Models
After being fit, models can be used to value contracts:
Additionally, ActuaryUtilities.jl offers a number of other methods that can be used, such as duration, convexity, price which can be used for analysis with the fitted models.
Rates are types that wrap scalar values to provide information about how to determine discount and accumulation factors.
There are two Frequency types:
• Periodic(m) for rates that compound m times per period (e.g. m times per year if working with annual rates).
• Continuous() for continuously compounding rates.
Continuous(0.05) # 5% continuously compounded
Periodic(0.05,2) # 5% compounded twice per period
These are both subtypes of the parent Rate type and are instantiated as:
Rate(0.05,Continuous()) # 5% continuously compounded
Rate(0.05,Periodic(2)) # 5% compounded twice per period
Rates can also be constructed by specifying the Frequency and then passing a scalar rate:
Convert rates between different types with convert. E.g.:
r = Rate(FinanceModels.Periodic(12),0.01) # rate that compounds 12 times per rate period (ie monthly)
convert(FinanceModels.Periodic(1),r) # convert monthly rate to annual effective
convert(FinanceModels.Continuous(),r) # convert monthly rate to continuous
Adding, substracting, multiplying, dividing, and comparing rates is supported.
Meeting your exposure calculation needs.
df = DataFrame(
policy_id = 1:3,
issue_date = [Date(2020,5,10), Date(2020,4,5), Date(2019, 3, 10)],
end_date = [Date(2022, 6, 10), Date(2022, 8, 10), Date(2022,12,31)],
status = ["claim", "lapse", "inforce"]
df.policy_year = exposure.(
df.status .== "claim"; # continued exposure
study_start = Date(2020, 1, 1),
study_end = Date(2022, 12, 31)
df = flatten(df, :policy_year)
df.exposure_fraction =
map(e -> yearfrac(e.from, e.to + Day(1), DayCounts.Thirty360()), df.policy_year)
# + Day(1) above because DayCounts has Date(2020, 1, 1) to Date(2021, 1, 1) as an exposure of 1.0
# here we end the interval at Date(2020, 12, 31), so we need to add a day to get the correct exposure fraction.
policy_id issue_date end_date status policy_year exposure_fraction
Int64 Date Date String @NamedTuple{from::Date, to::Date, policy\_timestep::Int64} Float64
1 2020-05-10 2022-06-10 claim (from = Date(โ 2020-05-10โ ), to = Date(โ 2021-05-09โ ), policy_timestep = 1) 1.0
1 2020-05-10 2022-06-10 claim (from = Date(โ 2021-05-10โ ), to = Date(โ 2022-05-09โ ), policy_timestep = 2) 1.0
1 2020-05-10 2022-06-10 claim (from = Date(โ 2022-05-10โ ), to = Date(โ 2023-05-09โ ), policy_timestep = 3) 1.0
2 2020-04-05 2022-08-10 lapse (from = Date(โ 2020-04-05โ ), to = Date(โ 2021-04-04โ ), policy_timestep = 1) 1.0
2 2020-04-05 2022-08-10 lapse (from = Date(โ 2021-04-05โ ), to = Date(โ 2022-04-04โ ), policy_timestep = 2) 1.0
2 2020-04-05 2022-08-10 lapse (from = Date(โ 2022-04-05โ ), to = Date(โ 2022-08-10โ ), policy_timestep = 3) 0.35
3 2019-03-10 2022-12-31 inforce (from = Date(โ 2020-01-01โ ), to = Date(โ 2020-03-09โ ), policy_timestep = 1) 0.191667
3 2019-03-10 2022-12-31 inforce (from = Date(โ 2020-03-10โ ), to = Date(โ 2021-03-09โ ), policy_timestep = 2) 1.0
3 2019-03-10 2022-12-31 inforce (from = Date(โ 2021-03-10โ ), to = Date(โ 2022-03-09โ ), policy_timestep = 3) 1.0
3 2019-03-10 2022-12-31 inforce (from = Date(โ 2022-03-10โ ), to = Date(โ 2022-12-31โ ), policy_timestep = 4) 0.808333
Available Exposure Basis
• ExperienceAnalysis.Anniversary(period) will give exposures periods based on the first date
• ExperienceAnalysis.Calendar(period) will follow calendar periods (e.g. month or year)
• ExperienceAnalysis.AnniversaryCalendar(period,period) will split into the smaller of the calendar or policy period.
Where period is a Period Type from the Dates standard library.
Calculate exposures with exposures(basis,from,to,continue_exposure).
• continue_exposures indicates whether the exposure should be extended through the full exposure period rather than terminate at the to date.
Easy-to-use scenario generation thatโ s FinanceModels.jl compatible.
Interest Rate Models
• Vasicek
• CoxIngersolRoss
• HullWhite
Interest Rate Model Examples
m = Vasicek(0.136,0.0168,0.0119,Continuous(0.01)) # a, b, ฯ , initial Rate
s = ScenarioGenerator(
1, # timestep
30, # projection horizon
m, # model
This can be iterated over, or you can collect all of the rates like:
rates = collect(s)
for r in s
# do something with r
And the package integrates with FinanceModels.jl:
will produce a yield curve object:
โ โ โ โ โ โ โ โ โ โ โ โ โ โ Yield Curve (FinanceModels.BootstrapCurve)โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
0.03 โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃ โ คโ คโ โ โ โ โ โ โ โ โ โ คโฃ โฃ โ Zero rates
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃ โ คโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โข โฃ โฃ โ โ โฃ โกคโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃ โ คโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โฃ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
Continuous โ โ โ โ โ โ โ โ โ โ โฃ โกคโ โ โ ฆโ คโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โขฐโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โฃ โ โ ขโก โกฐโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
0 โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ
โ 0โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ timeโ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ โ 30โ | {"url":"https://juliaactuary.org/packages","timestamp":"2024-11-09T07:39:42Z","content_type":"application/xhtml+xml","content_length":"65698","record_id":"<urn:uuid:e050eea4-539e-4dc6-8913-89397ba39329>","cc-path":"CC-MAIN-2024-46/segments/1730477028116.30/warc/CC-MAIN-20241109053958-20241109083958-00878.warc.gz"} |
Snowfall | R-bloggersSnowfall
[This article was first published on
Econometrics Beat: Dave Giles' Blog
, and kindly contributed to
]. (You can report issue about the content on this page
) Want to share your content on R-bloggers?
if you have a blog, or
if you don't.
Yesterday I had a short post reminding EViews users that their package (versions 7 or 8) will access all of the cores on a multi-core machine. I’ve been playing around with parallel processing in R
on my desktop machine at work over the last few days. It’s something I’ve been meaning to do for a while, and it proved to be well worth the time.
Before I share my results with you, let me make a couple of comments. First, parallel processing involves some costs in terms of communication overheads, so not all tasks are well-suited to this type
of processing. It’s easy to generate examples that are computationally intensive, but execute faster on a single processor than on a cluster (of cores, or machines).
Second, even when a task is suitable for parallel processing, don’t expect the reduction in elapsed time to be linearly related to the increase in the number of cores. Remember, there are overheads
Recently, there have been some posts out there that have illustrated the advantages of parallel processing in R. For example, WenSui Liu posted a piece describing some experiments run using the
Ubuntu O/S. Also,
Daniel Marcelino had
a post
that compared various “parallel” packages in R on a MacBook Pro. Nice choice of machine – it’s running UNIX beneath that pretty cover! And then, just as I was writing this post today, Arthur
Charpentier came out with
this related post
, also based on results using a Mac.
However, none of these posts deal with a Windows environment, or the sorts of Monte Carlo or bootstrap simulations that econometricians use all of the time. So, I felt that there was something more
to explore.
The first thing that I discovered, after a lot of digging around, is that although there’s a number of R packages to help with parallel processing, if you’re running Windows then your options are
limited. O.K., that’s no surprise, of course! Don’t write comments saying that I should be using a different O/S if I want to engage in fast computing. I know that!
However, let’s stick with Windows. In that case it seems that the
snowfall package
for R is the best choice, currently. That’s what the results below are based on.
Well, here are a couple of small examples, run on my DELL desktop. It has an Intel I7-3770 processor ([S:8:S] 4 cores + hyperthreading), and 12GB of RAM. I’m running Windows 7 (64 bit).
Test 1:
This test involves bootstrapping the sampling distribution of an OLS estimator. Of course, we know the answer – this is just an illustration of processing times!
There are 9,999 replications. The R script is on the code page for this blog, and it’s a slightly modified version of an example given by Knaus et al. (2009).
Test 2:This test involves a Monte Carlo simulation of the power of a paired t-test, using 1,999 replications, and sample sizes of n = 10 (5) 200. Again, the R script is on the code page for this
blog, and it’s a modified version of an example given by Spector (undated)
The results when we allow R to access different numbers of cores are:
BTW – this is what I really enjoyed seeing – all of the cores on my machine running at full steam!
Of course, these processing times could be improved a lot by moving to an environment other than Windows! The point of the exercise, though, is simply to show you the effect of grabbing more cores
when running a simulations of the type that we use a lot in econometrics.
Knaus, J., C. Porzelius, H. Binder, & G. Schwarzer, 2009. Easier parallel computing in R with snowfall and sfCluster. The R Journal, 1/1, 54-59.
Spector, P., undated. Using the snowfall library in R. Mimeo., Statistical Computing Facility, Department of Statistics, University of California, Berkeley.
© 2013, David E. Giles | {"url":"https://www.r-bloggers.com/2013/05/snowfall/","timestamp":"2024-11-06T05:32:27Z","content_type":"text/html","content_length":"122407","record_id":"<urn:uuid:00b1ad7c-c922-47f6-84c4-76630fb290a9>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00729.warc.gz"} |
CIDEC Estonian Winter Schools in Computer Science EWSCS 2005
ÜIK Eesti arvutiteaduse talvekoolid EATTK 2005
10^th Estonian Winter School in Computer Science (EWSCS)
X Eesti Arvutiteaduse Talvekool
Palmse, Estonia, February 27 - March 4, 2005
Peter Bro Miltersen
Dept. of Computer Science
University of Aarhus
Universal Methods for Derandomization
In these lectures, we consider the following related questions:
• How can we amplify the success probability of a randomized algorithm without using too many extra random bits?
• How can we run randomized algorithm using an imperfect random source?
• How can we turn our randomized algorithm into a deterministic one?
One can consider these questions both for specific randomized algorithms (using details of the analysis of the algorithms) and for arbitrary ones (i.e., without making assumption about the way the
algorithms use the randomness). We shall deal with the latter kind of universal methods for derandomization. In the last decade, significant progress has been made in finding optimal universal
methods. We shall give an introduction to these results.
Course materials
• P. B. Miltersen. Universal methods for derandomization. Lecture 1: Randomness in computation: five examples and a universal task. Slides. [pdf]
• P. B. Miltersen. Universal methods for derandomization. Lecture 2: Disperses and extractors. Slides. [pdf]
• P. B. Miltersen. Universal methods for derandomization. Lecture 3: Hitting set and pseudorandom generators. Slides. [pdf]
Background material
• P. B. Miltersen. Derandomizing complexity classes. Ch. 19 in S. Rajasekaran, P. M. Pardalos, J. H. Reif, J. D. Rolim, eds., Handbook of Randomized Computing, vols. 1-2, v. 9 of Combinatorial
Optimization. Kluwer Academic Publishers, 2001.
About the Lecturer
URL: http://www.daimi.au.dk/~bromille/
Modified Mar 11, 2005 14:52 by monika(at)cs.ioc.ee | {"url":"https://cs.ioc.ee/yik/schools/win2005/miltersen.php","timestamp":"2024-11-08T17:17:40Z","content_type":"text/html","content_length":"3781","record_id":"<urn:uuid:8a3b6945-2dcd-470e-a0e1-8300e9a3ea66>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00617.warc.gz"} |
Ultimates 3 #1-5 discussion [Loeb/Madureira] [spoilers]
May 3, 2005
Not a bad start for Loeb's first Ultimates issue the title lives up to the name in the first frame. "Sex, Lies, & DVD, as we see most of the team watching a video of Tony and Natalia having...
relations. And it seems as if the whole team is in a really bad mood.
So there they are Hawkeye, Jan, Tony, Thor, Black Panther (of which we don't see how he joins) the Scarlet Witch and Quicksilver watching this dvd thats being sold everywhere. and Venom comes
crashing in looking for quote n' quote her. So the fight begins. Hawkeye charges in without a thought while Jan scream at him to wait for backup. The Panther starts in and starts kicking tale so
Venom throws him arcoss town. Hawkeye jumps in and Venom... webs his face. All of a sudden Valkyrie jumps in and nearly slices Venom in half. Venom takes her sword away and is about to strike when
Thor sends a lighting bolt at him, rendering him unable to... collect himself. Hawkeye wants to just end Venom but Janet stops him by saying "What the hell is wrong with you Clint. Hawkeye then turns
his gun on her and says: "You call me That name in public one more time and I'll drop you right here in the street." Janet: "Get that gun out of my face. For good or for bad, I run this team. I tell
you to wait for backup, you wait for back up." Hawkeye: I went out to put the man down. That's my job. what else do you expect me to do?" What makes you think I'm trying to get killed." Jan relplys
and Hawkeye take off to look for Panther.
Back inside we se Wanda and Pietro talking Cap tells wanda that: I may be overstepping my boundes, but my feeling is that we should be setting an example." Pietro: and..? Cap: And, I'm of the opinion
that Wanda might want to wear something.. less revealing out in public. Pietro: You're right, Captain- Cap: I'm glad you- Pietro: -You have overstepped you pathetic boundaries." They tack off and Cap
is left to talk with Janet and Hawkeye. Cap doesn't get it and both Jan and Hawkeye explain that Wanda and Pietro are in love with each other.
Else where Jan goes to talk to Hank it seems like Hank is lost in his work when Jan goes to wake him and she sees that hank has O.D.ed. Back with the wonder twins we see them talking as people snap
pictures, a shot rings out and Pietro runs to stop it he thinks he has it but it went through his hand and into Wanda. Wanda falls Pietro holds her close, a Doctor in the crowd comes over and trys to
help but it's to late Wanda is Dead...
Don't expect me to take you with me when I go to s
Sep 15, 2004
Re: Ultimates 3 #1 spoiler/discussion
Seems like it's lost all subtlety Millar had (and Millar's not a subtle person...) :?
Re: Ultimates 3 #1 spoiler/discussion
wait wait
i only read the first line cuz i want to avoid spoilers, but did you say Natalia? isn't she dead? or is there a new Black Widow?
please tell me there isn't a continuity faux pah in the first page
Didn't **** any of those *****es
Jun 16, 2005
Re: Ultimates 3 #1 spoiler/discussion
... yep this sounds retarded
May 3, 2005
Re: Ultimates 3 #1 spoiler/discussion
wait wait
i only read the first line cuz i want to avoid spoilers, but did you say Natalia? isn't she dead? or is there a new Black Widow?
please tell me there isn't a continuity faux pah in the first page
No, it's a DVD made from the day the Liberators attacked in issue vol. 2 #9
Re: Ultimates 3 #1 spoiler/discussion
No, it's a DVD made from the day the Liberators attacked in issue vol. 2 #9
oh oh ooooooh
Mar 6, 2005
Re: Ultimates 3 #1 spoiler/discussion
wow. wanda dead, hank maybe dead, valkyrie? ugh ugh ugh. glad i stopped collecting.
May 8, 2006
Re: Ultimates 3 #1 spoiler/discussion
... yep this sounds retarded
I echo this sentiment.
Mar 30, 2006
Re: Ultimates 3 #1 spoiler/discussion
I enjoyed it. It immediately starts to shake things up, which is fun. The minute they started outing the Wanda/Pietro relationship I had a bad feeling. Killing Wanda? Wow.
I am still unsure about Wasp of all people being a GENERAL! Loeb will need a pretty darn cool story to make me buy that one. I am looking forward to the mystery being revealed about Valkeryie, Black
Panther, and some of the other change-ups, though.
Dec 24, 2004
Re: Ultimates 3 #1 spoiler/discussion
Its difficult to find too much fault really - all this did was open some storylines - its their continuation and resolution that will suggest the retroactive merit of the issue.
Interesting enough...the art was interesting, but distracting - kind of like Bachalo in Ultimate War
I'm still interested (and that's a victory, Loeb!)
May 27, 2004
Re: Ultimates 3 #1 spoiler/discussion
wait wait
i only read the first line cuz i want to avoid spoilers, but did you say Natalia? isn't she dead? or is there a new Black Widow?
please tell me there isn't a continuity faux pah in the first page
Nope, no continuity error. I'll spoil you this:
A sex tape of Tony and Natasha is leaked to the public, hence the title of the series "Sex, Lies, & Dvd" ,and thats why she was mentioned. At least as far as this issue is concerned, Natasha/Black
Widow is still dead
I hated the issue. Maybe its the fact thats its such a departure from Millar & Hitch. Maybe its the fact that the dialogue was forced and/or just plain stupid, but none of the characters said or did
anything worthwhile. It could be that I really don't care about any of the plotlines. The ending just didn't make sense to me. Quicksilver should've just matched speed with the bullet and flicked it
into a wall or something. Maybe he's been depowered some, or Loeb is just going in a different direction with the character and just wanted to write another greiving, semi-suicidal loner like
Hawkeye. Maybe its the fact that besides Thor, none of the big guns played much of a part (barely any Tony or Cap, no mention of Banner/Hulk). The art was nice in its own way, but was horrible in
terms of telling the story. Maybe it'll get better, but if the stupid, pointless blatter of the team members and villains is going to become the norm, I'm going to find it hard not to throw up when
reading the next issue.
I give this one a 1/5...at least Loeb (intentionally or unintentionally) kinda fixed a timeline issue. Like I said, I'll give the next couple issues a fair shot, but I'm not expecting much.
Dec 24, 2004
Re: Ultimates 3 #1 spoiler/discussion
The art was nice in its own way, but was horrible in terms of telling the story.
Exactly. Well put!
May 27, 2004
Re: Ultimates 3 #1 spoiler/discussion
Thank you. And I really hate that. Mad has a distinct style and thats well and good, but he didn't show me in this issue he could progress the story with his images, which truthfully are all pose
shots or fight scenes.
The whole stupid costume thing (I'm looking at you, Hawkeye, and that stupid ****ing bullseye on your forehead), the dumbass banter during the fights, the overloaded and unnecessary exposition
crammed into the dialogue bubbles...I really disliked this issue. A lot. Bleechh. I just threw up in my mouth a little thinking about it.
Last edited:
Dec 10, 2005
Re: Ultimates 3 #1 spoiler/discussion
I actualy liked this issue,then again anyone is better than millar.Also Pietro did catch the bullet it just went threw his hands.
Oh and of corse there was a drastic change.Happends every time theres a change in writers jesus cristo.
When did Valkery get powers?Thor is totally robing the cradle.
Don't expect me to take you with me when I go to s
Sep 15, 2004
Re: Ultimates 3 #1 spoiler/discussion
I actualy liked this issue,then again anyone is better than millar.
Re: Ultimates 3 #1 spoiler/discussion
I just want to make it known that i don't understand the big attraction of the Wasp
all this talk that "You can't have the Avengers without Jan" is rediculous, i mean she's about as interesting as Ms. Marvel (not exactly a compliment), and they put her in charge of the Ultimates
instead of say CAP or uh TONY, it's bullcrap
while she has demonstrated some competency in this incarnation she hasn't done anything noteworthy enough to put her in charge of this team
Dec 10, 2005
Re: Ultimates 3 #1 spoiler/discussion
I just want to make it known that i don't understand the big attraction of the Wasp
all this talk that "You can't have the Avengers without Jan" is rediculous, i mean she's about as interesting as Ms. Marvel (not exactly a compliment), and they put her in charge of the Ultimates
instead of say CAP or uh TONY, it's bullcrap
while she has demonstrated some competency in this incarnation she hasn't done anything noteworthy enough to put her in charge of this team
Well Tony is not leading becouse he's a rich drunk and cap.......doesnt have a myspace and he has a 1940's mentality(i.e. he's a racist).
and ms.marvel rocks what are you talking about?
Jan 24, 2005
Re: Ultimates 3 #1 spoiler/discussion
Well Tony is not leading becouse he's a rich drunk and cap.......doesnt have a myspace and he has a 1940's mentality(i.e. he's a racist).
and ms.marvel rocks what are you talking about?
Thor, then. Wasp is so insignificant and it just kind of annoys me when they make her more important then she should be.
May 17, 2004
Re: Ultimates 3 #1 spoiler/discussion
I actualy liked this issue,then again anyone is better than millar.Also Pietro did catch the bullet it just went threw his hands.
Oh and of corse there was a drastic change.Happends every time theres a change in writers jesus cristo.
When did Valkery get powers?Thor is totally robing the cradle.
This is why you're Venom Melendez.
Re: Ultimates 3 #1 spoiler/discussion
I actualy liked this issue,then again anyone is better than millar.
ms.marvel rocks what are you talking about?
while i respect your right to free speech and whatever, how do you keep a straight face while typing these things? | {"url":"https://www.thecomicboard.com/threads/ultimates-3-1-5-discussion-loeb-madureira-spoilers.10423/","timestamp":"2024-11-03T23:08:47Z","content_type":"text/html","content_length":"162424","record_id":"<urn:uuid:de576b75-41cb-4829-a19f-03b9157a0157>","cc-path":"CC-MAIN-2024-46/segments/1730477027796.35/warc/CC-MAIN-20241103212031-20241104002031-00423.warc.gz"} |
Column Plate Number and System Suitability
How suitable is the column plate number for system suitability testing?
I was recently contacted by a colleague inquiring about whether or not the column plate number, N, is a good measurement to include in system suitability testing (SST) for liquid chromatography (LC)
methods. He had noticed that monographs in the United States Pharmacopeia (USP) often had minimum plate number requirements as part of SST for those methods, but that often it was specified that N >
2000 was sufficient. He wondered about the origin of a N > 2000 requirement and if it really added any value to SST, especially in light of the performance of modern LC columns. This is a topic of
wide enough interest that I feel it is worth a discussion here.
The column plate number usually is part of the specification for new columns. Traditionally, N is calculated based on manual measurements of the retention time, t[R], and the baseline width, w[b],
measured between tangents to a chromatographic peak, as shown in Figure 1 using the formula
Figure 1: Calculation of column plate number. See equations 1 and 2 plus related discussion.
More commonly today, N is measured by data systems based on the more easily measured width at half the peak height, w^0.5, using
The method of equations 1 or 2 only applies to isocratic separations, where the mobile-phase composition is constant, and should not be used for gradient separations.
A well-packed, reversed-phase column containing totally porous particles tested under ideal conditions should generate plate numbers of
where L is the length of the column (in millimeters) and d[p] is the particle diameter (in micrometers). However, this requires an LC system with very low extracolumn volume, a small injection, a
well-behaved, small molecular weight test solute, an optimum flow rate, and other conditions that do not reflect typical application of the column in the routine laboratory. Also, if equation 3 were
used for acceptance criteria of a new column, column manufacturers would have to discard many otherwise satisfactory columns. As a result, equation 3 may be adjusted to use 2.5 d[p] (or some other
value) rather than 2 d[p] as the denominator for a quality control specification in the column manufacturing process.
When an LC column is used for a real LC method, usually the conditions are much less ideal than those used by the manufacturer for testing the column. The typical LC system will have more extracolumn
volume than the test system. The sample solutes will be less well-behaved, and other chromatographic conditions (temperature, flow rate, and so forth) may also degrade the column performance. It is
nice to have a rule of thumb for column performance under typical, “real” conditions to make sure the column is performing adequately. For totally porous particle columns I like to use the estimate
In Table I, I have compared the calculated plate numbers for two of the most popular column configurations today: a 150-mm, 5-µm d[p] column and a 100-mm, 3-µm column.
Some Practical Uses of N
Let’s consider some of the ways the knowledge of the plate number for a particular column can be useful. You can see from Table I that the plate number for the 150 mm, 5 µm column is approximately
the same as the 100 mm, 3 µm one. This means that these columns should give the same separation if the same column chemistry (brand and product line) and operating conditions are used. This can be
helpful in evaluating tradeoffs when changing column dimensions-if the 150-mm column had a 15-min run time, it would only take 10 min with the 100-mm column and would reduce solvent consumption by a
You also can monitor the column performance over time for help in deciding when to replace the column. I usually consider that the column should be replaced when you see ~30% loss in N relative to
the value measured when the column is new. So, if you measured N = 9000 for a new column and your method, when it dropped to N ≈ 6000, it probably would be time to replace it. The replacement
decision should also be based on other factors, such as changes in peak shape and back pressure.
The column plate number can be helpful in evaluating overall performance of the LC system. If you suspect that the cause of a drop in plate number is external to the column, you can replace the
column with a new one and compare the plate number (and other performance parameters) to historic values when the system was known to be operating well. If N is still low with a new column, look for
problems external to the column. Extracolumn band broadening can be a critical factor in such cases, especially with the increased popularity of ultrahigh-pressure LC (UHPLC), where 50 mm x 2.1 mm
columns packed with ≤2-µm diameter particles are popular. A fitting that has slipped a bit because of system over-pressure or poor assembly is not visible to the naked eye, but often will cause
noticeable deterioration of chromatographic performance, including N.
Resolution, R[s], is another parameter that is used to evaluate the suitability of a method for analysis. It is useful to compare the utility of N and R[s] and their roles in system suitability, so
let’s take a slight detour and review how we determine resolution.
Resolution can be calculated using the retention times of the two peaks (t[R1] and t[R2]) and the peak widths at baseline (w[b1] and w[b2]):
Alternatively, the widths at half-height (w[0.5,1] and w[0.5,2]) can be used:
As with the calculations of N at half the peak height, equation 6 (or 7) is a simpler measurement for the data system than peak widths at the baseline. A further advantage of the half-height
technique is that if R[s] < 1.5, baseline peak widths will be difficult or impossible to determine even for well-shaped peaks, so the method of equation 5 cannot be used. As you can see from
equations 5, 6, and 7, resolution is directly related to the peak width.
Another way to determine resolution is with the fundamental resolution equation:
where k is the retention factor and α is the selectivity, calculated as the ratio of retention factors for two adjacent peaks. Values of k, and thus α, are controlled by the chemistry of the mobile
and stationary phases as well as the temperature. So, if chemistry and temperature are constant, as is normally the case, resolution is a function of the square root of the plate number.
N as a System Suitability Parameter
Often the column plate number is included as part of system suitability testing, such as the N > 2000 requirement mentioned in the introduction. If, as is the case most of the time, one goal of an LC
method is to separate two compounds, the resolution (R[s]) of the two peaks will be included as part of SST. If resolution measurement is required, does measurement of N add any useful information to
To help compare and contrast R[s] and N, I’ve prepared Table II and Figure 2. Table II tabulates peak and chromatogram characteristics for different values of N. The various rows show data
corresponding to columns with plate numbers ranging from 10,000 down to 2000 in 20% increments (data in columns 1 and 2). This progression might occur when we start with a new, 150-mm, 5-µm d[p]
column (N ≈ 10,000 per Table I) and watch the plate number drop over the lifetime of the column. If the chemistry of the system stays the same, the retention will stay constant, so decreasing N will
increase the peak width as a square root function (see equations 1 or 2); the peak width is shown in the third column of Table II. For constant sample size and detector response, the peak height will
drop in proportion to the increase in peak width (column 4). As we saw above in equation 8, resolution will be reduced by the square root of the drop in N; resolution relative to the starting
condition is shown in column 5. Notice that it takes a fourfold drop in N to halve R[s]. Finally, columns 6 and 7 translate this relative loss of resolution into numeric values for separations
starting with R[s] = 2.0 or 1.7.
Figure 2: Simulated partial chromatograms to show the effect of changes in plate number on peak height, peak width, and resolution. Left to right, N = 10,000, 8000, 6000, 4000, and 2000. Top row R[s]
= 2.0 on left, progressing to 0.89 on the right corresponding to data shown in Table II. Bottom row, same as top, but ranging from R[s] = 1.7 on left to 0.76 on the right. See Table II and text for
more information.
Figure 2 comprises simulated partial chromatograms based on the data in Table II. In each chromatogram, a well-resolved peak precedes a peak pair whose resolution is tracked as follows. The top row
of chromatograms corresponds to column 6 of Table II where R[s] = 2.0 initially (left) and drops progressing to the right as the plate number drops to N = 2000 and R[s] = 0.89. The bottom row of
Figure 2 shows the corresponding chromatograms starting with R[s] = 1.7.
One thing that is quickly apparent when studying the chromatograms in Figure 2 is that a change in N may be less likely to be noticed than a change in R[s]. For example, the loss of ~25% in peak
resolution is obvious when comparing the chromatograms for N = 10,000 and N = 6000, whereas the loss of N, often noticed by increased peak width, is much less obvious without careful study. In Figure
2, the loss of peak height is obvious as N is decreased, but here all the chromatograms are on the same scale; if the peak heights were normalized to the tallest peak, a 25% change in peak height
might be overlooked in a visual evaluation of the chromatogram.
Another problem with the plate number as a measure of quality is that the absolute value of the plate number does not necessarily reflect the quality of the separation. For example, compare the
initial R[s] = 2.0 separation after the column degrades to N = 6000 to the original R[s] = 1.7 separation with the N = 8000 column. Both have the same resolution (R[s] = 1.5), but N differs by 25%.
Thus, setting some arbitrary minimum acceptable value of N, as was proposed in the introduction for N > 2000, makes no sense as a general requirement. Instead, the minimum acceptable plate number can
be determined only in light of the minimum acceptable value of resolution for each method. On the other hand, a specified value of resolution will always translate into the same visual appearance of
the chromatogram regardless of the method or the column plate number.
Does resolution always win out over the plate number as an effective SST parameter? Not necessarily. A value for resolution is of little practical use once two peaks are well separated at the
baseline (for example, R[s] > 3). This is apparent for the resolution between the first and second peaks of each partial chromatogram in Figure 2; in no case is a numeric value of resolution very
useful in alerting us to potential problems with the separation. The plate number, on the other hand, does give us an idea that something is not going well when the value drops precipitously.
The plate number also gives us information regarding how well the column performs relative to an ideal case (manufacturer’s test), a generic rule of thumb for “good” performance (equation 4), or the
same column when it was new. Tracking the plate number can help us determine when the column is nearing the end of its useful life so we can order a replacement. Following changes in the column plate
number can also help to anticipate problems with detection limits. As the plate number drops, the width increases and the peak height decreases (Table II) so that peak area remains constant. Whereas
most of us use peak area for quantification, peak height is the critical measurement when it comes to detection limits, which depend on the signal-to-noise ratio. Shorter peaks will have smaller
signal-to-noise ratios, and thus poorer detection limits. When methods are used for trace analysis, the plate number may serve as a very useful tool to help determine when to retire a column so as to
maintain suitable lower quantification limits.
A final consideration for the plate number is reflected in the speed of separation. Larger plate numbers (for the same column length) always translate into narrower peaks. Narrower peaks allow peaks
to be more closely eluted for any given resolution than for broader peaks. Therefore, larger plate numbers can be translated into faster separations. So the use of columns that generate substandard
values of N will require longer run times and thus increase analysis costs.
We have looked at the column plate number as a system suitability parameter to include in LC system suitability testing. We have seen that resolution is a much more useful than N if the peaks are
close together (for example, R[s] < 3), but its usefulness drops for very well separated peaks. Therefore, the plate number can be a useful way to predict method performance when the separation is
not challenging. It also can help to determine the overall health of the column and its performance compared to some specified reference condition. Because both N and R[s] can be calculated easily
and automatically by chromatographic data systems, I think it is a good idea to keep track of both measurements. Just be sure to use the data appropriately to evaluate current or potential problems
with an LC method. And an arbitrary lower limit of N > 2000? I don’t think this is a good idea.
So where did that N > 2000 requirement come from, anyway? As far as I can tell, it originated in a United States Food and Drug Administration (FDA) document (1). This document is designed to help FDA
inspectors evaluate chromatographic methods when they conduct laboratory audits. In the guidance, the recommendation is, “The theoretical plate number depends on the elution time but in general
should be >2000.” The document is dated 1994, and, although 5-µm d[p] columns were widely used by that time, older columns packed with 10-µm irregularly shaped particles were still common. There may
have been some justification for N > 2000 20 years ago, but not today, and certainly not as a generic requirement.
• U.S. Food and Drug Administration, Reviewer Guidance. Validation of Chromatographic Methods (USFDA-CDER, Rockville, Maryland, November 1994).
John W. Dolan “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 the
Vice President of and a principal instructor for LC Resources in Lafayette, California. He is also a member of LCGC’s editorial advisory board. Direct correspondence about this column via e-mail to | {"url":"https://www.chromatographyonline.com/view/column-plate-number-and-system-suitability","timestamp":"2024-11-13T16:27:06Z","content_type":"text/html","content_length":"573029","record_id":"<urn:uuid:a9f8c393-a711-4db3-b840-76d294e9da5d>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00411.warc.gz"} |
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When people ask me what my master thesis is about, I have no idea what to tell them. By people I mean, people with no background in pure math, no matter their intellect or education, from a shop
assistant up to an engineer. I just mumble the title of my thesis and the subject within math, which of course they don't know shit about, and I have no idea how to go on. Feels like explaining even
the simplest concepts is just unrealistic.
I don't know if it's an impossible task to talk about mathematics, or is it just my lack of communication skills... I spent so much time thinking about it but didn't get any closer to a solution.
Any suggestions? Or maybe any explanation to why it is impossible.
Personally, when I hear mathematicians appear in podcasts, tv-shows, etc, they make things sound so dumb, in order to make it more understandable that just makes my stomach turn. And also I don't
think it helps understanding, just makes it more relatable, perhaps? | {"url":"https://r-weld.vercel.app/r/puremathematics","timestamp":"2024-11-05T05:41:23Z","content_type":"text/html","content_length":"222313","record_id":"<urn:uuid:cd5289c6-2ad0-42d9-b312-c9b5e53dec45>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00561.warc.gz"} |
Hello, friends of good plane mathematics,
my problem is: I know some about analytical plane geometry, but I cannot really transmita plane equation in to a three-
dimensional Euclidean space, f.e.
E: 8x+y-13z=-8
Without a help of PC-software I cannot really imagine how to...
Hello, friends of good plane mathematics,
my problem is: I know some about analytical plane geometry, but I cannot really transmita plane equation in to a three-
dimensional Euclidean space, f.e.
E: 8x+y-13z=-8
Without a help of PC-software I cannot really imagine how to find an initial atttempt of a design.
Can you give me an hint, where I could study this problem?
Kind regards
Gerhard Pelzel | {"url":"http://image.absoluteastronomy.com/discussionpost/Hello_friends_of_good_plane_mathematicsmy_problem_is_I_know_some_about_analytical_plane_geometry_but_I_cannot_really_transmita_plane_equation__80938","timestamp":"2024-11-06T12:40:33Z","content_type":"text/html","content_length":"9271","record_id":"<urn:uuid:1364b4cd-56e7-49c1-8ecb-0ccabd0837c3>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00359.warc.gz"} |
Bilayer Graphene Inspires Two-Universe Cosmological Model | Joint Quantum Institute
Bilayer Graphene Inspires Two-Universe Cosmological Model
May 5, 2022
Physicists sometimes come up with crazy stories that sound like science fiction. Some turn out to be true, like how the curvature of space and time described by Einstein was eventually borne out by
astronomical measurements. Others linger on as mere possibilities or mathematical curiosities.
In a new paper in Physical Review Research, JQI Fellow Victor Galitski and JQI graduate student Alireza Parhizkar have explored the imaginative possibility that our reality is only one half of a pair
of interacting worlds. Their mathematical model may provide a new perspective for looking at fundamental features of reality—including why our universe expands the way it does and how that relates to
the most miniscule lengths allowed in quantum mechanics. These topics are crucial to understanding our universe and are part of one of the great mysteries of modern physics.
The pair of scientists stumbled upon this new perspective when they were looking into research on sheets of graphene—single atomic layers of carbon in a repeating hexagonal pattern. They realized
that experiments on the electrical properties of stacked sheets of graphene produced results that looked like little universes and that the underlying phenomenon might generalize to other areas of
physics. In stacks of graphene, new electrical behaviors arise from interactions between the individual sheets, so maybe unique physics could similarly emerge from interacting layers
elsewhere—perhaps in cosmological theories about the entire universe.
“We think this is an exciting and ambitious idea,” says Galitski, who is also a Chesapeake Chair Professor of Theoretical Physics in the Department of Physics. “In a sense, it's almost suspicious
that it works so well by naturally ‘predicting’ fundamental features of our universe such as inflation and the Higgs particle as we described in a follow up preprint.”
Stacked graphene’s exceptional electrical properties and possible connection to our reality having a twin comes from the special physics produced by patterns called moiré patterns. Moiré patterns
form when two repeating patterns—anything from the hexagons of atoms in graphene sheets to the grids of window screens—overlap and one of the layers is twisted, offset, or stretched.
The patterns that emerge can repeat over lengths that are vast compared to the underlying patterns. In graphene stacks, the new patterns change the physics that plays out in the sheets, notably the
electrons' behaviors. In the special case called “magic angle graphene,” the moiré pattern repeats over a length that is about 52 times longer than the pattern length of the individual sheets, and
the energy level that governs the behaviors of the electrons drops precipitously, allowing new behaviors, including superconductivity.
Galitski and Parhizkar realized that the physics in two sheets of graphene could be reinterpreted as the physics of two two-dimensional universes where electrons occasionally hop between universes.
This inspired the pair to generalize the math to apply to universes made of any number of dimensions, including our own four-dimensional one, and to explore if similar phenomenon resulting from moiré
patterns might pop up in other areas of physics. This started a line of inquiry that brought them face to face with one of the major problems in cosmology.
“We discussed if we can observe moiré physics when two real universes coalesce into one,” Parhizkar says. “What do you want to look for when you're asking this question? First you have to know the
length scale of each universe.”
A length scale—or a scale of a physical value generally—describes what level of accuracy is relevant to whatever you are looking at. If you’re approximating the size of an atom, then a ten-billionth
of a meter matters, but that scale is useless if you’re measuring a football field because it is on a different scale. Physics theories put fundamental limits on some of the smallest and largest
scales that make sense in our equations.
The scale of the universe that concerned Galitski and Parhizkar is called the Planck length, and it defines the smallest length that is consistent with quantum physics. The Planck length is directly
related to a constant—called the cosmological constant—that is included in Einstein’s field equations of general relativity. In the equations, the constant influences whether the universe—outside of
gravitational influences—tends to expand or contract.
This constant is fundamental to our universe. So to determine its value, scientists, in theory, just need to look at the universe, measure several details, like how fast galaxies are moving away from
each other, plug everything into the equations and calculate what the constant must be.
This straightforward plan hits a problem because our universe contains both relativistic and quantum effects. The effect of quantum fluctuations across the vast vacuum of space should influence
behaviors even at cosmological scales. But when scientists try to combine the relativistic understanding of the universe given to us by Einstein with theories about the quantum vacuum, they run into
One of those problems is that whenever researchers attempt to use observations to approximate the cosmological constant, the value they calculate is much smaller than they would expect based on other
parts of the theory. More importantly, the value jumps around dramatically depending on how much detail they include in the approximation instead of homing in on a consistent value. This lingering
challenge is known as the cosmological constant problem, or sometimes the “vacuum catastrophe.”
“This is the largest—by far the largest—inconsistency between measurement and what we can predict by theory,” Parhizkar says. “It means that something is wrong.”
Since moiré patterns can produce dramatic differences in scales, moiré effects seemed like a natural lens to view the problem through. Galitski and Parhizkar created a mathematical model (which they
call moiré gravity) by taking two copies of Einstein’s theory of how the universe changes over time and introducing extra terms in the math that let the two copies interact. Instead of looking at the
scales of energy and length in graphene, they were looking at the cosmological constants and lengths in universes.
Galitski says that this idea arose spontaneously when they were working on a seemingly unrelated project that is funded by the John Templeton Foundation and is focused on studying hydrodynamic flows
in graphene and other materials to simulate astrophysical phenomena.
Playing with their model, they showed that two interacting worlds with large cosmological constants could override the expected behavior from the individual cosmological constants. The interactions
produce behaviors governed by a shared effective cosmological constant that is much smaller than the individual constants. The calculation for the effective cosmological constant circumvents the
problem researchers have with the value of their approximations jumping around because over time the influences from the two universes in the model cancel each other out.
“We don't claim—ever—that this solves cosmological constant problem,” Parhizkar says. “That's a very arrogant claim, to be honest. This is just a nice insight that if you have two universes with huge
cosmological constants—like 120 orders of magnitude larger than what we observe—and if you combine them, there is still a chance that you can get a very small effective cosmological constant out of
In preliminary follow up work, Galitski and Parhizkar have started to build upon this new perspective by diving into a more detailed model of a pair of interacting worlds—that they dub “bi-worlds.”
Each of these worlds is a complete world on its own by our normal standards, and each is filled with matching sets of all matter and fields. Since the math allowed it, they also included fields that
simultaneously lived in both worlds, which they dubbed “amphibian fields.”
The new model produced additional results the researchers find intriguing. As they put together the math, they found that part of the model looked like important fields that are part of reality. The
more detailed model still suggests that two worlds could explain a small cosmological constant and provides details about how such a bi-world might imprint a distinct signature on the cosmic
background radiation—the light that lingers from the earliest times in the universe.
This signature could possibly be seen—or definitively not be seen—in real world measurements. So future experiments could determine if this unique perspective inspired by graphene deserves more
attention or is merely an interesting novelty in the physicists’ toy bin.
“We haven't explored all the effects—that's a hard thing to do, but the theory is falsifiable experimentally, which is a good thing,” Parhizkar says. “If it's not falsified, then it's very
interesting because it solves the cosmological constant problem while describing many other important parts of physics. I personally don't have my hopes up for that— I think it is actually too big to
be true.”
Story by Bailey Bedford
The research was supported by the Templeton Foundation and the Simons Foundation. | {"url":"https://jqi.umd.edu/news/bilayer-graphene-inspires-two-universe-cosmological-model?fbclid=IwAR2IS02vynZeBfnmX2tdgEr1TdLYb2OUN1E1vIXGUj1lDiLvbgFPl_LCzxs","timestamp":"2024-11-12T09:27:43Z","content_type":"text/html","content_length":"485917","record_id":"<urn:uuid:3a0ed329-684d-479a-b333-5e7c4b4fd230>","cc-path":"CC-MAIN-2024-46/segments/1730477028249.89/warc/CC-MAIN-20241112081532-20241112111532-00068.warc.gz"} |
Lesson 2
Introducing Geometric Sequences
2.1: Notice and Wonder: A Pattern in Lists (5 minutes)
This is the first Notice and Wonder activity in the course. Students are shown four sequences. The prompt to students is “What do you notice? What do you wonder?” Students are given a few minutes to
write down things they notice and things they wonder. After students have had a chance to write down their responses, ask several students to share things they noticed and things they wondered.
Record these for all to see. Often, the goal is to steer the conversation to wondering about something mathematical that the class is about to focus on. The purpose is to make a mathematical task
accessible to all students with these two approachable questions. By thinking about them and responding, students gain entry into the context and might get their curiosity piqued.
The purpose of this task is to re-introduce growth factor. (Students likely encountered it in an earlier course when they studied exponential functions.) Students notice and describe that each
sequence is characterized by the same type of relationship between consecutive terms.
When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or
precise language, and then restate their observation with more precise language in order to communicate more clearly.
The last two sequences may present a challenge since the growth factor is less than 1. The purpose of including these sequences is to encourage students to notice and make use of structure (MP7). If
they notice that in the first two sequences, each pair of consecutive terms has the same quotient, they could inspect the quotients in the last sequence. These two sequences also give opportunity to
point out that we still use "growth factor" even when the terms are decreasing.
Display the four sequences for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to
discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer to record what they notice and wonder.
Supports accessibility for: Language; Organization
Student Facing
What do you notice? What do you wonder?
• 40, 120, 360, 1080, 3240
• 2, 8, 32, 128, 512
• 1000, 500, 250, 125, 62.5
• 256, 192, 144, 108, 81
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the sequence. After all responses
have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification,
or point out contradicting information.
If the idea of each consecutive term in a sequence growing by the same factor or having a common ratio does not come up during the conversation, ask students to discuss this idea. Students may
describe how the “same thing” is happening with consecutive terms. Encourage students to use the word term, and to be specific when they describe what is happening, for example
• “In the first sequence, you always multiply a term by 3 to get the next term.”
• “In the last sequence, if you divide any term by the previous term, you always get \(\frac34\).”
Tell students that this “same thing” is called the growth factor or common ratio, and that we will use the first of these. For example, in the second list, the growth factor is 4 because \(8=4\
boldcdot2\), \(32=4\boldcdot8\), \(128=4\boldcdot32\), and \(512=4\boldcdot128\).
Emphasize that the growth factor is defined to be the multiplier from one term to the next; said another way, the quotient of a term and the previous term. For example, students may want to say that
the pattern of the third sequence is “divide by 2 each time.” This is true, but the growth factor is \(\frac12\), because it is the number you multiply by to get the next term.
In earlier courses, students may have learned that a ratio has two or more parts. In more advanced courses (like this one), ratio is sometimes used as a synonym for quotient.
2.2: Paper Slicing (20 minutes)
In this activity, students generate two geometric sequences from a mathematical situation. The purpose is to create representations of geometric sequences using tables and graphs. At this time,
students do not need to write equations for the situation since that work will be the focus of a future lesson.
Monitor for students sketching neat and accurate graphs to highlight during the whole-class discussion. Note that future lessons will focus on a reasonable domain for sequences when regarded as
functions. In this activity, students may “connect the dots” in the graphs with lines, but it’s not crucial to address at this time whether or not that’s a sensible thing to do.
Arrange students in groups of 2. Invite them to read the introduction to the task. If time allows, distribute scissors and blank paper (or copies of the blackline master, if using) and let students
work with their partner to carry out the paper cutting while completing the first few rows of the table. If time is limited, conduct a demonstration of the paper cutting.
Tell students to pause after completing the first few rows of the table and then ask,
• “What happens to the number of pieces after each cut?”
• “What happens to the area of each piece after each cut?”
Ask students to share their responses with a partner, and then invite a few groups to share their response with the class. Ensure students can articulate that as a result of a cut, the number of
pieces doubles, and the area of each piece is halved. Students then proceed with the remainder of the activity.
Speaking, Reading: MLR5 Co-Craft Questions. Use this routine to help students consider the context of this problem. Without revealing the questions that follow, display and read aloud only the task
statement that describes Clare's actions. Ask students to write down possible mathematical questions that could be asked about the situation. Invite 3–4 students to share their questions with the
class. Listen for and amplify any questions that include language related to looking for patterns.
Design Principle(s): Maximize meta-awareness; Support sense-making
Engagement: Provide Access by Recruiting Interest. Begin with a whole-class demonstration. If time does not allow for students to carry out paper cutting themselves, demonstrate the initial cuts so
that students can complete the first few rows of the table.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Clare takes a piece of paper, cuts it in half, then stacks the pieces. She takes the stack of two pieces, then cuts in half again to form four pieces, stacking them. She keeps repeating the process.
│number │ number │area in square inches │
│of cuts│of pieces │ of each piece │
│0 │ │ │
│1 │ │ │
│2 │ │ │
│3 │ │ │
│4 │ │ │
│5 │ │ │
1. The original piece of paper has length 8 inches and width 10 inches. Complete the table.
2. Describe in words how you can use the results after 5 cuts to find the results after 6 cuts.
3. On the given axes, sketch a graph of the number of pieces as a function of the number of cuts. How can you see on the graph how the number of pieces is changing with each cut?
4. On the given axes, sketch a graph of the area of each piece as a function of the number of cuts. How can you see how the area of each piece is changing with each cut?
Student Facing
Are you ready for more?
1. Clare has a piece of paper that is 8 inches by 10 inches. How many pieces of paper will Clare have if she cuts the paper in half \(n\) times? What will the area of each piece be?
2. Why is the product of the number of pieces and the area of each piece always the same? Explain how you know.
Anticipated Misconceptions
If students struggle to remember the meaning of area, ask them what they remember about the idea from their previous courses. A workable definition is “the number of one-inch squares covered by the
piece of paper.” Using the blackline master may assist in this understanding. (Length times width is how one computes the area of the region bounded by a rectangle, but it doesn’t explain what area
Activity Synthesis
The goal of this discussion is for students to identify the growth factor for each sequence and learn that sequences with a growth factor are called geometric sequences.
Invite previously identified students to share their graphs with the class. Display the two sequences in this activity (1, 2, 4, 8, 16, 32 and 80, 40, 20, 10, 5, 2.5) for all to see.
Here are some possible questions for discussion:
• “When did you stop cutting the paper and complete the table using a pattern?”
• “What was the pattern you noticed?” (To find the number of pieces, multiply the previous number by 2. To find the area, multiply the previous number by \(\frac12.\))
• “How did you find the results after six cuts?” (Multiply 32 by 2 to get 64, and divide 2.5 by 2 to get 1.25.)
• “What is the growth factor for each sequence?” (It’s 2 for the number of pieces and \(\frac12\) for the area.)
• “How can you see the growth factor in each graph?” (For the number of pieces, the height of each plotted point is twice the height of the previous plotted point. For the area, the height of each
plotted point is half the height of the previous plotted point.)
Tell students that the sequences they have seen today (in this activity and in the warm-up) have a special name: geometric sequences. Geometric sequences are characterized by a growth factor. In a
geometric sequence if you divide any term by the previous term, you always get the same value: the growth factor for the sequence. Reiterate that the growth factor for the area sequence is \(\frac12
\) because it’s what you multiply by to get the next term.
Some students may notice the similarity between a geometric sequence and an exponential function. Invite these students to share their observations, such as how both are defined by a growth factor.
Tell students that geometric sequences are a type of exponential function and that their knowledge of exponential functions will help them describe geometric sequences during this unit. If students
do not bring up the connection to exponential functions, ask "What do you remember about exponential functions?" Record student responses for all to see and invite comparisons between exponential
functions and geometric sequences.
2.3: Complete the Sequence (10 minutes)
The purpose of this task is to provide students with practice working with geometric sequences and identifying the growth factor of a sequence.
Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a calculator, or graphing software. Some students may benefit from a checklist or
list of steps to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
1. Complete each geometric sequence.
1. 1.5, 3, 6, ___, 24, ___
2. 40, 120, 360, ___, ___
3. 200, 20, 2, ___, 0.02, ___
4. \(\frac 1 7\), ___, \(\frac 9 7\), \(\frac {27} 7\), ___
5. 24, 12, 6, ___, ___
2. For each sequence, find its growth factor.
Activity Synthesis
In the lead up to writing recursive definitions for sequences, it is important for students to understand that for geometric sequences the growth factor is defined to be the multiplier from one term
to the next. Said another way, the growth factor is the quotient of a term and the previous term. For example, many students will want to say that the pattern of the third sequence is “divide by 10
each time.” This is true, but the growth factor is \(\frac{1}{10}\).
For each sequence, invite a student to share how they completed the sequence and determined the growth factor. Highlight the method of dividing any term by the previous term to find the growth
factor. Emphasize that the presence of a growth factor is what makes a sequence a geometric sequence.
Lesson Synthesis
Arrange students in groups of 2. Ask each group to come up with a new geometric sequence and be prepared to explain why it is a geometric sequence. After a brief work time, select 3–4 groups to share
their sequence and why it is a geometric sequence. Encourage students to use precise language as they share with the class, such as term, multiplier, and growth factor.
If time allows, ask students if they think the sequence 2, 2, 2, 2, 2 is a geometric sequence. (Yes, it has a growth factor of 1.)
2.4: Cool-down - A Possible Geometric Sequence (5 minutes)
Student Facing
Consider the sequence 2, 6, 18, . . . How would you describe how to calculate the next term from the previous?
In this case, each term in this sequence is 3 times the term before it.
A way to describe this sequence would be: the starting term is 2, and the \(\text{current term} = 3 \boldcdot \text{previous term}\).
This is an example of a geometric sequence. A geometric sequence is one where the value of each term is the value of the previous term multiplied by a constant. If you know the constant to multiply
by, you can use it to find the value of other terms.
This constant multiplier (the “3” in the example) is often called the sequence’s growth factor or common ratio. To find it, you can divide consecutive terms. This can also help you decide whether a
sequence is geometric.
The sequence 1, 3, 5, 7, 9 is not a geometric sequence because \(\frac31 \neq \frac53 \neq \frac75\). The sequence 100, 20, 4, 0.8, however, is because if you divide each term by the previous term
you get 0.2 each time: \(\frac{20}{100} = \frac{4}{20} = \frac{0.8}{4} = 0.2\). | {"url":"https://im.kendallhunt.com/HS/teachers/3/1/2/index.html","timestamp":"2024-11-06T15:43:48Z","content_type":"text/html","content_length":"104668","record_id":"<urn:uuid:38d86d9a-50ac-440d-b52e-b3157b6b32c2>","cc-path":"CC-MAIN-2024-46/segments/1730477027932.70/warc/CC-MAIN-20241106132104-20241106162104-00286.warc.gz"} |
Problem of trigonometry
Jul 8, 2024
Oct 6, 2005
Please follow the forum guidelines at the link below.
Post the instructions for this exercise.
Post what you've tried to do so far.
Explain any part(s) that you do not understand, or ask specific questions.
Welcome to our tutoring boards! :) This page summarizes main points from our posting guidelines. As our name implies, we provide math help (primarily to students with homework). We do not generally
post immediate answers or step-by-step solutions. We don't do your homework. We prefer to help...
Nov 12, 2017
As I understand it, you are given that B (a vertex) and G (the center) of the square lie on the circle, and you want to find 2 times the tangent of the marked angle. (I can't translate "mallar"
without knowing the language!)
What have you tried? Do you want to use coordinate geometry, or trigonometry directly, or what?
Jul 8, 2024
I want to use trigonometry, and what the problem asks is correct, find 2 times the tangent of the angle theta
Nov 12, 2017
I want to use trigonometry, and what the problem asks is correct, find 2 times the tangent of the angle theta
Again, please show what you have tried, so we can help you continue.
You might start by calling the side of the square 1, or perhaps the radius instead. You might also draw in some extra segments, such as radii. | {"url":"https://www.freemathhelp.com/forum/threads/problem-of-trigonometry.138589/","timestamp":"2024-11-13T12:57:10Z","content_type":"text/html","content_length":"53730","record_id":"<urn:uuid:f8ff2409-c947-4b21-b83b-d5880c40c782>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00260.warc.gz"} |
Can I make an excel to predict with new input after have a training model ANN (Neural net)
The Altair Community is migrating to a new platform to provide a better experience for you. In preparation for the migration, the Altair Community is on read-only mode from October 28 - November 6,
2024. Technical support via cases will continue to work as is. For any urgent requests from Students/Faculty members, please submit the form linked
Can I make an excel to predict with new input after have a training model ANN (Neural net)
Hi everyone,
I am in a problem to make an excel with the result after running and training model ANN Neural net in rapidminer.
I made an prediction task, using input with many numbers and labels, OUTPUT is a real number (REGRESSTION)
But when the result show the neural net with nodes and theirs weights, i dont know how to make an regresstion calculator by excel.
I see on the internet that when we use this for classifier rapidminer use active function (sigmoid) and when the predict result is regression, rapidminer will use linear .
I also think linear is this: Y (output) = W*x
w is the matrix of weights and x is the matrix of input.
had 6 years ago.
Here is what he/she did, same as me:
My process:
"Data" --> "x-validation"
In the "x-validation" is the "neural net" on the left side (training), "apply model" and "performance" is on the right side (testing).
In the end of x-validation is again "apply model" connected with multiply to the data.
Rapid miner shows in the input layer attribute 1, attribute 2 and a threshold node.
In the hidden layer are three noddes and a threshold node.
There is only one output node.
Descripted Result:
Node 1
1 att. -1.763
2 att. -1.144
Node 2
1 att. -1.776
2 att. -1.103
Bias -1.178
Node 3
1 att. -1.937
2 att. -1.937
Bias -0.996
1 Node -1.389
2 Node -1.376
3 Node -1.495
Threshold 0.112
Rapid miner gives me also a predicted value for every row.
Now I want to use the first row of my dataset (att. 1 and 2) to recalculate the predicted result.
The Idea is, if i know how to recalculate I can calculate new data. Or use the formula in a Excel sheet. In my exapmle it is the turnover of a neu grocery store.
My calculations (1att. 0.532, 2. att 0.089, the predicted result is 0.341)
Node 1
0.532 * (-1.763) + 0.089 * (-1.144) + (-1.173) = -2,212
after the sigmoid transformation I get 0,098
Node 2
0.532 * (-1.776) + 0.089 * (-1.103) + (-1.178) = -2,220
after the sigmoid transformation I get 0,097
Node 3
0.532 * (-1.937) + 0.089 * (-1.254) + -0.996 = -2,137
after the sigmoid transformation I get 0,105
Then I do the linear regression. Im not sure if it is right. I dont know how to use the threshold value.
0,098 (-1.389)
0,097 (-1.376)
0,105 (-1.495)
I get a correlation coefficient of 0,999
but I expect 0.341 ????
Do you understand my problem?
I hope somebody can help me. If you need detail please ask me.
How can rapid miner predict the turnover of a new grocery store (att. 1+ 2, but without a label/turnover)?
I appreciate your help.
Greeting Immo001
HELP ME ANY BRO? | {"url":"https://community.rapidminer.com/discussion/58830/can-i-make-an-excel-to-predict-with-new-input-after-have-a-training-model-ann-neural-net","timestamp":"2024-11-05T06:19:55Z","content_type":"text/html","content_length":"278928","record_id":"<urn:uuid:a4fa1deb-db7c-47c5-bb16-1a58d4d060db>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00439.warc.gz"} |
You have $10,000 to invest and are considering two mutual funds:
No-load Fund and Economy Fund....
You have $10,000 to invest and are considering two mutual funds: No-load Fund and Economy Fund....
You have $10,000 to invest and are considering two mutual funds: No-load Fund and Economy Fund. The No-load Fund charges a 12b-1 fee of 1% and maintains an expense ratio of 0.50%. The Economy Fund
charges a front-end load of 4%, but has no 12b-1 fee and an expense ratio of 0.50%. Assume that the rate of return on both funds’ portfolios (before any fees) is 8% per year.
1. If you plan to buy and hold for three years, which one should you invest? Please explain.
2. If you plan to buy and hold for ten years, which one should you invest? Please explain.
Answer :- 12b-1 is an annual marketing expense of an mutual fund it is considered as an operating expense of a mutual fund whereas front endoad is the fee paid for investing in mutual fund.
Situation 1:- If you plan to buy and hold for three years.
No-load fund :- The total cost of holding investment is 3%
Economy fund :- The total cost of holding investment is 4%
And 0.50% expense ratio is for both the funds which means that No load fund is better to invest.
Situation 2 :- If you plan to buy and hold for ten years.
No load fund = Total cost = 10%
Economy fund = Total cost = 4%
And 0.50% expense ratio for both the funds which means Economy fund is better to invest. | {"url":"https://justaaa.com/finance/262533-you-have-10000-to-invest-and-are-considering-two","timestamp":"2024-11-09T23:23:31Z","content_type":"text/html","content_length":"41670","record_id":"<urn:uuid:06aeb5c8-55c1-4a5d-b0f8-55d1086971d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00794.warc.gz"} |
Why It Matters: Perfect Competition
Why analyze a firm’s profit maximizing decisions under conditions of perfect competition?
This module is the second in the theory of the firm and the first of four modules examining models of market structure. Market structure means, in a nutshell, how competitive or monopolistic is a
particular industry. It should be clear that market structure influences how firms behave.
We start by looking at the ideal model of perfect competition. This model is a bit of a head scratcher since there are, actually, very few examples of industries like this in the real world. Why then
do we study it? Here’s a question for you to think about as you move through the module: What’s so perfect about perfect competition? Hint: the model has certain ideal features that you will learn.
Have you ever noticed that all the tomatoes of the same type in a farmer’s market cost about the same price? The same thing is true of roadside vegetable stands in the countryside. If one stall in a
locality has tomatoes for $3 per pound, they all do. Now the price may change from week to week, but it’s always the same across the different vendors in the market. You will soon learn why this is.
There are more similarities than differences between this and the following three modules. What you learn in this module will carry over and help you understand the next ones, so the more effort you
put into learning this one, the easier the next three modules will be.
• Define the characteristics of Perfect Competition
• Understand the difference between the firm and the industry
• Calculate and graph the firm’s fixed, variable, average, marginal and total costs
• Measure variable and total costs as the area under the average variable and average total cost curves
• Determine the break-even, and the shutdown points of production for a perfectly competitive firm
• Explain the difference between short run and long run equilibrium
• Understand why perfectly competitive markets are efficient | {"url":"https://courses.lumenlearning.com/atd-sac-microeconomics/chapter/why-it-matters-9/","timestamp":"2024-11-05T10:19:39Z","content_type":"text/html","content_length":"51863","record_id":"<urn:uuid:338df420-8ef0-44a3-9376-119ed228f57b>","cc-path":"CC-MAIN-2024-46/segments/1730477027878.78/warc/CC-MAIN-20241105083140-20241105113140-00286.warc.gz"} |
Missing Digit Division Worksheets
How Missing Digit Division Problems Help Students Think Critically - Who said mathematics has to be boring? Probably, mathematics is the only course that is all fun and interesting. As a whole is all
about finding relationships, numbers, and the missing figures to make the unknown known. When students solve problems, they compute the numbers, explain and understand the concepts, derive formulas,
and make assumptions. In its core, mathematics is mainly concerned with making assumptions with the help of prevailing definitions to solve a particular problem at hand. Students learn about
Pythagorean theorems and its related formulas. In short, mathematics has a direct impact on the cognitive abilities of the students. Students who derive equations, formulae, learn them and then use
them in different mathematical problems possess the ability to think critically and explain why a formula works a certain way. They possess the ability to trace the steps and define why one
mathematical concept that might fit in one problem can not fit in the other problem. Not only can they solve a problem, but they can also define the logic behind approaching that problem using a
particular process.
How Does That Go?
When you are doing division the top number (numerator) goes inside the "house" (long division symbol) and the bottom number (denominator) outside it. Switching these is a common error. To avoid
switching, say the numbers as you write out the equation.
Technically in long division, both the expression 63 ÷ 9 and its answer are called the quotient. You can call the expression the quotient and the answer can be called the quotient solved. However,
usually we just call the answer the quotient. | {"url":"https://www.mathworksheetscenter.com/mathskills/division/missingdigitdivision/","timestamp":"2024-11-13T12:57:51Z","content_type":"text/html","content_length":"21213","record_id":"<urn:uuid:39cba5e1-42c1-4e04-8edf-6bed5e0bcf13>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00887.warc.gz"} |
Entanglement entropy from tensor network states for stabilizer codes
In this paper, we present the construction of tensor network states (TNS) for some of the degenerate ground states of three-dimensional (3D) stabilizer codes. We then use the TNS formalism to obtain
the entanglement spectrum and entropy of these ground states for some special cuts. In particular, we work out examples of the 3D toric code, the X-cube model, and the Haah code. The latter two
models belong to the category of "fracton" models proposed recently, while the first one belongs to the conventional topological phases. We mention the cases for which the entanglement entropy and
spectrum can be calculated exactly: For these, the constructed TNS is a singular value decomposition (SVD) of the ground states with respect to particular entanglement cuts. Apart from the area law,
the entanglement entropies also have constant and linear corrections for the fracton models, while the entanglement entropies for the toric code models only have constant corrections. For the cuts we
consider, the entanglement spectra of these three models are completely flat. We also conjecture that the negative linear correction to the area law is a signature of extensive ground-state
degeneracy. Moreover, the transfer matrices of these TNSs can be constructed. We show that the transfer matrices are projectors whose eigenvalues are either 1 or 0. The number of nonzero eigenvalues
is tightly related to the ground-state degeneracy.
All Science Journal Classification (ASJC) codes
• Electronic, Optical and Magnetic Materials
• Condensed Matter Physics
Dive into the research topics of 'Entanglement entropy from tensor network states for stabilizer codes'. Together they form a unique fingerprint. | {"url":"https://collaborate.princeton.edu/en/publications/entanglement-entropy-from-tensor-network-states-for-stabilizer-co","timestamp":"2024-11-08T03:16:39Z","content_type":"text/html","content_length":"51650","record_id":"<urn:uuid:d3428052-9047-4c40-9a72-00c2d630f5ab>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00491.warc.gz"} |
Math Help Websites - 5 Myths and Facts | Thinkster Math
Last Updated on August 31, 2021 by Thinkster
Are you new to the world of math help websites? Perhaps you have a student who is struggling in math, or one who does well in math but you want them to work ahead and master more challenging topics.
Whatever the situation, you’ll find that math websites, like everything else on the internet, are abundant. It can be hard to know which ones are trustworthy and which ones will truly help your
We know you want the best for your children, especially when it comes to their education, so today we’ll review some myths about math help websites and the facts you need to know.
Knowledge is power, and this article will help you navigate the world of math websites with a little more insight, including what to look for and what to avoid.
5 Myths About Math Help Websites and Facts About Great Ones
Myth #1: All math help blogs and websites are written by qualified teachers and tutors.
It can be easy to fall into the mindset that anything written on the internet, including math help websites and blogs, is written by a qualified teacher or tutor. You hope that the websites you find
do their due diligence when it comes to their content, but you can’t always be sure.
The fact is that anyone can set up a website and claim to be an authority in math. But unless the blog is written by a vetted teacher or tutor, it’s possible that the information may not have been
fact-checked, and it could be inaccurate.
So as you’re seeking out new math resources and websites, look for sites that are well-established. Look for brands and names with good reputations, and look for sites that are professional,
well-written and have few errors. If you see basic errors in the written content, you may want to steer clear of their math problems.
We also suggest that you check the reviews to be sure that others are saying positive things about them. A simple Google search of the site name plus “reviews” can give you great insight into what
others have experienced with the site you are considering.
If you’re still unsure, you can ask your student’s teacher at school to take a look at the site and make sure some of their main principles are accurate. The teacher should be able to tell you if the
content on the site is good and reliable or not.
Myth #2: A math help website will simply answer my math problems or give me answers for my homework.
A great math help website will go far beyond simply giving you the answers to common problems. Your student should not expect that – and this will not help them learn anyway!
While some websites may have equation generators for when you’re really stuck, a great math help website will give your student the tools they need to solve not only that one problem but insight into
the method behind the problem.
This helps your student build a strong foundation that will give them confidence in their math skills for years to come – especially as math problems get more difficult.
A great website should not just solve math problems, but show you how to figure out the answer through written explanations or even video tutorials. An added bonus would be a site that offers a
variety of math worksheets, so your student can practice that type of problem as many times as they need to.
Myth #3: If you find what you need on a math website, you don’t need a tutor.
Math help websites are designed to give information, but are not designed to take the place of a math tutor. If you’re hesitant about committing to a math tutor, you may think that just by using a
math help website will be enough to help your student in math.
In reality, this is not an ideal way for your student to learn or get help on tough math concepts. Nothing can replace the experience of working one-on-one with a qualified math tutor! This could be
either an in-person tutor or an online tutor; either way, the expert help your child will receive is invaluable.
There are also many variables that come into play when your student needs math help. Your math tutor will take the time to get to know your student’s strengths and weaknesses, as well as their
learning style. If traditional methods aren’t working, your tutor should be able to come up with unique ideas that will help your student learn best.
Your tutor should also be familiar with your child’s specific curriculum and how the teacher is teaching it, so that your student can practice outside of the classroom and then take that knowledge
back into the classroom for success.
Finally, spending time with a tutor can teach your student other skills and help them build confidence in their abilities in a safe, comfortable environment. That extra personal touch, which you
can’t get from a website alone, can be an excellent motivator!
Myth #4: A math help website and online tutor are not as useful as an in-person tutor.
While some people believe that a website can replace a one-on-one tutor, which we have addressed above, others may be skeptical of the quality of an online tutor.
While you don’t want to rely just on a math help website, tutoring should be an important tool in your math arsenal. Fortunately, you can get many of the benefits mentioned above from either an
in-person or online tutor, so you can choose what works best for you and your child.
An online tutor will still give your student personalized attention, even though they’re not in the same room. In fact, for some students, online tutors can be more effective, as they allow your
student to learn and practice math in an environment where they’re comfortable – your home, in most cases.
Also, since you won’t be restricted to your specific area or city, you’ll have a larger pool of tutors to choose from — so you can find one who gets along well with your child. Not to mention, you
won’t have to worry about commuting anywhere, which is helpful if you and/or your child have a busy schedule.
The technology aspect should not be a deterrent when it comes to online tutoring, either. Most likely, your student is already used to using technological devices and online methods to communicate,
and online tutoring can take this one step further. Your student will likely find online tutoring and math worksheets more engaging, especially those that use a gamification model and prizes as
There is research to show that students learn just as well this way, if not better, and you’ll find that many classrooms have adopted online learning in some form or fashion anyway. Online tutoring
may be just the thing that your student needs to build on what they have learned in the classroom and excel.
Myth #5: Quality Math Tutoring & Websites Will Be Too Expensive
You may think that the only way to get quality math help is to spend a lot of money on an expensive tutor or tutoring program. Of course, you may find in-person tutoring expensive, and more expensive
websites do exist, but you don’t have to spend a fortune to get quality math help.
Tutors may charge anywhere from $30 to $85 an hour, depending on your location, as well as the tutor’s experience and levels covered. And as your student’s math level gets more difficult, you could
expect to pay more. This hourly rate strictly covers the tutor’s time, with anything extra — like materials or transportation reimbursement — adding to the cost. This is usually the case at math
tutoring centers, too.
Online tutoring typically works a little bit differently, especially if you are using a full online tutoring program or math app that comes with added tutoring support. With many of these online
tutoring programs, you are paying for tutoring time as well as any extra benefits the program may provide, such as math worksheets and tutorials, feedback on worksheets outside of tutoring sessions,
test prep time and more.
At Thinkster, our math program subscriptions start at $30 a month for access to math worksheets alone or $60 a month for plans that come with daily teacher grading and feedback. And we offer tiered
pricing so that you can choose the plan that works best for your student’s needs.
Try Our Math Website
When it comes to reputable math help websites, we recommend you take a look at Thinkster. Our blog is fact-checked and reviewed by knowledgeable math experts to give you a solid base of information.
To give some examples, we have received the Seal of Approval from the National Parenting Center and have also been covered by all the major media like The New York Times, CBS, NBC, Fox, ABC, Forbes,
Parent & Child and many more.
But our program is also so much more than a math help website and a math problem solving app. We provide online tutoring as well as a full program of unlimited math worksheets with video tutorials,
homework help, test preparation, and more.
Give our math program a try for free for 7 days and you’ll find you don’t need to look at any other websites for math help for your student. We strive to make lifelong learners and math champions out
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